# Satisficing and prior-free optimality in price competition.

I. INTRODUCTION

The rational choice approach to market interaction investigates price competition maintaining commonly known unbounded rationality of sellers. Undoubtedly, to explain how the reasoning of competing sellers can result in a mutually optimal constellation of sales strategies via solving a fixed-point problem is an interesting and inspiring intellectual exercise. However, this exercise needs to be supplemented with studies that do not provide only "as if"-explanations, but try to more realistically capture how sellers may mentally represent sales competition and generate sales choices based on such mental representation.

Drawing on Simon's (1947, 1955) work (see also Selten 1998, which partly relies on Sauermann and Selten 1962), this paper models sellers' behavior using a bounded rationality approach based on aspiration levels. Profit maximizing is replaced by the goal of making "satisfactory" profits: sellers have aspiration levels concerning their profits and search for sales policies that guarantee these aspirations. (1) Because of the interdependence of sellers, it is reasonable to suppose that the profits a seller aims to achieve (or his profit aspirations) depend on what he expects from the others. Accordingly, the present analysis assumes that aspiration levels capture a seller's uncertainty about the others' behavior. The way in which this uncertainty is dealt with represents a key distinguishing feature of the model outlined here. (2) While the standard game theoretical approach to market interaction suggests that sellers share a common and correct conjecture about the others' actions, we allow sellers to entertain multiple conjectures. (3) Previous theoretical studies motivate why this assumption may be taken as valid. In particular, the so-called multiple prior models, proposed by Gilboa and Schmeidler (1989) and Bewley (2002) for one-person decision problems, generalize expected utility theory by assuming that the decision maker has a set of priors, rather than a single one. The premise of these models is that the "Bayesian" tenet according to which any uncertainty can and should be summarized by a single probability measure is too strong and represents an inaccurate description of people's behavior. (4) Applied to our context, this means that asking sellers to hold a unique conjecture about the price charged by the others may be unrealistic.

The theories of decision making without a precise prior allow a representation of beliefs by a set of probabilities, rather than by a single probability, but they still operate in an expected utility framework. (5) As opposed to these theories, we assume true uncertainty in a Knightian sense: no probability distribution is assigned to events (Knight 1985). The intuition behind this assumption is that even if boundedly rational people may not want to exclude the possibility that an event occurs, they may still be unable to specify how likely the event is. To emphasize that the approach we adopt is non-probabilistic, we prefer the word "conjecture" to "expectation" or "belief." We do not preclude the possibility that the set-valued conjecture contains only one element. We simply claim that if the set is not a singleton, then no probabilities need to be attached to its various elements. Thus, we regard the conjecture as prior free.

The main contribution of the present paper is to propose a theory that allows satisficing sellers to make "optimal" decisions without being equipped with any prior. Specifically, we consider an oligopoly where each seller chooses a unique price level and forms a set-valued prior-free conjecture about the average price charged by the remaining sellers. We do not model the process by which the conjecture is formed. Rather, we assume that the set-valued conjecture is idiosyncratically generated by each of the competing sellers, who furthermore form a profit aspiration for each element of the conjecture.

In such a context, a seller is said to follow a satisficing mode of behavior if the unique price he chooses is satisficing in the sense that, for each element of the seller's conjecture, it yields profits not below the corresponding aspiration level. The chosen price is called rationalizable if it is a best response to some price belonging to the convex combination of the minimum and maximum elements in the seller's conjecture. The prior-free optimality theory that we propose requires a seller (1) to choose a rationalizable price and (2) for each element of his conjecture, to set the corresponding aspiration levels equal to the profits attainable at that rationalizable price. Condition 1 provides a weak constraint on choices, which must be somehow "reasonable." Condition 2 simply postulates that sellers should not leave profit potential unexhausted.

The experiment reported here is meant to test whether our notion of prior-free optimality provides an accurate description of behavior. (6) To this aim, we rely on the approach of Berninghaus et al. (forthcoming) and directly elicit profit aspirations, rather than following the tradition of revealed preference analysis and trying to infer aspirations from behavior. Our experiment implements a multi-period triopoly market where, in every period, each seller participant must (1) choose one price, (2) specify a finite set-valued conjecture about the average price of his two current competitors, and (3) form a profit aspiration for each conjectured price. To motivate participants not only to focus on realized profits, but also to predict the others' behavior as accurately as possible and to abide by satisficing, we monetarily incentivize all three tasks.

As we are also interested in exploring how participants react when being informed that their price is (not) satisficing, in every period we inform participants of whether or not their price is satisficing and allow them to revise any aspect of their decisions (conjectures, profit aspirations, and/or price) up to five times. This allows us to investigate whether the likelihood of revising depends on the received feedback and, if participants engage in revisions, what they revise more often (their conjectured prices, their profit aspirations, or their own price).

The paper proceeds as follows. Section II formalizes the characteristics of the oligopoly market and our notion of prior-free optimality. Section III illustrates the experimental procedures in detail. Section IV contains the data analysis. Section V concludes by summarizing and commenting the results.

II. THE MARKET MODEL AND THEORETICAL ANALYSIS

We study a multi-period heterogeneous oligopoly market with price competition. Quantity sold by individual firm i ([x.sub.i]) depends negatively on the firm's own price ([p.sub.i]) and positively on the average price of other firms ([bar.p]-i) in the market. For simplicity, we assume a linear relationship and constant marginal (production) costs. This allows us to equate revenues and profits by setting the price equal to the unit profit. These considerations give rise to a demand function for the ith firm of the following form:

(1) [x.sub.i](p)= [alpha] - [beta] [p.sub.i] - [gamma] ([p.sub.i] - [[bar.p].sub.-i])

where p = ([p.sub.1], ..., [p.sub.n]) is the vector of all sales prices (or unit profits), n is the number of firms in the market, [alpha], [beta], [gamma] > 0, and [[bar.p].sub.-i] = [[summation].sub.j[not equal to]i] [p.sub.j]/(n - 1).

We impose non-negativity constraints on price and quantity, implying 0 [less than or equal to ] [p.sub.i] [less than or equal to ] ([alpha] + [gamma][[bar.p].sub.p-i])/([beta] + [gamma]). A well-defined market game requires that the set of possible prices [p.sub.i] does not depend on the others' choices. In the case at hand, this can be obtained by imposing 0 [less than or equal to ] [p.sub.i] [less than or equal to ] [alpha]/([beta] + [gamma]).

Given the demand function specified in Equation (1), the profits for the ith firm (i = 1, ..., n) can be written as:

(2) [[pi].sub.i]([p.sub.i], [[bar.p].sub.-i]) = [p.sub.i]([alpha] - [beta][p.sub.i] - [gamma] ([p.sub.i] - [[bar.p].sub.-i]))"

If the ith firm pursues a noncooperative profit maximizing strategy, given [[bar.p].sub.-i], then i's reaction curve is [p.sub.i]([[bar.p].sub.-i)]) = ([alpha] + [gamma][[bar.p].sub.-i])/(2([beta] + [gamma])). The noncooperative symmetric equilibrium benchmark is given by

(3) [p.sup.*.sub.i] = [alpha]/(2[beta] + [gamma]) for all i,

yielding profits [[pi].sub.i]([p.sup.*]) = ([[alpha].sup.2]([beta] + [gamma]))/[(2[beta] + [gamma]).sup.2] for all i.

This equilibrium benchmark assumes common knowledge of rationality (in the sense of expected profit maximization) and correct beliefs. In this paper, we consider sellers who are "satisficing" rather than maximizing profits based on rational beliefs. We assume that each of the n sellers has aspirations with regard to profits. These profit aspirations reflect the seller's conjecture about the average price charged by the remaining sellers, where the conjecture is supposed to be a finite set. Thus, in line with existing models of boundedly rational behavior (Camerer et al. 2004; Costa-Gomes and Crawford 2006), we replace equilibrium beliefs with other beliefs, which, however, and in the tradition of the multiple prior models, do not need to be point-beliefs.

Formally, let [C.sub.i] C R be seller i's set-valued conjecture about his competitors' average price and let [c.sub.i] denote an element of this set. For each [c.sub.i] [member of] [C.sub.i], seller i forms a profit aspiration, denoted by [A.sub.i] ([c.sub.i]). A given sales price [p.sub.i] is said to be satisficing for seller i if

(4) [[pi].sub.i] ([p.sub.i], [c.sub.i]) [greater than or equal to] [A.sub.i]([c.sub.i]) for all [c.sub.i] [member of] [C.sub.i],

where [[pi].sub.i]([p.sub.i], [c.sub.i]) are i's attainable profits, i.e., the profits i can attain given his price [p.sub.i] and his conjecture [c.sub.i]. Profit aspirations abiding by requirement (4) will be called achievable aspirations.

We rely on nonprobabilistic conjectures. If [c.sub.i] [member of] [C.sub.i], this simply means that seller i does not want to exclude the event [[bar.p].sub.-i] = [c.sub.i] without necessarily being able to specify how likely the event is. Notwithstanding being nonprobabilistic, this approach does allow for optimality, which we qualify as "prior-free". In the following, we define the two conditions needed for prior-free optimality, and discuss how to classify and measure deviations from it. These two conditions are rather sensible: the first simply requires that sellers should best respond to their own set of conjectures; the second postulates that each specified aspiration must be achievable and not too moderate, that is, it must fully exhaust the profit potential allowed by the corresponding conjectured price and the chosen price.

More specifically, consider seller i with a set-valued conjecture [C.sub.i] and an aspiration profile [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Define [[c.bar].sub.i] = min{[c.sub.i] : [c.sub.i] [member of] [C.sub.i]} and [[bar.c].sub.i] = max{[c.sub.i] : [c.sub.i] [member of] [C.sub.i]} . Take the convex hull of the elements in i's conjecture, i.e.,

conv[C.sub.i] = {[c.sub.i] ([lambda]) = (1 - [lambda]) [[c.bar].sub.i] + [lambda][[bar.c].sub.i]: [lambda] [member of] [0, 1]}.

For any [c.sub.i] ([lambda]) [member of] conv [C.sub.i], seller i's best response to [c.sub.i] ([lambda]) is

[p.sup.*.sub.i] ([lambda]) = ([alpha] + [gamma][c.sub.i]([lambda]))/(2([beta] + [gamma])),

so that [p.sup.*.sub.i](0) = ([alpha] + [gamma][[c.bar].sub.i])/(2([beta] + [gamma])), [p.sup.*.sub.i](1) = ([alpha] + [gamma][[bar.c].sub.i])/(2([beta] + [gamma])), and [p.sup.*.sub.i] ([lambda]) increases continuously from [p.sup.*.sub.i] (0) to [p.sup.*.sub.i] (1) for [lambda] increasing from 0 to 1. Let us term a price [p.sub.i] rationalizable if it complies with [p.sub.i] [member of] [[p.sup.*.sub.i](0), [p.sup.*.sub.i](1)]. This delivers the first condition that seller i's choices must meet for being prior-free optimal.

Condition 1 Prior-free optimality requires seller i to specify a rationalizable price.

Price choices that fall outside the interval [[p.sup.*.sub.i](0), [p.sup.*.sub.i] (1)] represent a failure of prior-free optimality because they cannot be rationalized by any probability distribution over [C.sub.i]. We refer to this as type 1-deviation from prior-free optimality and measure it by the share of price choices [p.sub.i] such that [p.sub.i] [??] [[p.sup.*.sub.i](0), [p.sup.*.sub.i](1)].

Requiring [p.sub.i] to be rationalizable does not suffice for prior-free optimality. A second condition applies to i's aspiration profile.

Condition 2 Prior-free optimality requires seller i to form an aspiration profile [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [[pi].sub.i]([p.sub.i], [c.sub.i])= [A.sub.i] ([c.sub.i]) for each [c.sub.i] in [C.sub.i].

Thus, even if [p.sub.i] [member of] [[p.sup.*.sub.i](0), [p.sup.*.sub.i](1)], seller i may fail to comply with prior-free optimality if he forms an aspiration profile [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for which another achievable profile [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exists such that [[??].sub.i] ([c.sub.i]) > [A.sub.i]([c.sub.i]) for at least some [c.sub.i] [member of] [C.sub.i], and [[??].sub.i]([c.sub.i]) [greater than or equal to] [A.sub.i]([c.sub.i]) otherwise. We refer to this as type 2-deviation from prior-free optimality and measure it by the share of aspiration profiles [A.sub.i] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Conditions 1 and 2 define optimality in a more basic sense than that required by expected utility maximization because they do not entail the specification of any probability distribution over the set of conjectured prices. If seller i attaches probabilities to the various elements of [C.sub.i], maximization of expected profits will clearly imply prior-free optimality, but not necessarily vice versa because prior-free optimality usually defines a rather large set (Guth 2010; Guth et al. 2010). In our market model, common prior-free optimality (in the sense that all n sellers comply with conditions 1 and 2 stated earlier) will correspond to the standard equilibrium benchmark only if, for each seller i, [C.sub.i] = {[c.sub.i]}, that is, [C.sub.i] is a singleton, and [c.sub.i] = [[bar.p].sub.-i], that is, i expects from his competitors what they actually choose (see Aumann and Brandenburger 1995 for a characterization of equilibrium behavior by optimality and rational expectations). In this sense, common prior-free optimality is a coarsening of the equilibrium concept. (7)

III. EXPERIMENTAL PROTOCOL

The computerized experiment was conducted at the laboratory of the Max Planck Institute in Jena (Germany). The experiment was programmed via z-Tree (Fischbacher 2007). Overall, we ran three sessions with a total of 81 participants, all being students from various fields at the University of Jena. Participants were recruited using the ORSEE software (Greiner 2004). Each session needed about 2 hours, with about half of the time being used up for reading the instructions and answering some control questions. Money in the experiment was denoted in experimental currency unit (ECU) with 100 ECU = 1.00 [euro]. The average earnings per subject were 18.50 [euro] (including a 2.50 [euro] show-up fee).

Each experimental session consisted of nine periods. In each period, the 27 participants of a session were divided into nine groups of three sellers (i.e., n = 3) so as to form nine triopoly markets. New groups were randomly formed in each repetition (strangers design). (8) To collect more than one independent observation per session, subjects were rematched within matching groups of nine players, guaranteeing three independent observations per session and nine independent observations in total. In order to discourage repeated game effects, participants were not informed that random rematching of the groups had been restricted in such a way.

In the instructions (see the Appendix for an English translation), subjects were told that they would act as a firm which, together with two other firms, serves one market, and that in each period all three firms were to choose, independently, a price from 0 to 12. Choices were limited to numbers up to two decimals. Participants were informed that their period-profits would be determined via function (2), where we set [alpha] = 40, [beta] = 2, and [gamma] = 1. Given these parameters, from Equation (3), the noncooperative equilibrium price is [p.sup.*.sub.i] = 8 and the corresponding quantity is [x.sub.i]([p.sup.*]) = 24, implying profits [[pi].sub.i] (p*) = 192.

To check experimentally whether participants comply with the two conditions characterizing prior-free optimal behavior, in every period, besides choosing a sales price, each subject had to specify a set of the others' average price that he considered as possible and the profits he aimed to achieve for each conjectured price. Participants were allowed to provide a maximum of six conjectures per period, so that their aspiration profile could contain at the most six elements. Therefore, in each period, participants had to fulfill three tasks: (1) choose their own price, (2) predict at most six average prices that the others could charge, and (3) form their profit aspiration for each prediction.

As we want to explore also how people react to feedback of (non-)satisficing, after having completed the previous three tasks, participants were informed by the experimental software whether their price was satisficing or not (i.e., whether, given their conjectures, their own price guaranteed all their aspirations). Regardless of whether the specified price was satisficing or not, a participant could either confirm it or revise some aspects of his decisions. Based on former studies (Guth et al. 2009; Berninghaus et al. forthcoming), we expect most subjects to react to non-satisficing feedback by modifying their aspirations. To reduce the likelihood of noise in the decisions to revise, and in line with the work of Guth et al. (2009), a maximum of five revisions per period was warranted.

To incentivize all three tasks, in each period subjects could be paid according to realized profits, conjectures, or aspiration choices, with all three possibilities being equally likely. The three members of a group/market were paid according to the same mode. When payments were based on conjectured prices, the payoff of a seller participant was given by [W.sub.i] = 180 - 10[min.sub.[c.sub.i] [member of] [C.sub.i]] [absolute value of [[bar.p].sub.i] - [c.sub.i]]. Participants were informed about this rule, and they were told that the closer their best prediction to the actual average price of the others, the higher their earnings. When payments were based on aspirations, a subject earned his highest achieved aspiration, that is, the highest [A.sub.i]([c.sub.i]) complying with [[pi].sub.i]([p.sub.i], [[bar.p].sub.-i]]) [greater than or equal to] [A.sub.i]([c.sub.i]). If all the aspirations stated by the subject exceeded his actual profits, his earnings were nil. Thus, seller participant i's expected payoff, as determined by our payment procedure, was

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Could our payment procedure create incentives to distort "true" conjectures or own price choice? Clearly, if boundedly rational individuals consider the three parts of function (5) separately, the answer is no. But, even if a seller participant wants to maximize the expectation of his experimentally designed monetary payoff, he should (1) predict the others' average price as accurately as possible because his conjectures enter in the [W.sub.i] part of Equation (5), (2) given these conjectures, choose a rationalizable price because specifying a price that cannot be rationalized by any probability distribution over [C.sub.i], in the attempt of lowering one's own actual profit and satisfying the condition in the third addend of Equation (5), would not be beneficial, and (3) set each aspiration level equal to the profits attainable given the chosen price and the corresponding conjectured price. (9)

Because we paid only one of the three tasks, hedging opportunities--that is, best responding to one possible outcome (the others' average price) when choosing the price and predicting another outcome when stating conjectures in order to insure against small total payoffs--are eliminated or at least minimized (Blanco et al. 2008). Only if subjects entertained more than six conjectures about the average price of the others and wanted to play "safe," in the sense of guaranteeing themselves a positive outcome in case of payment based on aspirations, our payment procedure could induce deviations from prior-free optimality. We viewed this possibility as rather unlikely, however, and preferred fixing the cardinality of [C.sub.i] to either restricting the price choice set (so as to experimentally induce a complete set of conjectures) or allowing participants to predict as many [[bar.p].sub.-i] as they wished.

Seller participants had the possibility to use a "profit calculator" to compute their period-profits. The calculator was part of the experimental software and could be started by pressing the corresponding button on the screen. When provided with data regarding the others' average price and the own price, the calculator returned the resulting period-profits. Hence, the calculator allowed participants to try out the consequences of various price strategies.

At the end of each period, participants got feedback about the average price of the others, their own period-profits, their closest prediction to the actual average price of the others, their highest achieved profit aspiration, the mode of payment, and their resulting period experimental earnings.

IV. EXPERIMENTAL RESULTS

We present our results in several subsections. In the first subsection, we present an overview of observed price choices, set-valued conjectures, and aspiration profiles. Then, we turn to investigate some issues concerning participants' satisficing behavior. Finally, we check if participants conform to the two conditions defining prior-free optimality by measuring deviations from either condition.

A. Observed Prices, Conjectured Prices, and Aspiration Profiles

As a first step, we analyze how ambiguous the elicited conjectures about the others' average price are. Let [absolute value of [C.sub.i]] denote the cardinality of the elicited set [C.sub.i], that is, the number of player i's conjectured prices. In period 1, the mean and median [absolute value of [C.sub.i]] are close to 4. From period 2 on, most subjects provide the maximum allowed number of conjectures (i.e., 6). Given the increase in [absolute value of [C.sub.i]], the distribution of conjectures and corresponding aspirations may become more disperse. This issue seems important because subjects might have used the feature of our design that specifying more than one conjecture was costless to improve their chance of earning more when payments were based on conjectures or aspirations.

To measure the dispersion of conjectures and aspirations, we compute the coefficient of variation (ratio of the standard deviation to the mean) for each subject and each period. On average, the coefficient of variation of conjectures is 0.192 in the first period and 0.127 in the last period. The corresponding values for aspirations are 0.067 and 0.050. Wilcoxon signed rank tests (henceforth WSRT) comparing the coefficient of variation of conjectures as well as of aspirations in the first and the last period reveal a statistically significant difference (p < 0.01 in both cases). (10) Thus, the increase in the number of conjectured prices is associated with a decrease in the dispersion of conjectures and aspirations. This finding suggests that our seller participants tended to become more confident about their competitors' behavior.

The boxplots in Figure 1 provide descriptive statistics on the distributions of stated prices and average conjectured prices (i.e., ([[summation].sub.[c.sub.i] [member of] [C.sub.i]] [c.sub.i]) /[absolute value of [C.sub.i]]) over all periods. In both graphs, the x dots denote the means, and the horizontal lines indicate the theoretical equilibrium benchmark. Inspecting Figure 1A, we see that the median and the mean stated prices are both close to 7 in the first period and increase over time, with final values being, on average, significantly greater than initial ones (p = 0.012; WSRT). Although price choices converge to a value close to the noncooperative equilibrium benchmark (the mean price in the last period is 7.6), they are always lower (p = 0.074 in period 8; p = 0.055 in periods 2 and 5; p < 0.039 in the remaining periods; WSRT). Play was, therefore, mostly out of equilibrium with seller participants being more competitive than predicted by the equilibrium benchmark.

Turning to average conjectured prices (Figure 1B), the median and the mean values are, respectively, 6.50 and 6.69 in the first period, and 7.25 and 7.22 in the last period. The increase is statistically significant (p = 0.02; WSRT). A series of WSRT comparing observed average conjectured prices with the equilibrium benchmark reveal a statistically significant difference in all periods (p < 0.01 always). This indicates that most seller participants do not think, on average, that their competitors will behave in accordance with the equilibrium.

[FIGURE 1 OMITTED]

Although the average conjectured prices are, typically, lower than 8, are conjectures accurate? To address this question, we proceed in steps. First, we consider conv[C.sub.i], that is, the convex hull of the conjectured prices, and check whether the actual average price of i's competitors lies in it. By this means, we deliberately focus on subjects whose conjectures, being distributed around [[bar.p].sub.-i], are somehow "rationalizable." Then, for those subjects complying with [[bar.p].sub.-i] [member of] conv[C.sub.i] we check how their best conjecture compares with the others' actual average price. Let us call [[??].sub.i] i's closest conjecture to the actual [[bar.p].sub.-i].

Table 1 reports the percentage of subjects whose conjectured prices are such that [[bar.p].sub.-i] lies in their convex hull. The figure starts at 62.96% and is greater than 70% from the second period on. With two exceptions (periods 8 and 9), we have that [[??].sub.i] is never significantly different from [[bar.p].sub.-i] (p = 0.055 in period 8; p = 0.027 in period 9; p > 0.10 in all other periods; WSRT), thereby suggesting that, overall, subjects' conjectures are rather accurate. Taking, for each participant and each period, the squared deviation of the best conjecture from the others' actual average price, that is, [([[??].sub.i] - [[bar.p].sub.-i]).sup.2], as a measure of conjecture accuracy, which we call [[phi].sub.i], we find that the first quartile, the median, and the third quartile of the distribution of [[phi].sub.i] are 0.002, 0.026, and 0.198, respectively.

The boxplots in Figure 2 refer to realized profits and average aspirations ([Z.sub.[A.sub.i] [member of] [A.sub.i]] [A.sub.i]/[absolute value of [C.sub.i]). (11) From Figure 2A, we see that realized profits tend to increase over time. In fact, as compared to the average initial value (176.71), average realized profits in the last period (185.95) are significantly higher (p = 0.008). However, because of seller participants' competitive behavior, profits stay always significantly below the theoretical benchmark (p < 0.05 in each period; WSRT).

As to average aspirations (Figure 2B), mean and median values increase over the first four periods, and are rather stable (around 180) afterwards. Overall, average aspirations are significantly lower than actual profits (p < 0.01 in each period; WSRT). This already suggests that aspirations are, on average, more moderate than they could actually be given the others' observed price choices.

B. Evidence on Satisficing Behavior

The central questions in this subsection are: do participants choose a satisficing price, that is, a price complying with requirement (4)? Do the profit aspirations of the satisficing participants exhaust the full profit potential allowed by their chosen price and their conjectured prices? If subjects engage in revisions, what do they revise more often?

Table 2 presents some descriptive statistics about the participants' satisficing behavior in each of the nine periods. (12) Given our experimental protocol and our payment procedure, it is not surprising that the share of participants who choose a satisficing price at the end of each period is always above 96% and is rather stable over time (see row 1). The share of those immediately satisficing (i.e., who achieve all their aspirations at first attempt) ranges from 83.95% in period 1 to 97.53% in period 9 (see row 2). Over all periods, the percentage of subjects undertaking at least one revision is quite low (see row 3). The figure starts at 28.40%, and sharply declines over time.

[FIGURE 2 OMITTED]

As to the motivations underlying revisions, rows 4 and 5 suggest that the likelihood of revising depends on whether one chooses a satisficing price at first attempt or not. Although the figures in both rows tend to decrease over time, the propensity to revise is different depending on whether aspirations are achievable immediately or not. Taking averages over subjects and periods, the share of nonimmediately satisficing subjects who revise is far above that of immediately satisficing subjects who revise (85.46 vs. 8.01%). Finally, row 6 shows that, on average, those who revise engage in one revision (out of 5) in each period.

Overall, 76.54% of the satisficing participants specify at least one aspiration that is lower than the profits attainable given the chosen price and the corresponding conjectured price. For each satisficing participant and each period, define the average unexhausted profit potential relative to the attainable profits as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] . Averaging over subjects and periods, [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) is equal to 4.18%. The amount is significantly different from zero in each period (p < 0.001 always; WSRT), but it tends to decrease over time with values in the first period being, on average, significantly different from values in the last period (6.59 vs. 3.24%; p = 0.008 according to a WSRT). The relative shares of unexhausted profits are rather different across participants (with an overall standard deviation of 9.11). In particular, [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) is less than 1% for 32.77% of the subjects, it ranges from 1% to 10% for 60.61% of them, and it exceeds 10% for the remaining 6.62%. The observation that people tend to be "content" with a given choice and do not try to aspire to the maximum they may attain, although contradicting condition 2 for prior-free optimality, agrees with what has been termed "contentment factor" by Gilboa and Schmeidler (2001). In analyzing how satisficing consumers react to price changes, the authors postulate that aspiration levels tend to be below the experienced surplus. They justify this hypothesis by noticing that if aspiration levels were tending to the consumer surplus precisely, "the smallest shock (such as a minuscule price increase) would render the product unsatisfactory" (p. 218).

We now turn to explore what seller participants revise more often: their price, their conjectures, or their aspirations. In line with Guth et al. (2009) and Berninghaus et al. (forthcoming), we find that most of the within-period revisions concern aspirations. The finding that conjectures are barely modified is, in retrospect, quite reasonable: if a subject considers some strategies of the rivals as plausible, his conjectures should not vary unless new information comes in (which is not the case within a period). The further finding that aspirations are revised more often than stated prices may be because of the fact that, in our framework, experimenting a new price requires a careful reconsideration of the entire aspiration profile. Thus, if only some of the specified aspirations are not achievable, lowering the aspirations is cognitively less demanding than varying the price. Our data show some support for this explanation: for the nonimmediately satisficing subjects who engage in revisions, the overall ratio of non-achievable aspirations to provided aspirations is 38.2% and most of the revised aspirations (about 60%) were initially not achievable. (13)

In contrast to what is observed within each period, the percentage of revisions between two consecutive periods is quite high throughout the experiment. The observation that aspirations are adjusted more frequently than conjectured and stated prices applies also to across period-revisions. Specifically, on average, from one period to the next, 78.55% of the seller participants modify their aspirations, 65.43% their conjectures, and 49.85% their stated price. The highest average rate of revisions, in all the three possible dimensions, is observed from period 1 to period 2. In the other periods, revisions tend to be smaller but, with few exceptions, always positive in sign. This is in line with our earlier findings that stated prices, average conjectured prices, and average aspirations tend to increase over time. Because average aspirations are in each period below realized profits, it is not surprising that seller participants try more ambitious (although still too moderate) profit aspirations in the following period.

C. Compliance with Prior-Free Optimality

Finally, we investigate whether subjects follow prior-free optimality. Table 3 presents (1) the percentage of subjects who set a price that cannot be rationalized by any probability distribution over [C.sub.i] (type 1-deviation from prior-free optimality), (2) the percentage of subjects who choose a rationalizable price, but specify too moderate aspiration profiles (type 2-deviation from prior-free optimality), (3) the percentage of subjects who meet both conditions for prior-free optimality, and (4) for the satisficing subjects exhibiting a type 2- (but not a type 1-) deviation, the average relative unexhausted profit potential [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) in percentage terms.

Most seller participants fall within the type 1-deviation category in each of the nine periods, even though the percentage of nonrationalizable prices significantly decreases over time (p = 0.022; WSRT comparing the nine average independent shares of type 1-deviation in the first and the last period). (14) As conjectures are rather accurate, the observation that the chosen price is often not rationalizable suggests that subjects also fail to best respond to the (ex ante unknown) actual choice of the others. We find indeed that, overall, subjects choose a price that is a best response to the actual [[bar.p].sub.-i] in only 3.57% of the cases (the percentage rises from 0% in period 1 to 12.35% in period 9). However, deviations from the best response are not very large: for 67% of the observations the squared deviation of the best response to [[bar.p].sub.-i] from the chosen price is smaller than 1. It is worth noticing that conjecture accuracy does not help explain type 1-deviations: we observe no substantial difference in the percentage of nonrationalizable prices between subjects with [[phi].sub.i] < 0.026 (the median of the distribution on [[phi].sub.i]) and subjects with [[phi].sub.i] [greater than or equal to] 0.026 (59.89 vs. 67.32%).

The percentage of type 2-deviations done by those who state a rationalizable price ranges from 22.22% in period 1 to 33.33% in period 9. The difference between the two periods is statistically significant (p = 0.020; WSRT). At the outset of the experiment, only 3.70% of the participants meet the two conditions for prior-free optimality. Although this percentage increases over time, a WSRT does not allow rejecting the null hypothesis that the percentages (at the independent group level) in the first and the last period are the same (p = 0.181). Finally, the average relative unexhausted profit potential of those who choose a rationalizable price is significantly different from zero in all periods, and it is rather stable over time. (15) For 75.13% of the seller participants choosing a rationalizable price, [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) ranges from 1 to 10%; for 16.82% of them, [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) is less than 1%; and for the remaining 8.05%, [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) exceeds 10%.

These findings indicate that participants do not appear to comply with prior-free optimality. Most of them fail to report a rationalizable price, and the decline in type 1-deviations does not lead to an increase in prior-free optimal choices because type 2-deviations become more frequent over time.

The observation that often subjects do not best respond to their own conjectures is in line with the results of Costa-Gomes and Weizsacker (2008), who elicit subjects' point-beliefs in a set of 14 two-person one-shot 3x3 games. They find that subjects choose actions as if they expected their opponent to act randomly and, when asked which actions they expect their opponent to play, they transpose their own reasoning to the other, who is predicted to respond to monetary incentives but with the expectation that their own play is random. As we elicit a set-valued conjecture, the degree of rationality we require from our participants is much lower. Yet, we still observe a very high percentage of nonrationalizable prices. This finding questions the rational choice idea that people form beliefs about the others' actions and then optimally respond to these beliefs, thereby supporting the conclusions reached by Costa-Gomes and Weizsacker. In our view, this does not suggest any neoclassical repairing or game fitting via, for example, probabilistic-choice models (McKelvey and Palfrey 1995; Weizsacker 2003; Goeree and Holt 2004), but rather a bounded rationality approach that, although not necessarily excluding optimality, renders it an unlikely border case.

V. CONCLUSIONS

In this paper, we have applied the notion of satisficing to a repeated experimental triopoly market with price competition, where what one finds satisfactory depends on his conjectures about the others' average behavior. In every period, each seller participant had to choose a unique price, specify a possibly set-valued conjecture about the average price of his two current competitors, and form a profit aspiration for each of his conjectured prices. In this context, a seller participant is said to follow a satisficing mode of behavior if, for each conjectured price, the corresponding aspiration does not exceed the profits realizable from this conjectured price and the stated price.

We allow conjectures to be prior-free, that is, we do not require seller participants to specify a probability distribution over the set of conjectured prices. Thus, we can test optimality in a more basic sense than that required by expected utility maximization. More specifically, observed choices are compatible with our notion of prior-free optimality if they satisfy two testable conditions. Of these conditions, one is that the chosen price must be rationalizable, and the other is that, in case of satisficing, each specified aspiration must fully exhaust the profit potential allowed by the corresponding conjectured price and the chosen price.

In line with previous experiments on oligopoly markets (Dufwenberg and Gneezy 2000; Dufwenberg et al. 2007), we find that play is mostly out of equilibrium. Our seller participants behave, on average, more competitively than predicted by equilibrium play, and anticipate that their competitors will do the same. We also observe rather few prior-free optimal choices: overall, 35.53% of our seller participants report a rationalizable price, and only 9.19% meet both conditions for prior-free optimality. These findings suggest that decision makers have difficulties in pursuing optimal reasoning, even when optimality does not require the specification of a prior.

A further major result of our study is that 76.54% of the satisficing seller participants specify, on average, a too moderate aspiration profile: overall, they forego 4.18% of the profits they could aspire to given their chosen price and their conjectures about the others' average price. The latter finding is striking because subjects had access to a profit calculator allowing them to compute the profits corresponding to the own and the others' average price. Hence, computational limitations should not be held responsible for the observed moderate aspirations. One may argue that this is because of "safe" play by our participants, who wanted to guarantee themselves a positive outcome in case of payment based on aspirations. Yet, we are rather confident that this argument lacks relevance in our setting: in order to improve their chance of earning money, our seller participants could report several conjectures and aspirations without having to forego profits resulting from their conjectured prices. The claim that our participants did not play "safe" is supported by the observation that the increase in the number of conjectured prices is associated with a decrease in the dispersion of conjectures and aspirations.

Finally, the experiment shows that revisions within a period are more likely to be undertaken by those who do not satisfice at first attempt, and that most of the revisions concern profit aspirations. The latter finding is consistent with the results in Guth et al. (2009) and Berninghaus et al. (forthcoming). In our setting, it may be because of the fact that adjusting only the nonachievable aspirations requires less cognitive effort than revising the own unique price.

To conclude, our experiment is not designed to understand why subjects, although satisficing, aspire to less than they could, given their chosen price and their conjectured prices. Our primary goal here was to document relevant experimental evidence on our concept of prior-free optimality in market interaction. Identifying why people ask for too little could be interesting to look at in future research.

doi:10.1111/j.1465-7295.2010.00365.x

APPENDIX: TRANSLATED INSTRUCTIONS

Welcome and thanks for participating in this experiment. You will receive 2.50 [euro] for having shown up on time. Please read the instructions--which are identical for all participants--carefully. From now on any communication with other participants is forbidden. If you do not follow this rule, you will be excluded from the experiment and you will not receive any payment. Whenever you have a question, please raise your hand. An experimenter will then come to you and answer your question privately.

The experiment allows you to earn money. Money in the experiment will be denoted in ECU. Each ECU is worth 0.01 [euro]; this means that 100 ECU = 1 [euro]. How many ECU you will earn depends on your decisions and on the decisions of other participants matched with you. All your decisions will be treated in an anonymous manner and they will be gathered across a computer network. At the end of the experiment, the ECU you have earned will be converted to euros and paid out to you in cash together with the show-up fee of 2.50 [euro].

Detailed Information

In this experiment you will have to make decisions repeatedly. In every period you will be matched in groups of three persons. The composition of your group will randomly change after each period so that the other two members of your group will be different from one period to the next. The identity of the other participants you will interact with will not be revealed to you at any time.

In the experiment you have the role of a firm that, like two other firms (the participants you are matched with), produces and sells a certain good on a market. In each period you, as well as the other firms in your group, have to fulfill three tasks.

Task 1. Your first task is to decide at which price you wish to sell the good. Your price decision can be any number between 0 and 12. You can use up to two decimals. Thus, your choice of price can be: 0,0.01,0.02,..., 11.99, or 12.

In each period, your profit is given by the price you choose multiplied by the units of the good you sell at that price:

Your period-profit = (your price) x (number of units you sell).

The "number of units you sell" depends on your price and the average price of the other two firms (where the average is obtained by adding up the prices of the two other firms and dividing the resulting sum by two). In particular, the number of units you sell is given by:

[40 - 2 x (your price)] - [(your price) - (average price of the others)].

In words, two times your price is subtracted from 40; then the difference between your price and the average price of the others is

* subtracted from the resulting amount if the difference is positive (i.e., if your price is higher than the average price of the others),

* added to the resulting amount if the difference is negative (i.e., if your price is lower than the average price of the others).

Thus, the higher is your price compared to the average price of the others, the fewer units you sell. On the other hand, you sell more if the average price of the others is higher than your price.

Suppose, for example, that the prices of the other firms are 6 and 8 so that their average price is: (6 + 8)/2 = 14/2 = 7.

* If your price is 5 (<7), then the number of units you sell is: [40 - 2 x 5] - [5 - 7] = 30 + 2 = 32. Consequently, your period-profit is 5 x 32 = 160.

* If your price is 8 (>7), then the number of units you sell is: [40 - 2 x 8] - [8 - 7] = 24 - 1= 23. Consequently, your period-profit is 8 x 23 = 184.

* If your price is 10 (>7), then the number of units you sell is: [40 - 2 x 10] - [10 - 7] = 20 -3 = 17, and your period-profit is 10 x 17 = 170.

Task 2. Your second task in every period is to guess the average price of the other two firms in your current group. In every period, you must make at least one guess about their average price, and you can--if you wish to--make additional guesses. The maximum number of guesses you can make is six.

You should make as many guesses as the number of possible average prices of the others you do not want to exclude. Suppose, for instance, that you do not want to exclude that: (a) the average price of the others is 5, and (b) the average price of the others is 6.5. Then, you should make two guesses about the others' average price: (a) a first guess in which you expect the other two firms to choose, on average, 5; (b) a second guess in which you expect the other two firms to choose, on average, 6.5.

Your guesses about the average price of the others must be a number from 0 to 12. You can use up to two decimals.

Task 3. Your last task in every period is to specify the period-profit you wish to guarantee your self for each average price you guessed the others could choose.

Suppose, for instance, that you made two guesses about the others' average price. For each of these two guesses, you need to specify the period-profit you aspire to. Similarly, if you made tour guesses about the others' average price, you must specify the period-profit you aspire to for each of your four guesses.

In the following, we will refer to the period-profit you aspire to as your profit aspiration.

The Decision Aid

To help you make "satisfactory" decisions, that is, decisions achieving your aspired period-profit for each guess you made, we will provide you with a decision aid. In each period, after you have (1) chosen your price, (2) guessed the possible average prices of the others, and (3) specified your profit aspiration thr each of your guesses, the decision aid will inform you whether your stated profit aspiration(s) can be achieved or not. That is, you will learn whether, given your own price and your guesses about the others' average price, you can achieve the period-profit you aspire to for each of your guesses. The decision aid will then ask you if you want to revise your specifications in (1), (2), and/or (3).

* If you want to revise something, you have to click the "revise"-button. You will then move to a screen where you can modify your own price and/or your guesses about the others' average price and/or your profit aspirations.

* If you do not want to revise anything, you have to click the "not-revise"-button. After all participants have finished with their revisions, you will move on to the next period.

Notice that you can revise something even if your decisions were "satisfactory," that is, they allowed you to achieve your profit aspirations. In every period, you can make at most five revisions.

Period-Profit Calculator

Additionally, you have access to a period-profit calculator that calculates your period-profit for arbitrary price combinations. You can start the calculator by pressing the corresponding button on your screen. If you do so, a window will appear on your screen. Into this window you must enter two values: a price for yourself and an average price for the others. Given these figures, if you press the apposite button, you will know how much you would earn.

Your Experimental Earnings in Each Period

In each period, you can be paid according to your period-profit, your guesses about the others' average price, or your profit aspirations, where all the three modes of payment are equally likely. The randomly selected mode of payment applies to all three interacting participants, which means that you and the other two firms in your current group will be paid according to the same procedure.

If, by random choice, your payment is based on your "guesses," you will earn 180 minus 10 times the smallest difference between the average price you guessed the others could choose and the true average price of the others. In particular, the computer will

* consider your closest guess to the true average price of the others;

* take the numerical distance between your closest guess and the others' true average price;

* multiply this distance by 10;

* subtract the resulting amount from 180.

Hence, if your payment is based on your guesses, the closer your guesses are to the true average price of the others, the higher will be your period-payment.

Suppose that you made three guesses about the others' average price, which were 5, 6, and 6.5. If the true average price of the others is 7, your closest guess to 7 is 6.5. The numerical distance between 7 and 6.5 is 0.5 (i.e., 6.5 deviates from 7 by 0.5). Then, you will receive 180 - 10 x 0.5 = 175 ECU.

If, by random choice, your payment is based on your "profit aspirations," you will earn your highest achieved profit aspiration, that is, your highest aspiration that does not exceed your period-profit. In particular, the computer will check which of your profit aspirations are equal to or smaller than your period-profit. Among the profit aspirations that do not exceed your period-profit, you will earn the highest one. If all your profit aspirations exceed your period-profit, then you will earn 0 (zero) ECU.

Suppose that your period-profit is 162 ECU and you made three guesses about the others' average price so that you had to specify three profit aspirations. If your profit aspirations were 170, 160, and 150, then you earn 160 ECU because 160 is the highest aspiration that does not exceed your period-profit of 162 ECU. If, instead, your profit aspirations were 180, 172, and 170, then you earn 0 ECU because all your aspirations exceed 162 ECU.

The Information You Will Receive at the End of Each Period

At the end of each period you will be told: your price, the average price of the others, your own period-profit, your closest guess to the average price of the others, your highest achieved aspiration, your period experimental earnings.

Your Final Earnings

Your final earnings will be calculated by adding up your experimental earnings in all periods. The resulting sum will be converted to euros and paid out to you in cash in addition to the show-up fee of 2.50 [euro].

Before the experiment starts, you will have to answer some control questions to ensure your understanding of the experiment.

Please remain quiet until the experiment starts and switch off your mobile phone. If you have any questions, please raise your hand now.

REFERENCES

Abbink, K., and J. Brandts. "Collusion in Growing and Shrinking Markets: Empirical Evidence from Experimental Duopolies," in Experiments and Competition Policy, edited by J. Hinloopen and H.-T. Normann. Cambridge: Cambridge University Press, 2009, 34-60.

Aumann, R., and A. Brandenburger. "Epistemic Conditions for Nash Equilibrium." Econometrica, 63(5), 1995, 1161-80.

Aumann, R., and J. H. Dreze. "Rational Expectations in Games." American Economic Review, 98(1), 2008, 72-86.

Berninghaus, S., W. Guth, M. V. Levati, and J. Qiu. "Satisficing Search versus Aspiration Adaptation in Sales Competition: Experimental Evidence." International Journal of Game Theory, forthcoming.

Bewley, T. "Knightian Decision Theory: Part I." Decisions in Economics and Finance, 25, 2002, 79-110.

Blanco, M., D. Engelmann, A. Koch, and H.-T. Normann. "Belief Elicitation in Experiments: Is There a Hedging Problem?" IZA Discussion Paper No. 3517, Bonn, 2008.

Camerer, C., T. Ho, and J.-K. Chong. "A Cognitive Hierarchy Model of Games." Quarterly Journal of Economics, 119, 2004, 861-98.

Costa-Gomes, M. A., and V. Crawford. "Cognition and Behavior in Guessing Games: An Experimental Study." American Economic Review, 96, 2006, 1737-68.

Costa-Gomes, M. A., and G. Weizsacker. "Stated Beliefs and Play in Normal-Form Games." Review of Economic Studies, 75, 2008, 729-62.

Cyert, R., and J. March. "Organizational Factors in the Theory of Oligopoly." Quarterly Journal of Economics, 70(1), 1956, 44-64.

--. A Behavioral Theory of the Firm, Englewood Cliffs, NJ: Prentice-Hall, 1963.

Dixon, H. D. "Keeping Up with the Joneses: Competition and the Evolution of Collusion." Journal of Economic Behavior & Organization, 43, 2000, 223-38.

Dufwenberg, M., and U. Gneezy. "Price Competition and Market Concentration: An Experimental Study." International Journal of Industrial Organization, 18, 2000, 7-22.

Dufwenberg, M., U. Gneezy, J. K. Goeree, and R. Nagel. "Price Floors and Competition." Economic Theory, 33, 2007, 211-24.

Fischbacher, U. "Zurich Toolbox for Readymade Economic Experiments." Experimental Economics, 10, 2007, 171-78.

Gilboa, I. "Questions in Decision Theory." Annual Reviews its Economics. 2. 2010. 1-19.

Gilboa, I., F. Maccheroni, M. Marinacci, and D. Schmeidler. "Objective and Subjective Rationality in a Multiple Prior Model." Econometrica, 78, 2010, 755-70.

Gilboa, I., and D. Schmeidler. "Maxmin Expected Utility with a Non-Unique Prior." Journal of Mathematical Economics, 18, 1989, 141-53.

--. "Reaction to Price Changes and Aspiration Level Adjustments." Review of Economic Design, 6(2), 2001, 215-23.

Goeree, J., and C. Holt. "A Model of Noisy Introspection." Games and Economic Behavior, 46, 2004, 365-82.

Greiner, B. "An Online Recruitment System for Economic Experiments," in Forschung und wissenschaftliches Rechnen 2003. GWDG Bericht, Vol. 63, edited by K. Kremer and V. Macho. Gottingen: Gesellschaft fur Wissenschaftliche Datenverarbeitung, Ges. fur Wiss. Datenverarbeitung, 2004, 79-93.

Guth, W. "Satisficing in Portfolio Selection--Theoretical Aspects and Experimental Tests." Journal of SocioEconomics, 36, 2007, 505-22.

--. "Satisficing and (Un)Bounded Rationality: A Formal Definition and Its Experimental Validity." Journal of Economic Behavior & Organization, 73, 2010, 308-16.

Guth, W., M. V. Levati, and M. Ploner. "An Experimental Analysis of Satisficing in Saving Decisions." Journal of Mathematical Psychology, 53, 2009, 265-72.

Guth, W., M. V. Levati, and M. Ploner. "Satisficing in Strategic Environments: A Theoretical Approach and Experimental Evidence." Journal of Socio-Economics, 39, 2010, 554-561.

Huck, S., K. A. Konrad, W. Muller, and H.-T. Normann. "The Merger Paradox and Why Aspiration Levels Let It Fail in the Laboratory." Economic Journal, 117, 2007, 1073-95.

Knight, F. H. Risk, Uncertainty and Profit. Chicago: University of Chicago Press, 1985 (first printed in 1921).

Mckelvey, R., and T. Palfrey. "Quantal Response Equilibrium for Normal Form Games." Games and Economic Behavior, 10, 1995, 6-38.

Oechssler, J. "Cooperation as a Result of Learning with Aspiration Levels." Journal of Economic Behavior & Organization, 49, 2002, 405-09.

Sauermann, H., and R. Selten. "Anspruchsanpassungstheorie der Unternehmung." Zeitschrift fur die gesamte Staatswissenschaft, 118, 1962, 577-97.

Selten, R. "Aspiration Adaption Theory." Journal of Mathematical Psychology, 42, 1998, 191-214.

--. "What is Bounded Rationality'?" in Bounded Rationality: The Adaptive Toolbox, edited by G. Gigerenzer, and R. Selten. Cambridge, MA: MIT Press, 2001, 13-36.

Simon, H. A. Administrative Behavior: A Study of DecisionMaking Processes in Administrative Organization, New York: Macmillan, 1947.

--. "A Behavioral Model of Rational Choice." Quarterly Journal of Economics, 69, 1955, 99-118.

Van Damme, E. "Strategic Equilibrium," in Handbook of Game Theory, Vol. 3, Chapter 41, edited by R. Aumann and S. Hart. Amsterdam, The Netherlands: North Holland, 2002.

Weizsacker, G. "Ignoring the Rationality of Others: Evidence from Experimental Normal Form Games." Games and Economic Behavior, 44, 2003, 145-71.

(1.) Cyert and March (1956, 1963) were the first to apply the concept of aspiration level to oligopoly theory. More recent theoretical studies on aspiration-based models of firm behavior include Dixon (2000) and Oechssler (2002) who both investigate how behavioral rules based on aspiration levels can induce collusion in Cournot games.

(2.) See also Selten (2001) for a survey of heuristics and bounded rationality ideas.

(3.) The reason why we use the term "conjecture" instead of "expectation" or "belief' will be explained later.

(4.) For more recent discussions on this issue, see Gilboa (2010) and Gilboa et al. (2010).

(5.) Gilboa and Schmeidler (1989) couple the set of priors with a decision rule that chooses an action whose minimal expected utility (over all priors in the set) is the highest. Bewley (2002) uses the set of priors to define a partial order over actions: he suggests to prefer an action to another if and only if its expected utility is higher for each and every prior in the set.

(6.) To the best of our knowledge, the only previous experimental studies investigating satisficing behavior in oligopoly markets are by Huck et al. (2007) and Berninghaus et al. (forthcoming). Huck et al. (2007) analyze how aspirations may lead to a failure of the merger paradox in the laboratory without eliciting aspiration levels. Berninghaus et al. (forthcoming) test the absorption of satisficing in duopoly Cournot markets by directly asking participants to form profit aspirations.

(7.) Other solution concepts that coarsen the Nash equilibrium are objective correlated equilibrium, subjective correlated equilibrium, and rationalizability (Van Damme 2002; Aumann and Dreze 2008). The main difference between these concepts and prior-free optimality is that the latter does not require the specification of any probability distribution over the other players' actions.

(8.) This should isolate the effects of experience from the opportunities of tacit collusion that may occur in a repeated game. See, e.g., Abbink and Brandts (2009) for an experimental study of collusive behavior in a homogeneous market with partners design.

(9.) We do not intend to propose an incentive compatible method of eliciting aspirations, that is, a method that induces participants to truthfully reveal their aspirations. We simply want that one should not aspire to profits lower than those allowed by [p.sub.i] and [C.sub.i]. Asking subjects to specify aspirations without payoff consequences makes them more likely to invest little effort in the aspiration formation and adaptation tasks (see, e.g., the survey by Guth 2007 or Guth et al. 2009). Paying for the highest realized aspiration should encourage subjects to think carefully about the problem and to comply with satisficing. It also matches the implication of aspirations in the satisficing approach: one is "satisfied" if aspirations are met (aspired profits are not greater than realized ones) while one is "unsatisfied" if aspirations are not met (aspired profits are greater than realized ones).

(10.) All reported nonparametric tests are two-sided and (unless otherwise stated) rely on the averages over players for each matching group. Because of our rematching system. the numbers of statistically independent groups are 9 in each period.

(11.) The meaning of the x dots and the horizontal lines is as in Figure 1.

(12.) To avoid misclassifications originating in participants' rounding, the computations assessing satisficing behavior (as well as prior-free optimality) were performed by rounding numbers to the first integer.

(13.) of course, this is a hindsight-driven, ad hoc explanation. A priori, one could have expected seller participants to try out a new price strategy whenever some of their aspirations were not achievable.

(14.) The observed decrease in the dispersion of conjectured prices does not therefore lead to an increase in type 1-deviations. Actually one may suspect that subjects with less dispersed conjectures would be less likely to exhibit type 1-deviations than others because they are more confident about their competitors' average price. However, when seller participants specify a set-valued conjecture with a coefficient of variation lower than 0.133 (the median value), they choose a rationalizable price in 30.99% of the cases. When the coefficient of variation is equal or higher than 0.133, they choose a rationalizable price in 42.04% of the cases.

(15.) A WSRT comparing [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) in the first and the last period delivers p = 0.469.

WERNER GOTH, MARIA VITTORIA LEVATI, MATTEO PLONER *

* We thank Birendra Kumar Rai and Dominique Cappelletti for helpful comments.

Guth: Director, Max Planck Institute of Economics. Strategic Interaction Group, Kahlaische Strasse 10, Jena D07745, Germany. Phone 49-3641-686620, Fax 49-3641686667, E-mail gueth@econ.mpg.de

Levati: Research Group Leader, Max Planck Institute of Economics, Strategic Interaction Group, Kahlaische Strasse 10, Jena D-07745, Germany. Phone 49-3641686629. Fax 49-3641-686667, E-mail levati@econ. mpg.de

Ploner: Research Associate, Department of Economics-CEEL. University of Trento, Via Inama 5, Trento 38100, Italy. Phone 39-461-883139, Fax 39-461-882222, E-mail matteo.ploner@unitn.it

The rational choice approach to market interaction investigates price competition maintaining commonly known unbounded rationality of sellers. Undoubtedly, to explain how the reasoning of competing sellers can result in a mutually optimal constellation of sales strategies via solving a fixed-point problem is an interesting and inspiring intellectual exercise. However, this exercise needs to be supplemented with studies that do not provide only "as if"-explanations, but try to more realistically capture how sellers may mentally represent sales competition and generate sales choices based on such mental representation.

Drawing on Simon's (1947, 1955) work (see also Selten 1998, which partly relies on Sauermann and Selten 1962), this paper models sellers' behavior using a bounded rationality approach based on aspiration levels. Profit maximizing is replaced by the goal of making "satisfactory" profits: sellers have aspiration levels concerning their profits and search for sales policies that guarantee these aspirations. (1) Because of the interdependence of sellers, it is reasonable to suppose that the profits a seller aims to achieve (or his profit aspirations) depend on what he expects from the others. Accordingly, the present analysis assumes that aspiration levels capture a seller's uncertainty about the others' behavior. The way in which this uncertainty is dealt with represents a key distinguishing feature of the model outlined here. (2) While the standard game theoretical approach to market interaction suggests that sellers share a common and correct conjecture about the others' actions, we allow sellers to entertain multiple conjectures. (3) Previous theoretical studies motivate why this assumption may be taken as valid. In particular, the so-called multiple prior models, proposed by Gilboa and Schmeidler (1989) and Bewley (2002) for one-person decision problems, generalize expected utility theory by assuming that the decision maker has a set of priors, rather than a single one. The premise of these models is that the "Bayesian" tenet according to which any uncertainty can and should be summarized by a single probability measure is too strong and represents an inaccurate description of people's behavior. (4) Applied to our context, this means that asking sellers to hold a unique conjecture about the price charged by the others may be unrealistic.

The theories of decision making without a precise prior allow a representation of beliefs by a set of probabilities, rather than by a single probability, but they still operate in an expected utility framework. (5) As opposed to these theories, we assume true uncertainty in a Knightian sense: no probability distribution is assigned to events (Knight 1985). The intuition behind this assumption is that even if boundedly rational people may not want to exclude the possibility that an event occurs, they may still be unable to specify how likely the event is. To emphasize that the approach we adopt is non-probabilistic, we prefer the word "conjecture" to "expectation" or "belief." We do not preclude the possibility that the set-valued conjecture contains only one element. We simply claim that if the set is not a singleton, then no probabilities need to be attached to its various elements. Thus, we regard the conjecture as prior free.

The main contribution of the present paper is to propose a theory that allows satisficing sellers to make "optimal" decisions without being equipped with any prior. Specifically, we consider an oligopoly where each seller chooses a unique price level and forms a set-valued prior-free conjecture about the average price charged by the remaining sellers. We do not model the process by which the conjecture is formed. Rather, we assume that the set-valued conjecture is idiosyncratically generated by each of the competing sellers, who furthermore form a profit aspiration for each element of the conjecture.

In such a context, a seller is said to follow a satisficing mode of behavior if the unique price he chooses is satisficing in the sense that, for each element of the seller's conjecture, it yields profits not below the corresponding aspiration level. The chosen price is called rationalizable if it is a best response to some price belonging to the convex combination of the minimum and maximum elements in the seller's conjecture. The prior-free optimality theory that we propose requires a seller (1) to choose a rationalizable price and (2) for each element of his conjecture, to set the corresponding aspiration levels equal to the profits attainable at that rationalizable price. Condition 1 provides a weak constraint on choices, which must be somehow "reasonable." Condition 2 simply postulates that sellers should not leave profit potential unexhausted.

The experiment reported here is meant to test whether our notion of prior-free optimality provides an accurate description of behavior. (6) To this aim, we rely on the approach of Berninghaus et al. (forthcoming) and directly elicit profit aspirations, rather than following the tradition of revealed preference analysis and trying to infer aspirations from behavior. Our experiment implements a multi-period triopoly market where, in every period, each seller participant must (1) choose one price, (2) specify a finite set-valued conjecture about the average price of his two current competitors, and (3) form a profit aspiration for each conjectured price. To motivate participants not only to focus on realized profits, but also to predict the others' behavior as accurately as possible and to abide by satisficing, we monetarily incentivize all three tasks.

As we are also interested in exploring how participants react when being informed that their price is (not) satisficing, in every period we inform participants of whether or not their price is satisficing and allow them to revise any aspect of their decisions (conjectures, profit aspirations, and/or price) up to five times. This allows us to investigate whether the likelihood of revising depends on the received feedback and, if participants engage in revisions, what they revise more often (their conjectured prices, their profit aspirations, or their own price).

The paper proceeds as follows. Section II formalizes the characteristics of the oligopoly market and our notion of prior-free optimality. Section III illustrates the experimental procedures in detail. Section IV contains the data analysis. Section V concludes by summarizing and commenting the results.

II. THE MARKET MODEL AND THEORETICAL ANALYSIS

We study a multi-period heterogeneous oligopoly market with price competition. Quantity sold by individual firm i ([x.sub.i]) depends negatively on the firm's own price ([p.sub.i]) and positively on the average price of other firms ([bar.p]-i) in the market. For simplicity, we assume a linear relationship and constant marginal (production) costs. This allows us to equate revenues and profits by setting the price equal to the unit profit. These considerations give rise to a demand function for the ith firm of the following form:

(1) [x.sub.i](p)= [alpha] - [beta] [p.sub.i] - [gamma] ([p.sub.i] - [[bar.p].sub.-i])

where p = ([p.sub.1], ..., [p.sub.n]) is the vector of all sales prices (or unit profits), n is the number of firms in the market, [alpha], [beta], [gamma] > 0, and [[bar.p].sub.-i] = [[summation].sub.j[not equal to]i] [p.sub.j]/(n - 1).

We impose non-negativity constraints on price and quantity, implying 0 [less than or equal to ] [p.sub.i] [less than or equal to ] ([alpha] + [gamma][[bar.p].sub.p-i])/([beta] + [gamma]). A well-defined market game requires that the set of possible prices [p.sub.i] does not depend on the others' choices. In the case at hand, this can be obtained by imposing 0 [less than or equal to ] [p.sub.i] [less than or equal to ] [alpha]/([beta] + [gamma]).

Given the demand function specified in Equation (1), the profits for the ith firm (i = 1, ..., n) can be written as:

(2) [[pi].sub.i]([p.sub.i], [[bar.p].sub.-i]) = [p.sub.i]([alpha] - [beta][p.sub.i] - [gamma] ([p.sub.i] - [[bar.p].sub.-i]))"

If the ith firm pursues a noncooperative profit maximizing strategy, given [[bar.p].sub.-i], then i's reaction curve is [p.sub.i]([[bar.p].sub.-i)]) = ([alpha] + [gamma][[bar.p].sub.-i])/(2([beta] + [gamma])). The noncooperative symmetric equilibrium benchmark is given by

(3) [p.sup.*.sub.i] = [alpha]/(2[beta] + [gamma]) for all i,

yielding profits [[pi].sub.i]([p.sup.*]) = ([[alpha].sup.2]([beta] + [gamma]))/[(2[beta] + [gamma]).sup.2] for all i.

This equilibrium benchmark assumes common knowledge of rationality (in the sense of expected profit maximization) and correct beliefs. In this paper, we consider sellers who are "satisficing" rather than maximizing profits based on rational beliefs. We assume that each of the n sellers has aspirations with regard to profits. These profit aspirations reflect the seller's conjecture about the average price charged by the remaining sellers, where the conjecture is supposed to be a finite set. Thus, in line with existing models of boundedly rational behavior (Camerer et al. 2004; Costa-Gomes and Crawford 2006), we replace equilibrium beliefs with other beliefs, which, however, and in the tradition of the multiple prior models, do not need to be point-beliefs.

Formally, let [C.sub.i] C R be seller i's set-valued conjecture about his competitors' average price and let [c.sub.i] denote an element of this set. For each [c.sub.i] [member of] [C.sub.i], seller i forms a profit aspiration, denoted by [A.sub.i] ([c.sub.i]). A given sales price [p.sub.i] is said to be satisficing for seller i if

(4) [[pi].sub.i] ([p.sub.i], [c.sub.i]) [greater than or equal to] [A.sub.i]([c.sub.i]) for all [c.sub.i] [member of] [C.sub.i],

where [[pi].sub.i]([p.sub.i], [c.sub.i]) are i's attainable profits, i.e., the profits i can attain given his price [p.sub.i] and his conjecture [c.sub.i]. Profit aspirations abiding by requirement (4) will be called achievable aspirations.

We rely on nonprobabilistic conjectures. If [c.sub.i] [member of] [C.sub.i], this simply means that seller i does not want to exclude the event [[bar.p].sub.-i] = [c.sub.i] without necessarily being able to specify how likely the event is. Notwithstanding being nonprobabilistic, this approach does allow for optimality, which we qualify as "prior-free". In the following, we define the two conditions needed for prior-free optimality, and discuss how to classify and measure deviations from it. These two conditions are rather sensible: the first simply requires that sellers should best respond to their own set of conjectures; the second postulates that each specified aspiration must be achievable and not too moderate, that is, it must fully exhaust the profit potential allowed by the corresponding conjectured price and the chosen price.

More specifically, consider seller i with a set-valued conjecture [C.sub.i] and an aspiration profile [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Define [[c.bar].sub.i] = min{[c.sub.i] : [c.sub.i] [member of] [C.sub.i]} and [[bar.c].sub.i] = max{[c.sub.i] : [c.sub.i] [member of] [C.sub.i]} . Take the convex hull of the elements in i's conjecture, i.e.,

conv[C.sub.i] = {[c.sub.i] ([lambda]) = (1 - [lambda]) [[c.bar].sub.i] + [lambda][[bar.c].sub.i]: [lambda] [member of] [0, 1]}.

For any [c.sub.i] ([lambda]) [member of] conv [C.sub.i], seller i's best response to [c.sub.i] ([lambda]) is

[p.sup.*.sub.i] ([lambda]) = ([alpha] + [gamma][c.sub.i]([lambda]))/(2([beta] + [gamma])),

so that [p.sup.*.sub.i](0) = ([alpha] + [gamma][[c.bar].sub.i])/(2([beta] + [gamma])), [p.sup.*.sub.i](1) = ([alpha] + [gamma][[bar.c].sub.i])/(2([beta] + [gamma])), and [p.sup.*.sub.i] ([lambda]) increases continuously from [p.sup.*.sub.i] (0) to [p.sup.*.sub.i] (1) for [lambda] increasing from 0 to 1. Let us term a price [p.sub.i] rationalizable if it complies with [p.sub.i] [member of] [[p.sup.*.sub.i](0), [p.sup.*.sub.i](1)]. This delivers the first condition that seller i's choices must meet for being prior-free optimal.

Condition 1 Prior-free optimality requires seller i to specify a rationalizable price.

Price choices that fall outside the interval [[p.sup.*.sub.i](0), [p.sup.*.sub.i] (1)] represent a failure of prior-free optimality because they cannot be rationalized by any probability distribution over [C.sub.i]. We refer to this as type 1-deviation from prior-free optimality and measure it by the share of price choices [p.sub.i] such that [p.sub.i] [??] [[p.sup.*.sub.i](0), [p.sup.*.sub.i](1)].

Requiring [p.sub.i] to be rationalizable does not suffice for prior-free optimality. A second condition applies to i's aspiration profile.

Condition 2 Prior-free optimality requires seller i to form an aspiration profile [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [[pi].sub.i]([p.sub.i], [c.sub.i])= [A.sub.i] ([c.sub.i]) for each [c.sub.i] in [C.sub.i].

Thus, even if [p.sub.i] [member of] [[p.sup.*.sub.i](0), [p.sup.*.sub.i](1)], seller i may fail to comply with prior-free optimality if he forms an aspiration profile [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for which another achievable profile [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exists such that [[??].sub.i] ([c.sub.i]) > [A.sub.i]([c.sub.i]) for at least some [c.sub.i] [member of] [C.sub.i], and [[??].sub.i]([c.sub.i]) [greater than or equal to] [A.sub.i]([c.sub.i]) otherwise. We refer to this as type 2-deviation from prior-free optimality and measure it by the share of aspiration profiles [A.sub.i] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Conditions 1 and 2 define optimality in a more basic sense than that required by expected utility maximization because they do not entail the specification of any probability distribution over the set of conjectured prices. If seller i attaches probabilities to the various elements of [C.sub.i], maximization of expected profits will clearly imply prior-free optimality, but not necessarily vice versa because prior-free optimality usually defines a rather large set (Guth 2010; Guth et al. 2010). In our market model, common prior-free optimality (in the sense that all n sellers comply with conditions 1 and 2 stated earlier) will correspond to the standard equilibrium benchmark only if, for each seller i, [C.sub.i] = {[c.sub.i]}, that is, [C.sub.i] is a singleton, and [c.sub.i] = [[bar.p].sub.-i], that is, i expects from his competitors what they actually choose (see Aumann and Brandenburger 1995 for a characterization of equilibrium behavior by optimality and rational expectations). In this sense, common prior-free optimality is a coarsening of the equilibrium concept. (7)

III. EXPERIMENTAL PROTOCOL

The computerized experiment was conducted at the laboratory of the Max Planck Institute in Jena (Germany). The experiment was programmed via z-Tree (Fischbacher 2007). Overall, we ran three sessions with a total of 81 participants, all being students from various fields at the University of Jena. Participants were recruited using the ORSEE software (Greiner 2004). Each session needed about 2 hours, with about half of the time being used up for reading the instructions and answering some control questions. Money in the experiment was denoted in experimental currency unit (ECU) with 100 ECU = 1.00 [euro]. The average earnings per subject were 18.50 [euro] (including a 2.50 [euro] show-up fee).

Each experimental session consisted of nine periods. In each period, the 27 participants of a session were divided into nine groups of three sellers (i.e., n = 3) so as to form nine triopoly markets. New groups were randomly formed in each repetition (strangers design). (8) To collect more than one independent observation per session, subjects were rematched within matching groups of nine players, guaranteeing three independent observations per session and nine independent observations in total. In order to discourage repeated game effects, participants were not informed that random rematching of the groups had been restricted in such a way.

In the instructions (see the Appendix for an English translation), subjects were told that they would act as a firm which, together with two other firms, serves one market, and that in each period all three firms were to choose, independently, a price from 0 to 12. Choices were limited to numbers up to two decimals. Participants were informed that their period-profits would be determined via function (2), where we set [alpha] = 40, [beta] = 2, and [gamma] = 1. Given these parameters, from Equation (3), the noncooperative equilibrium price is [p.sup.*.sub.i] = 8 and the corresponding quantity is [x.sub.i]([p.sup.*]) = 24, implying profits [[pi].sub.i] (p*) = 192.

To check experimentally whether participants comply with the two conditions characterizing prior-free optimal behavior, in every period, besides choosing a sales price, each subject had to specify a set of the others' average price that he considered as possible and the profits he aimed to achieve for each conjectured price. Participants were allowed to provide a maximum of six conjectures per period, so that their aspiration profile could contain at the most six elements. Therefore, in each period, participants had to fulfill three tasks: (1) choose their own price, (2) predict at most six average prices that the others could charge, and (3) form their profit aspiration for each prediction.

As we want to explore also how people react to feedback of (non-)satisficing, after having completed the previous three tasks, participants were informed by the experimental software whether their price was satisficing or not (i.e., whether, given their conjectures, their own price guaranteed all their aspirations). Regardless of whether the specified price was satisficing or not, a participant could either confirm it or revise some aspects of his decisions. Based on former studies (Guth et al. 2009; Berninghaus et al. forthcoming), we expect most subjects to react to non-satisficing feedback by modifying their aspirations. To reduce the likelihood of noise in the decisions to revise, and in line with the work of Guth et al. (2009), a maximum of five revisions per period was warranted.

To incentivize all three tasks, in each period subjects could be paid according to realized profits, conjectures, or aspiration choices, with all three possibilities being equally likely. The three members of a group/market were paid according to the same mode. When payments were based on conjectured prices, the payoff of a seller participant was given by [W.sub.i] = 180 - 10[min.sub.[c.sub.i] [member of] [C.sub.i]] [absolute value of [[bar.p].sub.i] - [c.sub.i]]. Participants were informed about this rule, and they were told that the closer their best prediction to the actual average price of the others, the higher their earnings. When payments were based on aspirations, a subject earned his highest achieved aspiration, that is, the highest [A.sub.i]([c.sub.i]) complying with [[pi].sub.i]([p.sub.i], [[bar.p].sub.-i]]) [greater than or equal to] [A.sub.i]([c.sub.i]). If all the aspirations stated by the subject exceeded his actual profits, his earnings were nil. Thus, seller participant i's expected payoff, as determined by our payment procedure, was

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Could our payment procedure create incentives to distort "true" conjectures or own price choice? Clearly, if boundedly rational individuals consider the three parts of function (5) separately, the answer is no. But, even if a seller participant wants to maximize the expectation of his experimentally designed monetary payoff, he should (1) predict the others' average price as accurately as possible because his conjectures enter in the [W.sub.i] part of Equation (5), (2) given these conjectures, choose a rationalizable price because specifying a price that cannot be rationalized by any probability distribution over [C.sub.i], in the attempt of lowering one's own actual profit and satisfying the condition in the third addend of Equation (5), would not be beneficial, and (3) set each aspiration level equal to the profits attainable given the chosen price and the corresponding conjectured price. (9)

Because we paid only one of the three tasks, hedging opportunities--that is, best responding to one possible outcome (the others' average price) when choosing the price and predicting another outcome when stating conjectures in order to insure against small total payoffs--are eliminated or at least minimized (Blanco et al. 2008). Only if subjects entertained more than six conjectures about the average price of the others and wanted to play "safe," in the sense of guaranteeing themselves a positive outcome in case of payment based on aspirations, our payment procedure could induce deviations from prior-free optimality. We viewed this possibility as rather unlikely, however, and preferred fixing the cardinality of [C.sub.i] to either restricting the price choice set (so as to experimentally induce a complete set of conjectures) or allowing participants to predict as many [[bar.p].sub.-i] as they wished.

Seller participants had the possibility to use a "profit calculator" to compute their period-profits. The calculator was part of the experimental software and could be started by pressing the corresponding button on the screen. When provided with data regarding the others' average price and the own price, the calculator returned the resulting period-profits. Hence, the calculator allowed participants to try out the consequences of various price strategies.

At the end of each period, participants got feedback about the average price of the others, their own period-profits, their closest prediction to the actual average price of the others, their highest achieved profit aspiration, the mode of payment, and their resulting period experimental earnings.

IV. EXPERIMENTAL RESULTS

We present our results in several subsections. In the first subsection, we present an overview of observed price choices, set-valued conjectures, and aspiration profiles. Then, we turn to investigate some issues concerning participants' satisficing behavior. Finally, we check if participants conform to the two conditions defining prior-free optimality by measuring deviations from either condition.

A. Observed Prices, Conjectured Prices, and Aspiration Profiles

As a first step, we analyze how ambiguous the elicited conjectures about the others' average price are. Let [absolute value of [C.sub.i]] denote the cardinality of the elicited set [C.sub.i], that is, the number of player i's conjectured prices. In period 1, the mean and median [absolute value of [C.sub.i]] are close to 4. From period 2 on, most subjects provide the maximum allowed number of conjectures (i.e., 6). Given the increase in [absolute value of [C.sub.i]], the distribution of conjectures and corresponding aspirations may become more disperse. This issue seems important because subjects might have used the feature of our design that specifying more than one conjecture was costless to improve their chance of earning more when payments were based on conjectures or aspirations.

To measure the dispersion of conjectures and aspirations, we compute the coefficient of variation (ratio of the standard deviation to the mean) for each subject and each period. On average, the coefficient of variation of conjectures is 0.192 in the first period and 0.127 in the last period. The corresponding values for aspirations are 0.067 and 0.050. Wilcoxon signed rank tests (henceforth WSRT) comparing the coefficient of variation of conjectures as well as of aspirations in the first and the last period reveal a statistically significant difference (p < 0.01 in both cases). (10) Thus, the increase in the number of conjectured prices is associated with a decrease in the dispersion of conjectures and aspirations. This finding suggests that our seller participants tended to become more confident about their competitors' behavior.

The boxplots in Figure 1 provide descriptive statistics on the distributions of stated prices and average conjectured prices (i.e., ([[summation].sub.[c.sub.i] [member of] [C.sub.i]] [c.sub.i]) /[absolute value of [C.sub.i]]) over all periods. In both graphs, the x dots denote the means, and the horizontal lines indicate the theoretical equilibrium benchmark. Inspecting Figure 1A, we see that the median and the mean stated prices are both close to 7 in the first period and increase over time, with final values being, on average, significantly greater than initial ones (p = 0.012; WSRT). Although price choices converge to a value close to the noncooperative equilibrium benchmark (the mean price in the last period is 7.6), they are always lower (p = 0.074 in period 8; p = 0.055 in periods 2 and 5; p < 0.039 in the remaining periods; WSRT). Play was, therefore, mostly out of equilibrium with seller participants being more competitive than predicted by the equilibrium benchmark.

Turning to average conjectured prices (Figure 1B), the median and the mean values are, respectively, 6.50 and 6.69 in the first period, and 7.25 and 7.22 in the last period. The increase is statistically significant (p = 0.02; WSRT). A series of WSRT comparing observed average conjectured prices with the equilibrium benchmark reveal a statistically significant difference in all periods (p < 0.01 always). This indicates that most seller participants do not think, on average, that their competitors will behave in accordance with the equilibrium.

[FIGURE 1 OMITTED]

Although the average conjectured prices are, typically, lower than 8, are conjectures accurate? To address this question, we proceed in steps. First, we consider conv[C.sub.i], that is, the convex hull of the conjectured prices, and check whether the actual average price of i's competitors lies in it. By this means, we deliberately focus on subjects whose conjectures, being distributed around [[bar.p].sub.-i], are somehow "rationalizable." Then, for those subjects complying with [[bar.p].sub.-i] [member of] conv[C.sub.i] we check how their best conjecture compares with the others' actual average price. Let us call [[??].sub.i] i's closest conjecture to the actual [[bar.p].sub.-i].

Table 1 reports the percentage of subjects whose conjectured prices are such that [[bar.p].sub.-i] lies in their convex hull. The figure starts at 62.96% and is greater than 70% from the second period on. With two exceptions (periods 8 and 9), we have that [[??].sub.i] is never significantly different from [[bar.p].sub.-i] (p = 0.055 in period 8; p = 0.027 in period 9; p > 0.10 in all other periods; WSRT), thereby suggesting that, overall, subjects' conjectures are rather accurate. Taking, for each participant and each period, the squared deviation of the best conjecture from the others' actual average price, that is, [([[??].sub.i] - [[bar.p].sub.-i]).sup.2], as a measure of conjecture accuracy, which we call [[phi].sub.i], we find that the first quartile, the median, and the third quartile of the distribution of [[phi].sub.i] are 0.002, 0.026, and 0.198, respectively.

The boxplots in Figure 2 refer to realized profits and average aspirations ([Z.sub.[A.sub.i] [member of] [A.sub.i]] [A.sub.i]/[absolute value of [C.sub.i]). (11) From Figure 2A, we see that realized profits tend to increase over time. In fact, as compared to the average initial value (176.71), average realized profits in the last period (185.95) are significantly higher (p = 0.008). However, because of seller participants' competitive behavior, profits stay always significantly below the theoretical benchmark (p < 0.05 in each period; WSRT).

As to average aspirations (Figure 2B), mean and median values increase over the first four periods, and are rather stable (around 180) afterwards. Overall, average aspirations are significantly lower than actual profits (p < 0.01 in each period; WSRT). This already suggests that aspirations are, on average, more moderate than they could actually be given the others' observed price choices.

B. Evidence on Satisficing Behavior

The central questions in this subsection are: do participants choose a satisficing price, that is, a price complying with requirement (4)? Do the profit aspirations of the satisficing participants exhaust the full profit potential allowed by their chosen price and their conjectured prices? If subjects engage in revisions, what do they revise more often?

Table 2 presents some descriptive statistics about the participants' satisficing behavior in each of the nine periods. (12) Given our experimental protocol and our payment procedure, it is not surprising that the share of participants who choose a satisficing price at the end of each period is always above 96% and is rather stable over time (see row 1). The share of those immediately satisficing (i.e., who achieve all their aspirations at first attempt) ranges from 83.95% in period 1 to 97.53% in period 9 (see row 2). Over all periods, the percentage of subjects undertaking at least one revision is quite low (see row 3). The figure starts at 28.40%, and sharply declines over time.

[FIGURE 2 OMITTED]

As to the motivations underlying revisions, rows 4 and 5 suggest that the likelihood of revising depends on whether one chooses a satisficing price at first attempt or not. Although the figures in both rows tend to decrease over time, the propensity to revise is different depending on whether aspirations are achievable immediately or not. Taking averages over subjects and periods, the share of nonimmediately satisficing subjects who revise is far above that of immediately satisficing subjects who revise (85.46 vs. 8.01%). Finally, row 6 shows that, on average, those who revise engage in one revision (out of 5) in each period.

Overall, 76.54% of the satisficing participants specify at least one aspiration that is lower than the profits attainable given the chosen price and the corresponding conjectured price. For each satisficing participant and each period, define the average unexhausted profit potential relative to the attainable profits as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] . Averaging over subjects and periods, [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) is equal to 4.18%. The amount is significantly different from zero in each period (p < 0.001 always; WSRT), but it tends to decrease over time with values in the first period being, on average, significantly different from values in the last period (6.59 vs. 3.24%; p = 0.008 according to a WSRT). The relative shares of unexhausted profits are rather different across participants (with an overall standard deviation of 9.11). In particular, [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) is less than 1% for 32.77% of the subjects, it ranges from 1% to 10% for 60.61% of them, and it exceeds 10% for the remaining 6.62%. The observation that people tend to be "content" with a given choice and do not try to aspire to the maximum they may attain, although contradicting condition 2 for prior-free optimality, agrees with what has been termed "contentment factor" by Gilboa and Schmeidler (2001). In analyzing how satisficing consumers react to price changes, the authors postulate that aspiration levels tend to be below the experienced surplus. They justify this hypothesis by noticing that if aspiration levels were tending to the consumer surplus precisely, "the smallest shock (such as a minuscule price increase) would render the product unsatisfactory" (p. 218).

We now turn to explore what seller participants revise more often: their price, their conjectures, or their aspirations. In line with Guth et al. (2009) and Berninghaus et al. (forthcoming), we find that most of the within-period revisions concern aspirations. The finding that conjectures are barely modified is, in retrospect, quite reasonable: if a subject considers some strategies of the rivals as plausible, his conjectures should not vary unless new information comes in (which is not the case within a period). The further finding that aspirations are revised more often than stated prices may be because of the fact that, in our framework, experimenting a new price requires a careful reconsideration of the entire aspiration profile. Thus, if only some of the specified aspirations are not achievable, lowering the aspirations is cognitively less demanding than varying the price. Our data show some support for this explanation: for the nonimmediately satisficing subjects who engage in revisions, the overall ratio of non-achievable aspirations to provided aspirations is 38.2% and most of the revised aspirations (about 60%) were initially not achievable. (13)

In contrast to what is observed within each period, the percentage of revisions between two consecutive periods is quite high throughout the experiment. The observation that aspirations are adjusted more frequently than conjectured and stated prices applies also to across period-revisions. Specifically, on average, from one period to the next, 78.55% of the seller participants modify their aspirations, 65.43% their conjectures, and 49.85% their stated price. The highest average rate of revisions, in all the three possible dimensions, is observed from period 1 to period 2. In the other periods, revisions tend to be smaller but, with few exceptions, always positive in sign. This is in line with our earlier findings that stated prices, average conjectured prices, and average aspirations tend to increase over time. Because average aspirations are in each period below realized profits, it is not surprising that seller participants try more ambitious (although still too moderate) profit aspirations in the following period.

C. Compliance with Prior-Free Optimality

Finally, we investigate whether subjects follow prior-free optimality. Table 3 presents (1) the percentage of subjects who set a price that cannot be rationalized by any probability distribution over [C.sub.i] (type 1-deviation from prior-free optimality), (2) the percentage of subjects who choose a rationalizable price, but specify too moderate aspiration profiles (type 2-deviation from prior-free optimality), (3) the percentage of subjects who meet both conditions for prior-free optimality, and (4) for the satisficing subjects exhibiting a type 2- (but not a type 1-) deviation, the average relative unexhausted profit potential [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) in percentage terms.

Most seller participants fall within the type 1-deviation category in each of the nine periods, even though the percentage of nonrationalizable prices significantly decreases over time (p = 0.022; WSRT comparing the nine average independent shares of type 1-deviation in the first and the last period). (14) As conjectures are rather accurate, the observation that the chosen price is often not rationalizable suggests that subjects also fail to best respond to the (ex ante unknown) actual choice of the others. We find indeed that, overall, subjects choose a price that is a best response to the actual [[bar.p].sub.-i] in only 3.57% of the cases (the percentage rises from 0% in period 1 to 12.35% in period 9). However, deviations from the best response are not very large: for 67% of the observations the squared deviation of the best response to [[bar.p].sub.-i] from the chosen price is smaller than 1. It is worth noticing that conjecture accuracy does not help explain type 1-deviations: we observe no substantial difference in the percentage of nonrationalizable prices between subjects with [[phi].sub.i] < 0.026 (the median of the distribution on [[phi].sub.i]) and subjects with [[phi].sub.i] [greater than or equal to] 0.026 (59.89 vs. 67.32%).

The percentage of type 2-deviations done by those who state a rationalizable price ranges from 22.22% in period 1 to 33.33% in period 9. The difference between the two periods is statistically significant (p = 0.020; WSRT). At the outset of the experiment, only 3.70% of the participants meet the two conditions for prior-free optimality. Although this percentage increases over time, a WSRT does not allow rejecting the null hypothesis that the percentages (at the independent group level) in the first and the last period are the same (p = 0.181). Finally, the average relative unexhausted profit potential of those who choose a rationalizable price is significantly different from zero in all periods, and it is rather stable over time. (15) For 75.13% of the seller participants choosing a rationalizable price, [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) ranges from 1 to 10%; for 16.82% of them, [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) is less than 1%; and for the remaining 8.05%, [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) exceeds 10%.

These findings indicate that participants do not appear to comply with prior-free optimality. Most of them fail to report a rationalizable price, and the decline in type 1-deviations does not lead to an increase in prior-free optimal choices because type 2-deviations become more frequent over time.

The observation that often subjects do not best respond to their own conjectures is in line with the results of Costa-Gomes and Weizsacker (2008), who elicit subjects' point-beliefs in a set of 14 two-person one-shot 3x3 games. They find that subjects choose actions as if they expected their opponent to act randomly and, when asked which actions they expect their opponent to play, they transpose their own reasoning to the other, who is predicted to respond to monetary incentives but with the expectation that their own play is random. As we elicit a set-valued conjecture, the degree of rationality we require from our participants is much lower. Yet, we still observe a very high percentage of nonrationalizable prices. This finding questions the rational choice idea that people form beliefs about the others' actions and then optimally respond to these beliefs, thereby supporting the conclusions reached by Costa-Gomes and Weizsacker. In our view, this does not suggest any neoclassical repairing or game fitting via, for example, probabilistic-choice models (McKelvey and Palfrey 1995; Weizsacker 2003; Goeree and Holt 2004), but rather a bounded rationality approach that, although not necessarily excluding optimality, renders it an unlikely border case.

V. CONCLUSIONS

In this paper, we have applied the notion of satisficing to a repeated experimental triopoly market with price competition, where what one finds satisfactory depends on his conjectures about the others' average behavior. In every period, each seller participant had to choose a unique price, specify a possibly set-valued conjecture about the average price of his two current competitors, and form a profit aspiration for each of his conjectured prices. In this context, a seller participant is said to follow a satisficing mode of behavior if, for each conjectured price, the corresponding aspiration does not exceed the profits realizable from this conjectured price and the stated price.

We allow conjectures to be prior-free, that is, we do not require seller participants to specify a probability distribution over the set of conjectured prices. Thus, we can test optimality in a more basic sense than that required by expected utility maximization. More specifically, observed choices are compatible with our notion of prior-free optimality if they satisfy two testable conditions. Of these conditions, one is that the chosen price must be rationalizable, and the other is that, in case of satisficing, each specified aspiration must fully exhaust the profit potential allowed by the corresponding conjectured price and the chosen price.

In line with previous experiments on oligopoly markets (Dufwenberg and Gneezy 2000; Dufwenberg et al. 2007), we find that play is mostly out of equilibrium. Our seller participants behave, on average, more competitively than predicted by equilibrium play, and anticipate that their competitors will do the same. We also observe rather few prior-free optimal choices: overall, 35.53% of our seller participants report a rationalizable price, and only 9.19% meet both conditions for prior-free optimality. These findings suggest that decision makers have difficulties in pursuing optimal reasoning, even when optimality does not require the specification of a prior.

A further major result of our study is that 76.54% of the satisficing seller participants specify, on average, a too moderate aspiration profile: overall, they forego 4.18% of the profits they could aspire to given their chosen price and their conjectures about the others' average price. The latter finding is striking because subjects had access to a profit calculator allowing them to compute the profits corresponding to the own and the others' average price. Hence, computational limitations should not be held responsible for the observed moderate aspirations. One may argue that this is because of "safe" play by our participants, who wanted to guarantee themselves a positive outcome in case of payment based on aspirations. Yet, we are rather confident that this argument lacks relevance in our setting: in order to improve their chance of earning money, our seller participants could report several conjectures and aspirations without having to forego profits resulting from their conjectured prices. The claim that our participants did not play "safe" is supported by the observation that the increase in the number of conjectured prices is associated with a decrease in the dispersion of conjectures and aspirations.

Finally, the experiment shows that revisions within a period are more likely to be undertaken by those who do not satisfice at first attempt, and that most of the revisions concern profit aspirations. The latter finding is consistent with the results in Guth et al. (2009) and Berninghaus et al. (forthcoming). In our setting, it may be because of the fact that adjusting only the nonachievable aspirations requires less cognitive effort than revising the own unique price.

To conclude, our experiment is not designed to understand why subjects, although satisficing, aspire to less than they could, given their chosen price and their conjectured prices. Our primary goal here was to document relevant experimental evidence on our concept of prior-free optimality in market interaction. Identifying why people ask for too little could be interesting to look at in future research.

ABBREVIATIONS ECU: Experimental Currency Unit WSRT: Wilcoxon Signed Rank Tests

doi:10.1111/j.1465-7295.2010.00365.x

APPENDIX: TRANSLATED INSTRUCTIONS

Welcome and thanks for participating in this experiment. You will receive 2.50 [euro] for having shown up on time. Please read the instructions--which are identical for all participants--carefully. From now on any communication with other participants is forbidden. If you do not follow this rule, you will be excluded from the experiment and you will not receive any payment. Whenever you have a question, please raise your hand. An experimenter will then come to you and answer your question privately.

The experiment allows you to earn money. Money in the experiment will be denoted in ECU. Each ECU is worth 0.01 [euro]; this means that 100 ECU = 1 [euro]. How many ECU you will earn depends on your decisions and on the decisions of other participants matched with you. All your decisions will be treated in an anonymous manner and they will be gathered across a computer network. At the end of the experiment, the ECU you have earned will be converted to euros and paid out to you in cash together with the show-up fee of 2.50 [euro].

Detailed Information

In this experiment you will have to make decisions repeatedly. In every period you will be matched in groups of three persons. The composition of your group will randomly change after each period so that the other two members of your group will be different from one period to the next. The identity of the other participants you will interact with will not be revealed to you at any time.

In the experiment you have the role of a firm that, like two other firms (the participants you are matched with), produces and sells a certain good on a market. In each period you, as well as the other firms in your group, have to fulfill three tasks.

Task 1. Your first task is to decide at which price you wish to sell the good. Your price decision can be any number between 0 and 12. You can use up to two decimals. Thus, your choice of price can be: 0,0.01,0.02,..., 11.99, or 12.

In each period, your profit is given by the price you choose multiplied by the units of the good you sell at that price:

Your period-profit = (your price) x (number of units you sell).

The "number of units you sell" depends on your price and the average price of the other two firms (where the average is obtained by adding up the prices of the two other firms and dividing the resulting sum by two). In particular, the number of units you sell is given by:

[40 - 2 x (your price)] - [(your price) - (average price of the others)].

In words, two times your price is subtracted from 40; then the difference between your price and the average price of the others is

* subtracted from the resulting amount if the difference is positive (i.e., if your price is higher than the average price of the others),

* added to the resulting amount if the difference is negative (i.e., if your price is lower than the average price of the others).

Thus, the higher is your price compared to the average price of the others, the fewer units you sell. On the other hand, you sell more if the average price of the others is higher than your price.

Suppose, for example, that the prices of the other firms are 6 and 8 so that their average price is: (6 + 8)/2 = 14/2 = 7.

* If your price is 5 (<7), then the number of units you sell is: [40 - 2 x 5] - [5 - 7] = 30 + 2 = 32. Consequently, your period-profit is 5 x 32 = 160.

* If your price is 8 (>7), then the number of units you sell is: [40 - 2 x 8] - [8 - 7] = 24 - 1= 23. Consequently, your period-profit is 8 x 23 = 184.

* If your price is 10 (>7), then the number of units you sell is: [40 - 2 x 10] - [10 - 7] = 20 -3 = 17, and your period-profit is 10 x 17 = 170.

Task 2. Your second task in every period is to guess the average price of the other two firms in your current group. In every period, you must make at least one guess about their average price, and you can--if you wish to--make additional guesses. The maximum number of guesses you can make is six.

You should make as many guesses as the number of possible average prices of the others you do not want to exclude. Suppose, for instance, that you do not want to exclude that: (a) the average price of the others is 5, and (b) the average price of the others is 6.5. Then, you should make two guesses about the others' average price: (a) a first guess in which you expect the other two firms to choose, on average, 5; (b) a second guess in which you expect the other two firms to choose, on average, 6.5.

Your guesses about the average price of the others must be a number from 0 to 12. You can use up to two decimals.

Task 3. Your last task in every period is to specify the period-profit you wish to guarantee your self for each average price you guessed the others could choose.

Suppose, for instance, that you made two guesses about the others' average price. For each of these two guesses, you need to specify the period-profit you aspire to. Similarly, if you made tour guesses about the others' average price, you must specify the period-profit you aspire to for each of your four guesses.

In the following, we will refer to the period-profit you aspire to as your profit aspiration.

The Decision Aid

To help you make "satisfactory" decisions, that is, decisions achieving your aspired period-profit for each guess you made, we will provide you with a decision aid. In each period, after you have (1) chosen your price, (2) guessed the possible average prices of the others, and (3) specified your profit aspiration thr each of your guesses, the decision aid will inform you whether your stated profit aspiration(s) can be achieved or not. That is, you will learn whether, given your own price and your guesses about the others' average price, you can achieve the period-profit you aspire to for each of your guesses. The decision aid will then ask you if you want to revise your specifications in (1), (2), and/or (3).

* If you want to revise something, you have to click the "revise"-button. You will then move to a screen where you can modify your own price and/or your guesses about the others' average price and/or your profit aspirations.

* If you do not want to revise anything, you have to click the "not-revise"-button. After all participants have finished with their revisions, you will move on to the next period.

Notice that you can revise something even if your decisions were "satisfactory," that is, they allowed you to achieve your profit aspirations. In every period, you can make at most five revisions.

Period-Profit Calculator

Additionally, you have access to a period-profit calculator that calculates your period-profit for arbitrary price combinations. You can start the calculator by pressing the corresponding button on your screen. If you do so, a window will appear on your screen. Into this window you must enter two values: a price for yourself and an average price for the others. Given these figures, if you press the apposite button, you will know how much you would earn.

Your Experimental Earnings in Each Period

In each period, you can be paid according to your period-profit, your guesses about the others' average price, or your profit aspirations, where all the three modes of payment are equally likely. The randomly selected mode of payment applies to all three interacting participants, which means that you and the other two firms in your current group will be paid according to the same procedure.

If, by random choice, your payment is based on your "guesses," you will earn 180 minus 10 times the smallest difference between the average price you guessed the others could choose and the true average price of the others. In particular, the computer will

* consider your closest guess to the true average price of the others;

* take the numerical distance between your closest guess and the others' true average price;

* multiply this distance by 10;

* subtract the resulting amount from 180.

Hence, if your payment is based on your guesses, the closer your guesses are to the true average price of the others, the higher will be your period-payment.

Suppose that you made three guesses about the others' average price, which were 5, 6, and 6.5. If the true average price of the others is 7, your closest guess to 7 is 6.5. The numerical distance between 7 and 6.5 is 0.5 (i.e., 6.5 deviates from 7 by 0.5). Then, you will receive 180 - 10 x 0.5 = 175 ECU.

If, by random choice, your payment is based on your "profit aspirations," you will earn your highest achieved profit aspiration, that is, your highest aspiration that does not exceed your period-profit. In particular, the computer will check which of your profit aspirations are equal to or smaller than your period-profit. Among the profit aspirations that do not exceed your period-profit, you will earn the highest one. If all your profit aspirations exceed your period-profit, then you will earn 0 (zero) ECU.

Suppose that your period-profit is 162 ECU and you made three guesses about the others' average price so that you had to specify three profit aspirations. If your profit aspirations were 170, 160, and 150, then you earn 160 ECU because 160 is the highest aspiration that does not exceed your period-profit of 162 ECU. If, instead, your profit aspirations were 180, 172, and 170, then you earn 0 ECU because all your aspirations exceed 162 ECU.

The Information You Will Receive at the End of Each Period

At the end of each period you will be told: your price, the average price of the others, your own period-profit, your closest guess to the average price of the others, your highest achieved aspiration, your period experimental earnings.

Your Final Earnings

Your final earnings will be calculated by adding up your experimental earnings in all periods. The resulting sum will be converted to euros and paid out to you in cash in addition to the show-up fee of 2.50 [euro].

Before the experiment starts, you will have to answer some control questions to ensure your understanding of the experiment.

Please remain quiet until the experiment starts and switch off your mobile phone. If you have any questions, please raise your hand now.

REFERENCES

Abbink, K., and J. Brandts. "Collusion in Growing and Shrinking Markets: Empirical Evidence from Experimental Duopolies," in Experiments and Competition Policy, edited by J. Hinloopen and H.-T. Normann. Cambridge: Cambridge University Press, 2009, 34-60.

Aumann, R., and A. Brandenburger. "Epistemic Conditions for Nash Equilibrium." Econometrica, 63(5), 1995, 1161-80.

Aumann, R., and J. H. Dreze. "Rational Expectations in Games." American Economic Review, 98(1), 2008, 72-86.

Berninghaus, S., W. Guth, M. V. Levati, and J. Qiu. "Satisficing Search versus Aspiration Adaptation in Sales Competition: Experimental Evidence." International Journal of Game Theory, forthcoming.

Bewley, T. "Knightian Decision Theory: Part I." Decisions in Economics and Finance, 25, 2002, 79-110.

Blanco, M., D. Engelmann, A. Koch, and H.-T. Normann. "Belief Elicitation in Experiments: Is There a Hedging Problem?" IZA Discussion Paper No. 3517, Bonn, 2008.

Camerer, C., T. Ho, and J.-K. Chong. "A Cognitive Hierarchy Model of Games." Quarterly Journal of Economics, 119, 2004, 861-98.

Costa-Gomes, M. A., and V. Crawford. "Cognition and Behavior in Guessing Games: An Experimental Study." American Economic Review, 96, 2006, 1737-68.

Costa-Gomes, M. A., and G. Weizsacker. "Stated Beliefs and Play in Normal-Form Games." Review of Economic Studies, 75, 2008, 729-62.

Cyert, R., and J. March. "Organizational Factors in the Theory of Oligopoly." Quarterly Journal of Economics, 70(1), 1956, 44-64.

--. A Behavioral Theory of the Firm, Englewood Cliffs, NJ: Prentice-Hall, 1963.

Dixon, H. D. "Keeping Up with the Joneses: Competition and the Evolution of Collusion." Journal of Economic Behavior & Organization, 43, 2000, 223-38.

Dufwenberg, M., and U. Gneezy. "Price Competition and Market Concentration: An Experimental Study." International Journal of Industrial Organization, 18, 2000, 7-22.

Dufwenberg, M., U. Gneezy, J. K. Goeree, and R. Nagel. "Price Floors and Competition." Economic Theory, 33, 2007, 211-24.

Fischbacher, U. "Zurich Toolbox for Readymade Economic Experiments." Experimental Economics, 10, 2007, 171-78.

Gilboa, I. "Questions in Decision Theory." Annual Reviews its Economics. 2. 2010. 1-19.

Gilboa, I., F. Maccheroni, M. Marinacci, and D. Schmeidler. "Objective and Subjective Rationality in a Multiple Prior Model." Econometrica, 78, 2010, 755-70.

Gilboa, I., and D. Schmeidler. "Maxmin Expected Utility with a Non-Unique Prior." Journal of Mathematical Economics, 18, 1989, 141-53.

--. "Reaction to Price Changes and Aspiration Level Adjustments." Review of Economic Design, 6(2), 2001, 215-23.

Goeree, J., and C. Holt. "A Model of Noisy Introspection." Games and Economic Behavior, 46, 2004, 365-82.

Greiner, B. "An Online Recruitment System for Economic Experiments," in Forschung und wissenschaftliches Rechnen 2003. GWDG Bericht, Vol. 63, edited by K. Kremer and V. Macho. Gottingen: Gesellschaft fur Wissenschaftliche Datenverarbeitung, Ges. fur Wiss. Datenverarbeitung, 2004, 79-93.

Guth, W. "Satisficing in Portfolio Selection--Theoretical Aspects and Experimental Tests." Journal of SocioEconomics, 36, 2007, 505-22.

--. "Satisficing and (Un)Bounded Rationality: A Formal Definition and Its Experimental Validity." Journal of Economic Behavior & Organization, 73, 2010, 308-16.

Guth, W., M. V. Levati, and M. Ploner. "An Experimental Analysis of Satisficing in Saving Decisions." Journal of Mathematical Psychology, 53, 2009, 265-72.

Guth, W., M. V. Levati, and M. Ploner. "Satisficing in Strategic Environments: A Theoretical Approach and Experimental Evidence." Journal of Socio-Economics, 39, 2010, 554-561.

Huck, S., K. A. Konrad, W. Muller, and H.-T. Normann. "The Merger Paradox and Why Aspiration Levels Let It Fail in the Laboratory." Economic Journal, 117, 2007, 1073-95.

Knight, F. H. Risk, Uncertainty and Profit. Chicago: University of Chicago Press, 1985 (first printed in 1921).

Mckelvey, R., and T. Palfrey. "Quantal Response Equilibrium for Normal Form Games." Games and Economic Behavior, 10, 1995, 6-38.

Oechssler, J. "Cooperation as a Result of Learning with Aspiration Levels." Journal of Economic Behavior & Organization, 49, 2002, 405-09.

Sauermann, H., and R. Selten. "Anspruchsanpassungstheorie der Unternehmung." Zeitschrift fur die gesamte Staatswissenschaft, 118, 1962, 577-97.

Selten, R. "Aspiration Adaption Theory." Journal of Mathematical Psychology, 42, 1998, 191-214.

--. "What is Bounded Rationality'?" in Bounded Rationality: The Adaptive Toolbox, edited by G. Gigerenzer, and R. Selten. Cambridge, MA: MIT Press, 2001, 13-36.

Simon, H. A. Administrative Behavior: A Study of DecisionMaking Processes in Administrative Organization, New York: Macmillan, 1947.

--. "A Behavioral Model of Rational Choice." Quarterly Journal of Economics, 69, 1955, 99-118.

Van Damme, E. "Strategic Equilibrium," in Handbook of Game Theory, Vol. 3, Chapter 41, edited by R. Aumann and S. Hart. Amsterdam, The Netherlands: North Holland, 2002.

Weizsacker, G. "Ignoring the Rationality of Others: Evidence from Experimental Normal Form Games." Games and Economic Behavior, 44, 2003, 145-71.

(1.) Cyert and March (1956, 1963) were the first to apply the concept of aspiration level to oligopoly theory. More recent theoretical studies on aspiration-based models of firm behavior include Dixon (2000) and Oechssler (2002) who both investigate how behavioral rules based on aspiration levels can induce collusion in Cournot games.

(2.) See also Selten (2001) for a survey of heuristics and bounded rationality ideas.

(3.) The reason why we use the term "conjecture" instead of "expectation" or "belief' will be explained later.

(4.) For more recent discussions on this issue, see Gilboa (2010) and Gilboa et al. (2010).

(5.) Gilboa and Schmeidler (1989) couple the set of priors with a decision rule that chooses an action whose minimal expected utility (over all priors in the set) is the highest. Bewley (2002) uses the set of priors to define a partial order over actions: he suggests to prefer an action to another if and only if its expected utility is higher for each and every prior in the set.

(6.) To the best of our knowledge, the only previous experimental studies investigating satisficing behavior in oligopoly markets are by Huck et al. (2007) and Berninghaus et al. (forthcoming). Huck et al. (2007) analyze how aspirations may lead to a failure of the merger paradox in the laboratory without eliciting aspiration levels. Berninghaus et al. (forthcoming) test the absorption of satisficing in duopoly Cournot markets by directly asking participants to form profit aspirations.

(7.) Other solution concepts that coarsen the Nash equilibrium are objective correlated equilibrium, subjective correlated equilibrium, and rationalizability (Van Damme 2002; Aumann and Dreze 2008). The main difference between these concepts and prior-free optimality is that the latter does not require the specification of any probability distribution over the other players' actions.

(8.) This should isolate the effects of experience from the opportunities of tacit collusion that may occur in a repeated game. See, e.g., Abbink and Brandts (2009) for an experimental study of collusive behavior in a homogeneous market with partners design.

(9.) We do not intend to propose an incentive compatible method of eliciting aspirations, that is, a method that induces participants to truthfully reveal their aspirations. We simply want that one should not aspire to profits lower than those allowed by [p.sub.i] and [C.sub.i]. Asking subjects to specify aspirations without payoff consequences makes them more likely to invest little effort in the aspiration formation and adaptation tasks (see, e.g., the survey by Guth 2007 or Guth et al. 2009). Paying for the highest realized aspiration should encourage subjects to think carefully about the problem and to comply with satisficing. It also matches the implication of aspirations in the satisficing approach: one is "satisfied" if aspirations are met (aspired profits are not greater than realized ones) while one is "unsatisfied" if aspirations are not met (aspired profits are greater than realized ones).

(10.) All reported nonparametric tests are two-sided and (unless otherwise stated) rely on the averages over players for each matching group. Because of our rematching system. the numbers of statistically independent groups are 9 in each period.

(11.) The meaning of the x dots and the horizontal lines is as in Figure 1.

(12.) To avoid misclassifications originating in participants' rounding, the computations assessing satisficing behavior (as well as prior-free optimality) were performed by rounding numbers to the first integer.

(13.) of course, this is a hindsight-driven, ad hoc explanation. A priori, one could have expected seller participants to try out a new price strategy whenever some of their aspirations were not achievable.

(14.) The observed decrease in the dispersion of conjectured prices does not therefore lead to an increase in type 1-deviations. Actually one may suspect that subjects with less dispersed conjectures would be less likely to exhibit type 1-deviations than others because they are more confident about their competitors' average price. However, when seller participants specify a set-valued conjecture with a coefficient of variation lower than 0.133 (the median value), they choose a rationalizable price in 30.99% of the cases. When the coefficient of variation is equal or higher than 0.133, they choose a rationalizable price in 42.04% of the cases.

(15.) A WSRT comparing [[bar.[pi]].sup.U.sub.i]([p.sub.i], [c.sub.i]) in the first and the last period delivers p = 0.469.

WERNER GOTH, MARIA VITTORIA LEVATI, MATTEO PLONER *

* We thank Birendra Kumar Rai and Dominique Cappelletti for helpful comments.

Guth: Director, Max Planck Institute of Economics. Strategic Interaction Group, Kahlaische Strasse 10, Jena D07745, Germany. Phone 49-3641-686620, Fax 49-3641686667, E-mail gueth@econ.mpg.de

Levati: Research Group Leader, Max Planck Institute of Economics, Strategic Interaction Group, Kahlaische Strasse 10, Jena D-07745, Germany. Phone 49-3641686629. Fax 49-3641-686667, E-mail levati@econ. mpg.de

Ploner: Research Associate, Department of Economics-CEEL. University of Trento, Via Inama 5, Trento 38100, Italy. Phone 39-461-883139, Fax 39-461-882222, E-mail matteo.ploner@unitn.it

TABLE 1 Percentage of Subjects Complying with [[bar.p].sub.-1]; [member of] [convC.sub.I] in Each Period Period 1 2 3 4 5 6 7 % Subjects 62.96 71.61 79.01 70.37 80.25 76.54 72.84 Period 8 9 % Subjects 79.01 75.31 TABLE 2 Revisions and Satisficing Behavior Period 1 2 3 4 5 6 7 Subjects finally satisficing (%) i. 98.77 96.30 98.77 97.53 98.77 97.53 96.30 Subjects satisficing at first attempt (%) ii. 83.95 90.12 93.83 87.65 93.83 93.83 95.06 Subjects revising (%) iii. 28.40 24.69 16.05 14.82 11.11 14.82 6.17 Subjects revising among those not satisficing at first attempt (%) iv. 92.31 100.00 100.00 90.00 100.00 80.00 50.00 Subjects revising among those satisficing at first attempt (%) v. 16.18 16.44 10.53 4.23 5.26 10.53 3.90 Average number of revisions vi. 1.22 1.25 1.39 1.25 1.44 1.08 1.60 Period 8 9 Subjects finally satisficing (%) i. 97.53 97.53 Subjects satisficing at first attempt (%) ii. 96.30 97.53 Subjects revising (%) iii. 4.94 3.70 Subjects revising among those not satisficing at first attempt (%) iv. 66.67 0.00 Subjects revising among those satisficing at first attempt (%) v. 2.56 3.80 Average number of revisions vi. 1.25 1.00 TABLE 3 Deviations from and Compliance with Prior-Free Optimality Period 1 2 3 4 5 6 7 8 9 Type 1-deviation (% Subjects) i. 72.84 62.96 69.14 64.20 61.73 61.73 56.79 56.79 53.09 Type 2-deviation (% Subjects) ii. 22.22 27.16 23.46 25.93 25.93 23.46 27.16 28.40 33.33 Prior-free optimality (% Subjects) iii. 3.70 6.17 6.17 7.41 11.11 12.35 12.35 12.35 11.11 Average unexhausted profit potential in type 2-deviations iv. 5.50 4.70 3.60 4.30 5.60 5.70 4.50 4.30 4.00

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Author: | Guth, Werner; Levati, Maria Vittoria; Ploner, Matteo |
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Publication: | Economic Inquiry |

Geographic Code: | 1USA |

Date: | Apr 1, 2012 |

Words: | 10661 |

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