# Incomplete contracting: A laboratory experimental analysis.

I. INTRODUCTION

Williamson [1975] and Klein, Crawford, and Alchian [1978], among others, pioneered the notion that long-term contracts arise as a means of governing and protecting valuable investments in long-term trading relationships. Contract incompleteness results when some terms are left unspecified and usually is a result of practical difficulties in specifying contingent responses to unforeseen future states of the world. Macaulay [1963] finds that incomplete contracts are common in business dealings, and Holmstrom and Hart [1985] argue that "incompleteness is probably at least as important empirically as asymmetric information as an explanation for departures from 'ideal' Arrow-Debreu contingent contracts." A substantial formal literature has built on this pioneering research and includes work by Grossman and Hart [1986], Tirole [1986], Crawford [1988], Hart and Moore [1988], Fudenberg, Holmstrom, and Milgrom [1987], Milgrom and Roberts [1991], and Hart and Holmstrom [1987]. This formal literature has distilled the earlier notions down to a standard two-period model that is used as a foundation to much of contemporary research on contracting and the firm.(1)

To motivate this model, consider a single buyer and seller of some intermediate good, such as computer software. They anticipate innovative change in the design of the intermediate good, but cannot foresee the precise nature of the innovation. As a result the contract cannot be made contingent on the change in design. The benefit of the design change to the buyer is uncertain, but the buyer can make a transaction-specific investment in flexibility that increases the likelihood that the benefit will be large. Similarly the cost of implementing the design change is uncertain, but the seller can make a transaction-specific investment that increases the likelihood that it will be small. In the second stage the innovation is realized. The buyer and seller then enter negotiations, the outcome of which determines whether the innovation will be implemented and how the surplus added by the innovation will be divided.

This two-stage model is solved using the technique of backward induction. First a game theoretic solution is imposed on second-stage negotiations, and this solution determines how joint surplus is shared. This division of joint surplus is assumed to be foreseen in the first stage and is incorporated into each party's expected profit function. Equilibrium is then found when each (risk neutral) party chooses their own expected profit maximizing level of transaction-specific investment. One potentially testable implication of this model is that the future division of joint surplus directly affects the level of transaction-specific investment, but the level of transaction-specific investment does not affect the division of joint surplus.

The bargaining literature has progressed rapidly in the development of non-cooperative bargaining models with unique equilibria that are testable using laboratory procedures. Influential non-cooperative bargaining models include those of Rubinstein [1982], Binmore [1985], and Binmore, Rubinstein, and Wolinsky [1986]. Accumulated evidence from laboratory experiments on bargaining such as in Guth, Schmittberger, and Schwarz [1982], Guth and Tietz [1990], Ochs and Roth [1989], and Neelin, Sonnenshein, and Spiegel [1988], suggests that non-strategic factors may have a substantial effect on bargaining outcomes. Much of the experimental work has been with ultimatum games, where one player submits a share proposal to another, who can either accept or force breakdown. When breakdown results in zero payoffs, the (subgame-perfect) equilibrium proposal grants very nearly 100 percent for the person making the proposal and the residual to the person accepting the proposal. Laboratory experimental studies of these games are usually able to reject this hypothesis. These studies typically find that allocations tend to fall somewhere between equal splits and the hypothesized equilibrium allocation. It has been suggested that players may perceive these games as being inherently 'unfair,' and that this perception is reflected in the failure of the equilibrium prediction.

These experimental studies suggest that the bargaining process nested in incomplete contracts may be substantially different from that assumed in the standard bargaining models from which contract equilibria are derived. In particular, consider a situation in which one party has made a transaction-specific investment that is large relative to that made by the other party. Imposition of the subgame-perfect or Nash bargaining outcome implies that transaction-specific investments have no influence on the outcome of bargaining. Hence the party making the greater investment will receive a smaller net payoff. This is directly tested in the present experimental study by varying the relative investment incentives induced on buyers and sellers, and then seeing whether bargaining outcomes are influenced by differences in transaction-specific investment. As in ultimatum game studies, bargainers may instead choose outcomes that attenuate the difference in their net payoffs. The testable implication is that the greater is a player's relative investment level, the greater is his share of surplus.

The results reported here strongly reject the hypothesis that differences in transaction-specific investment have no influence on the outcome of bargaining in incomplete contracts. instead, bargaining outcomes are significantly influenced by parties' transaction-specific investment. The party that made the greater transaction-specific investment is found to receive a greater share of realized surplus in the bargaining process, although the effect is generally not strong enough to equate net payoffs. The results are highly consistent with the experimental studies of the ultimatum game. These results are also important to the theoretical literature on incomplete contracts; when the bargaining process is used to attenuate differences in net payoffs, the result can be an increase in expected joint surplus over that which would occur under subgame-perfect bargaining.

II. INCOMPLETE CONTRACTING IN THE LABORATORY

The Model

A simple two-stage incomplete contracting model was created for laboratory experimentation based on the archetypical model described by Holmstrom and Hart [1985], Tirole [1988], and Holmstrom and Tirole [1989]. Following Tirole [1988], consider a situation in which the buyer and the seller have agreed to trade one unit of some good in the future. They agree to a basic design for the good, but cannot contract over a quality-enhancing product improvement that is anticipated in the second stage. The joint return to the basic design is normalized to zero, and so the analysis that follows (as well as the subject's decision environment) is based on the added valuation and cost of the product improvement.

The incomplete contracting process is outlined in Figure 1 and described in detail below. In the first stage the buyer's induced value (V) of the product improvement is uncertain and can either be "high" ([V.sub.h]) or "low" ([V.sub.l). Similarly the seller's induced cost (C) of the product improvement can either be "low" ([C.sub.l]) or "high" ([C.sub.h]). The process begins with the buyer and seller non-cooperatively making sunk investments that increase expected value and cost. In the first stage the buyer chooses the likelihood that value is high by choosing a number 0 [is less than or equal to] X [is less than or equal to] 1 at (sunk) cost [X.sub.2]. Likewise the seller chooses the likelihood that cost is low by choosing a number 0 [is less than or equal to] Y [is less than or equal to] 1 at (sunk) cost [Y.sup.2].

After the buyer and seller choose X and Y in the first stage, the second stage begins with the realization of value and cost. Value is determined by comparing X to a random number uniformly distributed over the unit interval; if X is greater than or equal to the random number, then value is high, and otherwise value is low. Similarly, cost is determined by comparing Y to a random number uniformly distributed over the unit interval; if Y is greater than or equal to the random number, then cost is low, and otherwise cost is high. X, Y, V, and C then become common knowledge, and buyers and sellers must decide whether or not to bargain over shares of realized surplus (V-C).(2) If both the buyer and the seller agree, they bargain over how to share realized surplus. Successful bargaining yields a buyer share s and a seller share (1-s) of realized surplus, where 0 [is less than or equal to] s [is less than or equal to] 1. If one or both parties rejects bargaining, or if attempted bargaining breaks down, the disagreement surplus allocation gives V to the buyer, with C incurred by the seller.

After the bargaining decision either the buyer or the seller can decide to "veto" the product improvement, in which case both the value and the cost of the improvement fall to zero. The seller can credibly commit to a veto if the surplus realization is not shared, and this possibility motivates the buyer to bargain. To see this, note that as long as the (non-negative) surplus allocation is shared, neither the buyer nor the seller will choose to veto, as a veto would reduce either of their payoffs. On the other hand, if the surplus realization is not shared, the seller can avoid the cost of producing the improvement by choosing to veto. Knowing this, buyers will always agree to bargain, because failure to bargain will trigger a seller veto and so a loss of at least a share of the surplus realization. Were it not for the seller's ability to veto, the buyer would not have an incentive to bargain and share the surplus realization. This is the role of vertical integration in the influential Grossman and Hart [1986] model. The implication is that bargaining and no veto are the equilibrium of the second-stage sub-game.

Parties foreseeing the outcome of second-stage bargaining can then compute their expected payoff maximizing first-stage investment. The research question posed here is whether bargaining outcomes are independent of the relative level of transaction-specific investment ([X.sup.2] and [Y.sup.2]) made by the buyer and the seller. In order to test this hypothesis here one must first be specific about the bargaining institution and the equilibrium bargaining outcome that it implies.

The laboratory bargaining setting is based on a variant of the Rubinstein [1982] procedure made operational for laboratory research. In the Rubinstein procedure the parties alternate in making proposals indefinitely until a proposal is accepted or the negotiations break down. Some mechanism is required to give parties an incentive to reach early agreement. The particular mechanism employed here was most recently described in Binmore et al. [1991]. Upon rejection of a proposal, a random move determines whether another round of negotiations will be allowed or the bargaining game will end. This "forced breakdown" sequential bargaining institution was used because it has a unique subgame-perfect bargaining equilibrium that also corresponds to the Nash bargaining solution. An important property of this equilibrium is that the allocation of realized surplus is independent of the relative levels of sunk, transaction-specific investment committed to by the buyer and the seller.

To see this, suppose first that sunk costs are ignored and (V-C) is the object of bargaining with disagreement payoffs of zero and zero. Then as the probability [Delta] that bargaining will continue if a proposal is rejected approaches 1.0, the Rubinstein model implies that the buyer's share s of (V-C) approaches 0.50, which is also the Nash solution with status quo located at (0,0). The buyer's net payoff is then (V-C)/2 - [X.sup.2], and the seller's net payoff is (V-C)/2-[Y.sup.2]. Suppose instead that sunk costs are not ignored, so that the object of bargaining is (V-C-[X.sup.2]-[Y.sup.2]) and the buyer and seller's disagreement payoffs are [-X.sup.2] and [-Y.sup.2]. Following Binmore, Shaked, and Sutton [1989], the unique subgame-perfect equilibrium of this bargaining game has the first proposer (say the buyer) offering a share of (V-C-[X.sup.2]-[Y.sup.2]) equal to [[Delta](1-B) + S](1+[Delta]) to the other, who then accepts it (B = [-X.sup.2]/(V-C-[X.sup.2]-Y), and S = [-Y.sup.2]/(V-C-[X.sup.2] [-Y.sup.2]).(3) As [Delta] approaches 1.0 the buyer's equilibrium net payoff approaches (V-C)/2-[Y.sup.2], while the seller's approaches (V-C)/2-[Y.sup.2]. But these net payoffs are identical to the equilibrium of the original bargaining model that omits sunk costs. Hence the subgame-perfect equilibrium ignores sunk costs, as is assumed in the incomplete contracting model.

The subgame-perfect equilibrium share s* in turn affects investment incentives in the first stage of the incomplete contracting process. It is useful here to establish a theoretical benchmark. If V and C are independent random variables, and if the buyer and the seller are risk-neutral and capable of perfect foresight, then the buyer will choose X* to maximize (1) s* {X[Y([V.sub.h] - [C.sub.l]) + (1 - Y)([V.sub.h] - [C.sub.h])] + (1-X)[Y([V.sub.l] - [C.sub.l]) + (1- Y)([V.sub.1] - [C.sub.h])]} - [X.sup.2]

The first term is simply the buyer's share of the expected value of (V-C). The seller will similarly choose Y* to maximize (2) (1-s*) {Y[X([V.sub.h] - [C.sub.l]) + (1-X)([V.sub.1] - [C.sub.l])] + (1-Y)[X([V.sub.h] - [C.sub.l]) + (1-X)([V.sub.1] - [C.sub.h])} [Y.sup.2], where the first term is the seller's share of the expected value of (V-C). Maximizing equations (1) and (2) with respect to X and Y yields: (3) X* = s* ([V.sub.h] - [V.sub.l]) / 2, (4) Y* = (1-s*) ([C.sub.h] - [C.sub.l]) / 2.

This benchmark analysis illustrates the one-way influence that is a property of the incomplete contracting model. The bargaining solution s* directly affects the buyer's and the seller's investment decisions, but the investment decisions do not affect the bargaining solution.

Experimental studies of the ultimatum game have consistently found that the subgame-perfect bargaining equilibriurn fails to predict surplus allocations. In a one-round ultimatum game there is a valuable "cake" to be divided by two players. The payoffs from the game are derived from division of the cake K, which can most easily be thought of as a cash sum. The proposer is chosen from some exogenous selection process and will be referred to as player 1. Player 1 proposes a division of the cake to player 2, who can either accept or reject the proposal. If player 2 accepts, each receives the proposed share of the cake as their payoff. If player 2 rejects, the cake is destroyed and each player receives a zero payoff. Let k be the smallest positive unit of money. If player 2 would not accept a zero share, the optimal proposal by player 1 would yield player 2 a payoff of k, as it is optimal for player 2 to accept any positive share of the cake (otherwise a zero share can also be a subgame-perfect equilibrium).

In the ultimatum game described above, the subgame-perfect equilibrium proposal yields (K - k) to player 1, and k to player 2. Laboratory experiments on ultimatum bargaining by Guth, Schmittberger, and Schwarz [1982], Guth and Tietz [1990], Ochs and Roth [1989], and Neelin, Sonnenshein, and Spiegel [1988], have generally been able to reject the subgame-perfect prediction. The accumulated evidence suggests that non-strategic factors may have a substantial effect on bargaining outcomes. Observed allocations tend to locate somewhere between an equal split and the equilibrium prediction of nearly 100 percent to the proposer. Such behavior is consistent with players reducing the disparity in net payoffs. In terms of the present paper, suppose that (V.sub.h] - [V.sub.l]) > ([C.sub.h] - [C.sub.l]). If the hypothesis of subgame-perfect bargaining holds so that s = 0.5, then from equations (3) and (4) we have X* > Y*, which implies that both expected and actual buyer payoffs are smaller that of the seller. In fact the greater is the difference between X* and Y*, the more pronounced is this effect.

As in ultimatum games, bargainers here could attenuate this net payoff difference by allocating a greater share of realized surplus to the buyer. Using the bargaining process to attenuate differences in net payoffs implies a buyer share [s.sup.e]=.5 + f ([X.sup.2] - [Y.sup.2]), where f' (.) > 0, and f (0) = 0. The form that f (.) might take is an empirical question. While the ultimatum game studies do not generally find equal net payoffs, at the limit bargainers could equalize net payoffs by choosing a buyer share [s.sup.e] = .5 + ([X.sup.2] - [Y.sup.2])/2(V - C). A central focus of this paper is to test the subgame-perfect hypothesis when bargaining is preceded by strategically irrelevant sunk investments, as is the case in incomplete contracts.

Laboratory Parameterization of the Model

Of central importance here is that [s.sup.e] depends directly on ([X.sup.2] - [Y.sup.2]), while s* does not. If X = Y then the buyer's net payoff is equal to the seller's net payoff, and s* cannot be distinguished from [s.sup.e]. On the other hand, if X is not equal to Y, then the relative predictive power of the two hypotheses can be directly tested by evaluating the relationship between s and ([X.sup.2] - [Y.sup.2]). Three treatment conditions were devised that differed in the relative investment incentives induced on buyers and sellers. This is most easily done by varying the size of ([V.sub.h] - [V.sub.l]) relative to ([C.sub.h] - [C.sub.l]). To make the description of the treatment conditions that follows more manageable, let [D.sub.v] stand for ([V.sub.h] - [V.sub.l]), and let [D.sub.c] stand for ([C.sub.h] - [C.sub.l]).

A baseline condition was devised in which V can take on the values of $5 or $3, while C can take on the values of $2.50 or 50 cents. As a result [D.sub.v] = [D.sub.c] = $2. From equations (3) and (4) of the benchmark model X* = Y* = 0.50, implying that under the conditions of the benchmark model s* equalizes net payoffs (the benchmark buyer and seller net payoff is $1 each in expectation). Since the induced investment incentives are equal in this treatment, it will be referred to as the Equal Investment Incentives treatment.

Two additional treatment conditions were devised with the idea of inducing widely different investments on the part of the buyer and the seller. In one of these, relatively strong buyer incentives are induced by having [D.sub.v] > [D.sub.c]. In this treatment condition V can take on the values of $5.50 or $2.50, implying that [D.sub.v] = $3, while C can take on the values of $2.10 or $2, implying that [D.sub.c] = 10 cents. From equations (3) and (4) of the benchmark model X* = .75 and Y* = .025, implying that the benchmark expected net payoff is $0.76 for the buyer and $1.32 for the seller. At a buyer share of .64 the buyer will choose X = .96, the seller will choose Y = .019, and they will receive an expected payoff of $1.18 each. The implication is that in this Strong Buyer Incentives treatment the subgame-perfect share s* results in a greater payoff to the seller, and s* attempts to use the bargaining process to reduce the difference in net payoffs will require a buyer share [s.sup.e] > s*.

In the symmetric Strong Seller Incentives treatment condition, relatively strong seller incentives are induced by having [D.sub.c] > [D.sub.v]. In this treatment condition V can take on the values of $5 or $4.90, implying that [D.sub.v] = 10 cents, while C can take on the values of $4.50 or $1.50, implying that [D.sub.c] = $3. Under the subgame-perfect bargaining equilibrium hypothesis, s* =.5. From equations (3) and (4) of the benchmark model, X* = .025 and Y* = .75, and so the benchmark expected net payoff is $1.32 for the buyer and $0.76 for the seller. At a buyer share of .36 the buyer will choose X=.019, the seller will choose Y = .96, and each will receive an expected net payoff of $1.18. The implication is that s* yields greater net payoffs for the buyer, and so attempts to reduce this difference in net payoffs will require a buyer share [s.sup.e] < s*.

Discussion. The benchmark model suggests that the treatment conditions are likely to generate the desired variation in buyer and seller investment incentives. The goal is not to test the benchmark model, but instead to test the hypothesis that the outcome of bargaining is independent of the buyer's and seller's relative investments. An interesting implication of the incomplete contracting model is that expected joint surplus is greater under [s.sup.e] than under s*. To see this, consider the Strong Buyer Incentives condition (a parallel argument holds for the Strong Seller Incentives Condition). An increase in X (up to the maximum of 1.0) will always increase expected joint surplus more than an equal increase in Y, since [D.sub.v] > [D.sub.c]. Hence expected joint surplus is maximized if the buyer's share s = 1.00, in which case the buyer will choose X = 1 and the seller will choose Y = 0. As a result, when s rises from s* to [s.sup.e] in the Strong Buyer Incentives Condition, net payoff differences fall and expected joint surplus rises, but the change does not satisfy the Pareto criterion. These properties of the model were first described in the Grossman and Hart [1986] study of vertical integration.(4)

Participant's Decision Environment

Subjects were recruited from economics and business economics courses at Indiana University and had no prior experience with the experiment.(5) Eight subjects were used in each experiment, except for the first repetition of the baseline condition which used six subjects due to an unexpectedly high number of no-shows. The experimental setting was computerized and consisted of ten rounds of the two-stage incomplete contracting game played under one of the three treatment conditions described above.(6) Subjects were first seated at their computer terminals and read the experiment instructions. Subjects then answered a questionnaire that required a detailed understanding of the experiment, and were allowed to use their instructions as a reference. (The instructions and questionnaire for the baseline condition are provided as an appendix.) After completing the questionnaire, subjects were given a $4 initial cash balance and then participated in ten rounds of the incomplete contracting game. The experimenter was available for answering procedural questions. In this experiment inexperienced sellers can lose up to C + [Y.sup.2] per round if they are unfamiliar with the experimental procedures (e.g., they don't understand how to veto). To help subjects become familiar with the experimental procedures, and following Binmore et al. [1991], decisions were made without cash reward in early (the first two) rounds. Participants received total cash rewards ranging from $8 to $16, and the experiments lasted approximately two hours.

At the start of each round the computer program randomly and anonymously paired subjects. One was randomly assigned to have the identity of a buyer, and the other a seller.(7) After subjects were matched, the first stage began. Before subjects chose X or Y, they provided forecasts of (i) whether bargaining would occur, and if so, what the other party's investment and the outcome of bargaining would be, and (ii) whether a veto would occur. The software then used these forecasts to compute equation (1) for buyers, or (2) for sellers, for each possible X or Y value. Recall that parties are assumed to know their expected payoffs in the model, and by eliciting forecasts from subjects no beliefs are externally imposed regarding the other party's investment, bargaining outcomes, or veto decisions. Moreover, forecasts were not binding in any way; subjects were free to choose any value of X or Y between 0 and 100. As a result, subjects could see the investment cost ([X.sup.2 or [Y.sup.2]) and their forecast-contingent expected second-stage payoff for that round, for all possible X or Y values they could choose.

At the beginning of the second stage, V and C were determined after the random number n was realized and compared to X and Y. The n drawn in each of the ten rounds were

.04,.65,.38,.38,.02,.60,.72,.12,.87,.21. These were set in advance, were the same for each experiment, and given this sample one cannot statistically reject the hypothesis that n was randomly drawn from the unit interval. X, Y, n, V, and C were then revealed to both the buyer and the seller.(8) Next each subject was asked whether or not they wanted to bargain over (V-C). If either responded negatively, bargaining did not occur, and (V-C) was not shared. In this case the buyer gets V and the seller pays C.

If both parties agreed to bargain, they then proceeded into the sequential bargaining procedure described in Binmore et al. [1991]. Buyers and sellers alternated in making sharing rule proposals (0 [is less than or equal to] s [is less than or equal to] 1) indefinitely until a proposal is accepted or negotiations break down. In odd-numbered contracting rounds buyers made the first offer, while in even-numbered periods sellers offered first. Once an offer was made, the other party could then accept it or reject it. Rejection of a share offer leads to a chance that negotiations will exogenously be ended and no sharing will occur in that round. Binmore et al. used the technique of fixing the number of allowed rejections in advance, but not telling subjects what the length was.(9) The allowed number of rejections for each of the ten rounds both here and in their experiment were:

9, 2, 11, 2, 10, 7, 7, 16, 12, 8. Binmore and his co-authors state that the two short games at the start of each experiment were intended to convince subjects that a forced end to bargaining could occur, and that otherwise one could not statistically reject the hypothesis that upon rejection a breakdown occurs independently with probability 0.1. Given the discount rate of 0.1, the unique subgame perfect equilibrium has a first offer of .474 accepted, which approximates the .50 predicted as the discount rate converges to 1.00.

Subjects next independently decided whether or not to veto. Subjects were told that a veto implied that the hypothetical product improvement would not occur, and so the value V and cost C associated with implementing the product improvement became $0. After the veto decision, each subject realized their payoff for the round based on their investment, bargaining, and veto decisions. Cash payoffs from the round were then added to (or subtracted from, if negative) their previous cash balance. Finally, subjects were given feedback information that allowed them to compare their forecasts with the actual outcomes of the incomplete contracting process from that round.

III. ANALYSIS AND DISCUSSION OF EXPERIMENTAL RESULTS

The experimental design features three repetitions of each of the three parameter regimes described in section II above. To attenuate any added complication of learning effects, and following Binmore et al., I restrict the analysis reported below to data from later rounds of each experiment.(10) In particular, the analysis below is based on 175 observations generated in the last five rounds of each of the nine experimental sessions.

Recall that both the buyer and the seller must agree to enter the bargaining process; if one or both did not agree to bargain, the result is the same as when bargaining breaks down--the buyer retains V and the seller pays C. Bargaining is predicted because buyers know that sellers have a credible threat of vetoing, and so setting V = C = $0 if buyers do not agree to bargain. It is instructive to note that this threat appears to have been quite credible in the experiment, as shown in the contingency tables reported in Table I. Buyers failed to agree to bargain a total of ten times, and sellers subsequently vetoed in each case; none of these buyers who tested sellers' resolve repeated their test in later rounds. An excess of rejected offers lead to forced breakdown eight times, and again sellers consistently vetoed in each case. In the second replication of the Strong Seller Incentives treatment a buyer in round 9 agreed to bargain, but the seller refused ((V-C) was 40 cents), and subsequently vetoed. The reason for this seller's behavior is unknown. [TABULAR DATA I OMITTED]

A total of 156 successful bargaining outcomes remain, and in each case the seller did not veto. The results are consistent with buyers generally foreseeing that sellers would veto if bargaining did not occur, and with sellers vetoing when buyers occasionally decided to test sellers' resolve. Analysis of this data follows.

The Data

Investment and bargaining outcome data from each of the three parameter regimes are summarized in Figures 2 through 4. A statistical analysis follows below. It can be seen in Figure 2 that buyer and seller investment in the Equal Investment Incentives treatment is somewhat larger than 0.50 predicted by risk-neutral expected payoff maximization. The buyer's share of (V-C) is clustered around the subgame-perfect prediction of 0.50.

Investment and bargaining outcomes in the Strong Seller Incentives treatment are presented in Figure 3, and reflect sellers' relatively strong induced incentives for investment. Observed values of X never exceed 0.40, and the majority of observations fall between 0.00 and 0.10. On the other hand, only two observations on Y are less than 0.55, and the majority of observations fall between 0.65 and 0.90. In aggregate, the result is that sellers have made a much greater sunk investment relative to buyers. There is a corresponding downward bias in the buyer's share of (V-C); the majority of observations fall between a share of 0.30 and 0.50, with only three observations slightly exceeding .50.

In marked contrast, outcomes from the Strong Buyer Incentives treatment presented in Figure 4 reflect buyers' relatively strong induced incentives for investment. All observed values of X fall between 0.5 and 1.0, and the majority lie between 0.7 and 0.9. Symmetrically, most of the observed values of Y lie between 0.0 and 0.15. In this case nearly all observed s fall between 0.5 and 0.65, which is consistent with the equitable hypothesis, given the relationship between X and Y.

Statistical Analysis

Casual analysis of the data indicates that the investment and bargaining data differ across the three parameter regimes. The question is whether there is enough evidence to reject the hypothesis that the data sets from the three treatments are generated by the same statistical process. This is evaluated using two-sample, nonparametric testing methods, the results of which are reported in Table II. The null hypothesis is that samples of observations on X, Y, and s from any two of the treatments are generated by the same stochastic process. [TABULAR DATA II OMITTED]

Two tests commonly used in this context are the Wilcoxon and the Kolmogorov-Smirnov. In the Wilcoxon test procedure both samples are combined into a single ordered sample, and then ranks are assigned to the sample values from the smallest to the largest. The ranks are then summed for each of the two samples, and the difference is then taken between these sums. The null hypothesis is rejected if in absolute value the difference is sufficiently large. In the Kolmogorov-Smirnov test one first computes the sample cumulative distribution functions and then the maximum of the absolute value of the difference between the distribution functions. If this difference is sufficiently large in absolute value, the null hypothesis is rejected.

The nonparametric test reports are presented in Table II. The Wilcoxon value is given under the heading labeled "Z," and the significance level at which the null hypothesis of no difference in distribution is retained or rejected is given under the, heading "Probability > ~Z~." Similarly, the Kolmogorov-Smirnov value is listed under the heading "K," and the significance level at which the null is retained or rejected is listed under the heading "Probability > K."

The tests reject the null hypothesis of no difference in distribution of all three variables at well below the 1 percent level. Moreover, there is little meaningful difference in the results of the Wilcoxon and the Kolmogorov-Smirnov tests. The results for buyer investment in X and seller investment in Y reflect the variation in induced investment incentives across the three parameter regimes. In particular, larger X and smaller Y are observed in the ([Vh.sub.h] - [Vh.sub.l]) > ([C.sub.h] - [C.sub.l]) regime relative to the ([Vh.sub.h] - [Vh.sub.l]) < ([C.sub.l], - [C.sub.l]) regime. Investment in X and Y are decisions made under uncertainty, however, and so without knowing the subject's risk attitudes and beliefs one must be cautious in comparing outcomes to those predicted by the perfect foresight, risk-neutral model. Some discussion of the relationship between predicted and actual investment will, however, be made in the following section.

A central question addressed in this paper regards the appropriateness of imposing exogenous bargaining solutions on the nested bargaining process that occurs in incomplete contracts. The three treatment conditions feature widely different investment incentives, the purpose of which is to see if relative investment levels affect observed bargaining solutions. Both the Wilcoxon and the Kolmogorov-Smirnov tests strongly reject the null hypothesis that the sample of observations on X, Y, and s from different treatments are generated by the same stochastic process. In the Strong Seller Incentives treatment the mean s is .425, while in the baseline the mean s is .542, and in the Strong Buyer Incentives treatment the mean of s is .595. This variation in X, Y, and s across treatment condition suggests a possible relationship between relative investment levels and the sharing rule for realized surplus.

The relationship between s and relative investment levels is estimated using ordinary least squares regression analysis, the results of which are presented in Table III. Recall that buyers invest in X at cost [X.sup.2], and sellers invest in Y at cost [Y.sup.2]. The independent variable ([X.sup.2] - [Y.sup.2]) captures the difference in sunk costs from transaction-specific investment. The constant term establishes the point estimate on s when there is no difference in transaction-specific investment.

The regression analysis strongly supports the notion that the relative level of sunk, transaction-specific investment affects the outcome of future bargaining in incomplete contracts. The constant term is estimated to be .515, and has a remarkably small standard error of .0056. This is very close to the .50 Nash bargaining solution.(11) A central result of the analysis is that the difference in sunk cost ([X.sup.2] - [Y.sup.2]) has a significant and positive effect on the outcome of bargaining. In particular, the point estimate on ([X.sup.2] - [Y.sup.2]) is .142 with a standard error of .0104. The regression analysis strongly suggests that differences in sunk costs do indeed affect bargaining in the incomplete contracts investigated here. In fact, the joint null hypothesis that [Alpha] = .5 and [Beta] = 0 is rejected at well below the 1 percent significance level [F(2,154) 97.09; prob. > F = .00011.

In simple OLS regressions such as this where a distribution in two dimensions is fitted to a straight line, the estimators are quite sensitive to outlying points. Moreover, the data set analyzed here includes a number of outlying observations. Robust estimation techniques detect and weight outliers, and so yield a good fit to the majority of the data. Rousseauw and LeRoy [1987] suggests a least median squares (LMS) technique, which corresponds to finding the narrowest strip covering half of the observations. OLS estimates reweighted from the LMS technique are presented in Table III. LMS assigned a zero weight to twelve outlying observations. While the standard errors fell as expected under robust estimation, the point estimates differ only slightly from those of ordinary least squares, which suggests that the OLS estimators are remarkably robust.

One might also conjecture that substantial variation in individual experimental sessions may elicit strikingly different coefficient estimates on ([X.sup.2] - [Y.sup.2]). The hypothesis that the coefficient estimates on [X.sup.2] - [Y.sup.2]) are the same across experimental session cannot be rejected at commonly accepted significance levels [F(8,146) 1.344; prob. > F = .23].

Recall that sequential bargaining with a discount rate 6 of .9 creates a small first-mover advantage, in this case s =.526 rather than .50 in the subgame-perfect equilibrium. A first-mover advantage can be directly tested by creating separate samples for buyers and sellers as-first movers and then testing if the two samples of observations on s are generated by the same stochastic process. This was done, and neither the Wilcoxon nor the Kolmogorov-Smirnov tests were significant at even the 10 percent level.(12) Moreover, the first-mover advantage requires that the first proposal be accepted, and overall this never occurred more than one-third of the time in any of the treatments.

In summary, the analysis strongly rejects the hypothesis that the subgame-perfect bargaining equilibrium, s* predicts actual outcomes of bargaining in the incomplete contracts investigated here. Differences in transaction-specific investment have a significant effect on s, and the direction of influence is consistent with that of the equitable bargaining hypothesis [s.sup.e]; the marginal direct effect of [X.sup.2] - [Y.sup.2]) on s is approximately(13) with remarkably small standard error. It is important to note, however, that subjects did not generally divide surplus in such a way that net payoffs were equal. This is illustrated in Figure 5, which plots the cumulative percentage of agreements that deviate from equal net payoffs. If, hypothetically, all agreements had equalized net payoffs, then the "EP" plot shows that the cumulative percentage would be 100 at zero deviations from equal net payoffs. The thick line plots the cumulative percentage deviation of actual bargaining outcomes from equal net payoffs. Finally, in the hypothetical case in which all bargaining outcomes are subgame-perfect, the dotted line, labelled "SGP", plots the hypothetical cumulative percentage deviation from equal net payoffs that would have resulted.

Figure 5 shows that subjects managed to substantially reduce, but not eliminate, net payoff differentials in the two Strong Incentive treatments relative to what would occur if all bargains were the subgame-perfect 50-50 split. Since there is little difference between buyer and seller investments in the Equal Incentives treatment, the actual bargaining outcomes tended around the subgame-perfect share, and so these plots follow each other closely.

Backward Induction: The Influence of Bargaining Outcomes on Investment

A difficulty with assessing the role of bargaining outcomes on expected joint surplus in the laboratory is that transaction-specific investment here is a decision made under uncertainty over the marginal benefits of investment. As a result, assessing the effect of equitable bargaining outcomes requires empirically decomposing the constituent effects of risk preference and beliefs, among others, on investment.

Having pointed this out, consider the sample means for X and Y in Table Il. Results from the Strong Seller and Strong Buyer Incentives treatments are highly symmetric and are remarkably close to the X* and Y* predicted by the subgame-perfect analysis with risk neutral agents capable of perfect foresight. In the Strong Seller Incentives treatment, that model predicts X = .025 and Y = .75, whereas the sample means are X=.101 and Y=.760. In the Strong Buyer Incentives treatment the benchmark model predicts X=.75 and Y=.025, and the sample means are X = .774 and Y = .101. The greatest deviation from the model appears in the Equal Investment Incentives treatment, in which X and Y are predicted to be .5, and the sample means are X = .653 and Y = .596.

Rather than try to decompose the effects of risk preference and equitable bargaining outcomes on observed investment decisions, we can instead look directly at the bargaining outcomes forecasted by subjects. Recall that subjects provided forecasts of the outcome of bargaining over shares of (V-C) so that the computer could compute the subject's forecast-contingent expected payoff information in stage 1. Table IV presents nonparametric tests of the null hypothesis that forecast observations from different parameter regimes are generated by the same stochastic process. In nearly all cases this null hypothesis can be rejected at below the 1 percent significance level. The exception is in comparing the seller's forecast share, (1-s)[sup.f], for the Equal Investment incentives and the Strong Buyer Incentives paired samples, where the null can be retained at between the 12 and 18 percent level. Hence there is some support for the notion that subjects anticipate the influence of relative investment levels on the outcome of bargaining.

Backward induction implies that there should be a correlation between a subject's forecast of the bargaining outcome and the subject's level of investment. In fact, the correlation coefficient for [s.sup.f] and X is 0.48, and the correlation coefficient for (1-s)[sup.f] and Y is .42, and both of these coefficients are strongly significant. This correlation, together with the evidence that subjects anticipate the influence of relative investment on the bargaining outcome, suggests that [s.sup.e] was transmitted to the subjects' investment choices.

IV. SUMMARY AND CONCLUDING COMMENTS

The results of this experiment suggest that the bargaining process nested in incomplete contracts has properties that differ from those of the standard bargaining models. In particular, relative levels of sunk, transaction-specific investment influence the outcome of bargaining, which refutes the predictions of the subgame-perfect bargaining model. The party making the greater investment received the greater share of realized surplus. This finding is important to the way that economists model the incomplete contracting problem, as it implies a higher level of joint surplus than that which results from an exogenously imposed bargaining solution.

This research also identifies a situation in which people do not appear to ignore sunk costs. Phillips, Battalio and Kogut [19911 is another recent study which investigates the effects of sunk costs on valuation and decision making. They find that when subjects were required to make a sunk payment for a lottery ticket, the size of the sunk cost affected their subsequent valuation of the lottery. This effect was attenuated when subjects submitted bids to buy an auctioned lottery ticket. More research on the role of seemingly irrelevant commitments appears to be warranted.

This research is also related to the growing body of experimental bargaining literature. The ultimatum bargaining experiment literature surveyed by Guth and Tietz [1990] indicates that ultimatum proposals lying between equal splits and the subgame-perfect 100 percent allocation to the proposer predominate across a large number of studies. Similarly, here subjects use the bargaining process to attenuate but not fully equalize net payoffs. Hoffman and Spitzer [1982] survey the bargaining literature and find a number of factors that are associated with equal net payoff splits, including repeat face-to-face designs, being told their "task" is to divide a cash sum, the ability to make Pareto-improving moves, and knowledge of one-another's payoffs. Only the last condition is met here.

In their research on reward allocation Hoffman and Spitzer found at times that allocators, who were selected through a coin toss, accepted bargains that made them worse off than if disagreement had occurred. In Hoffman and Spitzer [1985] the allocator was chosen based on performance in tic-tac-toe rather than a coin toss. They found a large increase in self-regarding allocator behavior, and they speculate that it was because the allocator was perceived to have 'earned' the right to a larger payoff. Similarly in this study one might speculate that deviations from the subgame-perfect sharing rule occurred when a subject had 'earned' it by making a relatively larger sunk investment. While net payoff differences are reduced this way, full equality of net payoffs is generally not reached. This suggests that fairness notions based on net payoff equity cannot alone explain bargaining outcomes, as would also apply to ultimatum games.

Forsythe et al. [1991) specifically tested this by comparing laboratory bargaining outcomes from ultimatum games, where the other party can reject the ultimatum, to a game where the outcome is dictated by a single party.(13) If fairness alone explains allocations, then the outcomes of these games should be identical. Forsythe et al. can reject this hypothesis; it is interesting to note, however, that only 36 percent of the (paid) dictator game outcomes were consistent with subgame-perfection, and all (paid) ultimatum game observations were centered around the equal shares proposal. As with the ultimatum game experiments, this study of incomplete contracting points to the need for a more descriptive and testable theory of bargaining behavior.

APPENDIX

Instructions and Questionnaire for the Equal Investment Incentives Treatment

This is an experiment in the economics of decision making. The funding for this experiment has come from a number of sources, including Indiana University. The instructions are simple, and if you follow them carefully and make good decisions, you may earn a considerable amount of money that will be paid to you in cash at the end of the experiment. The experiment takes place on the computer, but you do not need to have any specialized knowledge of computers in order to successfully participate in this experiment. Any attempt at communicating with other participants verbally or in any way other than through the computer is grounds for being REMOVED from the experiment.

The experiment consists of a series of periods. Each period the computer will randomly match one participant with another to form independent decision-making pairs. Once participants have been matched, the computer will randomly choose one participant to have the status of a "buyer;" the other participant will then have the status of a "seller." You are equally likely to be chosen a buyer or a seller. You will not know the other party's identity.

In this experiment each buyer and seller decision-making pair must make a series of decisions regarding the implementation of a PRODUCT IMPROVEMENT. Your payoffs depend on the type of decisions that are made by you and by the participant that you have been matched with in a given period. The following is a step by step summary of the sequence of decisions you will be confronted with each period:

I. INVESTMENT DECISION

Buyers can invest some of their earnings in increasing the VALUE (V) of the product improvement; similarly, sellers can invest some of their earnings in decreasing the COST (C) of providing the product improvement. The investment process proceeds as follows.

At the time the investment decision is made, the cost C and the value V of the product improvement are not known. Specifically, the value of the product improvement will be either $5 or $3. Buyers will have an opportunity to invest some of their earnings to increase the PROBABILITY (X) that V will be $5 rather than $3. Similarly, the cost C will be either $0.50 or $2.50, and sellers will have an opportunity to invest some of their earnings to increase the PROBABILITY (Y) that C will be $0.50 rather than $2.50. The computer uses X and Y in the following way when it randomly chooses V and C:

Suppose a buyer decided to invest enough earnings to raise X up to 30-a 30% probability that V is $5 rather than $3. The computer will then randomly draw a number, and any number between 0 and 100 is equally likely to be drawn. If the number drawn by the computer is between 0 and 30, then V = $5; on the other hand, if the number drawn by the computer is between 31 and 100, then V is $3. The more earnings the buyer invests the larger is X, and so the larger is the probability that V = $5 rather than $3.

Similarly, suppose a seller decided to invest enough of her earnings to raise Y up to 75-a 75% probability that C is $0.50 rather than $2.50. If the number drawn by the computer is between 0 and 75, then C = $0.50; on the other hand, if the number drawn by the computer is between 76 and 100, then C is $2.50. The more earnings the seller invests the larger is Y, and so the larger is the probability that C = $0.50 rather than $2.50.

The amount of earnings needed to increase X and Y will be illustrated to you on the computer following these instructions. After the buyer chooses X and the seller chooses Y, the computer will show the values of X and Y, as well as the values of V and C, to both parties.

II. SHARING DECISION

Once V and C are made known to the buyer and the seller, they can choose to share the GROSS RETURN (V-C)--the difference between the value and the cost of the product improvement. Note that the expense incurred by the buyer in increasing X, and the expense incurred by the seller in increasing Y, are sunk costs that are paid regardless of how (V-C) is shared.

The gross return (V-C) can only be shared if both the buyer and the seller agree. If they both agree, then the buyer and the seller will alternate making share offers until EITHER an offer is accepted, OR negotiations break down. In odd-numbered periods buyers will offer first, while in even-numbered periods sellers will offer first.

When it is your turn to make a share offer, then the other person has the choice of accepting or rejecting your offer. The total number of share offers that can be made by you and the other party is limited, and has been fixed in advance. Each time an offer is rejected, there is a chance that no more share offers will be allowed, and negotiations will be ended without (V-C) having been shared. The maximum number of share offers that can be made may be large or small, and will usually be different from one period to the next. You should reckon that there is a 90 percent chance that at least one more share offer can be made.

Buyers and sellers can make any share offer which adds up to 100 percent. If negotiations break down, then the buyer gets V, and the seller pays C.

EXAMPLE: suppose that V = $5 and C = $0.50, so that (V-C) = $4.50. Unless BOTH the buyer and the seller agree to share (V-C), the buyer will get V = $5, and the seller will have C = $0.50 deducted from her earnings. Suppose instead that they agree to share, and the buyer made a share offer of 33-the buyer was willing to take 33 percent of (V-C) = $1.50, and so give the seller 67 percent of (V-C) = $3. If the seller accepts this offer, that is how the gross return will be shared.

III. VETO DECISION

After the sharing decision has been made, EITHER the buyer OR the seller can decide to stop ("veto") the product improvement. If either the buyer or the seller vetoes, then the value of the product improvement V = $0, and the cost of the product improvement C = $0.

IMPORTANT ISSUES:

1. FORECASTING

At the time you choose X or Y, you will always know how much it costs you to choose different levels of X or Y. On the other hand, the money payoff that you can expect to receive depends on your forecast of what the SHARING and VETO decisions will be. In order to provide you with your FORECASTED PAYOFF, you will be asked to enter your forecast of (i) whether you and the other party will SHARE (V-C), and if so, what share you will receive, and how much investment in X or Y the person you are matched with will make, and (ii) whether the product improvement will be VETOED. 2. NET PAYOFFS

Net payoffs each period are computed as follows: If the product improvement is NOT vetoed:

A. If (V-C) was shared:

BUYER PAYOFF =

share* $ (V-C) = $(investment in X)

SELLER PAYOFF =

B. If (V-C) was not shared:

BUYER PAYOFF =

$V - $(investment in X)

SELLER PAYOFF =

-$C - $(investment in Y)

If the product improvement IS vetoed:

BUYER PAYOFF =

-$(investment in X)

SELLER PAYOFF =

-$(investment in Y)

You will have a starting balance of $4 at the beginning of the experiment. Your new balance each period will be your previous balance, plus you net payoffs that period. Net payoffs can be negative, and if your balance falls below $0, you will be removed from the experiment.

If you have any questions, you can enter "R" re-read the instructions, or you can raise your hand.

This table illustrates how much of your earnings you must invest to increase the probability X or Y. Use this information for the questions below. 1. Pick an X for a buyer, and a Y for a seller.

X = --, cost of X = $--;

Y = --, and cost of Y --. 2. Suppose that the computer randomly picks

the number 47. If the X you picked is 47 or

larger, then V + $5; otherwise V = $3. If the

Y you picked is 47 or larger, then C = $0.50;

otherwise C = $2.50. 3. Calculate V, C, and (V-C) based on the information

in #2 above.

V = $--, C = $--,

(V-C) = $--. 4. a. Suppose that (V-C) is NOT shared, and

the product improvement is NOT vetoed.

Then the buyer's and the seller's net payoffs

are:

buyer: $-- seller: $--. b. Suppose that (V-C) is NOT shared, and

the product improvement IS vetoed. Then

the buyer's and the seller's net payoffs

are:

buyer: $-- seller: $--. c. Suppose that (V-C) IS shared, and the

product improvement is NOT vetoed.

Pick a way to split (V-C):

buyer share = --%

seller share = --%

buyer's net payoff = $--;

seller's net payoff = $--. d. Suppose that (V-C) IS shared as you

picked in c. above, but the product improvement

IS vetoed. Then calculate

buyer's net payoff = $--;

seller's net payoff = $--. (1.) Tirole [1988, 29-32] and Holmstrom and Tirole [1989, 66-67] offer accessible introductions to this model. There are certainly other important models of incomplete contracting, such as Wiggins [1990], which focuses on an effort-information sharing tradeoff. See Wiggins, Hackett, and Battalio [1989] for an experimental study based on this model. (2.) This step can be omitted with no violence to the properties of the model and does not affect the subgame-perfect bargaining equilibrium. This step was added for laboratory investigation in order to allow subjects to immediately express their desire not to bargain at all. If subjects were forced to bargain, the desire not to bargain could be expressed by rejecting all share proposals and making unacceptable proposals (e.g., 100,0), until breakdown of the bargaining process is realized. (3.) This can be written symmetrically for the seller making a time 0 offer to the buyer, which takes the form [[Delta](1-S) + B]/(1 + [Delta]). (4.) In the spirit of their model of vertical integration, one could argue that if [D.sub.v] > [D.sub.c] the maximum joint surplus is achieved when the buyer purchases the seller's veto right and sets s = 1. (5.) The subject pool was recruited from honors principles classes and courses taught at the junior, senior, and first-year graduate level. Students from game theory and experimental economics classes were not recruited. The subject pool included international students from China, Germany, India, Malaysia, Mexico, the Netherlands, and Spain. (6.) The programs controlling the information and message spaces of the experiment were executed on a VAX computer at Indiana University. (7.) The design of having ten game repetitions using anonymously and randomly matched inexperienced subjects under a single parameter regime was chosen to closely follow the procedures of Binmore et al. [1991]. Half of the time their subjects were buyers, and the other half they were sellers. Similarly, here subjects stand a 50 percent chance of being a buyer or a seller in any given period. Random, anonymous matching and establishment of buyer and seller identities greatly attenuates any reputation effects from the multi-period setting. In addition, as Binmore et al. point out, bargaining models assume parties are knowledgeable of the others' incentives, and this procedure provides subjects with the experience to develop this knowledge. (8.) Subjects were told that this information would be made known to both parties. By randomizing whether subjects were buyers or sellers in a given period, it quickly became clear to them that the announced common knowledge of these parameter values was in fact true. (9.) Binmore et al. report initial difficulty in communicating breakdown probabilities to subjects, which led them to fixing them in advance. (10.) Using two sample non-parametric techniques that compare samples from the first three and last five periods in which subjects were paid, the null hypotheses that X, Y, and s are generated by the same stochastic process can only be rejected in a few circumstances. In particular, s shows a significant downward shift in the last five periods of the Strong Seller Incentives treatment. Both s and X show a significant upward shift in the last five periods of the Strong Buyer Incentives treatment. These shifts in s that are associated with more experienced play are both in a direction that reinforces the argument for [s.sup.e]. (11.) The small amount that it exceed .5 is significant, however. This could be accounted for by the fact that mean X > mean Y in the baseline condition. (12.) Since buyers had this advantage in odd-numbered periods, a strong time trend from one period to the next can create a problem with these tests. Recall from footnote 10 that the sample of observations on s from the first three periods in which subjects were paid differ from those of the last five. This does not present a problem per se, however, as both the buyer and the seller had an opportunity to move first in both the first three and the last five periods. Linear and exponential decay trend variables are found to have no significant influence on s. (13.) Harrison and McCabe [1991] also challenge the fairness explanation. When first proposals are not generally accepted, so subjects have experience in bargaining subgame play, they find that behavior converges to the subgame-perfect prediction. Note that in this study first proposals are usually rejected, and so subjects have experience with subgame play, yet shares tend to move away from the subgame-perfect prediction.

REFERENCES

Binmore, K. "Bargaining and Coalitions, I." in Game Theoretic Models of Bargaining, edited by A. Roth. Cambridge: Cambridge University Press, 1985, 269-304. Binmore, K., Rubinstein and A. Wolinsky. "The Nash Bargaining Solution in Economic Modeling." Rand Journal of Economics, Summer 1986,176-88. Binmore, K., Shaked and J. Sutton. "An Outside Option Experiment." Quarterly Journal of Economics, November 1989, 753-70. Binmore, K., P. Morgan, A. Shaked and J. Sutton. "Do People Exploit Their Bargaining Power? An Experimental Study." Games and Economic Behavior, August 1991, 295-322. Crawford, V. "Long Term Relations Governed by Short Term Contracts." American Economic Review, June 1988,485-99. Forsythe, R., J. Horowitz, N. Savin and M. Sefton. "Fairness in Simple Bargaining Experiments." Photocopy, University of Iowa, 1991. Friedman, D., and R. McGuire. "Fairness as an Efficiency-Enhancing Constraint.' Photocopy, University of California, Santa Cruz, 1988. Fudenberg, D., B. Holmstrom and R Milgrom. "Short-Term Contracts and Long-Term Agency Relationships." Working Paper No. 488, MIT, 1987. Grossman, S., and O. Hart. "The Costs and Benefits of Ownership: A Theory of Vertical and Lateral Integration." Journal of Political Economy, August 1986, 691-719. Guth, W., R. Schmittberger and B. Schwarze. "An Experimental Study of Ultimatum Bargaining." Journal of Economic Behavior and Organization 3(3), 1982, 367-88. Guth, W., and R. Tietz. 'Ultimatum Bargaining Behavior." Journal of Economic Psychology, September 1990, 417-49. Harrison, G., and K. McCabe. "Testing Bargaining Theory in Experiments." Photocopy, University of South Carolina, 1991. Hart, O., and J. Moore. "Incomplete Contracts and Renegotiation." Econometrica, July 1988, 755-86. Hart, O., and B. Holmstrom. "The Theory of Contracts," in Advances in Economic Theory, edited by T. Bewley. Fifth World Congress, Cambridge: Cambridge University Press, 1987, 71-155. Hoffman, E., and M. Spitzer. "The Coase Theorem: Some Experimental Tests." Journal of Law and Economics, April 1982, 73-98. --. "Entitlements, Rights, and Fairness: An Experimental Examination of Subjects' Concepts of Distributive Justice." Journal of Legal Studies, June 1985, 259-97. Holmstrom, B., and O. Hart. "The Theory of Contracts." Photocopy, MIT, 1985. Holmstrom, B., and J. Tirole. "The Theory of The Firm," in Handbook of Industrial Organization, edited by R. Schmalensee and R. Willig. Amsterdam: North-Holland, 1989, 63-133. Klein, B., R. Crawford and A. Alchian. "Vertical Integration, Appropriable Rents, and the Competitive Bidding Process." Journal of Law and Economics, October 1978, 297-326. Macaulay, S. "Non-Contractual Relations in Business." American Sociological Review, February 1963, 55-70. Milgrom, P, and J. Roberts. "Bargaining and Influence Costs and the Organization of Economic Activity." in Positive Perspectives on Political Economy, edited by J. Alt and K. Shepsle. Cambridge: Cambridge University Press, 1991, 57-89. Neelin, J., H. Sonnenshein and M. Spiegel. "A Further Test of Noncooperative Bargaining Theory: Comment." American Economic Review, September 1988, 824-36. Ochs, J., and A. Roth. "An Experimental Study of Sequential Bargaining." American Economic Review, June 1989, 355-84. Phillips, O., R. Battalio and C. Kogut. "Sunk and Opportunity Costs in Valuation and Bidding." Soutern Economic Journal, July 1991, 835-56. Rousseauw, P, and A. Leroy. Robust Regression and Outlier Detection, New York: John Wiley and Sons, 1987. Rubinstein, A. "Perfect Equilibrium in a Bargaining Model." Econometrica, January 1982, 97-110. Tirole, J. "Procurement and Renegotiation." Journal of Political Economy, April 1986, 235-59. --. The Theory of Industrial Organization, Cambridge: MIT Press, 1988. White, H. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity." Econometrira, July 1980, 817-38. Wiggins, S. "The Comparative Advantage of Long Term Contracts and Firms." Journal of Law, Economics, and Organization, Spring 1990, 155-70. Wiggins, S., S. Hackett and R. Battalio. "Relational Contracts, Firms, and Reputation: An Experimental Study of Institutional Choke." Photocopy, Texas A&M University, 1989. Williamson, O. Markets and Hierarchies. New York: Free Press, 1975.

Department of Economics, Indiana University, Bloomington, IN 47405. This paper has benefitted from helpful comments by three anonymous referees and coeditors Thomas Borcherding and Rodney Smith. I would also like to thank Ray Battalio, Dave Besanko, Ken Binmore, Miguel Delgado, Roy Gardner, Ron Johnson, Charles Kahn, Tom Lyon, Jimmy Walker, Steve Wiggins, Arlie Williams, and Doug Young. Support from the Workshop in Political Theory and Policy Analysis at Indiana University is gratefully acknowledged. All data are stored on permanent disk files and are available on request.

Williamson [1975] and Klein, Crawford, and Alchian [1978], among others, pioneered the notion that long-term contracts arise as a means of governing and protecting valuable investments in long-term trading relationships. Contract incompleteness results when some terms are left unspecified and usually is a result of practical difficulties in specifying contingent responses to unforeseen future states of the world. Macaulay [1963] finds that incomplete contracts are common in business dealings, and Holmstrom and Hart [1985] argue that "incompleteness is probably at least as important empirically as asymmetric information as an explanation for departures from 'ideal' Arrow-Debreu contingent contracts." A substantial formal literature has built on this pioneering research and includes work by Grossman and Hart [1986], Tirole [1986], Crawford [1988], Hart and Moore [1988], Fudenberg, Holmstrom, and Milgrom [1987], Milgrom and Roberts [1991], and Hart and Holmstrom [1987]. This formal literature has distilled the earlier notions down to a standard two-period model that is used as a foundation to much of contemporary research on contracting and the firm.(1)

To motivate this model, consider a single buyer and seller of some intermediate good, such as computer software. They anticipate innovative change in the design of the intermediate good, but cannot foresee the precise nature of the innovation. As a result the contract cannot be made contingent on the change in design. The benefit of the design change to the buyer is uncertain, but the buyer can make a transaction-specific investment in flexibility that increases the likelihood that the benefit will be large. Similarly the cost of implementing the design change is uncertain, but the seller can make a transaction-specific investment that increases the likelihood that it will be small. In the second stage the innovation is realized. The buyer and seller then enter negotiations, the outcome of which determines whether the innovation will be implemented and how the surplus added by the innovation will be divided.

This two-stage model is solved using the technique of backward induction. First a game theoretic solution is imposed on second-stage negotiations, and this solution determines how joint surplus is shared. This division of joint surplus is assumed to be foreseen in the first stage and is incorporated into each party's expected profit function. Equilibrium is then found when each (risk neutral) party chooses their own expected profit maximizing level of transaction-specific investment. One potentially testable implication of this model is that the future division of joint surplus directly affects the level of transaction-specific investment, but the level of transaction-specific investment does not affect the division of joint surplus.

The bargaining literature has progressed rapidly in the development of non-cooperative bargaining models with unique equilibria that are testable using laboratory procedures. Influential non-cooperative bargaining models include those of Rubinstein [1982], Binmore [1985], and Binmore, Rubinstein, and Wolinsky [1986]. Accumulated evidence from laboratory experiments on bargaining such as in Guth, Schmittberger, and Schwarz [1982], Guth and Tietz [1990], Ochs and Roth [1989], and Neelin, Sonnenshein, and Spiegel [1988], suggests that non-strategic factors may have a substantial effect on bargaining outcomes. Much of the experimental work has been with ultimatum games, where one player submits a share proposal to another, who can either accept or force breakdown. When breakdown results in zero payoffs, the (subgame-perfect) equilibrium proposal grants very nearly 100 percent for the person making the proposal and the residual to the person accepting the proposal. Laboratory experimental studies of these games are usually able to reject this hypothesis. These studies typically find that allocations tend to fall somewhere between equal splits and the hypothesized equilibrium allocation. It has been suggested that players may perceive these games as being inherently 'unfair,' and that this perception is reflected in the failure of the equilibrium prediction.

These experimental studies suggest that the bargaining process nested in incomplete contracts may be substantially different from that assumed in the standard bargaining models from which contract equilibria are derived. In particular, consider a situation in which one party has made a transaction-specific investment that is large relative to that made by the other party. Imposition of the subgame-perfect or Nash bargaining outcome implies that transaction-specific investments have no influence on the outcome of bargaining. Hence the party making the greater investment will receive a smaller net payoff. This is directly tested in the present experimental study by varying the relative investment incentives induced on buyers and sellers, and then seeing whether bargaining outcomes are influenced by differences in transaction-specific investment. As in ultimatum game studies, bargainers may instead choose outcomes that attenuate the difference in their net payoffs. The testable implication is that the greater is a player's relative investment level, the greater is his share of surplus.

The results reported here strongly reject the hypothesis that differences in transaction-specific investment have no influence on the outcome of bargaining in incomplete contracts. instead, bargaining outcomes are significantly influenced by parties' transaction-specific investment. The party that made the greater transaction-specific investment is found to receive a greater share of realized surplus in the bargaining process, although the effect is generally not strong enough to equate net payoffs. The results are highly consistent with the experimental studies of the ultimatum game. These results are also important to the theoretical literature on incomplete contracts; when the bargaining process is used to attenuate differences in net payoffs, the result can be an increase in expected joint surplus over that which would occur under subgame-perfect bargaining.

II. INCOMPLETE CONTRACTING IN THE LABORATORY

The Model

A simple two-stage incomplete contracting model was created for laboratory experimentation based on the archetypical model described by Holmstrom and Hart [1985], Tirole [1988], and Holmstrom and Tirole [1989]. Following Tirole [1988], consider a situation in which the buyer and the seller have agreed to trade one unit of some good in the future. They agree to a basic design for the good, but cannot contract over a quality-enhancing product improvement that is anticipated in the second stage. The joint return to the basic design is normalized to zero, and so the analysis that follows (as well as the subject's decision environment) is based on the added valuation and cost of the product improvement.

The incomplete contracting process is outlined in Figure 1 and described in detail below. In the first stage the buyer's induced value (V) of the product improvement is uncertain and can either be "high" ([V.sub.h]) or "low" ([V.sub.l). Similarly the seller's induced cost (C) of the product improvement can either be "low" ([C.sub.l]) or "high" ([C.sub.h]). The process begins with the buyer and seller non-cooperatively making sunk investments that increase expected value and cost. In the first stage the buyer chooses the likelihood that value is high by choosing a number 0 [is less than or equal to] X [is less than or equal to] 1 at (sunk) cost [X.sub.2]. Likewise the seller chooses the likelihood that cost is low by choosing a number 0 [is less than or equal to] Y [is less than or equal to] 1 at (sunk) cost [Y.sup.2].

After the buyer and seller choose X and Y in the first stage, the second stage begins with the realization of value and cost. Value is determined by comparing X to a random number uniformly distributed over the unit interval; if X is greater than or equal to the random number, then value is high, and otherwise value is low. Similarly, cost is determined by comparing Y to a random number uniformly distributed over the unit interval; if Y is greater than or equal to the random number, then cost is low, and otherwise cost is high. X, Y, V, and C then become common knowledge, and buyers and sellers must decide whether or not to bargain over shares of realized surplus (V-C).(2) If both the buyer and the seller agree, they bargain over how to share realized surplus. Successful bargaining yields a buyer share s and a seller share (1-s) of realized surplus, where 0 [is less than or equal to] s [is less than or equal to] 1. If one or both parties rejects bargaining, or if attempted bargaining breaks down, the disagreement surplus allocation gives V to the buyer, with C incurred by the seller.

After the bargaining decision either the buyer or the seller can decide to "veto" the product improvement, in which case both the value and the cost of the improvement fall to zero. The seller can credibly commit to a veto if the surplus realization is not shared, and this possibility motivates the buyer to bargain. To see this, note that as long as the (non-negative) surplus allocation is shared, neither the buyer nor the seller will choose to veto, as a veto would reduce either of their payoffs. On the other hand, if the surplus realization is not shared, the seller can avoid the cost of producing the improvement by choosing to veto. Knowing this, buyers will always agree to bargain, because failure to bargain will trigger a seller veto and so a loss of at least a share of the surplus realization. Were it not for the seller's ability to veto, the buyer would not have an incentive to bargain and share the surplus realization. This is the role of vertical integration in the influential Grossman and Hart [1986] model. The implication is that bargaining and no veto are the equilibrium of the second-stage sub-game.

Parties foreseeing the outcome of second-stage bargaining can then compute their expected payoff maximizing first-stage investment. The research question posed here is whether bargaining outcomes are independent of the relative level of transaction-specific investment ([X.sup.2] and [Y.sup.2]) made by the buyer and the seller. In order to test this hypothesis here one must first be specific about the bargaining institution and the equilibrium bargaining outcome that it implies.

The laboratory bargaining setting is based on a variant of the Rubinstein [1982] procedure made operational for laboratory research. In the Rubinstein procedure the parties alternate in making proposals indefinitely until a proposal is accepted or the negotiations break down. Some mechanism is required to give parties an incentive to reach early agreement. The particular mechanism employed here was most recently described in Binmore et al. [1991]. Upon rejection of a proposal, a random move determines whether another round of negotiations will be allowed or the bargaining game will end. This "forced breakdown" sequential bargaining institution was used because it has a unique subgame-perfect bargaining equilibrium that also corresponds to the Nash bargaining solution. An important property of this equilibrium is that the allocation of realized surplus is independent of the relative levels of sunk, transaction-specific investment committed to by the buyer and the seller.

To see this, suppose first that sunk costs are ignored and (V-C) is the object of bargaining with disagreement payoffs of zero and zero. Then as the probability [Delta] that bargaining will continue if a proposal is rejected approaches 1.0, the Rubinstein model implies that the buyer's share s of (V-C) approaches 0.50, which is also the Nash solution with status quo located at (0,0). The buyer's net payoff is then (V-C)/2 - [X.sup.2], and the seller's net payoff is (V-C)/2-[Y.sup.2]. Suppose instead that sunk costs are not ignored, so that the object of bargaining is (V-C-[X.sup.2]-[Y.sup.2]) and the buyer and seller's disagreement payoffs are [-X.sup.2] and [-Y.sup.2]. Following Binmore, Shaked, and Sutton [1989], the unique subgame-perfect equilibrium of this bargaining game has the first proposer (say the buyer) offering a share of (V-C-[X.sup.2]-[Y.sup.2]) equal to [[Delta](1-B) + S](1+[Delta]) to the other, who then accepts it (B = [-X.sup.2]/(V-C-[X.sup.2]-Y), and S = [-Y.sup.2]/(V-C-[X.sup.2] [-Y.sup.2]).(3) As [Delta] approaches 1.0 the buyer's equilibrium net payoff approaches (V-C)/2-[Y.sup.2], while the seller's approaches (V-C)/2-[Y.sup.2]. But these net payoffs are identical to the equilibrium of the original bargaining model that omits sunk costs. Hence the subgame-perfect equilibrium ignores sunk costs, as is assumed in the incomplete contracting model.

The subgame-perfect equilibrium share s* in turn affects investment incentives in the first stage of the incomplete contracting process. It is useful here to establish a theoretical benchmark. If V and C are independent random variables, and if the buyer and the seller are risk-neutral and capable of perfect foresight, then the buyer will choose X* to maximize (1) s* {X[Y([V.sub.h] - [C.sub.l]) + (1 - Y)([V.sub.h] - [C.sub.h])] + (1-X)[Y([V.sub.l] - [C.sub.l]) + (1- Y)([V.sub.1] - [C.sub.h])]} - [X.sup.2]

The first term is simply the buyer's share of the expected value of (V-C). The seller will similarly choose Y* to maximize (2) (1-s*) {Y[X([V.sub.h] - [C.sub.l]) + (1-X)([V.sub.1] - [C.sub.l])] + (1-Y)[X([V.sub.h] - [C.sub.l]) + (1-X)([V.sub.1] - [C.sub.h])} [Y.sup.2], where the first term is the seller's share of the expected value of (V-C). Maximizing equations (1) and (2) with respect to X and Y yields: (3) X* = s* ([V.sub.h] - [V.sub.l]) / 2, (4) Y* = (1-s*) ([C.sub.h] - [C.sub.l]) / 2.

This benchmark analysis illustrates the one-way influence that is a property of the incomplete contracting model. The bargaining solution s* directly affects the buyer's and the seller's investment decisions, but the investment decisions do not affect the bargaining solution.

Experimental studies of the ultimatum game have consistently found that the subgame-perfect bargaining equilibriurn fails to predict surplus allocations. In a one-round ultimatum game there is a valuable "cake" to be divided by two players. The payoffs from the game are derived from division of the cake K, which can most easily be thought of as a cash sum. The proposer is chosen from some exogenous selection process and will be referred to as player 1. Player 1 proposes a division of the cake to player 2, who can either accept or reject the proposal. If player 2 accepts, each receives the proposed share of the cake as their payoff. If player 2 rejects, the cake is destroyed and each player receives a zero payoff. Let k be the smallest positive unit of money. If player 2 would not accept a zero share, the optimal proposal by player 1 would yield player 2 a payoff of k, as it is optimal for player 2 to accept any positive share of the cake (otherwise a zero share can also be a subgame-perfect equilibrium).

In the ultimatum game described above, the subgame-perfect equilibrium proposal yields (K - k) to player 1, and k to player 2. Laboratory experiments on ultimatum bargaining by Guth, Schmittberger, and Schwarz [1982], Guth and Tietz [1990], Ochs and Roth [1989], and Neelin, Sonnenshein, and Spiegel [1988], have generally been able to reject the subgame-perfect prediction. The accumulated evidence suggests that non-strategic factors may have a substantial effect on bargaining outcomes. Observed allocations tend to locate somewhere between an equal split and the equilibrium prediction of nearly 100 percent to the proposer. Such behavior is consistent with players reducing the disparity in net payoffs. In terms of the present paper, suppose that (V.sub.h] - [V.sub.l]) > ([C.sub.h] - [C.sub.l]). If the hypothesis of subgame-perfect bargaining holds so that s = 0.5, then from equations (3) and (4) we have X* > Y*, which implies that both expected and actual buyer payoffs are smaller that of the seller. In fact the greater is the difference between X* and Y*, the more pronounced is this effect.

As in ultimatum games, bargainers here could attenuate this net payoff difference by allocating a greater share of realized surplus to the buyer. Using the bargaining process to attenuate differences in net payoffs implies a buyer share [s.sup.e]=.5 + f ([X.sup.2] - [Y.sup.2]), where f' (.) > 0, and f (0) = 0. The form that f (.) might take is an empirical question. While the ultimatum game studies do not generally find equal net payoffs, at the limit bargainers could equalize net payoffs by choosing a buyer share [s.sup.e] = .5 + ([X.sup.2] - [Y.sup.2])/2(V - C). A central focus of this paper is to test the subgame-perfect hypothesis when bargaining is preceded by strategically irrelevant sunk investments, as is the case in incomplete contracts.

Laboratory Parameterization of the Model

Of central importance here is that [s.sup.e] depends directly on ([X.sup.2] - [Y.sup.2]), while s* does not. If X = Y then the buyer's net payoff is equal to the seller's net payoff, and s* cannot be distinguished from [s.sup.e]. On the other hand, if X is not equal to Y, then the relative predictive power of the two hypotheses can be directly tested by evaluating the relationship between s and ([X.sup.2] - [Y.sup.2]). Three treatment conditions were devised that differed in the relative investment incentives induced on buyers and sellers. This is most easily done by varying the size of ([V.sub.h] - [V.sub.l]) relative to ([C.sub.h] - [C.sub.l]). To make the description of the treatment conditions that follows more manageable, let [D.sub.v] stand for ([V.sub.h] - [V.sub.l]), and let [D.sub.c] stand for ([C.sub.h] - [C.sub.l]).

A baseline condition was devised in which V can take on the values of $5 or $3, while C can take on the values of $2.50 or 50 cents. As a result [D.sub.v] = [D.sub.c] = $2. From equations (3) and (4) of the benchmark model X* = Y* = 0.50, implying that under the conditions of the benchmark model s* equalizes net payoffs (the benchmark buyer and seller net payoff is $1 each in expectation). Since the induced investment incentives are equal in this treatment, it will be referred to as the Equal Investment Incentives treatment.

Two additional treatment conditions were devised with the idea of inducing widely different investments on the part of the buyer and the seller. In one of these, relatively strong buyer incentives are induced by having [D.sub.v] > [D.sub.c]. In this treatment condition V can take on the values of $5.50 or $2.50, implying that [D.sub.v] = $3, while C can take on the values of $2.10 or $2, implying that [D.sub.c] = 10 cents. From equations (3) and (4) of the benchmark model X* = .75 and Y* = .025, implying that the benchmark expected net payoff is $0.76 for the buyer and $1.32 for the seller. At a buyer share of .64 the buyer will choose X = .96, the seller will choose Y = .019, and they will receive an expected payoff of $1.18 each. The implication is that in this Strong Buyer Incentives treatment the subgame-perfect share s* results in a greater payoff to the seller, and s* attempts to use the bargaining process to reduce the difference in net payoffs will require a buyer share [s.sup.e] > s*.

In the symmetric Strong Seller Incentives treatment condition, relatively strong seller incentives are induced by having [D.sub.c] > [D.sub.v]. In this treatment condition V can take on the values of $5 or $4.90, implying that [D.sub.v] = 10 cents, while C can take on the values of $4.50 or $1.50, implying that [D.sub.c] = $3. Under the subgame-perfect bargaining equilibrium hypothesis, s* =.5. From equations (3) and (4) of the benchmark model, X* = .025 and Y* = .75, and so the benchmark expected net payoff is $1.32 for the buyer and $0.76 for the seller. At a buyer share of .36 the buyer will choose X=.019, the seller will choose Y = .96, and each will receive an expected net payoff of $1.18. The implication is that s* yields greater net payoffs for the buyer, and so attempts to reduce this difference in net payoffs will require a buyer share [s.sup.e] < s*.

Discussion. The benchmark model suggests that the treatment conditions are likely to generate the desired variation in buyer and seller investment incentives. The goal is not to test the benchmark model, but instead to test the hypothesis that the outcome of bargaining is independent of the buyer's and seller's relative investments. An interesting implication of the incomplete contracting model is that expected joint surplus is greater under [s.sup.e] than under s*. To see this, consider the Strong Buyer Incentives condition (a parallel argument holds for the Strong Seller Incentives Condition). An increase in X (up to the maximum of 1.0) will always increase expected joint surplus more than an equal increase in Y, since [D.sub.v] > [D.sub.c]. Hence expected joint surplus is maximized if the buyer's share s = 1.00, in which case the buyer will choose X = 1 and the seller will choose Y = 0. As a result, when s rises from s* to [s.sup.e] in the Strong Buyer Incentives Condition, net payoff differences fall and expected joint surplus rises, but the change does not satisfy the Pareto criterion. These properties of the model were first described in the Grossman and Hart [1986] study of vertical integration.(4)

Participant's Decision Environment

Subjects were recruited from economics and business economics courses at Indiana University and had no prior experience with the experiment.(5) Eight subjects were used in each experiment, except for the first repetition of the baseline condition which used six subjects due to an unexpectedly high number of no-shows. The experimental setting was computerized and consisted of ten rounds of the two-stage incomplete contracting game played under one of the three treatment conditions described above.(6) Subjects were first seated at their computer terminals and read the experiment instructions. Subjects then answered a questionnaire that required a detailed understanding of the experiment, and were allowed to use their instructions as a reference. (The instructions and questionnaire for the baseline condition are provided as an appendix.) After completing the questionnaire, subjects were given a $4 initial cash balance and then participated in ten rounds of the incomplete contracting game. The experimenter was available for answering procedural questions. In this experiment inexperienced sellers can lose up to C + [Y.sup.2] per round if they are unfamiliar with the experimental procedures (e.g., they don't understand how to veto). To help subjects become familiar with the experimental procedures, and following Binmore et al. [1991], decisions were made without cash reward in early (the first two) rounds. Participants received total cash rewards ranging from $8 to $16, and the experiments lasted approximately two hours.

At the start of each round the computer program randomly and anonymously paired subjects. One was randomly assigned to have the identity of a buyer, and the other a seller.(7) After subjects were matched, the first stage began. Before subjects chose X or Y, they provided forecasts of (i) whether bargaining would occur, and if so, what the other party's investment and the outcome of bargaining would be, and (ii) whether a veto would occur. The software then used these forecasts to compute equation (1) for buyers, or (2) for sellers, for each possible X or Y value. Recall that parties are assumed to know their expected payoffs in the model, and by eliciting forecasts from subjects no beliefs are externally imposed regarding the other party's investment, bargaining outcomes, or veto decisions. Moreover, forecasts were not binding in any way; subjects were free to choose any value of X or Y between 0 and 100. As a result, subjects could see the investment cost ([X.sup.2 or [Y.sup.2]) and their forecast-contingent expected second-stage payoff for that round, for all possible X or Y values they could choose.

At the beginning of the second stage, V and C were determined after the random number n was realized and compared to X and Y. The n drawn in each of the ten rounds were

.04,.65,.38,.38,.02,.60,.72,.12,.87,.21. These were set in advance, were the same for each experiment, and given this sample one cannot statistically reject the hypothesis that n was randomly drawn from the unit interval. X, Y, n, V, and C were then revealed to both the buyer and the seller.(8) Next each subject was asked whether or not they wanted to bargain over (V-C). If either responded negatively, bargaining did not occur, and (V-C) was not shared. In this case the buyer gets V and the seller pays C.

If both parties agreed to bargain, they then proceeded into the sequential bargaining procedure described in Binmore et al. [1991]. Buyers and sellers alternated in making sharing rule proposals (0 [is less than or equal to] s [is less than or equal to] 1) indefinitely until a proposal is accepted or negotiations break down. In odd-numbered contracting rounds buyers made the first offer, while in even-numbered periods sellers offered first. Once an offer was made, the other party could then accept it or reject it. Rejection of a share offer leads to a chance that negotiations will exogenously be ended and no sharing will occur in that round. Binmore et al. used the technique of fixing the number of allowed rejections in advance, but not telling subjects what the length was.(9) The allowed number of rejections for each of the ten rounds both here and in their experiment were:

9, 2, 11, 2, 10, 7, 7, 16, 12, 8. Binmore and his co-authors state that the two short games at the start of each experiment were intended to convince subjects that a forced end to bargaining could occur, and that otherwise one could not statistically reject the hypothesis that upon rejection a breakdown occurs independently with probability 0.1. Given the discount rate of 0.1, the unique subgame perfect equilibrium has a first offer of .474 accepted, which approximates the .50 predicted as the discount rate converges to 1.00.

Subjects next independently decided whether or not to veto. Subjects were told that a veto implied that the hypothetical product improvement would not occur, and so the value V and cost C associated with implementing the product improvement became $0. After the veto decision, each subject realized their payoff for the round based on their investment, bargaining, and veto decisions. Cash payoffs from the round were then added to (or subtracted from, if negative) their previous cash balance. Finally, subjects were given feedback information that allowed them to compare their forecasts with the actual outcomes of the incomplete contracting process from that round.

III. ANALYSIS AND DISCUSSION OF EXPERIMENTAL RESULTS

The experimental design features three repetitions of each of the three parameter regimes described in section II above. To attenuate any added complication of learning effects, and following Binmore et al., I restrict the analysis reported below to data from later rounds of each experiment.(10) In particular, the analysis below is based on 175 observations generated in the last five rounds of each of the nine experimental sessions.

Recall that both the buyer and the seller must agree to enter the bargaining process; if one or both did not agree to bargain, the result is the same as when bargaining breaks down--the buyer retains V and the seller pays C. Bargaining is predicted because buyers know that sellers have a credible threat of vetoing, and so setting V = C = $0 if buyers do not agree to bargain. It is instructive to note that this threat appears to have been quite credible in the experiment, as shown in the contingency tables reported in Table I. Buyers failed to agree to bargain a total of ten times, and sellers subsequently vetoed in each case; none of these buyers who tested sellers' resolve repeated their test in later rounds. An excess of rejected offers lead to forced breakdown eight times, and again sellers consistently vetoed in each case. In the second replication of the Strong Seller Incentives treatment a buyer in round 9 agreed to bargain, but the seller refused ((V-C) was 40 cents), and subsequently vetoed. The reason for this seller's behavior is unknown. [TABULAR DATA I OMITTED]

A total of 156 successful bargaining outcomes remain, and in each case the seller did not veto. The results are consistent with buyers generally foreseeing that sellers would veto if bargaining did not occur, and with sellers vetoing when buyers occasionally decided to test sellers' resolve. Analysis of this data follows.

The Data

Investment and bargaining outcome data from each of the three parameter regimes are summarized in Figures 2 through 4. A statistical analysis follows below. It can be seen in Figure 2 that buyer and seller investment in the Equal Investment Incentives treatment is somewhat larger than 0.50 predicted by risk-neutral expected payoff maximization. The buyer's share of (V-C) is clustered around the subgame-perfect prediction of 0.50.

Investment and bargaining outcomes in the Strong Seller Incentives treatment are presented in Figure 3, and reflect sellers' relatively strong induced incentives for investment. Observed values of X never exceed 0.40, and the majority of observations fall between 0.00 and 0.10. On the other hand, only two observations on Y are less than 0.55, and the majority of observations fall between 0.65 and 0.90. In aggregate, the result is that sellers have made a much greater sunk investment relative to buyers. There is a corresponding downward bias in the buyer's share of (V-C); the majority of observations fall between a share of 0.30 and 0.50, with only three observations slightly exceeding .50.

In marked contrast, outcomes from the Strong Buyer Incentives treatment presented in Figure 4 reflect buyers' relatively strong induced incentives for investment. All observed values of X fall between 0.5 and 1.0, and the majority lie between 0.7 and 0.9. Symmetrically, most of the observed values of Y lie between 0.0 and 0.15. In this case nearly all observed s fall between 0.5 and 0.65, which is consistent with the equitable hypothesis, given the relationship between X and Y.

Statistical Analysis

Casual analysis of the data indicates that the investment and bargaining data differ across the three parameter regimes. The question is whether there is enough evidence to reject the hypothesis that the data sets from the three treatments are generated by the same statistical process. This is evaluated using two-sample, nonparametric testing methods, the results of which are reported in Table II. The null hypothesis is that samples of observations on X, Y, and s from any two of the treatments are generated by the same stochastic process. [TABULAR DATA II OMITTED]

Two tests commonly used in this context are the Wilcoxon and the Kolmogorov-Smirnov. In the Wilcoxon test procedure both samples are combined into a single ordered sample, and then ranks are assigned to the sample values from the smallest to the largest. The ranks are then summed for each of the two samples, and the difference is then taken between these sums. The null hypothesis is rejected if in absolute value the difference is sufficiently large. In the Kolmogorov-Smirnov test one first computes the sample cumulative distribution functions and then the maximum of the absolute value of the difference between the distribution functions. If this difference is sufficiently large in absolute value, the null hypothesis is rejected.

The nonparametric test reports are presented in Table II. The Wilcoxon value is given under the heading labeled "Z," and the significance level at which the null hypothesis of no difference in distribution is retained or rejected is given under the, heading "Probability > ~Z~." Similarly, the Kolmogorov-Smirnov value is listed under the heading "K," and the significance level at which the null is retained or rejected is listed under the heading "Probability > K."

The tests reject the null hypothesis of no difference in distribution of all three variables at well below the 1 percent level. Moreover, there is little meaningful difference in the results of the Wilcoxon and the Kolmogorov-Smirnov tests. The results for buyer investment in X and seller investment in Y reflect the variation in induced investment incentives across the three parameter regimes. In particular, larger X and smaller Y are observed in the ([Vh.sub.h] - [Vh.sub.l]) > ([C.sub.h] - [C.sub.l]) regime relative to the ([Vh.sub.h] - [Vh.sub.l]) < ([C.sub.l], - [C.sub.l]) regime. Investment in X and Y are decisions made under uncertainty, however, and so without knowing the subject's risk attitudes and beliefs one must be cautious in comparing outcomes to those predicted by the perfect foresight, risk-neutral model. Some discussion of the relationship between predicted and actual investment will, however, be made in the following section.

A central question addressed in this paper regards the appropriateness of imposing exogenous bargaining solutions on the nested bargaining process that occurs in incomplete contracts. The three treatment conditions feature widely different investment incentives, the purpose of which is to see if relative investment levels affect observed bargaining solutions. Both the Wilcoxon and the Kolmogorov-Smirnov tests strongly reject the null hypothesis that the sample of observations on X, Y, and s from different treatments are generated by the same stochastic process. In the Strong Seller Incentives treatment the mean s is .425, while in the baseline the mean s is .542, and in the Strong Buyer Incentives treatment the mean of s is .595. This variation in X, Y, and s across treatment condition suggests a possible relationship between relative investment levels and the sharing rule for realized surplus.

The relationship between s and relative investment levels is estimated using ordinary least squares regression analysis, the results of which are presented in Table III. Recall that buyers invest in X at cost [X.sup.2], and sellers invest in Y at cost [Y.sup.2]. The independent variable ([X.sup.2] - [Y.sup.2]) captures the difference in sunk costs from transaction-specific investment. The constant term establishes the point estimate on s when there is no difference in transaction-specific investment.

The regression analysis strongly supports the notion that the relative level of sunk, transaction-specific investment affects the outcome of future bargaining in incomplete contracts. The constant term is estimated to be .515, and has a remarkably small standard error of .0056. This is very close to the .50 Nash bargaining solution.(11) A central result of the analysis is that the difference in sunk cost ([X.sup.2] - [Y.sup.2]) has a significant and positive effect on the outcome of bargaining. In particular, the point estimate on ([X.sup.2] - [Y.sup.2]) is .142 with a standard error of .0104. The regression analysis strongly suggests that differences in sunk costs do indeed affect bargaining in the incomplete contracts investigated here. In fact, the joint null hypothesis that [Alpha] = .5 and [Beta] = 0 is rejected at well below the 1 percent significance level [F(2,154) 97.09; prob. > F = .00011.

In simple OLS regressions such as this where a distribution in two dimensions is fitted to a straight line, the estimators are quite sensitive to outlying points. Moreover, the data set analyzed here includes a number of outlying observations. Robust estimation techniques detect and weight outliers, and so yield a good fit to the majority of the data. Rousseauw and LeRoy [1987] suggests a least median squares (LMS) technique, which corresponds to finding the narrowest strip covering half of the observations. OLS estimates reweighted from the LMS technique are presented in Table III. LMS assigned a zero weight to twelve outlying observations. While the standard errors fell as expected under robust estimation, the point estimates differ only slightly from those of ordinary least squares, which suggests that the OLS estimators are remarkably robust.

One might also conjecture that substantial variation in individual experimental sessions may elicit strikingly different coefficient estimates on ([X.sup.2] - [Y.sup.2]). The hypothesis that the coefficient estimates on [X.sup.2] - [Y.sup.2]) are the same across experimental session cannot be rejected at commonly accepted significance levels [F(8,146) 1.344; prob. > F = .23].

Recall that sequential bargaining with a discount rate 6 of .9 creates a small first-mover advantage, in this case s =.526 rather than .50 in the subgame-perfect equilibrium. A first-mover advantage can be directly tested by creating separate samples for buyers and sellers as-first movers and then testing if the two samples of observations on s are generated by the same stochastic process. This was done, and neither the Wilcoxon nor the Kolmogorov-Smirnov tests were significant at even the 10 percent level.(12) Moreover, the first-mover advantage requires that the first proposal be accepted, and overall this never occurred more than one-third of the time in any of the treatments.

In summary, the analysis strongly rejects the hypothesis that the subgame-perfect bargaining equilibrium, s* predicts actual outcomes of bargaining in the incomplete contracts investigated here. Differences in transaction-specific investment have a significant effect on s, and the direction of influence is consistent with that of the equitable bargaining hypothesis [s.sup.e]; the marginal direct effect of [X.sup.2] - [Y.sup.2]) on s is approximately(13) with remarkably small standard error. It is important to note, however, that subjects did not generally divide surplus in such a way that net payoffs were equal. This is illustrated in Figure 5, which plots the cumulative percentage of agreements that deviate from equal net payoffs. If, hypothetically, all agreements had equalized net payoffs, then the "EP" plot shows that the cumulative percentage would be 100 at zero deviations from equal net payoffs. The thick line plots the cumulative percentage deviation of actual bargaining outcomes from equal net payoffs. Finally, in the hypothetical case in which all bargaining outcomes are subgame-perfect, the dotted line, labelled "SGP", plots the hypothetical cumulative percentage deviation from equal net payoffs that would have resulted.

Figure 5 shows that subjects managed to substantially reduce, but not eliminate, net payoff differentials in the two Strong Incentive treatments relative to what would occur if all bargains were the subgame-perfect 50-50 split. Since there is little difference between buyer and seller investments in the Equal Incentives treatment, the actual bargaining outcomes tended around the subgame-perfect share, and so these plots follow each other closely.

Backward Induction: The Influence of Bargaining Outcomes on Investment

A difficulty with assessing the role of bargaining outcomes on expected joint surplus in the laboratory is that transaction-specific investment here is a decision made under uncertainty over the marginal benefits of investment. As a result, assessing the effect of equitable bargaining outcomes requires empirically decomposing the constituent effects of risk preference and beliefs, among others, on investment.

Having pointed this out, consider the sample means for X and Y in Table Il. Results from the Strong Seller and Strong Buyer Incentives treatments are highly symmetric and are remarkably close to the X* and Y* predicted by the subgame-perfect analysis with risk neutral agents capable of perfect foresight. In the Strong Seller Incentives treatment, that model predicts X = .025 and Y = .75, whereas the sample means are X=.101 and Y=.760. In the Strong Buyer Incentives treatment the benchmark model predicts X=.75 and Y=.025, and the sample means are X = .774 and Y = .101. The greatest deviation from the model appears in the Equal Investment Incentives treatment, in which X and Y are predicted to be .5, and the sample means are X = .653 and Y = .596.

Rather than try to decompose the effects of risk preference and equitable bargaining outcomes on observed investment decisions, we can instead look directly at the bargaining outcomes forecasted by subjects. Recall that subjects provided forecasts of the outcome of bargaining over shares of (V-C) so that the computer could compute the subject's forecast-contingent expected payoff information in stage 1. Table IV presents nonparametric tests of the null hypothesis that forecast observations from different parameter regimes are generated by the same stochastic process. In nearly all cases this null hypothesis can be rejected at below the 1 percent significance level. The exception is in comparing the seller's forecast share, (1-s)[sup.f], for the Equal Investment incentives and the Strong Buyer Incentives paired samples, where the null can be retained at between the 12 and 18 percent level. Hence there is some support for the notion that subjects anticipate the influence of relative investment levels on the outcome of bargaining.

Backward induction implies that there should be a correlation between a subject's forecast of the bargaining outcome and the subject's level of investment. In fact, the correlation coefficient for [s.sup.f] and X is 0.48, and the correlation coefficient for (1-s)[sup.f] and Y is .42, and both of these coefficients are strongly significant. This correlation, together with the evidence that subjects anticipate the influence of relative investment on the bargaining outcome, suggests that [s.sup.e] was transmitted to the subjects' investment choices.

IV. SUMMARY AND CONCLUDING COMMENTS

The results of this experiment suggest that the bargaining process nested in incomplete contracts has properties that differ from those of the standard bargaining models. In particular, relative levels of sunk, transaction-specific investment influence the outcome of bargaining, which refutes the predictions of the subgame-perfect bargaining model. The party making the greater investment received the greater share of realized surplus. This finding is important to the way that economists model the incomplete contracting problem, as it implies a higher level of joint surplus than that which results from an exogenously imposed bargaining solution.

This research also identifies a situation in which people do not appear to ignore sunk costs. Phillips, Battalio and Kogut [19911 is another recent study which investigates the effects of sunk costs on valuation and decision making. They find that when subjects were required to make a sunk payment for a lottery ticket, the size of the sunk cost affected their subsequent valuation of the lottery. This effect was attenuated when subjects submitted bids to buy an auctioned lottery ticket. More research on the role of seemingly irrelevant commitments appears to be warranted.

This research is also related to the growing body of experimental bargaining literature. The ultimatum bargaining experiment literature surveyed by Guth and Tietz [1990] indicates that ultimatum proposals lying between equal splits and the subgame-perfect 100 percent allocation to the proposer predominate across a large number of studies. Similarly, here subjects use the bargaining process to attenuate but not fully equalize net payoffs. Hoffman and Spitzer [1982] survey the bargaining literature and find a number of factors that are associated with equal net payoff splits, including repeat face-to-face designs, being told their "task" is to divide a cash sum, the ability to make Pareto-improving moves, and knowledge of one-another's payoffs. Only the last condition is met here.

In their research on reward allocation Hoffman and Spitzer found at times that allocators, who were selected through a coin toss, accepted bargains that made them worse off than if disagreement had occurred. In Hoffman and Spitzer [1985] the allocator was chosen based on performance in tic-tac-toe rather than a coin toss. They found a large increase in self-regarding allocator behavior, and they speculate that it was because the allocator was perceived to have 'earned' the right to a larger payoff. Similarly in this study one might speculate that deviations from the subgame-perfect sharing rule occurred when a subject had 'earned' it by making a relatively larger sunk investment. While net payoff differences are reduced this way, full equality of net payoffs is generally not reached. This suggests that fairness notions based on net payoff equity cannot alone explain bargaining outcomes, as would also apply to ultimatum games.

Forsythe et al. [1991) specifically tested this by comparing laboratory bargaining outcomes from ultimatum games, where the other party can reject the ultimatum, to a game where the outcome is dictated by a single party.(13) If fairness alone explains allocations, then the outcomes of these games should be identical. Forsythe et al. can reject this hypothesis; it is interesting to note, however, that only 36 percent of the (paid) dictator game outcomes were consistent with subgame-perfection, and all (paid) ultimatum game observations were centered around the equal shares proposal. As with the ultimatum game experiments, this study of incomplete contracting points to the need for a more descriptive and testable theory of bargaining behavior.

APPENDIX

Instructions and Questionnaire for the Equal Investment Incentives Treatment

This is an experiment in the economics of decision making. The funding for this experiment has come from a number of sources, including Indiana University. The instructions are simple, and if you follow them carefully and make good decisions, you may earn a considerable amount of money that will be paid to you in cash at the end of the experiment. The experiment takes place on the computer, but you do not need to have any specialized knowledge of computers in order to successfully participate in this experiment. Any attempt at communicating with other participants verbally or in any way other than through the computer is grounds for being REMOVED from the experiment.

The experiment consists of a series of periods. Each period the computer will randomly match one participant with another to form independent decision-making pairs. Once participants have been matched, the computer will randomly choose one participant to have the status of a "buyer;" the other participant will then have the status of a "seller." You are equally likely to be chosen a buyer or a seller. You will not know the other party's identity.

In this experiment each buyer and seller decision-making pair must make a series of decisions regarding the implementation of a PRODUCT IMPROVEMENT. Your payoffs depend on the type of decisions that are made by you and by the participant that you have been matched with in a given period. The following is a step by step summary of the sequence of decisions you will be confronted with each period:

I. INVESTMENT DECISION

Buyers can invest some of their earnings in increasing the VALUE (V) of the product improvement; similarly, sellers can invest some of their earnings in decreasing the COST (C) of providing the product improvement. The investment process proceeds as follows.

At the time the investment decision is made, the cost C and the value V of the product improvement are not known. Specifically, the value of the product improvement will be either $5 or $3. Buyers will have an opportunity to invest some of their earnings to increase the PROBABILITY (X) that V will be $5 rather than $3. Similarly, the cost C will be either $0.50 or $2.50, and sellers will have an opportunity to invest some of their earnings to increase the PROBABILITY (Y) that C will be $0.50 rather than $2.50. The computer uses X and Y in the following way when it randomly chooses V and C:

Suppose a buyer decided to invest enough earnings to raise X up to 30-a 30% probability that V is $5 rather than $3. The computer will then randomly draw a number, and any number between 0 and 100 is equally likely to be drawn. If the number drawn by the computer is between 0 and 30, then V = $5; on the other hand, if the number drawn by the computer is between 31 and 100, then V is $3. The more earnings the buyer invests the larger is X, and so the larger is the probability that V = $5 rather than $3.

Similarly, suppose a seller decided to invest enough of her earnings to raise Y up to 75-a 75% probability that C is $0.50 rather than $2.50. If the number drawn by the computer is between 0 and 75, then C = $0.50; on the other hand, if the number drawn by the computer is between 76 and 100, then C is $2.50. The more earnings the seller invests the larger is Y, and so the larger is the probability that C = $0.50 rather than $2.50.

The amount of earnings needed to increase X and Y will be illustrated to you on the computer following these instructions. After the buyer chooses X and the seller chooses Y, the computer will show the values of X and Y, as well as the values of V and C, to both parties.

II. SHARING DECISION

Once V and C are made known to the buyer and the seller, they can choose to share the GROSS RETURN (V-C)--the difference between the value and the cost of the product improvement. Note that the expense incurred by the buyer in increasing X, and the expense incurred by the seller in increasing Y, are sunk costs that are paid regardless of how (V-C) is shared.

The gross return (V-C) can only be shared if both the buyer and the seller agree. If they both agree, then the buyer and the seller will alternate making share offers until EITHER an offer is accepted, OR negotiations break down. In odd-numbered periods buyers will offer first, while in even-numbered periods sellers will offer first.

When it is your turn to make a share offer, then the other person has the choice of accepting or rejecting your offer. The total number of share offers that can be made by you and the other party is limited, and has been fixed in advance. Each time an offer is rejected, there is a chance that no more share offers will be allowed, and negotiations will be ended without (V-C) having been shared. The maximum number of share offers that can be made may be large or small, and will usually be different from one period to the next. You should reckon that there is a 90 percent chance that at least one more share offer can be made.

Buyers and sellers can make any share offer which adds up to 100 percent. If negotiations break down, then the buyer gets V, and the seller pays C.

EXAMPLE: suppose that V = $5 and C = $0.50, so that (V-C) = $4.50. Unless BOTH the buyer and the seller agree to share (V-C), the buyer will get V = $5, and the seller will have C = $0.50 deducted from her earnings. Suppose instead that they agree to share, and the buyer made a share offer of 33-the buyer was willing to take 33 percent of (V-C) = $1.50, and so give the seller 67 percent of (V-C) = $3. If the seller accepts this offer, that is how the gross return will be shared.

III. VETO DECISION

After the sharing decision has been made, EITHER the buyer OR the seller can decide to stop ("veto") the product improvement. If either the buyer or the seller vetoes, then the value of the product improvement V = $0, and the cost of the product improvement C = $0.

IMPORTANT ISSUES:

1. FORECASTING

At the time you choose X or Y, you will always know how much it costs you to choose different levels of X or Y. On the other hand, the money payoff that you can expect to receive depends on your forecast of what the SHARING and VETO decisions will be. In order to provide you with your FORECASTED PAYOFF, you will be asked to enter your forecast of (i) whether you and the other party will SHARE (V-C), and if so, what share you will receive, and how much investment in X or Y the person you are matched with will make, and (ii) whether the product improvement will be VETOED. 2. NET PAYOFFS

Net payoffs each period are computed as follows: If the product improvement is NOT vetoed:

A. If (V-C) was shared:

BUYER PAYOFF =

share* $ (V-C) = $(investment in X)

SELLER PAYOFF =

(1-share)*$(V-C) = $(investment in Y)

B. If (V-C) was not shared:

BUYER PAYOFF =

$V - $(investment in X)

SELLER PAYOFF =

-$C - $(investment in Y)

If the product improvement IS vetoed:

BUYER PAYOFF =

-$(investment in X)

SELLER PAYOFF =

-$(investment in Y)

You will have a starting balance of $4 at the beginning of the experiment. Your new balance each period will be your previous balance, plus you net payoffs that period. Net payoffs can be negative, and if your balance falls below $0, you will be removed from the experiment.

If you have any questions, you can enter "R" re-read the instructions, or you can raise your hand.

Questionnaire COST OF X or Y 0 0 $0.00 10 10 $0.01 20 20 $0.04 30 30 $0.09 40 40 $0.16 50 50 $0.25 60 60 $0.36 70 70 $0.49 80 80 $0.64 90 90 $0.81 100 100 $1.00

This table illustrates how much of your earnings you must invest to increase the probability X or Y. Use this information for the questions below. 1. Pick an X for a buyer, and a Y for a seller.

X = --, cost of X = $--;

Y = --, and cost of Y --. 2. Suppose that the computer randomly picks

the number 47. If the X you picked is 47 or

larger, then V + $5; otherwise V = $3. If the

Y you picked is 47 or larger, then C = $0.50;

otherwise C = $2.50. 3. Calculate V, C, and (V-C) based on the information

in #2 above.

V = $--, C = $--,

(V-C) = $--. 4. a. Suppose that (V-C) is NOT shared, and

the product improvement is NOT vetoed.

Then the buyer's and the seller's net payoffs

are:

buyer: $-- seller: $--. b. Suppose that (V-C) is NOT shared, and

the product improvement IS vetoed. Then

the buyer's and the seller's net payoffs

are:

buyer: $-- seller: $--. c. Suppose that (V-C) IS shared, and the

product improvement is NOT vetoed.

Pick a way to split (V-C):

buyer share = --%

seller share = --%

buyer's net payoff = $--;

seller's net payoff = $--. d. Suppose that (V-C) IS shared as you

picked in c. above, but the product improvement

IS vetoed. Then calculate

buyer's net payoff = $--;

seller's net payoff = $--. (1.) Tirole [1988, 29-32] and Holmstrom and Tirole [1989, 66-67] offer accessible introductions to this model. There are certainly other important models of incomplete contracting, such as Wiggins [1990], which focuses on an effort-information sharing tradeoff. See Wiggins, Hackett, and Battalio [1989] for an experimental study based on this model. (2.) This step can be omitted with no violence to the properties of the model and does not affect the subgame-perfect bargaining equilibrium. This step was added for laboratory investigation in order to allow subjects to immediately express their desire not to bargain at all. If subjects were forced to bargain, the desire not to bargain could be expressed by rejecting all share proposals and making unacceptable proposals (e.g., 100,0), until breakdown of the bargaining process is realized. (3.) This can be written symmetrically for the seller making a time 0 offer to the buyer, which takes the form [[Delta](1-S) + B]/(1 + [Delta]). (4.) In the spirit of their model of vertical integration, one could argue that if [D.sub.v] > [D.sub.c] the maximum joint surplus is achieved when the buyer purchases the seller's veto right and sets s = 1. (5.) The subject pool was recruited from honors principles classes and courses taught at the junior, senior, and first-year graduate level. Students from game theory and experimental economics classes were not recruited. The subject pool included international students from China, Germany, India, Malaysia, Mexico, the Netherlands, and Spain. (6.) The programs controlling the information and message spaces of the experiment were executed on a VAX computer at Indiana University. (7.) The design of having ten game repetitions using anonymously and randomly matched inexperienced subjects under a single parameter regime was chosen to closely follow the procedures of Binmore et al. [1991]. Half of the time their subjects were buyers, and the other half they were sellers. Similarly, here subjects stand a 50 percent chance of being a buyer or a seller in any given period. Random, anonymous matching and establishment of buyer and seller identities greatly attenuates any reputation effects from the multi-period setting. In addition, as Binmore et al. point out, bargaining models assume parties are knowledgeable of the others' incentives, and this procedure provides subjects with the experience to develop this knowledge. (8.) Subjects were told that this information would be made known to both parties. By randomizing whether subjects were buyers or sellers in a given period, it quickly became clear to them that the announced common knowledge of these parameter values was in fact true. (9.) Binmore et al. report initial difficulty in communicating breakdown probabilities to subjects, which led them to fixing them in advance. (10.) Using two sample non-parametric techniques that compare samples from the first three and last five periods in which subjects were paid, the null hypotheses that X, Y, and s are generated by the same stochastic process can only be rejected in a few circumstances. In particular, s shows a significant downward shift in the last five periods of the Strong Seller Incentives treatment. Both s and X show a significant upward shift in the last five periods of the Strong Buyer Incentives treatment. These shifts in s that are associated with more experienced play are both in a direction that reinforces the argument for [s.sup.e]. (11.) The small amount that it exceed .5 is significant, however. This could be accounted for by the fact that mean X > mean Y in the baseline condition. (12.) Since buyers had this advantage in odd-numbered periods, a strong time trend from one period to the next can create a problem with these tests. Recall from footnote 10 that the sample of observations on s from the first three periods in which subjects were paid differ from those of the last five. This does not present a problem per se, however, as both the buyer and the seller had an opportunity to move first in both the first three and the last five periods. Linear and exponential decay trend variables are found to have no significant influence on s. (13.) Harrison and McCabe [1991] also challenge the fairness explanation. When first proposals are not generally accepted, so subjects have experience in bargaining subgame play, they find that behavior converges to the subgame-perfect prediction. Note that in this study first proposals are usually rejected, and so subjects have experience with subgame play, yet shares tend to move away from the subgame-perfect prediction.

REFERENCES

Binmore, K. "Bargaining and Coalitions, I." in Game Theoretic Models of Bargaining, edited by A. Roth. Cambridge: Cambridge University Press, 1985, 269-304. Binmore, K., Rubinstein and A. Wolinsky. "The Nash Bargaining Solution in Economic Modeling." Rand Journal of Economics, Summer 1986,176-88. Binmore, K., Shaked and J. Sutton. "An Outside Option Experiment." Quarterly Journal of Economics, November 1989, 753-70. Binmore, K., P. Morgan, A. Shaked and J. Sutton. "Do People Exploit Their Bargaining Power? An Experimental Study." Games and Economic Behavior, August 1991, 295-322. Crawford, V. "Long Term Relations Governed by Short Term Contracts." American Economic Review, June 1988,485-99. Forsythe, R., J. Horowitz, N. Savin and M. Sefton. "Fairness in Simple Bargaining Experiments." Photocopy, University of Iowa, 1991. Friedman, D., and R. McGuire. "Fairness as an Efficiency-Enhancing Constraint.' Photocopy, University of California, Santa Cruz, 1988. Fudenberg, D., B. Holmstrom and R Milgrom. "Short-Term Contracts and Long-Term Agency Relationships." Working Paper No. 488, MIT, 1987. Grossman, S., and O. Hart. "The Costs and Benefits of Ownership: A Theory of Vertical and Lateral Integration." Journal of Political Economy, August 1986, 691-719. Guth, W., R. Schmittberger and B. Schwarze. "An Experimental Study of Ultimatum Bargaining." Journal of Economic Behavior and Organization 3(3), 1982, 367-88. Guth, W., and R. Tietz. 'Ultimatum Bargaining Behavior." Journal of Economic Psychology, September 1990, 417-49. Harrison, G., and K. McCabe. "Testing Bargaining Theory in Experiments." Photocopy, University of South Carolina, 1991. Hart, O., and J. Moore. "Incomplete Contracts and Renegotiation." Econometrica, July 1988, 755-86. Hart, O., and B. Holmstrom. "The Theory of Contracts," in Advances in Economic Theory, edited by T. Bewley. Fifth World Congress, Cambridge: Cambridge University Press, 1987, 71-155. Hoffman, E., and M. Spitzer. "The Coase Theorem: Some Experimental Tests." Journal of Law and Economics, April 1982, 73-98. --. "Entitlements, Rights, and Fairness: An Experimental Examination of Subjects' Concepts of Distributive Justice." Journal of Legal Studies, June 1985, 259-97. Holmstrom, B., and O. Hart. "The Theory of Contracts." Photocopy, MIT, 1985. Holmstrom, B., and J. Tirole. "The Theory of The Firm," in Handbook of Industrial Organization, edited by R. Schmalensee and R. Willig. Amsterdam: North-Holland, 1989, 63-133. Klein, B., R. Crawford and A. Alchian. "Vertical Integration, Appropriable Rents, and the Competitive Bidding Process." Journal of Law and Economics, October 1978, 297-326. Macaulay, S. "Non-Contractual Relations in Business." American Sociological Review, February 1963, 55-70. Milgrom, P, and J. Roberts. "Bargaining and Influence Costs and the Organization of Economic Activity." in Positive Perspectives on Political Economy, edited by J. Alt and K. Shepsle. Cambridge: Cambridge University Press, 1991, 57-89. Neelin, J., H. Sonnenshein and M. Spiegel. "A Further Test of Noncooperative Bargaining Theory: Comment." American Economic Review, September 1988, 824-36. Ochs, J., and A. Roth. "An Experimental Study of Sequential Bargaining." American Economic Review, June 1989, 355-84. Phillips, O., R. Battalio and C. Kogut. "Sunk and Opportunity Costs in Valuation and Bidding." Soutern Economic Journal, July 1991, 835-56. Rousseauw, P, and A. Leroy. Robust Regression and Outlier Detection, New York: John Wiley and Sons, 1987. Rubinstein, A. "Perfect Equilibrium in a Bargaining Model." Econometrica, January 1982, 97-110. Tirole, J. "Procurement and Renegotiation." Journal of Political Economy, April 1986, 235-59. --. The Theory of Industrial Organization, Cambridge: MIT Press, 1988. White, H. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity." Econometrira, July 1980, 817-38. Wiggins, S. "The Comparative Advantage of Long Term Contracts and Firms." Journal of Law, Economics, and Organization, Spring 1990, 155-70. Wiggins, S., S. Hackett and R. Battalio. "Relational Contracts, Firms, and Reputation: An Experimental Study of Institutional Choke." Photocopy, Texas A&M University, 1989. Williamson, O. Markets and Hierarchies. New York: Free Press, 1975.

Department of Economics, Indiana University, Bloomington, IN 47405. This paper has benefitted from helpful comments by three anonymous referees and coeditors Thomas Borcherding and Rodney Smith. I would also like to thank Ray Battalio, Dave Besanko, Ken Binmore, Miguel Delgado, Roy Gardner, Ron Johnson, Charles Kahn, Tom Lyon, Jimmy Walker, Steve Wiggins, Arlie Williams, and Doug Young. Support from the Workshop in Political Theory and Policy Analysis at Indiana University is gratefully acknowledged. All data are stored on permanent disk files and are available on request.

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Author: | Hackett, Steven C. |
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Publication: | Economic Inquiry |

Date: | Apr 1, 1993 |

Words: | 10509 |

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