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Risk aversion and the Nash solution in stochastic bargaining experiments
Economics Letters North-Holland
RISK AVERSION IN STOCHASTlC Glenn
321
24 (1987) 321-326
AND THE NASH SOLUTION BARGAINING EXPERIMENTS
W. HARRISON
*
Unmemtyof Western Ontario. London, Ont., Canada N6A SC2 Received Accepted
3 December 1986 20 April 1987
It has been shown theoretically that the utility which the Nash solution assigns agents in a bargaining game with stochastic outcomes depends in a predictable manner on the risk attitudes of the agents. We control explicitly for the risk aversion of subjects in laboratory experiments designed to test these predictions. Our results provide support for the predicted risk-sensitivity of the Nash solution.
In this paper we examine the predictive power of the Nash solution concept in stochastic bargaining contexts in which its testable predictions are risk-sensitive. By controlling explicitly for the risk aversion of agents in these experiments we are able to provide direct tests of the risk-sensitivity of the Nash (1950) solution concept: ‘Nash taken neat’, as it were. Consider bargaining games in which the non-disagreement outcomes may only be attained as a risky outcome. Specifically, we shall study the class of games introduced by Roth and Malouf (1979) in which the players are bargaining over the division of a given number of ‘lottery tickets’. Each ticket entitles the holder to partake in a random lottery in which he may win a given positive monetary amount or receive nothing. The original bargaining game therefore amounts to bargaining over the probability that each player will receive his positive monetary amount. A convenient operationalization of this binary lottery is as follows. Let there be 100 tickets to divide, and let the outcome be decided by a random drawing from a uniform distribution with endpoints of 0 and 100. Each player will win his positive monetary payout if this random drawing is less than or equal to the number of tickets he holds, and he receives nothing otherwise. Roth and Rothblum (1982, Theorem 3) have shown that the utility which the Nash solution assigns to a player decreases as he becomes more risk averse if he prefers his opponent’s best (random) agreement outcome to the (non-random) disagreement outcome. Conversely, the utility assigned to a player by the Nash solution increases as he becomes more risk averse if he prefers the (non-random) disagreement outcome to his opponent’s best (random) agreement outcome. Finally, the utility assigned by the Nash solution is incariant to the risk aversion of a player if and only if he is indifferent to the (non-random) disagreement outcome and his opponent’s best (random) agreement outcome. The importance of these results is that it is not necessarily an advantage to be playing * I am grateful to Alvin Roth and E.E. Rutstriim Council of Canada for research support.
0165.1765/87/$3.50
for helpful
0 1987. Elsevier Science Publishers
comments,
and to the Social Sciences
B.V. (North-Holland)
and Humanities
Research
322
G. W. Harrison
against
a more risk averse opponent
/ Stochastic bargaining experiments
in games with risky agreement
outcomes.
The intuition
is clear:
‘for bargaining games in which potential agreements involve lotteries which have a positive probability of leaving one of the players worse off than if a disagreement had occurred, the more risk averse a player, the better the terms of the agreement which he must be offered in order to induce him to reach an agreement, and to compensate him for the risk involved’. [Roth and Rothblum (1982, p. 639).] It is important to note that the disagreement outcome is non-random throughout. The basic game proposed by Roth and Malouf (1979) elegantly ensures that each player is indifferent between the disagreement outcome and his opponent’s best outcome: the two outcomes are identical when each player is allowed to bargain freely over all possible divisions of the 100 lottery tickets. 1 To ensure that any player prefers his opponent’s best outcome to the disagreement outcome, one simply imposes a maximum number of lottery tickets (strictly less than 100) that his opponent may end up with. 2 To ensure that any player prefers the disagreement outcome to his opponent’s best outcome, one defines the disagreement outcome as the certainty-equivalent of some positive allocation of lottery tickets (strictly less than 100) to the player. It is not essential that this certainty-equivalent be the same for the two players, although with prior independent knowledge of the utility of income function of each player it would be possible to ensure this. Note again that the disagreement outcomes in all three types of games described above are non-stochastic. A simple and direct test of these predictions is possible if we can independently determine the risk aversion of the subjects in the bargaining game. We obtained subjects with known attitudes to risk by drawing on a subject pool that had participated in the risk aversion pre-test experiments described in Harrison (1986b). This pre-test is a variant of a procedure proposed by Becker, DeGroot and Marschak (1964) and evaluated in Harrison and Rutstrom (1986). Consider the experimental design embodied in the following instructions: You are about to participate in an experiment in decision making and bargaining. If you follow the instructions carefully you might earn considerable amounts of money, which will be paid to you at the end of the experimental series. Bargaining This experiment will consist of a number of fiue-minute sessions (we will conduct no more than ten sessions; the exact number depends on the availability of time). In each session you will be bargaining with one other person over the division between you of 100 fictitious ‘lottery tickets’. Each lottery ticket you end up with entitles you to participate in a lottery in which you may win $2 (in each session). You may not physically threaten your bargaining counterpart (a monitor will be listening in to your discussion to ensure that this does not happen; if you do try to physically threaten somebody you will be dismissed from the experiment and will receive no monetary reward). If you come to an agreement over the lottery ticket division you must both sign a written agreement to this effect (the monitor will also co-sign the agreement). The lottery The lottery operates as follows. A number will be drawn at random between a uniform probability distribution. All values between 0 and 100 are equally receive $2 if this random number is less than or equal to the number of lottery for that session. If the random number is greater than the number of lottery ’ If I allow my opponent to have all 700 tickets my lottery payoff is ‘zero’ (with certainty) monetary payoff (also with certainty). * Such a restriction is indeed imposed in Games 2 and 4 of Roth and Malouf (1979).
0 and likely. tickets tickets
and my opponent’s
100 from You will you hold you hold
is his positive
G. W. Harmon
/ Stochastic
bargaining experiments
323
you will receive nothing for that session. Thus, if you hold 100 lottery tickets in any session you are certain of receiving $2; if you hold no tickets you are equally certain of receiving nothing in that session. If you hold 28 tickets and the random number for that session is 26, you will win $2; if it is 29 or 91 you will win nothing. We will read out the series of random numbers at the end of all sessions (they are in the sealed envelope that is being displayed now, and are determined completely independently of your choices today). Remember that these numbers are drawn at random: you may therefore experience some bad luck, but you are just as likely to have some good luck. Disagreement outcome If you cannot agree on a division of the lottery tickets a ‘disagreement outcome’ applies. In the basic game this disagreement outcome is for both participants to receive no lottery tickets (thereby receiving no monetary payoff in that session). Three different games There are three types of games that differ in one simple respect from each other. In the basic game (type 1) you may divide the 100 lottery tickets any way you wish and the disagreement outcome is a zero lottery ticket allocation for both players. In the type 2 game PLAYER 2 may not receive more than 75 lottery tickets; otherwise the game is the same as the basic game. Which participant is PLAYER 2 in this game is determined by a flip of the coin at the beginning of each session. In the type 3 game the disagreement outcome is for both players to receive $0.75 (instead of zero in the basic game); otherwise the game is the same as the basic game. These differences are the same as the basic game. These differences are summarized on the blackboard. You will be informed by the monitor which type of game you are playing in each session. Aversion to risk In previous experiments you were offered a series of wagers and asked to specify a price that you would ‘sell’ the wager for. These experiments were designed to allow us to determine how averse to risk you were. We have summarized your behavior in that experiment by a single measure of your aversion to risk. This is the percentage confidence we have that you are ‘risk neutral’. These percentages are the numbers that we mentioned above. Thus, if one person has a 75% confidence level and another person has a 25% confidence level, it is reasonable to infer that the first person is less averse to risk than the second person. The notions of risk aversion and risk neutrality may be familiar to you. One textbook author explains ‘ . . when confronted with gambles with equal expected monetary values, [risk averters] prefer a gamble with a more certain outcome to one with a less certain outcome’. A risk-neutral person will be indifferent between receiving $50 for certain and a wager offering $100 with probability 0.5 and nothing with probability 0.5; a risk-averse person will always prefer the certain $50 in such a case. Consider two wagers: (A) $100 with probability 0.5 and $0 with probability 0.5; and (B) $60 with probability 0.5 and $40 with probability 0.5. A risk-neutral person would be indifferent between the two wagers (since they each have an expected payoff of $50); a risk-averse person, on the other hand, would prefer wager B since it is less risky than wager A. Final points Feel free to make as much money as you can by taking advantage of the special features of each game and the information provided. You will not play more than one game with any other person.
324
G. W. Harrison / Stochastic hargainrng experiments
The subjects were informed that their ‘officially determined’ aversion to risk would be on public display on a blackboard. 3 The notion of risk aversion is explained in as simple a fashion as possible, designed to avoid any explicit notions of ‘risk premia’ or ‘certainty equivalents’ on the grounds that these may be loaded expressions. 4 The values given to subjects were derived from F-test probability values such as those reported in Harrison (1986a), and were equal to those values times one hundred. All subjects were honors economics majors, allowing use of expressions such as ‘indifference’ in the explanation of risk aversion. The twelve subjects employed in the bargaining games with deterministic outcomes reported in Harrison (1986b) were also used in the present experiments (albeit in sessions separated by several weeks). 5 Six sessio ns of each type of bargaining game were conducted. Tables A.1 and A.2 in an appendix (available from the author upon request) detail the allocation of subjects to each game and the risk aversion measures of each subject. The type 1 bargaining game corresponds to the case in which the Nash outcome is invariant to the risk attitudes of the two players. Evidence of an equal-division (of lottery tickets) is therefore evidence consistent with the Nash prediction. It is worth noting that these subjects displayed a keen awareness in the experiments in Harrison (1986b) of the risk-sensitivity of the Nash solution in the context of deterministic outcomes. In the type 2 game PLAYER 1 is at a bargaining disadvantage as he becomes more risk averse, and PLAYER 2 is at an advantage as his opponents become more risk averse. We deliberately used a given subject in several successive type 2 games (obviously with different opponents). It would have been extremely convenient if PLAYER 2 was always a ‘risk-neutral’ subject, but this would have involved pre-allocating subjects in a way that might have encouraged them to ‘second-guess’ the experimental design sequence. We allocated subjects by chance (a flip of the coin), and as it happens ‘chance’ was reasonably kind to us in the pairings that eventuated: over 50% of the type 2 pairings involved a risk-neutral PLAYER 2. Focussing on those pairings, the Nash prediction is that PLAYER 2 will always end up with more lottery tickets than his risk-averse opponent and will end up with even more tickets if he faces an opponent that is even more risk averse. The type 3 game offers the more risk-averse player a bargaining advantage: he can threaten to disagree and ensure himself $0.75 unless he is ‘compensated’ in utility terms for undertaking the risky return of an agreement outcome. This ‘compensation’ should take the form of extracting more than one-half of the 100 lottery tickets, according to the prediction of the Nash solution. As in the type 2 game we attempted to pair a risk-neutral subject with a risk-averse subject and to allow subjects to gain some experience in this type of game (albeit with different opponents). Table 1 presents the results of these experiments, in the form of a statistic that is simply the (agreed) allocation of lottery tickets for the more risk-averse player as a ratio of the allocation of tickets to the less risk-averse player; table A.2 in the appendix (available upon request) lists the raw results. Where the latter player is risk-neutral, according to the classification procedure developed in
’ Preference distortion strategies, studied theoretically by Crawford and Varian (1979) and experimentally by Harrison and McKee (1985) and Harrison (1986b), were therefore severely restricted in credibility. 4 It is an open behavioral issue if the use of these expressions would significantly alter the outcomes of the experiments reported here. Casual introspection suggests strongly that they would, albeit in favor of the relevant Nash prediction. 5 These subjects were drawn from two undergraduate honors classes at the University of Western Ontario, and were used for a wide range of paid and unpaid experiments over a period of several months. These included risk pre-tests, job search, public goods, contract law. asset markets, regulated and unregulated monopoly. contestable duopolies and triopolies. Coasian bargaining, sealed-bid auctions, and computerized markets. Rarely were subjects informed, implicitly or explicitly, before any experiment of the particular group of subjects to be employed. Moreover, they were often provided only a vague notion of the subject matter for planned experiments. For these reasons it is extremely doubtful if any group of subjects was able to communicate in any disruptive way prior to any experimental session.
G. W. Harmon Table 1 Experimental
results:
Ratio of agreed
allocation
/ Stochastic barguming
of more risk-averse
player
325
experments
to allocation
of less risk-averse
Session
Type 1 game
Type 2 game
Type 3 game
1
1 .oo 0.92
0.82 d.C 1.00 h.c
1.00 h 1.00 a
2
1.00 3 0.96 a
0.96 ‘.’ 1.08 :’
1.00 a 1.08 ‘l
3
1.00 b 1.00 h
0.92 a.~ 0.96 a
1.00 1.04 a
4
1 .oo 1.00
1.04 d 1 .oo a.c
1.00 b 1.00 h
5
1.00 a 0.96 *
0.85 d.C 0.92 ‘.’
1.00 il 1.04 a
6
1.00 a 1.00 h
1 .oo 1.00 *
1.13 a 1.17 *
0.987 0.984 _
0.962 0.954 0.924
1.038 1.057 _
Average Average
(’ only)
Average
(’ only)
a The least risk averse player is risk-neutral. ’ Both players are risk-neutral. ’ Outcome corresponds to the risk-neutral player
being chosen
as PLAYER
player.
2.
Harrison (1986a) and employed in Harrison (1986b), this ratio is marked with an asterisk; where both players were risk-neutral we use two asterisks. In the type 2 games we are also interested in those pairings in which PLAYER 2 was risk-neutral (this eventuated in seven of the twelve possible cases). The results generally support the predictions of the Nash solution. ’ Those predictions, in terms of the ratio in table 1, are for values of one, less than one, and greater than one in the type 1, 2, and 3 games, respectively. The results for the type 1 games are the least favorable to this prediction, three of the twelve outcomes leaving the less risk-averse player with greater allocations than predicted. It is arguable that these games were viewed by subjects as a counterpart to the type II games with A deterministic outcomes reported in Harrison (1986b), in which risk aversion was a disadvantage. binomial test of the statistical significance of observing three ‘failures’ out of twelve trials indicates that we can reject the null hypothesis of nine ‘successes’ occurring by chance at a 0.093 critical probability level. ’ The outcomes of the type 2 game, in which risk aversion is a disadvantage, are clearly in support of the Nash prediction. A strict interpretation of our prediction (ratio values strictly less than one) implies a critical value of only 0.310 in a binomial test, since there are five ‘failures’ in twelve trials. However, a weak interpretation of the prediction (ratio values less than or equal to one) implies a 0.057 critical value. We adopt the latter interpretation and accept the Nash prediction. This ’ These results are also broadly consistent with the results in Murnighan, Roth and Schoumaker (1985). who used a very different experimental design. ’ This test procedure is described in Conover (1980, p. 96ff). Our test is one-tailed, with the null hypothesis that the probability of a success or a failure on any trial is 0.5; a ‘success’ is defined in terms of our predictions of the ratio in table 1. We report exact critical probability levels for the benefit of readers not willing to accept our ten-percent rejection level.
326
G. W. Harrison
/ Stochastic bargaining experiments
conclusion is strengthened by just examining the (bracketed) risk-neutral player was randomly chosen as PLAYER 2. ’ Similarly, the outcomes of the type 3 games, in which advantage, are consistent with the Nash solution prediction. binomial test of the hypothesis that the (weak) Nash prediction is merely 0.074, with no failures in twelve trials. 9
results for those cases in which the risk aversion confers a bargaining The critical probability value of the occurred by chance in the experiment
References Becker, G.M., M.H. DeGroot and J. Marschak, 1964, Measuring utility by a single-response sequential method, Behavioral Science 9, 226-232. Conover, W.J., 1980, Practical nonparametric statistics, second ed. (Wiley, New York). Crawford, V.P. and H.R. Varian, 1979, Distortion of preferences and the Nash theory of bargaining, Economics Letters 3, 203-206. Harrison, G.W., 1986a, An experimental test for risk aversion, Economics Letters 21. 7-11. Harrison, G.W., 1986b, Risk aversion and preference distortion in deterministic bargaining experiments, Economics Letters 22, 191-196. Harrison, G.W. and M.J. McKee, 1985, Experimental evaluation of the Coase theorem, Journal of Law and Economics 28, Oct., 653-670. Harrison, G.W. and E.E. Rutstrom, 1986, Experimental measurement of utility by a sequential method, Unpublished manuscript (Department of Economics, University of Western Ontario, London). Mumighan, J.K., A.E. Roth and F. Schoumaker, 1985, Risk aversion in bargaining; An experimental study, Unpublished manuscript, July (Department of Economics, University of Illinois, Chicago, IL). Nash, J.F., 1950, The bargaining problem, Econometrica 18, April, 115-162. Roth, A.E. and M.W.K. Malouf, 1979, Game-theoretic models and the role of information in bargaining, Psychological Review 86, Nov., 574-594. Roth, A.E. and U.G. Rothblum, 1982, Risk aversion and Nash’s solution for bargaining games with risky outcomes. Econometrica 50, May, 639-647. a In this critical 9 In this have a
case we have seven trials and observe two (zero) failures according to the strict (weak) interpretation, implying a probability level of 0.202 (0.117). case the data does not support the strong form of the Nash prediction: with seven ‘failures’ out of twelve trials we critical probability level of 0.690.
RISK AVERSION IN STOCHASTlC Glenn
321
24 (1987) 321-326
AND THE NASH SOLUTION BARGAINING EXPERIMENTS
W. HARRISON
*
Unmemtyof Western Ontario. London, Ont., Canada N6A SC2 Received Accepted
3 December 1986 20 April 1987
It has been shown theoretically that the utility which the Nash solution assigns agents in a bargaining game with stochastic outcomes depends in a predictable manner on the risk attitudes of the agents. We control explicitly for the risk aversion of subjects in laboratory experiments designed to test these predictions. Our results provide support for the predicted risk-sensitivity of the Nash solution.
In this paper we examine the predictive power of the Nash solution concept in stochastic bargaining contexts in which its testable predictions are risk-sensitive. By controlling explicitly for the risk aversion of agents in these experiments we are able to provide direct tests of the risk-sensitivity of the Nash (1950) solution concept: ‘Nash taken neat’, as it were. Consider bargaining games in which the non-disagreement outcomes may only be attained as a risky outcome. Specifically, we shall study the class of games introduced by Roth and Malouf (1979) in which the players are bargaining over the division of a given number of ‘lottery tickets’. Each ticket entitles the holder to partake in a random lottery in which he may win a given positive monetary amount or receive nothing. The original bargaining game therefore amounts to bargaining over the probability that each player will receive his positive monetary amount. A convenient operationalization of this binary lottery is as follows. Let there be 100 tickets to divide, and let the outcome be decided by a random drawing from a uniform distribution with endpoints of 0 and 100. Each player will win his positive monetary payout if this random drawing is less than or equal to the number of tickets he holds, and he receives nothing otherwise. Roth and Rothblum (1982, Theorem 3) have shown that the utility which the Nash solution assigns to a player decreases as he becomes more risk averse if he prefers his opponent’s best (random) agreement outcome to the (non-random) disagreement outcome. Conversely, the utility assigned to a player by the Nash solution increases as he becomes more risk averse if he prefers the (non-random) disagreement outcome to his opponent’s best (random) agreement outcome. Finally, the utility assigned by the Nash solution is incariant to the risk aversion of a player if and only if he is indifferent to the (non-random) disagreement outcome and his opponent’s best (random) agreement outcome. The importance of these results is that it is not necessarily an advantage to be playing * I am grateful to Alvin Roth and E.E. Rutstriim Council of Canada for research support.
0165.1765/87/$3.50
for helpful
0 1987. Elsevier Science Publishers
comments,
and to the Social Sciences
B.V. (North-Holland)
and Humanities
Research
322
G. W. Harrison
against
a more risk averse opponent
/ Stochastic bargaining experiments
in games with risky agreement
outcomes.
The intuition
is clear:
‘for bargaining games in which potential agreements involve lotteries which have a positive probability of leaving one of the players worse off than if a disagreement had occurred, the more risk averse a player, the better the terms of the agreement which he must be offered in order to induce him to reach an agreement, and to compensate him for the risk involved’. [Roth and Rothblum (1982, p. 639).] It is important to note that the disagreement outcome is non-random throughout. The basic game proposed by Roth and Malouf (1979) elegantly ensures that each player is indifferent between the disagreement outcome and his opponent’s best outcome: the two outcomes are identical when each player is allowed to bargain freely over all possible divisions of the 100 lottery tickets. 1 To ensure that any player prefers his opponent’s best outcome to the disagreement outcome, one simply imposes a maximum number of lottery tickets (strictly less than 100) that his opponent may end up with. 2 To ensure that any player prefers the disagreement outcome to his opponent’s best outcome, one defines the disagreement outcome as the certainty-equivalent of some positive allocation of lottery tickets (strictly less than 100) to the player. It is not essential that this certainty-equivalent be the same for the two players, although with prior independent knowledge of the utility of income function of each player it would be possible to ensure this. Note again that the disagreement outcomes in all three types of games described above are non-stochastic. A simple and direct test of these predictions is possible if we can independently determine the risk aversion of the subjects in the bargaining game. We obtained subjects with known attitudes to risk by drawing on a subject pool that had participated in the risk aversion pre-test experiments described in Harrison (1986b). This pre-test is a variant of a procedure proposed by Becker, DeGroot and Marschak (1964) and evaluated in Harrison and Rutstrom (1986). Consider the experimental design embodied in the following instructions: You are about to participate in an experiment in decision making and bargaining. If you follow the instructions carefully you might earn considerable amounts of money, which will be paid to you at the end of the experimental series. Bargaining This experiment will consist of a number of fiue-minute sessions (we will conduct no more than ten sessions; the exact number depends on the availability of time). In each session you will be bargaining with one other person over the division between you of 100 fictitious ‘lottery tickets’. Each lottery ticket you end up with entitles you to participate in a lottery in which you may win $2 (in each session). You may not physically threaten your bargaining counterpart (a monitor will be listening in to your discussion to ensure that this does not happen; if you do try to physically threaten somebody you will be dismissed from the experiment and will receive no monetary reward). If you come to an agreement over the lottery ticket division you must both sign a written agreement to this effect (the monitor will also co-sign the agreement). The lottery The lottery operates as follows. A number will be drawn at random between a uniform probability distribution. All values between 0 and 100 are equally receive $2 if this random number is less than or equal to the number of lottery for that session. If the random number is greater than the number of lottery ’ If I allow my opponent to have all 700 tickets my lottery payoff is ‘zero’ (with certainty) monetary payoff (also with certainty). * Such a restriction is indeed imposed in Games 2 and 4 of Roth and Malouf (1979).
0 and likely. tickets tickets
and my opponent’s
100 from You will you hold you hold
is his positive
G. W. Harmon
/ Stochastic
bargaining experiments
323
you will receive nothing for that session. Thus, if you hold 100 lottery tickets in any session you are certain of receiving $2; if you hold no tickets you are equally certain of receiving nothing in that session. If you hold 28 tickets and the random number for that session is 26, you will win $2; if it is 29 or 91 you will win nothing. We will read out the series of random numbers at the end of all sessions (they are in the sealed envelope that is being displayed now, and are determined completely independently of your choices today). Remember that these numbers are drawn at random: you may therefore experience some bad luck, but you are just as likely to have some good luck. Disagreement outcome If you cannot agree on a division of the lottery tickets a ‘disagreement outcome’ applies. In the basic game this disagreement outcome is for both participants to receive no lottery tickets (thereby receiving no monetary payoff in that session). Three different games There are three types of games that differ in one simple respect from each other. In the basic game (type 1) you may divide the 100 lottery tickets any way you wish and the disagreement outcome is a zero lottery ticket allocation for both players. In the type 2 game PLAYER 2 may not receive more than 75 lottery tickets; otherwise the game is the same as the basic game. Which participant is PLAYER 2 in this game is determined by a flip of the coin at the beginning of each session. In the type 3 game the disagreement outcome is for both players to receive $0.75 (instead of zero in the basic game); otherwise the game is the same as the basic game. These differences are the same as the basic game. These differences are summarized on the blackboard. You will be informed by the monitor which type of game you are playing in each session. Aversion to risk In previous experiments you were offered a series of wagers and asked to specify a price that you would ‘sell’ the wager for. These experiments were designed to allow us to determine how averse to risk you were. We have summarized your behavior in that experiment by a single measure of your aversion to risk. This is the percentage confidence we have that you are ‘risk neutral’. These percentages are the numbers that we mentioned above. Thus, if one person has a 75% confidence level and another person has a 25% confidence level, it is reasonable to infer that the first person is less averse to risk than the second person. The notions of risk aversion and risk neutrality may be familiar to you. One textbook author explains ‘ . . when confronted with gambles with equal expected monetary values, [risk averters] prefer a gamble with a more certain outcome to one with a less certain outcome’. A risk-neutral person will be indifferent between receiving $50 for certain and a wager offering $100 with probability 0.5 and nothing with probability 0.5; a risk-averse person will always prefer the certain $50 in such a case. Consider two wagers: (A) $100 with probability 0.5 and $0 with probability 0.5; and (B) $60 with probability 0.5 and $40 with probability 0.5. A risk-neutral person would be indifferent between the two wagers (since they each have an expected payoff of $50); a risk-averse person, on the other hand, would prefer wager B since it is less risky than wager A. Final points Feel free to make as much money as you can by taking advantage of the special features of each game and the information provided. You will not play more than one game with any other person.
324
G. W. Harrison / Stochastic hargainrng experiments
The subjects were informed that their ‘officially determined’ aversion to risk would be on public display on a blackboard. 3 The notion of risk aversion is explained in as simple a fashion as possible, designed to avoid any explicit notions of ‘risk premia’ or ‘certainty equivalents’ on the grounds that these may be loaded expressions. 4 The values given to subjects were derived from F-test probability values such as those reported in Harrison (1986a), and were equal to those values times one hundred. All subjects were honors economics majors, allowing use of expressions such as ‘indifference’ in the explanation of risk aversion. The twelve subjects employed in the bargaining games with deterministic outcomes reported in Harrison (1986b) were also used in the present experiments (albeit in sessions separated by several weeks). 5 Six sessio ns of each type of bargaining game were conducted. Tables A.1 and A.2 in an appendix (available from the author upon request) detail the allocation of subjects to each game and the risk aversion measures of each subject. The type 1 bargaining game corresponds to the case in which the Nash outcome is invariant to the risk attitudes of the two players. Evidence of an equal-division (of lottery tickets) is therefore evidence consistent with the Nash prediction. It is worth noting that these subjects displayed a keen awareness in the experiments in Harrison (1986b) of the risk-sensitivity of the Nash solution in the context of deterministic outcomes. In the type 2 game PLAYER 1 is at a bargaining disadvantage as he becomes more risk averse, and PLAYER 2 is at an advantage as his opponents become more risk averse. We deliberately used a given subject in several successive type 2 games (obviously with different opponents). It would have been extremely convenient if PLAYER 2 was always a ‘risk-neutral’ subject, but this would have involved pre-allocating subjects in a way that might have encouraged them to ‘second-guess’ the experimental design sequence. We allocated subjects by chance (a flip of the coin), and as it happens ‘chance’ was reasonably kind to us in the pairings that eventuated: over 50% of the type 2 pairings involved a risk-neutral PLAYER 2. Focussing on those pairings, the Nash prediction is that PLAYER 2 will always end up with more lottery tickets than his risk-averse opponent and will end up with even more tickets if he faces an opponent that is even more risk averse. The type 3 game offers the more risk-averse player a bargaining advantage: he can threaten to disagree and ensure himself $0.75 unless he is ‘compensated’ in utility terms for undertaking the risky return of an agreement outcome. This ‘compensation’ should take the form of extracting more than one-half of the 100 lottery tickets, according to the prediction of the Nash solution. As in the type 2 game we attempted to pair a risk-neutral subject with a risk-averse subject and to allow subjects to gain some experience in this type of game (albeit with different opponents). Table 1 presents the results of these experiments, in the form of a statistic that is simply the (agreed) allocation of lottery tickets for the more risk-averse player as a ratio of the allocation of tickets to the less risk-averse player; table A.2 in the appendix (available upon request) lists the raw results. Where the latter player is risk-neutral, according to the classification procedure developed in
’ Preference distortion strategies, studied theoretically by Crawford and Varian (1979) and experimentally by Harrison and McKee (1985) and Harrison (1986b), were therefore severely restricted in credibility. 4 It is an open behavioral issue if the use of these expressions would significantly alter the outcomes of the experiments reported here. Casual introspection suggests strongly that they would, albeit in favor of the relevant Nash prediction. 5 These subjects were drawn from two undergraduate honors classes at the University of Western Ontario, and were used for a wide range of paid and unpaid experiments over a period of several months. These included risk pre-tests, job search, public goods, contract law. asset markets, regulated and unregulated monopoly. contestable duopolies and triopolies. Coasian bargaining, sealed-bid auctions, and computerized markets. Rarely were subjects informed, implicitly or explicitly, before any experiment of the particular group of subjects to be employed. Moreover, they were often provided only a vague notion of the subject matter for planned experiments. For these reasons it is extremely doubtful if any group of subjects was able to communicate in any disruptive way prior to any experimental session.
G. W. Harmon Table 1 Experimental
results:
Ratio of agreed
allocation
/ Stochastic barguming
of more risk-averse
player
325
experments
to allocation
of less risk-averse
Session
Type 1 game
Type 2 game
Type 3 game
1
1 .oo 0.92
0.82 d.C 1.00 h.c
1.00 h 1.00 a
2
1.00 3 0.96 a
0.96 ‘.’ 1.08 :’
1.00 a 1.08 ‘l
3
1.00 b 1.00 h
0.92 a.~ 0.96 a
1.00 1.04 a
4
1 .oo 1.00
1.04 d 1 .oo a.c
1.00 b 1.00 h
5
1.00 a 0.96 *
0.85 d.C 0.92 ‘.’
1.00 il 1.04 a
6
1.00 a 1.00 h
1 .oo 1.00 *
1.13 a 1.17 *
0.987 0.984 _
0.962 0.954 0.924
1.038 1.057 _
Average Average
(’ only)
Average
(’ only)
a The least risk averse player is risk-neutral. ’ Both players are risk-neutral. ’ Outcome corresponds to the risk-neutral player
being chosen
as PLAYER
player.
2.
Harrison (1986a) and employed in Harrison (1986b), this ratio is marked with an asterisk; where both players were risk-neutral we use two asterisks. In the type 2 games we are also interested in those pairings in which PLAYER 2 was risk-neutral (this eventuated in seven of the twelve possible cases). The results generally support the predictions of the Nash solution. ’ Those predictions, in terms of the ratio in table 1, are for values of one, less than one, and greater than one in the type 1, 2, and 3 games, respectively. The results for the type 1 games are the least favorable to this prediction, three of the twelve outcomes leaving the less risk-averse player with greater allocations than predicted. It is arguable that these games were viewed by subjects as a counterpart to the type II games with A deterministic outcomes reported in Harrison (1986b), in which risk aversion was a disadvantage. binomial test of the statistical significance of observing three ‘failures’ out of twelve trials indicates that we can reject the null hypothesis of nine ‘successes’ occurring by chance at a 0.093 critical probability level. ’ The outcomes of the type 2 game, in which risk aversion is a disadvantage, are clearly in support of the Nash prediction. A strict interpretation of our prediction (ratio values strictly less than one) implies a critical value of only 0.310 in a binomial test, since there are five ‘failures’ in twelve trials. However, a weak interpretation of the prediction (ratio values less than or equal to one) implies a 0.057 critical value. We adopt the latter interpretation and accept the Nash prediction. This ’ These results are also broadly consistent with the results in Murnighan, Roth and Schoumaker (1985). who used a very different experimental design. ’ This test procedure is described in Conover (1980, p. 96ff). Our test is one-tailed, with the null hypothesis that the probability of a success or a failure on any trial is 0.5; a ‘success’ is defined in terms of our predictions of the ratio in table 1. We report exact critical probability levels for the benefit of readers not willing to accept our ten-percent rejection level.
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/ Stochastic bargaining experiments
conclusion is strengthened by just examining the (bracketed) risk-neutral player was randomly chosen as PLAYER 2. ’ Similarly, the outcomes of the type 3 games, in which advantage, are consistent with the Nash solution prediction. binomial test of the hypothesis that the (weak) Nash prediction is merely 0.074, with no failures in twelve trials. 9
results for those cases in which the risk aversion confers a bargaining The critical probability value of the occurred by chance in the experiment
References Becker, G.M., M.H. DeGroot and J. Marschak, 1964, Measuring utility by a single-response sequential method, Behavioral Science 9, 226-232. Conover, W.J., 1980, Practical nonparametric statistics, second ed. (Wiley, New York). Crawford, V.P. and H.R. Varian, 1979, Distortion of preferences and the Nash theory of bargaining, Economics Letters 3, 203-206. Harrison, G.W., 1986a, An experimental test for risk aversion, Economics Letters 21. 7-11. Harrison, G.W., 1986b, Risk aversion and preference distortion in deterministic bargaining experiments, Economics Letters 22, 191-196. Harrison, G.W. and M.J. McKee, 1985, Experimental evaluation of the Coase theorem, Journal of Law and Economics 28, Oct., 653-670. Harrison, G.W. and E.E. Rutstrom, 1986, Experimental measurement of utility by a sequential method, Unpublished manuscript (Department of Economics, University of Western Ontario, London). Mumighan, J.K., A.E. Roth and F. Schoumaker, 1985, Risk aversion in bargaining; An experimental study, Unpublished manuscript, July (Department of Economics, University of Illinois, Chicago, IL). Nash, J.F., 1950, The bargaining problem, Econometrica 18, April, 115-162. Roth, A.E. and M.W.K. Malouf, 1979, Game-theoretic models and the role of information in bargaining, Psychological Review 86, Nov., 574-594. Roth, A.E. and U.G. Rothblum, 1982, Risk aversion and Nash’s solution for bargaining games with risky outcomes. Econometrica 50, May, 639-647. a In this critical 9 In this have a
case we have seven trials and observe two (zero) failures according to the strict (weak) interpretation, implying a probability level of 0.202 (0.117). case the data does not support the strong form of the Nash prediction: with seven ‘failures’ out of twelve trials we critical probability level of 0.690.