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Prices in experimental asset markets under uncertainty.

1. Introduction

A growing body of experimental evidence argues that asset market anomalies are grounded in individual cognitive biases (Hirshleifer, 2001). This result contrasts with the neoclassical argument that market discipline induces traders to learn from experience, and from more rational players. Individual errors, therefore, should not be reflected on market prices.

Among many judgmental errors, reaction to uncertainty is a probability judgment bias of particular relevance for decision making in markets. The theory of asset markets accepted by most economists is built on the basis of Subjected Expected Utility theory (Savage, 1954), according to which agents' decisions should be represented by additive subjective probability distributions. Choice under risk (where probabilities of outcomes are known) should be equivalent to choice under uncertainty (where probabilities are vague or unknown), provided the possible outcomes and the expected probabilities of those outcomes are the same. Since Ellsberg's (1961) seminal experiment, however, substantial experimental evidence has shown that for many individuals uncertainty matters. Either ambiguity aversion--the preference to bet on a risky prospect rather than on an equivalent uncertain one, or ambiguity proneness--the preference for uncertainty as opposed to risk, are exhibited by many experimental subjects. Camerer & Weber (1992) provide a comprehensive review of experiments on ambiguity until the early 1990s. Reaction to ambiguity has been observed not only in experiments that do not adopt incentive-compatible elicitation procedures, but also in settings using preference-revealing devices such the Becket-De Groot-Marshak mechanism (Halevy, 2007).

The intuition that aversion/preference for uncertainty may have effects on the functioning of asset markets goes back to Keynes (1921). More recently, theoretical research (Dow & da Costa Werlang, 1992; Epstein & Wang, 1994; Mukerji & Tallon, 2001, among others) has shown that ambiguity aversion affects prices and trades, determining phenomena generally attributed to the presence of market incompleteness, asymmetric information, or transaction costs.

Most experimental tests of asset markets (Sunder, 1995) have been conducted under the assumption of risk, and there are few experimental tests of market behaviour in the presence of unknown or vague probabilities. In Camerer & Kunreuther's (1989) double auction market on insurance coverage, the presence of ambiguity did not have any significant effect on prices. Sarin & Weber (1993) built a series of double oral auctions and of sealed bid auctions in which both risky and uncertain certificates were traded either independently or simultaneously. Prices for risky and uncertain assets converged when assets were traded independently but not when simultaneous trade occurred. Di Mauro (2008) finds that ambiguity reaction (measured by the differential valuation of assets based on familiar versus unfamiliar events) is reflected in mean prices in a pronounced fashion.

This paper presents an experiment that investigates the impact of ambiguity on market prices. The experiment innovates the existing literature in two respects. The first is that equilibrium pricing predictions are the same whether the certificates traded are characterized by known or vague probabilities of payoffs (Rietz, 1999; Weber et al., 2000). The second concerns the investigation of the role of arbitrage as a market mechanism potentially capable of reducing price distortions due to uncertainty. In existing experiments (for instance, the aforementioned Camerer & Kunreuther, 1989; Sarin & Weber, 1993), the expected ability of markets to correct judgment errors relies on the repetition of the trading experience, and the feedback provided by exchange prices and outstanding bids and asks. With respect to experience and feedback, arbitrage is a source of discipline on prices that is truly market-specific. Experimental methods have a great potential for exploring this issue because in real markets arbitrage is rarely risk-less and costless, whereas the laboratory offers a controlled environment in which these conditions can be realized.

The design of the experiment and the hypotheses tested are presented in the next section. Section 3 analyses the experimental results obtained, while section 4 concludes with a discussion of implications for the functioning of markets.

2. The experiment

2.1. Design

The experimental markets are organized as computerized double auctions built using Z-Tree (Fischbacher, 2007). Consider two single-period state contingent certificates, Yellow and Red, traded in two separate markets. The Yellow certificate pays a liquidating dividend of 10,000 francs (1) if a yellow chip is drawn from an urn containing 50 yellow and 50 red chips. No dividend is paid if Red is drawn. The Red certificate yields a dividend of 10,000 francs if Red is drawn and nothing otherwise. Accordingly, the probability of drawing either Red or Yellow is equal to 0.5. Let us refer to the Yellow and the Red as the risky certificates given that the probability of drawing either colour is objectively known.

Next, consider the markets for two certificates denominated Blue and Green, and an urn containing 100 blue and green chips in unknown proportion. Since the probability of green/blue is not known, traders have to form subjective probabilities. The Blue/Green certificate yields a dividend of 10,000 francs if blue/green is drawn from the urn, and zero otherwise. Let us denote the Green and the Blue as the uncertain certificates. Ambiguity averse/neutral/prone traders will consider that the probability of drawing any of the two colours from the urn is less/equal/greater than 0.5. This implies that in the presence of aversion/preference for ambiguity, the Green and Blue certificates will trade at lower/higher prices than the Red and Yellow certificates.

This leads to the formulation of the following hypothesis:

Hypothesis 1. If traders are ambiguity averse/prone the sum of prices for Green and Blue certificates will be lower/higher than the sum of prices for Red and Yellow certificates.

In order to test Hypothesis 1, prices for risky certificates must be compared with those for uncertain certificates. Twelve experimental sessions were run in which risk and uncertainty were manipulated between-subject. In six sessions (1, 2, 5, 6, 9, 10) participants traded the risky certificates, while in six others (3, 4, 7, 8, 11, 12) subjects traded the uncertain certificates. Subjects who signed up for the experiment were randomly assigned to one of the two types of sessions.

Since the Yellow and Red certificates are perfectly and negatively correlated, an individual holding one unit of both certificates can simultaneously assure himself a gain of 10,000 francs. Arbitrage opportunities will arise every time the sum of buying/selling prices for the two certificates exceeds/is less than 10,000 francs. If the sum of prices for complementary certificates is below/above the certificate's payoff, traders should buy/ dump certificates and gain a risk-less profit. Efficient markets imply that in equilibrium the sum of prices for the two certificates equals 10,000 francs, irrespective of risk attitudes. Likewise, since ownership of an equal number of Blues and Greens gives a sure gain of 10,000 francs times the pairs of certificates owned, in equilibrium the sum of prices of Blue and Green certificates should equal 10,000 francs, irrespective of whether agents form subjective probabilities that differ from the objective probabilities of the risky certificates. The above discussion leads to the second hypothesis.

Hypothesis 2. Arbitrage drives the sum of prices for complementary assets to the equilibrium value of 10,000 francs.

A corollary to this hypothesis is that arbitrage forces the sum of prices of complementary certificates under risk and uncertainty to be equal.

The above design allows for risk-less cross-arbitrage between the markets for the complementary certificates. However, there are two caveats that may make arbitrage difficult. First, the idea that arbitrage forces the price to be equal to the dividend relies on the opportunity to sell or buy the two complementary assets at current prices. In the setting described above, a trader has to buy one asset (say Yellow) at the current market price before being able to buy the complementary asset (Red). In the meantime, the Red asset may already be sold and no longer available at the planned price. In order to overcome this problem, a third market was introduced in sessions 5, 6, 7, and 8, in addition to the two independent markets. In this new market the 'bundling' of Red/Yellow (Green/Blue) was permitted, i.e. the two complementary certificates traded as one object (Rietz, 1999). Since in the bundling setting traders can buy/sell two assets at the same time, arbitrage is greatly facilitated.

The second caveat comes from the fact that understanding and exploiting arbitrage opportunities may not come naturally (Schleifer & Vishny, 1997). For instance, the exploitation of arbitrage opportunities may require the presence of a professional arbitrager or the provision of specific information. In order to verify this, in four other sessions (9,10,11,12, referred to as 'bundling and information" henceforth) traders were allowed to trade bundles and further received oral explanations during the dry runs concerning how they could profit from arbitraging. Here, the following is a description of the type of information provided:
 Imagine that the sum of outstanding prices for the Red and Yellow
 (Green and Blue) certificates in the independent markets or in the
 third market were above 10,000. Since the dividend that each couple
 fetches is 10,000 francs, for each couple of certificates you own
 you can gain a sure profit by selling them at the market price.
 Your sure gain will be the difference between the sum of sale
 prices and 10,000.

 On the contrary, imagine that the sum of outstanding prices for
 the Red and Yellow (Green and Blue) certificates in the independent
 markets or in the third market were below 10,000. Since the
 dividend that each couple fetches is 10,000 francs, if you have
 enough cash, you can gain a sure profit by buying them at the
 market price. Your sure gain will be the difference between 10,000
 and the sum of purchase prices.

 Finally, if the sum of outstanding prices for the Red and Yellow
 (Green and Blue) certificates in the independent markets is
 different from the market price of the portfolio, you can profit by
 buying where the purchase price is lower and reselling them where
 the price is higher.


Therefore, Hypothesis 2 may be weakened by allowing for the fact that the sum of the prices of complementary assets will be closer to the dividend when bundling is possible, and even closer when information about arbitrage is provided.

Hypothesis 3. The difference between the sum of mean prices under risk and uncertainty is smallest when both bundling is allowed and information on how to conduct arbitrage is provided. It is highest when the exchange of bundles is not allowed.

2.2. Organization of experimental sessions

Twelve experimental sessions with 12 subjects each for a total of 144 participants were run at the University of Calabria and at the University of Catania in Italy. Subjects were undergraduates in either Economics, or Law or Political Sciences who had already taken one introductory course in Economics. Subjects were all first time participants in experiments.

Markets were run for 10 periods (except for three sessions that were run for eight periods due to a software crash in the ninth period), plus two dry runs that subjects used to make themselves familiar with the experimental procedure. Upon arrival, each subject sat in front of a computer screen and received a copy of the experiment's instructions. The experimenter then re-read the instructions aloud and encouraged subjects to ask questions. After the trial runs, subjects were again asked for questions and encouraged to take their time to understand the instructions and the arrangement of information on the computer screen. The full set of instructions across treatments can be found in the appendix.

At the beginning of each period, traders received an initial cash endowment and a risky position, which was either made up of 1 Yellow + 5 Red or 1 Red + 5 Yellow certificates. (2)

The initial asset endowment was inverted after five periods of trading. The cash endowment was large enough to allow traders to buy the entire supply of certificates at equilibrium prices. State contingent certificates lived one period and therefore expired at the end of each market round. Traders could both sell (i.e. announce offers for) and buy (i.e. announce bids for) certificates, with the only limitation that a new bid had to be higher than outstanding bids and a new offer had to be lower than outstanding offers. No short-selling was allowed, so certificates on hand could not be negative. Traders were remunerated on the basis of accumulated earnings.

The dividend on all types of certificates traded was determined at the end of each market round according to the mechanism explained in Section 2.1 above. In the uncertainty sessions, the proportion of blue and green chips was modified for each period using the table of random numbers. Hence, traders could not learn the exact composition of the urn.

Sessions lasted about an hour and a half each. At the end of the experiment, subjects received their earning in euros. Average earnings were about 17.5 [euro]. No out-of-pocket loss was possible except for the participation fee (5.00 [euro]) which players received upon entering the laboratory.

3. Results

In this section, the experimental findings are discussed, first by means of a descriptive analysis of prices and then by presenting the results of a panel data estimation.

Table 2 presents period means and standard deviations of the sum of prices for complementary assets for each market session.

In order to assess the effect of ambiguity on prices (Hypothesis 1) and whether this effect depends on the possibility of bundling or on the availability of information (Hypotheses 2 and 3), prices in the sessions for risky certificates are compared with those under uncertainty under the three treatments, 'no bundling', 'bundling', 'bundling and information'. In the 'no bundling' treatment, which did not encompass the third market (sessions 1-4), the sum of prices for the risky certificates is sensibly lower than that for the uncertain certificates in all periods. In the treatment in which bundling is allowed (sessions 5-8) evidence on the effect of ambiguity is unclear: the lowest prices are observed in one of the risky sessions (session 5), whereas the highest are observed in session 8 in which uncertain certificates were traded. However, prices for risky assets are not always lower than those for uncertain assets. Finally, in the 'bundling and information' treatment, in which bundling is permitted and oral information on how to conduct arbitrage is provided to traders, the difference between prices under risk and prices under ambiguity becomes smaller.

Figure 1 offers a more compact way to assess the impact of probability uncertainty on mean prices. In the figure, the ambiguity premium, calculated as the proportional change in the mean uncertain price with respect to the risky one, is shown for each treatment. Prices for the two risk/uncertainty sessions of each treatment have been pooled. The horizontal line with intercept at zero represents the ambiguity neutrality benchmark (no impact of ambiguity). The dotted line ([triangle]), which represents the 'no bundling' treatment lies above the ambiguity neutrality line, indicating ambiguity preference. The solid line ([circle]) represents the premium under the bundling treatment. Except for the first two periods, in which the extreme outliers in session 8 determine very high mean prices, the ambiguity premium tracks quite closely that observed in the no bundling treatment. Finally, the dashed line ([square]) shows that when information on how to arbitrate is provided, ambiguity reaction tends to disappear.

[FIGURE 1 OMITTED]

Summing up, this preliminary descriptive analysis tells us that the answer to the question 'does ambiguity affect market prices?' is a qualified yes: in the 'no bundling" treatment, ambiguity reaction arises, even if there are complete markets for certificates in which risk can be fully hedged. Likewise, in the 'bundling' treatment, the possibility of exchanging these assets in bundles does not eliminate the effect of ambiguity. However, in the 'bundling and information' treatment, the ambiguity premium tends to disappear. Once traders get a sounder understanding of how to arbitrate between markets, the effect of ambiguity is washed out.

The presence of an ambiguity premium implies that the possibility to arbitrate per se does not drive prices down to the equilibrium value of 10,000. Thus, Hypothesis 2 seems to be rejected by the data. However, Table 2 shows that the extent of over-pricing, i.e. the distance of the sum of mean prices from the no arbitrage value of 10,000, declines in the 'bundling' and 'bundling and information' sessions. This confirms the third hypothesis: as the way in which arbitrage must be exercised becomes clearer, over-pricing declines. Especially in the 'bundling and information' treatment, this contributes to reduce the impact of ambiguity. (3)

As it is standard in many market experiments, intra-session learning also occurred. Market experience reduced the extent of over-pricing and the ambiguity premium. As Figure 1 shows, the ambiguity premium declines through time. Table 2 further shows that the sum of mean prices in the last market period is lower than in the first one in all sessions but three.

Although prices and the ambiguity premium decline through time, over-pricing remains a stable feature in all sessions. The persistent evidence of overpricing may be due either to insufficient market experience (Smith et al., 1988) or to irrational trades by some of the players (Lei et al., 2001; Haruvy and Noussair, 2006). For instance, in three sessions with high over-pricing (namely sessions 3, 4, and 12) four subjects out of 12 bought at prices exceeding 10,000 francs. Two of these players made purchases at excess prices only in the initial periods of the game and quickly turned to lower bids. The remaining two, however, did not learn and continued to buy at high prices, thus preventing prices from falling. In session 8, five subjects bought at excess prices in the initial three periods, and two of them continued to do so later in the session.

A random effect panel data analysis has been carried out to allow for heterogeneity across sessions and dependence between periods. The mean sum of prices per period has been regressed on the following treatment variables:

(1) AMBIGUITY, a dummy taking value 1 in the sessions with vague probability of dividends;

(2) BUNDLING, a dummy taking value 1 in the sessions in which the exchange of certificates in bundles is allowed for;

(3) BUNDLING AND INFORMATION, a dummy taking value 1 in the sessions in which subjects both received oral information on how to conduct arbitrage and were allowed to exchange bundles.

A time trend (PERIOD) was also added to the regression. Results are reported in Table 3.

Two models were estimated. In the first, the effect of the time trend, of ambiguity and of bundling are estimated. The signs of the variables are as expected: the possibility of trading in bundles significantly reduces the mean sum of prices, whereas uncertain certificates trade at significantly higher prices. In addition, market experience significantly reduces the mean sum of prices. The second model differs from the first in that the dummy for the third treatment is introduced. The variable BUNDLING AND INFORMATION bears a negative sign as expected. However, it is not statistically related to the sum of prices.

4. Discussion

This experiment has explored two different issues. The first is whether the sum of the prices of two complementary certificates under risk is the same as that under uncertainty. The second issue is if arbitrage drives the two sums of prices equal to each other. The investigation of these two issues is relevant as it can shed light on the consequences of reaction to uncertainty for equilibria in real life asset markets.

The main results of the experiment can be thus summarised.

(1) Ambiguity matters for market prices. Assets whose payoff occurs with an objective probability of 0.5 exchange at lower prices with respect to equivalent assets having vague probabilities. This finding entails that the uncertainty setting creates greater opportunities for arbitrage. The higher degree of overpricing under uncertainty reflects the beliefs of ambiguity-seeking traders. Preference for uncertainty can be reconciled with some models of behaviour under uncertainty (for instance Hogarth & Einhorn, 1990). In particular, it may be explained by a psychological attitude of traders who tend to hope that the more favourable outcomes occur when the probabilities of those outcomes are not objectively known.

In a market, attitudes towards uncertainty may be heterogeneous across the traders' population. If so, one might question why the beliefs of ambiguity-averse traders are not reflected in market prices observed in this experiment. Bossaert et al. (2007) find that agents who are sufficiently ambiguity averse choose not to be exposed to ambiguity, i.e. they prefer not to engage in trade, so their beliefs about ambiguous states are not reflected in prices.

(2) The divergence of the sum of mean prices for complementary assets from the no arbitrage value diminishes with market repetitions and with the introduction of the possibility of trading complementary assets in bundles.

Since the possibility of bundling allows traders to buy/sell two assets at the same time, arbitrage is greatly facilitated. Further, the provision of oral explanations of arbitrage mechanisms, together with the introduction of bundling makes arbitrage opportunities even more evident to traders, thereby mitigating the judgmental bias due to ambiguity. We conjecture that the provision of explanations also managed to convey information to subjects participating in the uncertain markets about the fact that the likelihoods of drawing either of the two colours summed to one. This may justify why risky and uncertain prices in the treatment with bundling and information were actually closer than when just bundling was allowed.

(3) Overpricing persists in the final market period, although mean prices decay in the course of the market session. Several mutually consistent explanations for this can be identified. First, it is likely that the fact that traders were first time participants in experiments would have required longer market experience to determine convergence to fundamental value (Smith et al., 1988). Second, the experimental design that entailed a very large cash endowment and the absence of real out-of-pocket losses for traders may have biased prices upwards. Third, prices may have been pushed up by the presence of irrational traders. Finally, since subjects could not short-sell, their ability to exploit arbitrage opportunities was in principle limited. Haruvy & Noussair (2006) report experiments in which short-selling did reduce over-pricing but did not drive prices to the fundamental value.

To conclude, the experimental results suggest that the empirical relevance of uncertainty reaction is weakened when traders learn to arbitrate across markets. Thus, ambiguity may become irrelevant in markets in which hedging opportunities are clear to traders, or where professional arbitrageurs operate. Further tests of the robustness of these findings are of course needed.

Acknowledgements

The author thanks the editors and two anonymous referees for stimulating comments and criticism. Thanks are also due to Urs Fischbacher, David Grether, Anna Maffioletti, and participants in various seminars at which material drawn from this experiment has been presented. Benedetto Bruno and Sebastiano Scire helped computerize and run the experiment.

Appendix

General Instructions (translated from Italian)

You are about to participate in an experiment in the economics of asset market decision making.

The goal of the experiment is to gain insight into some features of economic behaviour in a market setting. The experiment is part of a research project financed by the Italian Ministry of Scientific and Technological Research.

The instructions are simple. If you follow them carefully and make good decisions, you might earn a considerable amount of money which will be paid to you in cash at the end of the experiment.

In this experiment we are going to conduct markets in which you will buy and sell two types of certificates in a sequence of 10 market periods. At the beginning of each market period you shall receive an endowment made up of certificates and cash. You can use the cash to buy and sell certificates or you can save it.

In the experiment earnings come from two sources: (1) dividends on certificates you hold at the end of each market period; (2) capital gains on certificates bought and sold.

The currency used in these markets is francs. All prices and profits will be denominated in francs. At the end of the experimental session, all profits in francs will be converted to euros at the exchange rate of 1 euro per 12000 francs, and you will be paid in euros. Notice that the more francs you earn, the more euros you will receive at the end of the experiment.

The experimental session lasts about one hour and thirty minutes.

During the experiment you are not allowed to communicate with the other participants. If you do, you will be asked to leave the lab and will be excluded from the list of the Experimental Economics Lab for the future.

At the end of the experiment, if you wish to, you can leave your comments and suggestions to the experimenters using the sheet provided.

Thank you for your collaboration.

Specific Instructions

Treatment 1 (Treatment 2 and 3 in italics)

In this experiment, you will participate in markets in which you can buy and sell two types of certificates: the 'Yellow' certificate and the 'Red' certificate [In the sessions fi)r uncertain certificates: the 'Green' certificate and the 'Blue' certificate]. There will be 10 market periods. At the beginning of each market period, you are endowed with both certificates and cash denominated in francs. The cash you receive at the beginning of each market period will be subtracted at the end of that same period as a 'fixed cost' for your participation in the market. The number of certificates and the amount of cash you are endowed with are written at the top of your computer screen. You can either sell the certificates you have been endowed with, or use your cash to buy more certificates from other traders, or you can save both certificates and cash until the end of each market period.

During each of the ten market periods, you are allowed to buy and sell if you wish to, provided your cash on hand, i.e. the amount of cash you hold every moment, is non negative.

Prior to the ten market periods, there will be two trial periods which are meant to make you familiar with the experiment and the market procedure.

Try to think of the experiment as if it involved a real situation.

The computer screen in front of you is divided into two halves: (Treatments 2 and 3. The computer screen in front of you is" divided into three parts) the left-hand side is yellow while the right-hand side is red. In the yellow part of the screen you will see all the information concerning the market for the yellow certificate, while in the red part of the screen you will see the information concerning the red certificate. Therefore, there will be separate markets for Yellow and Red certificates.

(Treatments 2 and 3: In addition to the two independent markets for the two certificates, there is a third market, denominated portfolio market, in which you can buy and sell the yellow and the red certificate simultaneously. This market is located in the lower part of the screen)

Information for each of the two certificate markets consists of: prices at which the other market participants are willing to buy or sell, prices at which previous exchanges have been concluded. For each market, you will be able to give your bids or your offers by indicating them inside the appropriate spaces.

Your profits during this experiment come from two sources: (1) dividends on the certificates you hold at the end of each market period; (2) capital gains on certificates bought and sold.

1) DIVIDENDS

For each certificate of each type you hold at the end of each market period you will earn a dividend. The size of the dividend is either 10,000 francs or zero francs. The size of the dividend you will receive depends on the certificate type and on the outcome of a random event. At the end of each market day, the experimenter will ask one of the participants to draw a chip out of an urn containing 50 yellow and 50 red chips. If a red chip is drawn, you shall receive 10,000 francs for each red certificate you own, and no dividend on the yellow certificates. For instance, if a red chip is drawn, and you hold 3 red certificates and 2 yellow ones, you will receive a dividend equal to 30,000 francs. If on the contrary, a yellow chip is drawn, you will receive 10,000 francs for each yellow certificate you hold and zero francs for each red certificate you hold. For instance, if a yellow chip is drawn, and you hold 3 red certificates and 2 yellow ones, you will receive a dividend equal to 20,000 francs.

[For sessions with uncertain certificates: The size of the dividend you will receive depends on the certificate type and on the outcome of a random event. Specifically, we have an urn containing 100 green and blue chips. The proportion of blue and green chips is unknown to you and will be determined at the end of each market day by means of a random device which will be explained in detail by the experimenter prior to the beginning of the experiment. At the end of each market period, one of the participants will draw a chip from the urn without looking. If the drawn chip is green, you will receive 10,000 francs for each green certificate you hold and zero francs for each blue certificate. For instance, if a green chip is drawn, and you hold 3 green certificates and 2 blue ones, you will receive a dividend equal to 30,000 francs. If on the contrary, a blue chip is drawn, you will receive 10,000 francs for each blue certificate you hold and zero francs for each green certificate you hold. For instance, if a blue chip is drawn, and you hold 3 green certificates and 2 blue ones, you will receive a dividend equal to 20,000 francs.]

Notice that for each couple of yellow and red certificates you hold (1 yellow and 1 red certificate), at the end of each market period you will receive a sure dividend of 10,000 francs, regardless of the colour of the chip drawn. For instance: if you hold 3 yellow and 3 red certificates, the dividend you will receive at the end of the market period will be 30,000 francs, regardless of whether the chip drawn is yellow or red.

2) CAPITAL GAINS

Every time you sell a certificate you hold, your cash on hand will increase by an amount equal to the selling price of the certificate. Likewise, every time you buy a certificate, your cash on hand will decrease by an amount equal to your buying price. Therefore, you can earn or lose money by buying and selling certificates and gaining/losing the difference between selling price and buying price.

Summing up, at the beginning of each market period you will be endowed with both cash and certificates (denominated in francs). During each market period you will be able to buy more certificates or sell those you hold. At the end of each market period, the experimenter will determine the dividends on the certificates you hold by means of a random draw.

Summing these dividends to the cash on hand at the end of each market period you will obtain your total gross earnings in francs.

For instance, if at the end of the market period you hold 460,000 francs and two red certificates, and a red chip is drawn, your gross earnings for that market day will be 480,000 francs. By subtracting the fixed cost of 360,000 francs, you get a profit of 120,000 francs.

Notice that the certificates you accumulate in the course of a market day, will expire after the dividend has been determined. This means that you will not be able to carry them over to the next market period.

TRADING RULES

We will now try to explain how you can sell and buy certificates on the markets. If you want to buy a certificate:

(a) You can accept the offer price made by other participants. These offers appear on the screen under the column denominated 'SELLING PRICE' for each market. Press the 'BUY' button to confirm your purchase.

(b) Or, you can indicate your bid on the active field of the appropriate market (for instance the market for the yellow certificate) and press the BUY button to confirm your purchase. Be careful! Any new bid you introduce must be higher than outstanding bids. The computer will issue a warning if your bid is too low.

Once you have confirmed your purchase, the computer will record an increase in the number of the certificates you hold by one unit, while it will record a reduction of your cash on hand equal to the buying price for the certificate.

If you want to sell a certificate you hold:

(a) You can accept the bids made by other participants. These bids appear on the screen under the column denominated 'BUYING PRICE' for each market. Press the 'SELL' button to confirm your sale.

(b) Or, you can indicate your offer on the active field of the appropriate market (for instance the market for the yellow certificate) and press the 'SELL' button to confirm your sale. Be careful! Any new offer you introduce must be lower than outstanding offers. The computer will issue a warning if your offer is too high.

Once you have confirmed your sale, the computer will record a reduction in the number of the certificates you hold by one unit, while it will record an increase of your cash on hand equal to the selling price for the certificate.

Notice that you are not allowed to hold a negative amount of certificates or cash. If your cash on hand goes to zero, you will no longer be allowed to submit bids unless you sell some of the certificates in your portfolio first.

(Treatments 2 and 3. If you want to buy simultaneously one red and one yellow certificate, you can add a new bid or accept the offer price of other participants. Likewise, if you want to sell simultaneously one red and one yellow certificate, you can add a new offer or accept the bid of other participants. Outstanding bids and offers for the couple of certificates appear in the lower part of the screen denominated portfolio market.

In order to be able to sell simultaneously one yellow and one red certificate, you must own one of each certificate.)

SUMMARY OF TRADING RULES

We summarize the trading rules for the two certificate markets 'YELLOW' and 'RED'.

(1) Every sale or purchase concerns one certificate at a time.

(2) To make a new offer or a new bid you must write the offer of bid in the appropriate field and confirm by pressing the button 'SELL' in case of an offer or the button 'BUY' in the case of a bid.

(3) New bids must be higher than outstanding bids, while new offers must be lower than outstanding offers.

(4) If you want to sell a certificate by accepting an outstanding bid, just move the cursor on the bid and confirm your sale by pressing the button 'SELL'.

(5) If you want to buy a certificate by accepting an outstanding offer, just move the cursor on the offer and confirm your purchase by pressing the button 'BUY'.

(6) You are not allowed to hold a negative amount of cash. If your cash on hand goes to zero you can try to increase it by selling some of the certificates you hold.

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Notes

(1.) Francs is the experimental currency used throughout the experiment.

(2.) Similarly, in the uncertain markets the certificate endowment was made up either of 1 Green and 5 Blues, or by 5 Blues and 1 Green.

(3.) Since there are four independent observations per treatment, two with risky and two with uncertain draws, t-tests cannot be used to compare mean prices under risk and uncertainty. The non-parametric Wilcoxon sign-rank test has been applied to test the one-tailed hypothesis that prices under ambiguity are stochastically larger than under risk. Considering the four pairs of sessions where no oral information is provided, the null hypothesis is rejected (both for periods 1-5 and 6-10: W = 24, p[W > 24] = 0.0571). When bundling is allowed, the null hypothesis cannot be rejected (periods 1-5: W = 22, p[W > 22] = 0.1714: periods 6-10: W = 21, p[W > 21] = 0.2429).

Carmela Di Mauro, Email: cdimauro@unict.it

D.A.P.P.S.I., Universita di Catania, Via Vittorio Emanuele 8, I-95131 Catania, Italy
Table 1. Experimental sessions.

 Probability Bundling Information
Session of payoff allowed on arbitrage

1 known No No
2 known No No
3 ambiguous No No
4 ambiguous No No
5 known Yes No
6 known Yes No
7 ambiguous Yes No
8 ambiguous Yes No
9 known Yes Yes
10 known Yes Yes
11 ambiguous Yes Yes
12 ambiguous Yes Yes

 No. No.
Session periods traders

1 10 12
2 8 12
3 8 12
4 10 12
5 8 12
6 10 12
7 10 12
8 10 12
9 10 12
10 10 12
11 10 12
12 10 12

Table 2. Mean sum of prices per session
(standard deviation in italics).

 PERIOD

 1 2 3 4

no bundling
 risk S1 30,991 23,179 22,861 21,358
 9,280 6,706 4,945 6,827
 risk S2 17,814 26,185 20,790 35,657
 6,357 12,346 6,321 7,475
 uncertainty S3 38,453 34,805 33,510 32,784
 12,631 5,143 5,329 3,265
 uncertainty S4 34,475 47,377 39,666 44,556
 15,730 21,430 9,238 8,011
bundling
 risk SS 7,900 10,442 11,026 10,622
 5,849 4,082 6,595 5,077
 risk S6 20,272 19,072 25,248 24,642
 16,942 8,767 12,861 10,899
 uncertainty S7 11,620 11,960 16,534 12,470
 4,643 4,017 11,880 3,349
 uncertainty S8 91,098 51,242 37,804 37,000
 70,634 20,154 17,037 13,625
bundling and information
 risk S9 16,594 16,691 15,147 14,015
 1,533 3,072 4,301 892
 risk S10 19,792 17,751 17,669 17,253
 7,412 2,185 1,247 1,096
 uncertainty S11 16,489 14,621 14,509 14,824
 818 1,187 306 487
 uncertainty S12 19,577 35,988 35,727 20,450
 11,132 333,695 25,110 10,971

 PERIOD

 5 6 7

no bundling
 risk 20,642 20,095 18,989
 5,969 7,670 4,483
 risk 23,628 28,355 16,838
 7,338 20,362 1,637
 uncertainty 33,625 32,064 28,023
 4,148 3,798 2,579
 uncertainty 40,807 28,216 28,619
 5,246 3,327 3,940
bundling
 risk 12,750 10,844 9,638
 7,163 3,370 2,972
 risk 21,924 21,836 18,832
 8,143 9,750 3,511
 uncertainty 12,650 13,140 11,328
 2,071 3,711 3,492
 uncertainty 23,870 28,790 27,952
 9,221 14,508 8,912
bundling and information
 risk 13,990 16,357 12,188
 1,366 5,557 771
 risk 18,356 17,651 20,568
 6,679 826 7,719
 uncertainty 14,834 13,770 13,076
 4,074 548 851
 uncertainty 16,592 15,532 15,247
 5,446 5,708 6,179

 PERIOD

 8 9 10

no bundling
 risk 18,605 18,335 18,018
 1,707 1,959 867
 risk 18,057
 3,177
 uncertainty 21,316
 2,345
 uncertainty 20,589 26,271 20,348
 3,327 3,939 1,693
bundling
 risk 11,754 11,950 13,498
 6,535 6,333 7,250
 risk 18,418
 4,122
 uncertainty 11,838 11,776 11,288
 2,697 3,482 3,383
 uncertainty 27,348 21,750 21,740
 13,834 7,697 7,791
bundling and information
 risk 12,616 14,661 12,148
 822 1,041 458
 risk 20,407 19,098 17,507
 7,868 6,810 6,568
 uncertainty 14,761 12,436 12,007
 6,951 336 959
 uncertainty 16,768 12,174 14,932
 8,886 1,538 4,368

Table 3. Random effects panel estimation (114 observations).

IND. VARIABLE Model 1 Model 2

Constant 29,327.670 29,328.120
 4,331.354 4,592.948
 (0.000) (0.000)
Ambiguity 8,392.497 8,393.134
 4,163.624 4,435.138
 (0.044) (0.058)
Bundling -9,365.695 -9,337.884
 4,423.271 5,438.942
 (0.029) (0.086)
Bundling and information -- -595.2042
 5,425.131
 (0.913)
Period -1,221.969 -1,222.231
 236.0786 236.1168
 (0.000) (0.000)
Wald chi-square 35.86 (0.00) 34.82 (0.00)
R-square 0.3753 0.3761
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Date:Aug 1, 2009
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