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The ultimatum game and non-selfish utility functions
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ELSEVIER
Journal of Economic Psychology 17 (1996) 259-271
The ultimatum game and non-selfish utility functions Judy Bethwaite *, Paul Tompkinson Faculty of Commerce and Administration, Victoria University of Wellington P.O. Box 600, Wellington, New Zealand Received 27 April 1993; accepted 28 November 1995
Abstract This paper reports on the results of a questionnaire which asked respondents how much they would offer as an allocator, or be prepared to accept as a recipient, in an ultimatum game with a $10 stake. This permitted an analysis of the respondents' reported offers and minimum acceptances as matched pairs, which has the potential to yield more information about players' utility functions than observation of player behaviour in experimental trials of the ultimatum game. The results suggest that over half of the respondents have a concern for fairness, which is a much greater proportion than those who could be considered to be motivated by envy or altruisim. Only about one quarter of the sample possess a selfish utility function of the type conventionally assumed by economists.
PsyclNFO classification." 2300 JEL classification: C72; C90 Keywords: Ultimatum game; Non-selfish utility functions
* Corresponding author. Fax: +64 4 471-2200, Tel.: +64 4 472-1000.
0167-4870/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0 167-4870(96)00006-2
260
Jud Bethwaite, P. Tompkinson / Journal of Economic Psychology 17 (1996) 259-271
1. Introduction In the simplest version of the ultimatum game, one player, the allocator, is directed to divide a sum of money (the stake) between him or herself and a second player, the recipient. If the recipient accepts the proposed division of the stake then both receive the amounts proposed by the allocator. If the recipient rejects the proposed division then both players receive nothing. The recipient knows the size of the stake, but the players do not know each other. The roles of allocator and recipient are typically determined by the toss of a coin. Thaler (1988) has reviewed the outcomes of experimental trials of the ultimatum game, a key feature of which is that many of the allocators make offers which are half of the total stake. This is at variance with the conventional assumption that individuals are rational and have selfish tastes. By rational we mean that they aim to maximise expected utility, and a player's tastes are selfish if they depend only on his or her payoff. The economists' assumption that rationality and selfish tastes prevail in human behaviour leads to the game theory prediction that the allocator should offer the smallest possible amount and that the recipient should accept this. This follows from backward induction: for the recipient any positive amount is better than zero and therefore should be accepted. The allocator knows this and so should make the smallest positive offer. The empirical evidence casts serious doubt as to whether the conventional specification of individual tastes is the correct one. Thaler argues that the prediction based on selfish tastes is falsified because the players are primarily motivated not by selfishness but by a concern for fairness. Binmore (1990, pp. 10-11), commenting on the same results, interprets the behaviour of the players as being like that of stimulus-response machines. Behaviour which may be appropriate in certain circumstances is applied without thought to a problem in which it is not appropriate. This explanation is consistent with sociological models of behaviour as being rule guided, and has the implication that the players are not optimising. On the basis of these experimental results Thaler argues that it may be useful to distinguish between three kinds of game theory: normative, prescriptive and descriptive. Currently much of game theory is normative, analysing how a rational individual should behave if this individual reasoned on the basis that rationality and selfish tastes were universal. Prescriptive theory would provide advice to a player as to how to play, recognising his or her own cognitive limitations and accepting that others may not be rational and may have non-selfish tastes. Descriptive game theory would describe how individuals actually play.
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In this paper, and in the light of the above summary, we wish to focus on the following questions: 1. What motivates players of the ultimatum game? Is it fairness, envy, altruism or selfishness? 2. For our sample, what advice can be given to players who wish to maximise their utility, for various utility functions? and 3. What monetary payoffs might players expect from playing particular strategies, consistent with these utility functions? Our approach to answering these questions is to assume that the players are utility maximisers for various specifications of the utility function. We consider briefly, at the end of the paper, whether this is a productive approach.
2. Procedures
A questionnaire was constructed which first outlined the ultimatum game and then asked respondents how much they would offer as an allocator, and how much they would be prepared to accept as a recipient, for a stake of $10 (New Zealand currency) where the minimum monetary unit was $1. Using a questionnaire, rather than conducting direct trials of the ultimatum game, allowed us to consider respondents' offers and minimum acceptances as matched pairs, which has the potential to yield more information about their utility functions. The main criticism of our research methodology must be that respondents have no incentive to report how they would actually behave in the game situation. But equally, we have no reason to believe that they have any incentive to misrepresent their behaviour, and in so far as it is possible to compare, the responses we obtained using the questionnaire approach are similar to the results reported in experiments where the game is played with real prizes (Giith et al., 1982). Finally, there is evidence from other contexts that answers to hypothetical questions provide good indicators of behaviour, see for example Wardman (1988). The participants in our study were a group of lawyers in a large private legal practice. We chose a group of lawyers for several reasons. Firstly, previous studies had used student samples and we were interested to know whether other groups would behave in similar ways. Secondly, we believed prior to analysing our results, that lawyers were more likely to behave 'rationally' than students. The evidence from earlier studies of the ultimatum game suggests that the selfishly rational individual of economic theory is a rare creature. If the same conclusion can be established for our sample then this would be significant
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given our prior as to the 'rationality' of lawyers. Others however, did not share our prior belief: they supposed that the lawyers may have thought that we were testing their sense of justice, and as a consequence, they expected them to behave less selfishly than students. As it tums out the lawyers bahave in much the same way as students. We also conducted our survey on a sample of students and results from the two groups were very similar; this being the case we have reported only the results from the lawyer sample. Questionnaires were mailed to 72 potential respondents and 43 were returned, representing a response rate of 60%. In Section 3 of this paper we postulate a number of utility functions which may motivate the behaviour of players of the ultimatum game. We specify functions which distinguish between individuals who are envious, have a taste for fairness, are altruistic and are selfish. We also distinguish between individuals who are instrumentally rational and those who are expressively rational (Hargreaves Heap, 1989). For our sample we determine the optimal behaviour for players with these utility functions. This allows us to provide prescriptions for how to play the game and thus an answer to our question 2. In Section 4, we calculate the monetary payoffs that could be expected from playing particular strategies and thus answer question 3. In Section 5 we investigate whether the offers and acceptances reported by the respondents are consistent with any of the utility functions we have postulated. In this way we are able to give some answers as to the motivation of our respondents and thus an answer to our question 1.
3. Utility functions and optimal strategies We postulate the following utility functions as motivating the behaviour of players of the ultimatum game:
Ui=Pi+b(Pi-Pj),
b>O,
G = P , - c[Pi- Pjl,
c > O,
(2)
1,
(3)
U i = P i + d P j,
O
(1)
where Pi is the payoff to player i and Pj is the payoff to player j. In Eq. (1), utility is a function of a player's own payoff and his or her relative payoff; this is one way of introducing envy into the utility function. Eq. (2) specifies utility as a function of the player's own payoff and the equity of the allocation, thus introducing a concern for fairness into the utility function. Eq.
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263
(3) is a possible specification for an altruistic individual, with the restriction that the weight given to the player's own payoff is greater than that given to the other player's payoff. Selfishness is a special case of all of these functions, obtained by setting b, c or d equal to zero. Using Eq. (1) we can determine for any minimum acceptance the range of values that b can take. For example if the minimum acceptance is $1, and noting that Pi + Pj = 10 for a $10 stake, it follows from Eq. (1) that: l+b(1-9)>0orthatb<0.125. Incorporating the assumption that b > 0, for $1 to be the minimum acceptance, we must have 0 < b < 0.125. Proceeding in this way we can calculate the range of values of b which are consistent with all possible minimum acceptances. It is straightforward to show that the minimum acceptance must be less than or equal to $5. Again for Eq. (1) if we assume that the allocators are risk neutral and know the distribution of minimum acceptances, we can also determine the range of values of b which are consistent with any given offer. For example if $1 is the optimal offer then this implies the following set of inequalities: E[ Pi(1)] + for
0i=
bE[ P i ( 1 ) -
Pj(1)] > E[
ei(oi)]
+
bE[ Pi(Oi)
-- ej(oi)],
2 to 9,
where E[P~(1)] is the expected payoff to player i if he or she makes an offer of $1 and E[Pi(Oi)] is the expected payoff to player i if he or she makes an offer of $O~ 4: 1. E[PflOi)] is the expectation held by player i as to what player j will receive if she makes an offer of O~. Each inequality in the above set of inequalities can be solved for b. By computing the range of values which satisfies all of these inequalities, we can find the range of values of b for which $1 is the optimal offer. If there is no range of values for b which satisfies all of these inequalities then this means that it can never be optimal to offer $1. This process is then repeated using the following set of inequalities: E[ Pi(2)] +
bE[ P i ( 2 ) -
Py(2)] > E[
Pi(Oi)] + bE[ P i ( O i ) - Pj(Oi)],
for O~= 1, 3 to 9. From this set of inequalities we can find the range of values of b, if any, which ensure that the optimal offer is $2. Proceeding in the same way, we can find for any offer the range of values of b, if any, which make that particular offer optimal. It turns out that for our sample the only offers that can be optimal are $1 or $5. The minimum acceptances and optimal offers, regarded as functions of b, for Eq. (1) are given in Table 1.
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Table 1 Minimum acceptances and optimal offers for Eq. (1) Value of b
$ Min. acceptance
$ Optimal offer
0.00 < b < 0.125 0.125 < b < 0.33 0.33 < b < 0.75 0.75 < b < 1.06 1.06 < b < 2.00 b>2
1 2 3 4 4 5
5 5 5 5 1 1
For individuals concerned only with the size of their own payoff, that is when b = 0, Table 1 shows that the optimal way to play the game is to offer half the stake and be prepared to accept the smallest positive offer. Calculations of the same sort can be carried out for Eqs. (2) and (3). For Eq. (2), the minimum acceptances are related to c in the same way that the minimum acceptances are related to b in Table 1. However regardless of the value of c in Eq. (2), the optimal offer is always $5. The optimal way for the altruist (Eq. (3)) to play is to accept $1 and to offer $5 for any value of d. Paradoxically, the optimal way for an altruist to play is identical to that of the selfish economic maximiser. So far we have assumed that utility is purely a function of the expected or actual outcomes. Alternatively, utility may be expressed as a function of the actions of the players. The utility functions defined over actions for the allocators are:
U~= ( T - Oi) + b ' ( T - 20~), Ui = ( Z -
0i) - ctlZ -
20,1,
U i = ( T - O i ) + d ' O i, 0 < d ' <
b ' > O,
(4)
c'> O,
(5)
1,
(6)
where T is the total stake and O i is the offer of player i. One justification for this approach is that the players are expressively rational. This will occur when players, in making their offers and deciding their acceptances, are primarily concerned to make a statement about their beliefs as to what is an appropriate division of the stake, rather than being concerned with the outcomes of their decisions. An alternative justification for these functions can be made by appealing to the concept of bounded rationality. For these functions, the allocators do not have to make any strong assumptions concerning the distribution of acceptances, and this in turn simplifies the computation of the optimal offer. So for example, for Eq. (4), U~ is a decreasing function of 0 i for
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any value of b' > 0 and so the function is maximised at O i = 0. If the allocator believes that an offer of zero is almost certain to be rejected then his or her best offer is $1. This is a much more straightforward computation for the allocator than finding the offer which maximises the expected payoff for some assumed distribution of minimum acceptances. The computation is the same for Eqs. (5) and (6). For Eq. (5) it is easy to establish that the offer would never exceed $5, and that if c' < 0.5 the optimal offer is $1. If c' = 0.5 then any offer between $1 to $5 will give the allocator the same level of utility. If c' > 0.5 then the optimal offer is $5. The behaviour for Eq. (6) is straightforward, it is optimal to offer $1. This is the prescription of conventional game theory, thus the behaviour of an expressive altruist is indistinguishable from that of an expressively selfish allocator. As the recipient has only to decide whether to accept or reject the offer and hence faces no uncertainty, Eqs. (1), (2) and (3) are, for the recipient, equivalent to Eqs. (4), (5) and (6) repectively. The material presented in this section is of the type that is necessary for developing a prescriptive game theory. It also demonstrates that while the behaviour of two players may be identical, their motivation may be quite different. For example, an individual who has a minimum acceptance of $1 and who offers $5 may have any of the utility functions represented by Eqs. (1), (2) or (3) for appropriate values of the various parameters. However, not all pairings of minimum acceptances and offers are compatible with all of the utility functions. For example, if an individual had a minimum acceptance of $5 and offered $1, then this behaviour is only consistent with Eqs. (1) and (4). This property is exploited in Section 5 where we use these results to try and say something more definitive about what motivates our respondents.
4. Expected monetary payoffs The material in the last section provides a prescription for players who wish to maximise their utility. In this section we present the monetary payoffs that different types of players would receive. We could not present all of the possible cases so we have selected a number of player strategies, and calculated the monetary payoffs that an individual would expect to receive, assuming that the probability of being cast as an allocator or recipient is one half (see Appendix A). By a 'strategy' we mean a specification of a player's minimum acceptance and offer.
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Table 2 Expected monetary payoffs from various strategies Strategy Altruistic/money maximising Equality Game theory Envious
$ (offer $5, accept $1) (offer $5, accept $5) (offer $1, accept $1) (offer $1, accept $5)
4.63 4.15 3.41 2.94
The 'monetary maximising' strategy is defined as an offer of $5 and a minimum acceptance of $1. This is optimal for Eqs. (1), (2) and (3) if respectively b -- 0, c = 0 and d = 0. The 'altruistic' strategy is also to offer $5 and accept $1. This is the optimal strategy for Eq. (3). The 'game theory' strategy is to offer $1 and accept $1. This is optimal for Eq. (4) when b = 0. The 'equality' strategy is to offer $5 and accept $5. This is optimal for Eq. (2), when c > 1.06 and for Eq. (5) when c' > 0.5. The 'envious' strategy is to offer $1 and accept $5. This is optimal if 1.06 < b < 2 in Eq. (1). Table 2 shows the monetary payoffs that players adopting these strategies would expect to receive. An appeal to the idea of natural selection suggests that over time only individuals playing the 'monetary maximising/altruist' strategy would survive. What is interesting is that in games like this there would be no selection against altruists. The proportion of altrustic and selfish individuals in the population in the long run would be basically undetermined. Sutherland (1992) has argued that one reason why irrationality is so pervasive is because it is possible to survive without being rational. Applying that argument here, if the return that ensured survival was not too high, it is possible that individuals playing the 'equality' strategy may also survive as the payoff they receive is reasonably close to the best payoff. An individual playing the 'game theory' strategy could be someone who has selfish preferences and who assumes that the selection process has already weeded out all non-selfish individuals. What Table 2 indicates is that reliance on reasoning based on what may happen in the long run will generate relatively poor payoffs. Table 2 also demonstrates that a strong concern with relative income will generate a low relative payoff.
5. Are the responses consistent with any of the postulated utility functions? We now investigate whether any of the matched pairs of offers and minimum acceptances given by our respondents are consistent with the utility functions we have postulated.
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Table 3 Consistency of offers and minimum acceptances with three utility functions Reported offer minus offer consistent with utility function
Proportions of sample Eq.(l)
Eq.(2)
Eq.(5)
Minus Zero Plus Minus/Zero Plus/Zero Plus/Minus
0.19 0.17 0.48 0.00 0.02 O. 14
0.36 0.60 0.05 0.00 0.00 0.00
0.17 0.55 0.29 0.00 0.00 0.00
Bold figures are those observations that are or could be consistent with the relevant equation. The Table is constructed as follows: If the offer reported by a respondent was consistent as in the example given in the text, this would appear in the zero row in Table 3. If the respondent offered less than the offer which is consistent with Eq. (1) this would appear in the minus row in Table 3. Therefore, the minus row shows the proportions of the sample for which the reported offers were less than the offers consistent with the stated minimum acceptances. The Zero row shows the proportion of reported offers which were consistent with the stated minimum acceptances. The Plus row shows the proportion of reported offers which were greater than the offers consistent with the stated minimum acceptances. The Plus/Zero row indicates that proportion of the sample for which the reported offer is either greater than, or equal to, the offer which would be consistent with the stated minimum acceptance. This ambiguity can arise if the minimum acceptance is $4. As Table 1 shows, the reported offer consistent with Eq. (1) is either $1 or $5. Hence if the player offers $5 it is not possible to decide whether it equals or exceeds the offer that would be consistent with the minimum acceptance. The Plus/Minus row indicates the proportion of the sample that had a minimum acceptance of $4 but who did not offer either $5 or $1.
Table 1 shows for Eq. (1) which offer(s) are consistent with any minimum acceptance. So for any particular respondent we are able to determine from her minimum acceptance what her offer should be if she is maximising Eq. (1). Her behaviour would be consistent with maximising Eq. (1) if her reported offer matched that given in Table 1. For example, if she stated a minimum acceptance of $2 then she would also have to have an offer of $5 if she was maximising Eq. (1). If the offer reported by the respondent was indeed $5, then the offer/acceptance pairing is consistent with Eq. (1). The material presented in Section 3 of the paper also allows us to check whether the offer/acceptance pairs reported by our sample are consistent with the other five utility functions. The results for Eqs. (1), (2) and (5) are given in Table 3. The proportion of offer/acceptance pairs which are consistent with any of the other utility functions postulated above are so small that we have not reported the results. If we adopt as the criteria for explanatory success, the proportion of our sample for which the offer/acceptance pairs are consistent with each of the utility functions, then Table 3 suggests a ranking of Eq. (2), Eq. (5) and Eq. (1). Both equations which include a term for fairness perform significantly better
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than Eq. (1), hence our results support the idea that fairness is an important factor in explaining the behaviour of our respondents. The figures in Table 3 for Eqs. (1) and (2) were constructed on the assumption that players know the distribution of minimum acceptances and that allocators are risk neutral. A more refined test would require an examination of these factors. A commonsense defence of the procedure we have used is that these factors will probably influence the computations for both equations in a similar way, and a more complex test would be unlikely to influence the rankings. Selfish tastes are a special case of all our utility functions, so the above results, even if accepted at face value, do not rule out the possibility that a significant proportion of our respondents have selfish tastes. An estimate of the prevalence of selfish tastes can be obtained from the proportions of the samples who reported the optimal strategy of the payoff maximiser (offer $5, accept $1). The proportion of our sample who reported this pairing was 0.14. It does not require much thought on the part of a selfish maximiser to know that a minimum acceptance of $1 is optimal. However, it is possible that misconceptions over the distribution of acceptances may lead selfish individuals to offer less than $5. Hence, if we re-classify all those who were prepared to accept $1 as selfish maximisers, the maximum proportion of selfish individuals in our sample is 0.29. We believe the true proportion is probably much less than this for two reasons. Firstly the individuals classified in this group could be altruists. Secondly the misconceptions required for an offer of less than $5 to be optimal are very large. The details are given in Table 4. In constructing this table we have assumed that the allocator believes that an offer of $5 will be accepted with certainty. What this table shows is that for $3 to be an optimal offer, the cumulative probability that $3 will be accepted has to be greater than or equal to 0.71, whereas the corresponding proportion from our sample was 0.33. The conclusion then is that at most, about one quarter of our sample possess the types of preferences that are routinely assumed by economists. Table 4 The cumulative probability, that an offer will be accepted, required to make such an offer optimal for a selfish maximiser, and the actual sample proportions Offers ($)
Required probabilities
Sample proportions
1 2 3 4
0.556 0.625 0.714 0.833
0.286 0.310 0.333 0.476
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6. Conclusion We began the paper by asking: What motivates players of the ultimatum game? Is it fairness, envy, altruism or selfishness? Of course our respondents probably have all of these motivations, however our broad conclusion is that the dominant motivation is a desire for fairness. Our results suggest that over half the participants in our study have a concern for fairness, which considerably exceeds the proportion who exhibit envy or altruism; and at most, only a quarter of the participants could be said to be motivated by selfishness. Our second question asked whether it was possible to advise players how to play the game in order to maximise their utility. From the responses given by our sample, we were able to determine the offer and the acceptance which would maximise a player's utility and these results are presented in Section 3. Finally, we asked: What monetary payoffs might be expected from playing strategies consistent with these utility functions? Assuming the chance of being an allocator or a recipient is one half, and that individuals play the strategies which would maximise their utility functions, the expected monetary payoff would be highest for the money maximiser, but only a little less for individuals motivated by fairness. Those who play the conventional game theory strategy would fare considerably worse. It may be argued that our respondents would have behaved selfishly if they had played the game for real prizes. However this argument is not supported by trials of the ultimatum game where individuals receive actual monetary payoffs. The advantage of the questionnaire is that we were able to elicit from respondents matched pairs of offers and acceptances, which allowed us to see if the pairings were consistent with non-selfish utility functions. Based on the data we have presented, we conclude that a concern for fairness is more pervasive than economists are generally prepared to recognise. As a methodological point, some have commented that our approach seems to be aimed at rescuing the economists' utility maximising model. Presumably they have in mind what Popper would call an 'immunising strategem'. However, we would argue that what we have demonstrated supports the proposition: If behaviour is to be viewed as utility maximising, then we must recognise that the determinants of utility are more complex than is conventionally recognised by economists. The major difficulty lies with the conventional assumption that utility functions do not contain interdependence terms. Given the resistance to including such terms, and given that economists are not going to abandon the model of utility maximising; as a tactical matter we believe that it is more productive to
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try and convince those who accept the utility maximising model that, if this approach is to have any relevance for understanding behaviour, they will have to drop the assumption that individuals are unconnected social atoms. This seems to us to be the key point and is one of widespread relevance. For example, Tompkinson (1994) has argued that the attempt by Solow (1990) to explain the emergence of the norm which prohibits unemployed workers from undercutting the wages of those in work, is not persausive because it fails to model human beings as social animals. As a final point, using the utility maximising model does not prevent anyone from trying to explain why it is that utility functions contain interdependence terms like fairness.
Appendix A The expected return to recipients is calculated as follows: 10
E(PR) =
E%E[PRIj], j=O
where 10
E[ PRIJ] = ~-'~Pii, i=j
and qj = probability the recipient will receive an offer of $j, and Pi = probability that a recipient will accept an offer of $i. It is important to note that the recipient does not have to make these calculations when deciding his or her minimum acceptable offer. In deciding on the minimum acceptable offer the recipient has only to decide what minimum offer is preferable to him or her than both players receiving zero. The expected payoff to allocators is calculated as follows: 10
E[ Pa] ~- E qjE[ PAIJ] , j=0
where J
E[ PAJJ] = ~".p,(lO-j). i=0
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References Binmore, K.G., 1990. Essays on the Foundations of Game Theory. Cambridge, MA: Blackwell. Giith, W., R. Schmittberger and B. Schwarze, 1982. An experimental analysis of ultimatum bargaining. Journal of Economic Behaviour and Organization 3, 367-388. Hargreaves Heap, S., 1989. Rationality in Economics. Oxford: Basil Blackwell. Solow, R.M., 1990. The Labour Market as a Social Institution. Oxford: Basil Blackwell. Sutherland, S., 1992. Irrationality, the Enemy Within. London: Constable. Thaler, R.H., 1988. Anomalies; the ultimatum game. Journal of Economic Perspectives 2, 195-206. Tompkinson, P., 1994. Review of R.M. Solow, 1990, The labour market as a social institution. New Zealand Economic Papers 28, 101-105. Wardman, M., 1988. A comparison of revealed preference and stated preference models. Journal of Transport Economics and Policy 22, 71-91.
ELSEVIER
Journal of Economic Psychology 17 (1996) 259-271
The ultimatum game and non-selfish utility functions Judy Bethwaite *, Paul Tompkinson Faculty of Commerce and Administration, Victoria University of Wellington P.O. Box 600, Wellington, New Zealand Received 27 April 1993; accepted 28 November 1995
Abstract This paper reports on the results of a questionnaire which asked respondents how much they would offer as an allocator, or be prepared to accept as a recipient, in an ultimatum game with a $10 stake. This permitted an analysis of the respondents' reported offers and minimum acceptances as matched pairs, which has the potential to yield more information about players' utility functions than observation of player behaviour in experimental trials of the ultimatum game. The results suggest that over half of the respondents have a concern for fairness, which is a much greater proportion than those who could be considered to be motivated by envy or altruisim. Only about one quarter of the sample possess a selfish utility function of the type conventionally assumed by economists.
PsyclNFO classification." 2300 JEL classification: C72; C90 Keywords: Ultimatum game; Non-selfish utility functions
* Corresponding author. Fax: +64 4 471-2200, Tel.: +64 4 472-1000.
0167-4870/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0 167-4870(96)00006-2
260
Jud Bethwaite, P. Tompkinson / Journal of Economic Psychology 17 (1996) 259-271
1. Introduction In the simplest version of the ultimatum game, one player, the allocator, is directed to divide a sum of money (the stake) between him or herself and a second player, the recipient. If the recipient accepts the proposed division of the stake then both receive the amounts proposed by the allocator. If the recipient rejects the proposed division then both players receive nothing. The recipient knows the size of the stake, but the players do not know each other. The roles of allocator and recipient are typically determined by the toss of a coin. Thaler (1988) has reviewed the outcomes of experimental trials of the ultimatum game, a key feature of which is that many of the allocators make offers which are half of the total stake. This is at variance with the conventional assumption that individuals are rational and have selfish tastes. By rational we mean that they aim to maximise expected utility, and a player's tastes are selfish if they depend only on his or her payoff. The economists' assumption that rationality and selfish tastes prevail in human behaviour leads to the game theory prediction that the allocator should offer the smallest possible amount and that the recipient should accept this. This follows from backward induction: for the recipient any positive amount is better than zero and therefore should be accepted. The allocator knows this and so should make the smallest positive offer. The empirical evidence casts serious doubt as to whether the conventional specification of individual tastes is the correct one. Thaler argues that the prediction based on selfish tastes is falsified because the players are primarily motivated not by selfishness but by a concern for fairness. Binmore (1990, pp. 10-11), commenting on the same results, interprets the behaviour of the players as being like that of stimulus-response machines. Behaviour which may be appropriate in certain circumstances is applied without thought to a problem in which it is not appropriate. This explanation is consistent with sociological models of behaviour as being rule guided, and has the implication that the players are not optimising. On the basis of these experimental results Thaler argues that it may be useful to distinguish between three kinds of game theory: normative, prescriptive and descriptive. Currently much of game theory is normative, analysing how a rational individual should behave if this individual reasoned on the basis that rationality and selfish tastes were universal. Prescriptive theory would provide advice to a player as to how to play, recognising his or her own cognitive limitations and accepting that others may not be rational and may have non-selfish tastes. Descriptive game theory would describe how individuals actually play.
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261
In this paper, and in the light of the above summary, we wish to focus on the following questions: 1. What motivates players of the ultimatum game? Is it fairness, envy, altruism or selfishness? 2. For our sample, what advice can be given to players who wish to maximise their utility, for various utility functions? and 3. What monetary payoffs might players expect from playing particular strategies, consistent with these utility functions? Our approach to answering these questions is to assume that the players are utility maximisers for various specifications of the utility function. We consider briefly, at the end of the paper, whether this is a productive approach.
2. Procedures
A questionnaire was constructed which first outlined the ultimatum game and then asked respondents how much they would offer as an allocator, and how much they would be prepared to accept as a recipient, for a stake of $10 (New Zealand currency) where the minimum monetary unit was $1. Using a questionnaire, rather than conducting direct trials of the ultimatum game, allowed us to consider respondents' offers and minimum acceptances as matched pairs, which has the potential to yield more information about their utility functions. The main criticism of our research methodology must be that respondents have no incentive to report how they would actually behave in the game situation. But equally, we have no reason to believe that they have any incentive to misrepresent their behaviour, and in so far as it is possible to compare, the responses we obtained using the questionnaire approach are similar to the results reported in experiments where the game is played with real prizes (Giith et al., 1982). Finally, there is evidence from other contexts that answers to hypothetical questions provide good indicators of behaviour, see for example Wardman (1988). The participants in our study were a group of lawyers in a large private legal practice. We chose a group of lawyers for several reasons. Firstly, previous studies had used student samples and we were interested to know whether other groups would behave in similar ways. Secondly, we believed prior to analysing our results, that lawyers were more likely to behave 'rationally' than students. The evidence from earlier studies of the ultimatum game suggests that the selfishly rational individual of economic theory is a rare creature. If the same conclusion can be established for our sample then this would be significant
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given our prior as to the 'rationality' of lawyers. Others however, did not share our prior belief: they supposed that the lawyers may have thought that we were testing their sense of justice, and as a consequence, they expected them to behave less selfishly than students. As it tums out the lawyers bahave in much the same way as students. We also conducted our survey on a sample of students and results from the two groups were very similar; this being the case we have reported only the results from the lawyer sample. Questionnaires were mailed to 72 potential respondents and 43 were returned, representing a response rate of 60%. In Section 3 of this paper we postulate a number of utility functions which may motivate the behaviour of players of the ultimatum game. We specify functions which distinguish between individuals who are envious, have a taste for fairness, are altruistic and are selfish. We also distinguish between individuals who are instrumentally rational and those who are expressively rational (Hargreaves Heap, 1989). For our sample we determine the optimal behaviour for players with these utility functions. This allows us to provide prescriptions for how to play the game and thus an answer to our question 2. In Section 4, we calculate the monetary payoffs that could be expected from playing particular strategies and thus answer question 3. In Section 5 we investigate whether the offers and acceptances reported by the respondents are consistent with any of the utility functions we have postulated. In this way we are able to give some answers as to the motivation of our respondents and thus an answer to our question 1.
3. Utility functions and optimal strategies We postulate the following utility functions as motivating the behaviour of players of the ultimatum game:
Ui=Pi+b(Pi-Pj),
b>O,
G = P , - c[Pi- Pjl,
c > O,
(2)
1,
(3)
U i = P i + d P j,
O
(1)
where Pi is the payoff to player i and Pj is the payoff to player j. In Eq. (1), utility is a function of a player's own payoff and his or her relative payoff; this is one way of introducing envy into the utility function. Eq. (2) specifies utility as a function of the player's own payoff and the equity of the allocation, thus introducing a concern for fairness into the utility function. Eq.
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(3) is a possible specification for an altruistic individual, with the restriction that the weight given to the player's own payoff is greater than that given to the other player's payoff. Selfishness is a special case of all of these functions, obtained by setting b, c or d equal to zero. Using Eq. (1) we can determine for any minimum acceptance the range of values that b can take. For example if the minimum acceptance is $1, and noting that Pi + Pj = 10 for a $10 stake, it follows from Eq. (1) that: l+b(1-9)>0orthatb<0.125. Incorporating the assumption that b > 0, for $1 to be the minimum acceptance, we must have 0 < b < 0.125. Proceeding in this way we can calculate the range of values of b which are consistent with all possible minimum acceptances. It is straightforward to show that the minimum acceptance must be less than or equal to $5. Again for Eq. (1) if we assume that the allocators are risk neutral and know the distribution of minimum acceptances, we can also determine the range of values of b which are consistent with any given offer. For example if $1 is the optimal offer then this implies the following set of inequalities: E[ Pi(1)] + for
0i=
bE[ P i ( 1 ) -
Pj(1)] > E[
ei(oi)]
+
bE[ Pi(Oi)
-- ej(oi)],
2 to 9,
where E[P~(1)] is the expected payoff to player i if he or she makes an offer of $1 and E[Pi(Oi)] is the expected payoff to player i if he or she makes an offer of $O~ 4: 1. E[PflOi)] is the expectation held by player i as to what player j will receive if she makes an offer of O~. Each inequality in the above set of inequalities can be solved for b. By computing the range of values which satisfies all of these inequalities, we can find the range of values of b for which $1 is the optimal offer. If there is no range of values for b which satisfies all of these inequalities then this means that it can never be optimal to offer $1. This process is then repeated using the following set of inequalities: E[ Pi(2)] +
bE[ P i ( 2 ) -
Py(2)] > E[
Pi(Oi)] + bE[ P i ( O i ) - Pj(Oi)],
for O~= 1, 3 to 9. From this set of inequalities we can find the range of values of b, if any, which ensure that the optimal offer is $2. Proceeding in the same way, we can find for any offer the range of values of b, if any, which make that particular offer optimal. It turns out that for our sample the only offers that can be optimal are $1 or $5. The minimum acceptances and optimal offers, regarded as functions of b, for Eq. (1) are given in Table 1.
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Table 1 Minimum acceptances and optimal offers for Eq. (1) Value of b
$ Min. acceptance
$ Optimal offer
0.00 < b < 0.125 0.125 < b < 0.33 0.33 < b < 0.75 0.75 < b < 1.06 1.06 < b < 2.00 b>2
1 2 3 4 4 5
5 5 5 5 1 1
For individuals concerned only with the size of their own payoff, that is when b = 0, Table 1 shows that the optimal way to play the game is to offer half the stake and be prepared to accept the smallest positive offer. Calculations of the same sort can be carried out for Eqs. (2) and (3). For Eq. (2), the minimum acceptances are related to c in the same way that the minimum acceptances are related to b in Table 1. However regardless of the value of c in Eq. (2), the optimal offer is always $5. The optimal way for the altruist (Eq. (3)) to play is to accept $1 and to offer $5 for any value of d. Paradoxically, the optimal way for an altruist to play is identical to that of the selfish economic maximiser. So far we have assumed that utility is purely a function of the expected or actual outcomes. Alternatively, utility may be expressed as a function of the actions of the players. The utility functions defined over actions for the allocators are:
U~= ( T - Oi) + b ' ( T - 20~), Ui = ( Z -
0i) - ctlZ -
20,1,
U i = ( T - O i ) + d ' O i, 0 < d ' <
b ' > O,
(4)
c'> O,
(5)
1,
(6)
where T is the total stake and O i is the offer of player i. One justification for this approach is that the players are expressively rational. This will occur when players, in making their offers and deciding their acceptances, are primarily concerned to make a statement about their beliefs as to what is an appropriate division of the stake, rather than being concerned with the outcomes of their decisions. An alternative justification for these functions can be made by appealing to the concept of bounded rationality. For these functions, the allocators do not have to make any strong assumptions concerning the distribution of acceptances, and this in turn simplifies the computation of the optimal offer. So for example, for Eq. (4), U~ is a decreasing function of 0 i for
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any value of b' > 0 and so the function is maximised at O i = 0. If the allocator believes that an offer of zero is almost certain to be rejected then his or her best offer is $1. This is a much more straightforward computation for the allocator than finding the offer which maximises the expected payoff for some assumed distribution of minimum acceptances. The computation is the same for Eqs. (5) and (6). For Eq. (5) it is easy to establish that the offer would never exceed $5, and that if c' < 0.5 the optimal offer is $1. If c' = 0.5 then any offer between $1 to $5 will give the allocator the same level of utility. If c' > 0.5 then the optimal offer is $5. The behaviour for Eq. (6) is straightforward, it is optimal to offer $1. This is the prescription of conventional game theory, thus the behaviour of an expressive altruist is indistinguishable from that of an expressively selfish allocator. As the recipient has only to decide whether to accept or reject the offer and hence faces no uncertainty, Eqs. (1), (2) and (3) are, for the recipient, equivalent to Eqs. (4), (5) and (6) repectively. The material presented in this section is of the type that is necessary for developing a prescriptive game theory. It also demonstrates that while the behaviour of two players may be identical, their motivation may be quite different. For example, an individual who has a minimum acceptance of $1 and who offers $5 may have any of the utility functions represented by Eqs. (1), (2) or (3) for appropriate values of the various parameters. However, not all pairings of minimum acceptances and offers are compatible with all of the utility functions. For example, if an individual had a minimum acceptance of $5 and offered $1, then this behaviour is only consistent with Eqs. (1) and (4). This property is exploited in Section 5 where we use these results to try and say something more definitive about what motivates our respondents.
4. Expected monetary payoffs The material in the last section provides a prescription for players who wish to maximise their utility. In this section we present the monetary payoffs that different types of players would receive. We could not present all of the possible cases so we have selected a number of player strategies, and calculated the monetary payoffs that an individual would expect to receive, assuming that the probability of being cast as an allocator or recipient is one half (see Appendix A). By a 'strategy' we mean a specification of a player's minimum acceptance and offer.
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Table 2 Expected monetary payoffs from various strategies Strategy Altruistic/money maximising Equality Game theory Envious
$ (offer $5, accept $1) (offer $5, accept $5) (offer $1, accept $1) (offer $1, accept $5)
4.63 4.15 3.41 2.94
The 'monetary maximising' strategy is defined as an offer of $5 and a minimum acceptance of $1. This is optimal for Eqs. (1), (2) and (3) if respectively b -- 0, c = 0 and d = 0. The 'altruistic' strategy is also to offer $5 and accept $1. This is the optimal strategy for Eq. (3). The 'game theory' strategy is to offer $1 and accept $1. This is optimal for Eq. (4) when b = 0. The 'equality' strategy is to offer $5 and accept $5. This is optimal for Eq. (2), when c > 1.06 and for Eq. (5) when c' > 0.5. The 'envious' strategy is to offer $1 and accept $5. This is optimal if 1.06 < b < 2 in Eq. (1). Table 2 shows the monetary payoffs that players adopting these strategies would expect to receive. An appeal to the idea of natural selection suggests that over time only individuals playing the 'monetary maximising/altruist' strategy would survive. What is interesting is that in games like this there would be no selection against altruists. The proportion of altrustic and selfish individuals in the population in the long run would be basically undetermined. Sutherland (1992) has argued that one reason why irrationality is so pervasive is because it is possible to survive without being rational. Applying that argument here, if the return that ensured survival was not too high, it is possible that individuals playing the 'equality' strategy may also survive as the payoff they receive is reasonably close to the best payoff. An individual playing the 'game theory' strategy could be someone who has selfish preferences and who assumes that the selection process has already weeded out all non-selfish individuals. What Table 2 indicates is that reliance on reasoning based on what may happen in the long run will generate relatively poor payoffs. Table 2 also demonstrates that a strong concern with relative income will generate a low relative payoff.
5. Are the responses consistent with any of the postulated utility functions? We now investigate whether any of the matched pairs of offers and minimum acceptances given by our respondents are consistent with the utility functions we have postulated.
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Table 3 Consistency of offers and minimum acceptances with three utility functions Reported offer minus offer consistent with utility function
Proportions of sample Eq.(l)
Eq.(2)
Eq.(5)
Minus Zero Plus Minus/Zero Plus/Zero Plus/Minus
0.19 0.17 0.48 0.00 0.02 O. 14
0.36 0.60 0.05 0.00 0.00 0.00
0.17 0.55 0.29 0.00 0.00 0.00
Bold figures are those observations that are or could be consistent with the relevant equation. The Table is constructed as follows: If the offer reported by a respondent was consistent as in the example given in the text, this would appear in the zero row in Table 3. If the respondent offered less than the offer which is consistent with Eq. (1) this would appear in the minus row in Table 3. Therefore, the minus row shows the proportions of the sample for which the reported offers were less than the offers consistent with the stated minimum acceptances. The Zero row shows the proportion of reported offers which were consistent with the stated minimum acceptances. The Plus row shows the proportion of reported offers which were greater than the offers consistent with the stated minimum acceptances. The Plus/Zero row indicates that proportion of the sample for which the reported offer is either greater than, or equal to, the offer which would be consistent with the stated minimum acceptance. This ambiguity can arise if the minimum acceptance is $4. As Table 1 shows, the reported offer consistent with Eq. (1) is either $1 or $5. Hence if the player offers $5 it is not possible to decide whether it equals or exceeds the offer that would be consistent with the minimum acceptance. The Plus/Minus row indicates the proportion of the sample that had a minimum acceptance of $4 but who did not offer either $5 or $1.
Table 1 shows for Eq. (1) which offer(s) are consistent with any minimum acceptance. So for any particular respondent we are able to determine from her minimum acceptance what her offer should be if she is maximising Eq. (1). Her behaviour would be consistent with maximising Eq. (1) if her reported offer matched that given in Table 1. For example, if she stated a minimum acceptance of $2 then she would also have to have an offer of $5 if she was maximising Eq. (1). If the offer reported by the respondent was indeed $5, then the offer/acceptance pairing is consistent with Eq. (1). The material presented in Section 3 of the paper also allows us to check whether the offer/acceptance pairs reported by our sample are consistent with the other five utility functions. The results for Eqs. (1), (2) and (5) are given in Table 3. The proportion of offer/acceptance pairs which are consistent with any of the other utility functions postulated above are so small that we have not reported the results. If we adopt as the criteria for explanatory success, the proportion of our sample for which the offer/acceptance pairs are consistent with each of the utility functions, then Table 3 suggests a ranking of Eq. (2), Eq. (5) and Eq. (1). Both equations which include a term for fairness perform significantly better
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than Eq. (1), hence our results support the idea that fairness is an important factor in explaining the behaviour of our respondents. The figures in Table 3 for Eqs. (1) and (2) were constructed on the assumption that players know the distribution of minimum acceptances and that allocators are risk neutral. A more refined test would require an examination of these factors. A commonsense defence of the procedure we have used is that these factors will probably influence the computations for both equations in a similar way, and a more complex test would be unlikely to influence the rankings. Selfish tastes are a special case of all our utility functions, so the above results, even if accepted at face value, do not rule out the possibility that a significant proportion of our respondents have selfish tastes. An estimate of the prevalence of selfish tastes can be obtained from the proportions of the samples who reported the optimal strategy of the payoff maximiser (offer $5, accept $1). The proportion of our sample who reported this pairing was 0.14. It does not require much thought on the part of a selfish maximiser to know that a minimum acceptance of $1 is optimal. However, it is possible that misconceptions over the distribution of acceptances may lead selfish individuals to offer less than $5. Hence, if we re-classify all those who were prepared to accept $1 as selfish maximisers, the maximum proportion of selfish individuals in our sample is 0.29. We believe the true proportion is probably much less than this for two reasons. Firstly the individuals classified in this group could be altruists. Secondly the misconceptions required for an offer of less than $5 to be optimal are very large. The details are given in Table 4. In constructing this table we have assumed that the allocator believes that an offer of $5 will be accepted with certainty. What this table shows is that for $3 to be an optimal offer, the cumulative probability that $3 will be accepted has to be greater than or equal to 0.71, whereas the corresponding proportion from our sample was 0.33. The conclusion then is that at most, about one quarter of our sample possess the types of preferences that are routinely assumed by economists. Table 4 The cumulative probability, that an offer will be accepted, required to make such an offer optimal for a selfish maximiser, and the actual sample proportions Offers ($)
Required probabilities
Sample proportions
1 2 3 4
0.556 0.625 0.714 0.833
0.286 0.310 0.333 0.476
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6. Conclusion We began the paper by asking: What motivates players of the ultimatum game? Is it fairness, envy, altruism or selfishness? Of course our respondents probably have all of these motivations, however our broad conclusion is that the dominant motivation is a desire for fairness. Our results suggest that over half the participants in our study have a concern for fairness, which considerably exceeds the proportion who exhibit envy or altruism; and at most, only a quarter of the participants could be said to be motivated by selfishness. Our second question asked whether it was possible to advise players how to play the game in order to maximise their utility. From the responses given by our sample, we were able to determine the offer and the acceptance which would maximise a player's utility and these results are presented in Section 3. Finally, we asked: What monetary payoffs might be expected from playing strategies consistent with these utility functions? Assuming the chance of being an allocator or a recipient is one half, and that individuals play the strategies which would maximise their utility functions, the expected monetary payoff would be highest for the money maximiser, but only a little less for individuals motivated by fairness. Those who play the conventional game theory strategy would fare considerably worse. It may be argued that our respondents would have behaved selfishly if they had played the game for real prizes. However this argument is not supported by trials of the ultimatum game where individuals receive actual monetary payoffs. The advantage of the questionnaire is that we were able to elicit from respondents matched pairs of offers and acceptances, which allowed us to see if the pairings were consistent with non-selfish utility functions. Based on the data we have presented, we conclude that a concern for fairness is more pervasive than economists are generally prepared to recognise. As a methodological point, some have commented that our approach seems to be aimed at rescuing the economists' utility maximising model. Presumably they have in mind what Popper would call an 'immunising strategem'. However, we would argue that what we have demonstrated supports the proposition: If behaviour is to be viewed as utility maximising, then we must recognise that the determinants of utility are more complex than is conventionally recognised by economists. The major difficulty lies with the conventional assumption that utility functions do not contain interdependence terms. Given the resistance to including such terms, and given that economists are not going to abandon the model of utility maximising; as a tactical matter we believe that it is more productive to
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try and convince those who accept the utility maximising model that, if this approach is to have any relevance for understanding behaviour, they will have to drop the assumption that individuals are unconnected social atoms. This seems to us to be the key point and is one of widespread relevance. For example, Tompkinson (1994) has argued that the attempt by Solow (1990) to explain the emergence of the norm which prohibits unemployed workers from undercutting the wages of those in work, is not persausive because it fails to model human beings as social animals. As a final point, using the utility maximising model does not prevent anyone from trying to explain why it is that utility functions contain interdependence terms like fairness.
Appendix A The expected return to recipients is calculated as follows: 10
E(PR) =
E%E[PRIj], j=O
where 10
E[ PRIJ] = ~-'~Pii, i=j
and qj = probability the recipient will receive an offer of $j, and Pi = probability that a recipient will accept an offer of $i. It is important to note that the recipient does not have to make these calculations when deciding his or her minimum acceptable offer. In deciding on the minimum acceptable offer the recipient has only to decide what minimum offer is preferable to him or her than both players receiving zero. The expected payoff to allocators is calculated as follows: 10
E[ Pa] ~- E qjE[ PAIJ] , j=0
where J
E[ PAJJ] = ~".p,(lO-j). i=0
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References Binmore, K.G., 1990. Essays on the Foundations of Game Theory. Cambridge, MA: Blackwell. Giith, W., R. Schmittberger and B. Schwarze, 1982. An experimental analysis of ultimatum bargaining. Journal of Economic Behaviour and Organization 3, 367-388. Hargreaves Heap, S., 1989. Rationality in Economics. Oxford: Basil Blackwell. Solow, R.M., 1990. The Labour Market as a Social Institution. Oxford: Basil Blackwell. Sutherland, S., 1992. Irrationality, the Enemy Within. London: Constable. Thaler, R.H., 1988. Anomalies; the ultimatum game. Journal of Economic Perspectives 2, 195-206. Tompkinson, P., 1994. Review of R.M. Solow, 1990, The labour market as a social institution. New Zealand Economic Papers 28, 101-105. Wardman, M., 1988. A comparison of revealed preference and stated preference models. Journal of Transport Economics and Policy 22, 71-91.