Social norms or social preferences?

Abstract

Some behavioral economists argue that the honoring of social norms can be adequately modeled as the optimization of social utility functions in which the welfare of others appears as an explicit argument. This paper suggests that the large experimental claims made for social utility functions are premature at best, and that social norms are better studied as equilibrium selection devices that evolved for use in games that are seldom studied in economics laboratories.

Appendices

Appendix 1: Coordination games

Figure 1 shows two coordination games. The Driving Game is a game of pure coordination, because the players have precisely the same aims. If they could discuss the game before playing it, they would therefore have no difficulty in resolving their coordination problem. But the game must be played without any pre-play negotiation. The Battle of the Sexes is a game of impure coordination. The players face a more difficult coordination problem in this game, because they have different preferences over its possible solutions.

Fig. 1

Two coordination games. A politically incorrect story accompanies the Battle of the Sexes. Adam and Eve are a newly married couple honeymooning in New York. At breakfast, they discuss whether to go to a boxing match or the ballet in the evening, but fail to make a decision. They later get separated in the crowds and now each has to decide independently where to go in the evening

The two players in the games of Fig. 1 are called Adam and Eve. Adam chooses a row and Eve independently chooses a column. Adam’s payoff is written in the south-west corner of each cell and Eve’s payoff in the north-east corner. The stars indicate best replies. For example, the fact that ballet is Eve’s best reply to Adam’s choice of ballet in the Battle of the Sexes is indicated by starring the payoff of 2 that she will receive if these strategies are played. Cells in which both payoffs are starred correspond to (Nash) equilibria in pure strategies.

At a Nash equilibrium, both players are simultaneously making best replies to the strategy chosen by their opponent. If an adaptive process that always moves in the direction of higher payoffs converges at all, it must converge on a Nash equilibrium, which is therefore a minimal requirement for evolutionary stability (whether biological or cultural). Aside from their two Nash equilibria in pure strategies, each game also has a Nash equilibrium in mixed (or randomized) strategies. For example, it is a Nash equilibrium in the Driving Game for each player to toss a fair coin to decide on which side of the road to drive. Both mixed equilibria are inefficient, in that each player prefers either of the two pure equilibria to the mixed alternative.

Coordination games usually have many equilibria. The games of Fig. 1 are exceptional in only having two equilibria (in pure strategies). If the people playing such a game are to succeed in coordinating their behavior, they need to aim for the same equilibrium. In the Driving Game, they must both drive on the left or both drive on the right. Social norms exist to solve such equilibrium selection problems. Different social norms sometimes evolve in different societies to solve the same problem. ¹⁰ For example, the Japanese drive on the left and the French on the right.

Appendix 2: The evolution of a social norm

An experiment I carried out with colleagues at the University of Michigan can be used to illustrate both the two levels of adaptation mentioned in Sect. 3 (Binmore 2007, chapter 2). Since the adaptation took place within a laboratory, it illustrates how subjects can change their behavior over time if offered the opportunity. As in experiments on the one-shot Prisoners’ Dilemma, such changes in behavior over time are standard. When one fails to see such adaptation in the laboratory, as in the Ultimatum Game, some explanation is required. However, it is the second level of adaptation that is perhaps more interesting. This is the level at which social norms emerge in a society as solutions of real-life coordination problems.

The coordination game used was the Nash Demand Game, which is a primitive bargaining game in which Adam and Eve can both gain if they can agree on how to cooperate (Nash 1950). Each player simultaneously makes a demand. An agreement is deemed to have been reached if the demands made are jointly feasible. The shaded region in Fig. 2 is the set of pairs of demands that count as feasible. If a pair of demands lies outside this region, each player gets nothing.

Fig. 2

Evolution of a social norm. The shaded region is the feasible set for an experimental version of the Nash Demand Game. Groups of subjects were conditioned to play one of four putative social norms, labeled E, K, N, and U. The subjects initially played ten trials against robots. Their screens showed small boxes indicating the demands a particular robot made when last occupying the role of player I or player II. The demands shown in the figure are each box’s initial position. During the ten trials, different groups saw the boxes gradually converge on different social norms. This device proved adequate to condition the subjects on a norm. When they knowingly began to play each other, they began by making demands close to the norm on which they had been conditioned. However, after thirty repetitions, all groups were playing one of the efficient Nash equilibria of the game, irrespective of their initial conditioning. In the figure, the efficient Nash equilibria lie on the frontier of the feasible set within the large box. At least one group got close to each of these efficient equilibria

The pure version of the game poses the equilibrium selection problem in an acute form, because every efficient outcome is a Nash equilibrium of the game. (There are also inefficient equilibria, including one in which both players get nothing because they both make demands that cannot be met however reasonable the other player’s demand.) As in the Battle of the Sexes, Adam and Eve are not indifferent between the efficient equilibria. If one of a pair of equilibria assigns Adam a higher payoff, then it assigns Eve a lower payoff.

In the experiment, we paid tribute to Nash by using a version of the game in which he chose to fuzz the boundary of the feasible set slightly. As a result, the set of efficient equilibria is reduced from being all points on the outer boundary of the feasible set to the subset of the boundary enclosed in a box in Fig. 2. ¹¹ It turned out that this modification to the design generated much more interesting results than would otherwise have been possible.

It is not normally possible to control the social conditioning with which subjects enter a laboratory, but the conditioning the subjects have received on what counts as fair in their society is clearly important when they are asked to play a game that everybody can see is a kind of bargaining game. However, we did our best to frame the experiment in a manner unlikely to trigger any particular fairness norm brought in from the outside. Instead, we sought to provide a substitute for outside social conditioning within the laboratory by asking the subjects to knowingly play ten “practice” games against against robots programmed to converge on one of four outcomes E, N, K, and U. The putative social norm E (for egalitarian) gives each player an equal payoff, and hence instantiates Rawls’ (1972) difference principle. The putative social norm U (for utilitarian) maximizes the sum of the two players’ payoffs. The two other putative social norms correspond to bargaining solutions from the game theory literature.

We found it surprisingly easy to condition different groups of subjects on any of these norms. After ten trials playing the conditioning robots, they then played thirty trials with randomly chosen partners from the same experimental group. At the beginning of the thirty trials, the subjects played as they had been conditioned. At the end of the thirty trials, each group ended up at one of the efficient Nash equilibria of the game. Neither the egalitarian outcome E nor the utilitarian outcome U were Nash equilibria in the experiment, and so our attempt to impose them as social norms on our subjects was a failure. ¹²

Different groups ended up at different Nash equilibria of the game. Indeed, the collection of experimental outcomes at the end of the game covered the whole set of Nash equilibria. If each group is regarded as a laboratory minisociety, then their learning to coordinate on a particular equilibrium can be regarded as exemplifying the process of cultural evolution by means of which social norms come to be accepted in real societies.

After the experiment, we asked subjects what they regarded as fair in the game they had just played. It turned out that the equilibrium on which a subject’s group had converged was a good predictor of what that subject afterwards said was fair in the game. That is to say, the subjects were willing to treat the social norm that they had observed evolving in the laboratory as a legitimate fairness norm. In a situation that does not match anything to which they were habituated, our subjects therefore showed little sign of having some universal fairness stereotype built into their utility functions.

Appendix 3: Ultimatum game

Not only does the Ultimatum Game have many Nash equilibria, but computer simulations show that models of adaptive learning can easily converge on one of the infinite number of Nash equilibria other than the equilibrium that behavioralists say is the unique neoclassical prediction (Binmore et al. 1995).

The same computer simulations show that one must expect any convergence that takes place to be very slow. (See also Roth and Erev 1995). Figure 3 shows one of the very large number of computer simulations reported by Binmore et al. (1995).

Fig. 3

Convergence in the Ultimatum Game. The figure shows one of many simulations of a perturbed version of the replicator dynamics from Binmore et al. (1995). Pairs of players are chosen at random from a population of proposers and a population of responders to play the Ultimatum Game. Proposers are characterized by how much of the available $40 they will offer if chosen to play. Responders are characterized by the least amount they will accept. The upper and lower diagrams respectively show the effect of evolution in the two populations. The vertical axis shows the proportion of various types in each population. The original distribution of types is concentrated about a notional social norm in which proposers offer $33, which responders accept, planning to refuse anything less. The system would not move from this norm if there were no noise, because (7, 33) is a Nash equilibrium outcome (like any other split of the $40), but the existence of noise destabilizes the system. Eventually, it converges on the outcome (30, 10) (which is not a subgame-perfect outcome), but the immediate point is that it takes some 6,000 iterations before much movement from the original norm is discernible

The original sum of money is $40 and the simulation begins with Alice offering Bob about $33, leaving $7 for herself. One has to imagine that the operant social norm in the society from which Alice and Bob are drawn selects this Nash equilibrium outcome from all those available when ultimatum situations arise in their repeated game of life. However, this split (like any other split) is also a Nash equilibrium outcome in the one-shot Ultimatum Game.

The figure shows our (perturbed replicator) dynamic ¹³ leading the system away from the vicinity of this (7, 33) equilibrium. The system eventually ends up at a (30, 10) equilibrium.

This final equilibrium does not yield the split (40, 0), which behavioralists insist is the unique neoclassical prediction. But this fact is not the point of drawing attention to the simulation. What is important here is that it takes some 6,000 periods before our simulated adaptive process moves the system any significant distance from the vicinity of the original (7, 33) equilibrium. This very large number of periods has to be compared with the 10 or so trials commonly considered ample for adaptive learning to take place in the laboratory.

More generally, if a society’s social norms lead inexperienced players to start playing close to a Nash equilibrium of a one-shot laboratory game, then, if there is any movement away from the original Nash equililibrium at all due to adaptive learning, we must expect such movement to be slow at the outset, although the Ultimatum Game is presumably exceptional in taking such an enormously long time for the adaptation process to converge