نوع دوستی با حداقل قابل قبول و بازی اولتیماتوم

I suppose that people react with anger when others show themselves not to be minimally altruistic. With heterogeneous agents, this can account for the experimental results of ultimatum and dictator games. Moreover, it can account for the surprisingly large fraction of individuals who offer an even split with parameter values that are more plausible than those that are required to explain outcomes in these experiments with the models of Levine [Levine, D.K., 1998. Modeling altruism and spitefulness in experiments. Review of Economic Dynamics 1, 593–622], Fehr and Schmidt [Fehr, E., Schmidt, K.M., 1999. A theory of fairness, competition and cooperation. Quarterly Journal of Economics 114, 817–868], Dickinson [Dickinson, D.L., 2000. Ultimatum decision making: a test of reciprocal kindness. Theory and Decision 48, 151–177] and Bolton and Ockenfels [Bolton, G.E., Ockenfels, A., 2000. ERC: a theory of equity, reciprocity, and competition. American Economic Review 90, 166–193].

This paper presents a model of individual preferences where individuals are mildly altruistic towards others while also expecting others to be mildly altruistic. If an individual encounters evidence that another is less altruistic than he finds acceptable, he becomes angry and derives pleasure from harming the excessively selfish individual. These preferences are shown to be capable of explaining the experimental outcomes of the ultimatum game of Güth et al. (1982) as well as those of an important variant, namely the dictator game of Forsythe et al. (1994). The main advantage of the model proposed here is that, unlike other models of social preferences that have been proposed to explain these experiments, it does not lead individuals to take unrealistic actions outside these experimental settings. Because the experimental outcomes of ultimatum and dictator games are in such sharp conflict with the predictions of standard economic models, many experimental variations have been considered, and this experimental literature is vast. Still, it is worth recalling the settings and some of the findings. Both games involve two players. The first player, who is called the proposer, offers to split a pie with the second, who is called the responder. In the ultimatum game, the responder can either accept or reject the proposer's offer. If the responder rejects it, neither player gets anything. Otherwise, the pie is split in the way suggested by the proposer. In the dictator game, the responder must passively accept the proposer's offer. The modal offer in the ultimatum game is to split the pie 50–50. The actual fraction of even splits varies somewhat from experiment to experiment, and sometimes varies across rounds of play within a given experiment. In Forsythe et al., about half the proposers offer an even split while Levine (1998) reports that about 28 percent of proposers offered an even split in the late rounds of the Roth et al. (1991) experiments. Splits that are less favorable to responders are often rejected. In several experiments (see Fig. 6 of Roth et al. as well as Harrison and McCabe, 1996), such rejections are so common that average earnings of proposers actually decline as they make offers that are less favorable to responders. With this behavior of responders, proposers should offer even splits if they wish to maximize their own expected payoffs. However, even in experiments where proposers earn less by making less generous offers and even after subjects have learned the game by playing it several times, some proposers make less generous offers.1 In the dictator game, expected monetary payoffs obviously rise when less generous offers are made. Not surprisingly, this implies that offers of even splits are observed less frequently. Still, Forsythe et al. report that about 20 percent of their proposers in the dictator game offered even splits. The model presented below is closely related to Levine who also supposes that agents’ altruism for others depends on their assessment of how altruistic others are in return. Unlike Levine, an agent's altruism is not assumed to depend linearly on that agent's perception of the altruism of the person he is interacting with. This allows the model to avoid Levine's conclusion that most people derive pleasure from seeing others suffer. His baseline parameters imply that 72 percent of the population would be willing to give up more than 25 cents to ensure that a stranger loses a dollar while 20 percent are ready to give up over 95 cents to bring about this outcome. In his model, this nastiness (or spitefulness) is important because it explains both why many responders reject uneven offers and why some proposers reduce their expected earnings by making offers that are less generous than even splits. One difficulty with the preferences implied by Levine's analysis is that, given the limitations of actual law enforcement institutions, they ought to lead to massive amounts of vandalism. The experimental evidence in other games also casts doubt on the ubiquitousness of spite. Using variants of dictator games, Charness and Rabin (2002) show that most people would actually sacrifice some of their own resources to induce small gains by others whereas essentially no one is willing to sacrifice significantly to hurt others. They show, in particular, that 73 percent of their subjects are willing to give up 100 of their units to cause another agent to gain 400. Not only would all the agents in Levine turn this down, because even his altruists are not sufficiently altruistic for this, but 20 percent of his agents prefer an outcome where they and another player both receive nothing to an outcome where the other player receives 80 while they get 20. In the Charness and Rabin experiments, almost all the players choose the 80/20 outcome over the one where neither gets anything.2 In an Appendix, Charness and Rabin present a model where agents who have acted in ways that are not consistent with maximizing social preferences accumulate “demerits” and where these demerits make agents less liked by others. While it features agents whose preferences are all identical, this model is similar in spirit to Levine and to the present paper. However, Charness and Rabin (2002, p. 851) make it clear that, in part because of the assumption of homogeneous preferences, which rules out different proposers making different offers or different responders responding differently, they do not intend this model to be “useful in its current form for calibrating experimental data.”3 Many of the existing models of fairness that are spelled out in sufficient detail so that it is possible to know the parameter values they require to explain laboratory phenomena do not explicitly suppose that individual utilities depend on the parameters of other agents’ utility functions. They suppose instead that people maximize a function that depends on the payoffs of several agents. There are two related ways in which this has been done. In the pioneering paper of Rabin (1993), an agent regards another as fair if he expects his actions to be “kind”, where kindness of the second player towards the first is defined as the difference between the payoffs that the first expects to receive from the second and the payoffs that it would have been “equitable” for the second to give to the first. Rabin then defines equitable payoffs as the average of the highest and the lowest payoffs the second agent can give to the first under the assumption that the player acts efficiently. Lastly, he supposes that agents maximize the sum of their own material payoffs and the product of their own kindness times the kindness they expect to receive from the other agent. This model has been extended by both Dickinson (2000) and Falk and Fischbacher (2006) to account for the results of ultimatum games. One important modification in Dickinson and in Falk and Fischbacher is that they allow an agent's utility function to put lower weight on the agent's own material payoffs than on “reciprocity” (i.e., on the product of the agent's own kindness and the kindness he receives from the other player). Offers of even splits can be obtained as equilibria in both papers, but only when agents care exclusively about reciprocity and ignore their own material payoffs. For obvious reasons, this limit is not particularly attractive as an empirical description of preferences. The alternative approach of Fehr and Schmidt (1999) and Bolton and Ockenfels (2000) supposes that agents care about how their own payoffs compare to those of others. These widely cited papers rationalize high offers in the ultimatum game by the unwillingness of responders to accept offers that give them payoffs that are low relative to those received by the proposer. Taken literally, these models imply that rejections do not hinge on the alternate courses of action that were available to the proposer and depend only on how the allocation of resources embodied in the proposer's offer compares to the allocation that results from rejection. However, the results of Kagel and Wolfe (2001) cast doubt on the importance of inequality in leading to rejections. They consider a variant of the ultimatum game where rejection by the responder leads to a payment to a third party so that inequality can be increased by rejection. In Kagel and Wolfe's experiments, the rate of rejection does not depend significantly on the amount that this third party receives when the proposer's offer is rejected. These models are also inconsistent with the evidence showing that the alternatives available to the proposer are an important determinant of responder behavior. Blount (1995), in particular, shows that the actions of responders depend on whether proposers themselves make the offer or whether the offer is randomly chosen by a computer. Similarly, Falk and Fischbacher as well as Charness and Rabin show that responders’ reactions depend not only on the final offer but also on the initial choices available to the proposer. Even leaving this aside, the models of Fehr and Schmidt and of Bolton and Ockenfels seem incapable of explaining the outcomes in ultimatum games except under implausible assumptions concerning parameters. As shown in Section 1, these models can predict even splits only when responders are unrealistically mean or when proposers are unrealistically kind-hearted (or both). The model of Section 2 supposes instead that, unless they feel provoked by evidence that someone is blameworthy, most people are modestly altruistic towards those around them. In principle, they are thus willing to incur small costs if this leads others to obtain large gains. Individuals are also assumed to care intensely about whether others are altruistic. This is broadly consistent with the evidence that responders care more about the actions of proposers than they do about the allocations that result from acceptance or rejection, because this evidence suggests that the proposer's intentions matter to responders. I suppose in particular that there is a minimal level of altruism that each person expects from those whom he interacts with and that demonstrations that one is more selfish than this are met with strong disapproval, and even anger. By contrast, demonstrating a level of altruism above the minimal level has a much smaller effect.4 The idea that people expect a minimal level of altruism from others fits with the notion that people are expected to be considerate and, more generally, to have manners.5 The notion that people change their preferences drastically when they feel that someone's actions demonstrate insufficient altruism is consistent with Pillutla and Murnighan's (1996) evidence that rejections in ultimatum games are associated with anger. One also observes rapid angry responses in the field when people feel mistreated by strangers. These emotions seem particularly salient in “road rage”, the sometimes violent reaction of drivers who feel that other drivers have acted badly.6 Anger at insufficient altruism can explain the tendency of responders to reject uneven offers. This leaves the question of how one can rationalize the observation that some proposers actually reduce their expected earnings by making uneven offers. These lower expected earnings also raise questions about the even-handedness of responders, since it would appear that these are punishing proposers whose actions are irrational. It turns out, however, that both the making of low offers and the recurrent rejections of such offers can be rationalized if experimental subjects sometimes act as if they were risk-loving. For a proposer, offers that are less favorable to the responder than an even split constitute a gamble in the sense that such offers are rejected with positive probability. Risk loving proposers with relatively low altruism levels would thus seek such gambles even if the expected returns of these gambles were slightly below those obtained from offering an even split. For this explanation to be plausible, one must believe that experimental subjects can act in risk-loving ways. As I discuss below, risk-loving behavior has been observed in several different experimental settings. At the same time, it must be recognized that the evidence is mixed in the sense that other experiments have found subjects to be risk-averse.7 The paper proceeds as follows. The next section briefly reviews the Fehr and Schmidt model and discusses outcome-based preferences more generally. Preferences with variable altruism are introduced in Section 2, where they are used to explain outcomes in ultimatum games. Because agents are assumed to be heterogeneous, the ultimatum game is a signaling game where proposers signal their altruism. As is common with signaling games there are many equilibria, though I focus on the one where all possible offers, including the 50–50 division of the pie, are observed. Section 3 is devoted to the dictator game and Section 4 concludes.

This paper has presented a model of preferences that can account for the experimental findings of ultimatum and dictator games without imposing extreme parameter values. It supposes that people feel mildly altruistic towards one another in most circumstances. Their normal altruism is mild enough that they would not transfer a dollar from their pocket to someone with a similar marginal utility of income, though they would transfer resources to people whose marginal utility of income they perceive as being much higher. They would also be willing to give up a dollar if someone else thereby gained substantially more than a dollar, as in the experiments of Charness and Rabin. The reason why preferences that differ so little from the selfish ones that form the baseline of economic analysis can fit these experiments is that there is a trigger that leads people to have very different preferences. In particular, people get upset with people who demonstrate extreme selfishness. One way of thinking about this is that the model formalizes the Camerer and Thaler (1995) insight that people get angry at individuals who have “poor manners”. Once this reaction has been triggered, people actually enjoy hurting those whom they regard as excessively selfish. What makes the ultimatum and dictator games different from more normal economic interactions is that they are good litmus tests for the extent to which people are selfish because actions in these games signal little else. By contrast, the moves people make in other economic interactions typically also signal their tastes for different commodity bundles as opposed to the extent of their altruism. These interactions are thus less prone to trigger anger. To see this, consider two individuals who wish to buy the same good. Suppose first that the price is fixed, that there is only one unit of the good left, and that the first manages to purchase the unit before the second. Even if the second covets the good as well, the purchase by the first does not necessarily prove that this individual is ungenerous. Thus, the second individual's disappointment is unlikely to turn to anger. Alternatively, imagine that the two individuals are bidding for the single unit that they both desire. When the first individual raises his bid, this raises the cost to the second of obtaining the unit. Even so, the second is not entitled to see this as purely reflecting the first individual's selfishness. This is so not only because the first individual might desire the good intensely but also because the first individual might be equally altruistic towards the seller as towards the second individual, and such even-handedness does not seem as subject to censure. This even-handedness would imply that, if the second individual ends up with the good, the vicarious gain to the first individual from the resources gained by the seller equals the vicarious losses he experiences from the reduction in the resources available to the buyer. There is thus no altruistic gain to the first individual from keeping his bid low. This can explain why selling prices end up being close to the reservation price of buyers in the “market” experiment of Roth et al.19 In this experiment there are several potential buyers, so the capacity of the subjects to experience anger does not lead to price restraint. Matters are different when there is a single buyer and a single seller. In this case, a seller who posts a high price demonstrates that he is selfish and this can lead to punishment. Hoffman et al. display an ultimatum game with this structure and show that, indeed, sellers end up charging considerably less than the reservation price of the buyer. Still, unlike the case of the ultimatum game with full information, buyers do not typically know the cost conditions faced by sellers, and this means that a seller's price is a less accurate measure of the seller's altruism than is the proposer's offer in an ultimatum game. Thus, price setting is more similar to the ultimatum game with incomplete information introduced by Mitzkewitz and Nagel (1993). I consider a price-setting situation of this sort in Rotemberg (2005) and show that, under plausible conditions, a single-shot price can be a quite poor signal of the seller's altruism. This would suggest that producers have a great deal of flexibility regarding the prices they charge. However, the ability of producers to change their prices without triggering anger is substantially lessened if buyers do not believe that cost conditions have changed significantly from the time that prices were changed last.