Yaozhen Huang Wei Xiong

Abstract.Reasoning about group rationality is a very important research topic in the areas of social choice,game theory,social norms,and among others.In this paper we develop strategic games with the concept of Rawls group rationality and argue the existence of solutions in such games.Further,we use the established strategic games to analyze some social dilemmas,which in turn helps us understand the social norm of cooperation.

1 Introduction

In traditional game theory,players are assumed to be rational individuals who always pursue the maximization of their own interests when they make decisions.[14])Based on this assumption,we can find Nash equilibria as solutions for a game.Yet,when analyzing such solutions of games,we might face a situation in which there is a conflict between individual and collective interest.That is,we will face a social dilemma in which the Nash equilibrium results in outcomes below the Pareto optimal.Particularly,the prisoner’s dilemma and the public goods game are well-known examples of social dilemmas.Indeed,in these games all individuals are all better off if all cooperate than if all defect,the action being rational for each player.

Currently,there has been an increasing research focus on social norms involving dilemmas in the literature.One way for exploring the problem of social dilemmas is to develop a norm for social cooperation under the concept of collective rationality.In [3,11,15],the researchers provide some valuable ideas concerning the question of how to identify a group utility and thus offer some approaches to studying social norms and team reasoning.

Rawls’ theory of justice ([12])is another one,being worthy of consideration here,to guide for studying social norms.This theory,an alternative to classical utilitarianism,is a monumental work in moral philosophy and philosophy of political science.One of main contributions of the theory is that it offers us an essential understanding of fairness.([6])In particular,Rawls proposes a maximin criterion,a wellknown decision rule,which embodies his basic idea about social fairness.

As such the purpose of the paper is to develop a strategic game based on Rawls’maximin criterion and use the model to analyze some social dilemmas.Our basic idea is that the preference of each player in a game can be represented by Rawls’maximin criterion.With which,we can define a game model and its solutions,social welfare equilibria ([11]).We can also show that Rawls’ maximin criterion for the fairness norm can induce a cooperative behavior in some canonical social dilemmas,in contrast to the result of reasoning with traditional concept of rationality.

The remainder of this paper is organized as follows.In Section 2,we recall Rawls’maximin criterion and define the concept of Rawls group rationality.Based on the concept,we establish strategic games with Rawls group rationality,and argue the existence of social welfare equilibria,solutions for such games.Section 3 uses our game model to analyze the prisoner’s dilemma and public goods games,showing that Rawls’maximin criterion can induce a cooperative behavior in these social dilemmas.Section 4 concludes the paper and suggests some possible future work.

2 Strategic Games with Rawls Group Rationality

In this section,we will establish strategic games based on the concept of Rawls group rationality.To this end,let us first recall some notions of Rawls’ theory of justice.

2.1 Rawls Group Rationality

In his seminal book([12]),Rawls argues a well-known opinion in which a fair society should be organized so as to admit economic inequalities to the extent that they are beneficial to the less advantaged agents.With this,he proposes a maximin criterion for the fairness norm.The main idea of this criterion is that we should make the least happy agent as happy as possible when considering a fairness norm.More specifically,the social preference should maximize the collective utility that is represented by the individual’s utility of the less advantaged agents.Formally,we have the following definition.

Definition 1(Rawls’maximin criterion)Given a set of possible outcomesW,and a groupNof decision makers.LetUi:W →R stand for the(expected)utility function of individualirepresenting her preference over the setW,i∈N.We say that a social preferenceof the group satisfies Rawls’maximin criterion if,for anyω,ω′∈W,

LetI(ω)=Then Formula(1)can also be represented as

In fact,I(ω)can be interpreted as the group(or collective,or social welfare)utility of the outcomeω.In the literature,I(ω)is often called the degree of ideality of the outcomeω(with Rawls’maximin criterion).([9])

To illustrate Rawls’ maximin criterion,let us consider the following example.Suppose that the set of outcomesW={ω1,ω2},the set of decision makersN={1,2,3},and the utilities of the decision makers are as follows:

Then,we haveI(ω1)=U1(ω1)=1,andI(ω2)=U1(ω2)=2.By Definition 1,the social preference of the group isω1≺ω2.

Definition 2(Rawls group rationality)A group is Rawls group rational if the social preference of the group satisfies Rawls’maximin criterion.

An individual of a group is not a self-interested agent when making decision with Rawls group rationality([8,5]).Instead,she will maximaize the degree of ideality if the group is Rawls group rational.

2.2 Game Model

This subsection provides strategic games based on the concept of Rawls group rationality.

Definition 3(Strategic games with Rawls group rationality)A strategic game with Rawls group rationality is a tupleG=〈N,(Ai)i∈N,(Ui)i∈N,I〉,where:

·N={1,...,n}is a finite set of players;

·for everyi∈N,Aiis playeri’s finite set of actions(pure strategies);

·for everyi∈N,Uiis the expected utility function of playerirepresenting her preference over(mixed)strategy profiles1As usual, a mixed strategy of a player, denoted as a Greek alphabet, is defined as a probability distribution over her set of actions.;

·Iis a function mapping every strategy profile to a real number measuring the degree of ideality of the strategy profile.

It can be seen that a strategic game with Rawls group rationality is established by adding the functionIthat represents the social preference of the group of the players in this game to a traditional strategic game.

Before presenting the solution concept for a strategic game with Rawls group rationality,let us recall some notions first and provide a formula for determining the degree of ideality to a mixed strategy profile.We writeσi(ai)to stand for the probability assigned by playeri’s mixed strategyσito her actionai.Given a strategic gameG=〈N,(Ai)i∈N,(Ui)i∈N,I〉,a mixed strategy profileσ=(σi,σ-i).Then,the expected utility of playerito a strategy profileσis a weighted average of her expected utilities to all actions when other players use the mixed strategy profileσ-i.Hence,we have

Definition 4(Rawls’maximin strategy)Given a strategic game with Rawls group rationalityG=〈N,(Ai)i∈N,(Ui)i∈N,I〉,a mixed strategy is the Rawls’ maximin strategy of playeriif it solves the problemwhere Σiis the strategic space of playeri.

There are two measurements in a strategic game with Rawls group rationality.One is the utility functions representing the preferences of players in the game,while another is the function of the degree of ideality(i.e.,the group utility function)representing the social preference of the group of players.Without the latter one,we shall obtain a traditional strategic game.As usual,we can obtain Nash equilibria as solutions for the game.If each player in a strategic game employs Rawls’maximin criterion as the decision rule in the game,then we shall obtain social welfare equilibria as solutions for such a game.

Definition 5(Social welfare equilibrium)Given a strategic game with Rawls group rationalitya strategy profileis a social welfare equilibrium of the game if for any strategyσiof playeri,

Given a strategic game with Rawls group rationality

Proposition 6Any strategic game with Rawls group rationality has a social welfare equilibrium.

ProofGiven a strategic game with Rawls group rationalityG,we can construct its traditional strategic formΓ.It follows from Definition 5 and the definition of Nash equilibrium in traditional game theory that a Nash equilibrium inΓis a social welfare equilibrium inG,and vise versa.Since any traditional strategic game in which each player has finite actions has a mixed strategy Nash equilibrium,any strategic game with Rawls group rationality also has a social welfare equilibrium. □

Proposition 7The strategy chosen by each player in a social welfare equilibrium of a strategic game with Rawls group rationality is her Rawls’maximin strategy,and vise versa.

ProofGiven a gameG=〈N,(Ai)i∈N,(Ui)i∈N,I〉,if the strategy profileσ*=is a social welfare equilibrium of this game,then for anyσi∈Σi,where Σiis playeri’s mixed strategy space,we haveHence,solves the problemwhich shows that the strategyis a Rawls’maximin strategy of playeri.

On the other direction,suppose thatis a Rawls’maximin strategy of playeri.Then for anysolves the problemwhereHence,we haveBy Definition 5,the strategy profileis a social welfare equilibrium of the game. □

To find social wefare equilibria in a game with Rawls group rationality,each player uses the functionIrather than the expected utilities to represent her preference over strategy profiles.With this function,we can define a best response of a player to other players’strategies as follow.

Definition 8(Best response function)Given a strategic game with Rawls group rationality

the playeri’s best response functionBiis identified as

Following Definition 5 and Definition 8,we can verify the following proposition.

Proposition 9Given a strategic game with Rawls group rationalityG,the mixed strategy profileis a social welfare equilibrium of gameGif and only if for each playeri,

3 Analysis of Social Norms

Social norms,the unplanned results of social interactions,are efficient means to achieve social welfare.([1,16])As a formal framework for modeling strategic interactions,game theory provides us a powerful tool to study social norms,which can help us understand some seemingly puzzling behaviors.

Studying cooperation in social dilemmas is one of the main research topics in the area of social norms.In this section,we will use the strategic games we established in the previous section to analyze some social dilemmas.

3.1 Prisoner’s Dilemma Game

The prisoner’s dilemma is a well-known example for studying social cooperation.The payoff matrix of the prisoner’s dilemma is shown in Figure 1,whereCdenotes the action“cooperation”,andDdenotes the action“defection”.The“dilemma”faced by two prisoners of the game is that,whatever the other does,each is better off defecting than cooperating;That is,for each player,the cooperation action is strictly dominated by the defection action.Actually,there is a unique Nash equilibrium(D,D)in the prisoner’s dilemma by traditional game theory.However,the outcome yielded when both defect is worse for each than that they would have obtained when both cooperate.

Let us now analyze this social dilemma with our model.In fact,we have the following proposition.

Proposition 10The strategy profiles(D,D)and(C,C)are the social welfare equilibria in the strategic game with Rawls group rationality for the prisoner’s dilemma.

ProofWe first construct a model with Rawls group rationality for the prisoner’s dilemma game

where

·the set of playersN={1,2};

·the players’set of actionsA1=A2={C,D};

·under the payoff matrix shown in Figure 1,the players’ expected payoffsU1andU2to the strategy profileσ=(σ1,σ2),whereσ1=(p,1-p)andσ2=(q,1-q),are as follows

·the functionIis given by Formula(3)as follows

We shall now find the social welfare equilibria in gameGPD.By Definition 8,we have

Hence,by Proposition 9 there are four possible candidates for the social welfare equilibria in the gameGPD,i.e.,((0,1)(0,1)),((0,1)(1,0)),((1,0)(1,0)),and((1,0)(0,1)),in the case ofp=q.Under the conditionp=q,we shall obtain two strategy profiles((0,1)(0,1))and((1,0)(1,0)),i.e.,(D,D)and(C,C),are the social welfare equilibria in this game. □

Both players’best response functions can be illustrated by Figure 2,where each intersection might correspond a social welfare equilibrium in the game.More specifically,the intersections of the first two figures in Figure 2 determine the social welfare equilibria(C,C)and(D,D),respectively,for the prisoner’s deilemma.By contrast,the intersections (D,C)and (C,D)in the latter two figures do not yield the social welfare equilibria because they are in contradiction with the conditionp=q.

This proposition therefore establishes that

(i)Rawls’ maximin criterion for the fairness norm can induce a cooperative behavior in the prisoner’s dilemma,in contrast to the result of reasoning with traditional concept of rationality;

(ii)A strictly dominated action in traditional strategic games can be used with positive probability in the social welfare equilibria for the games under the concept of Rawls group rationality.

3.2 Public Goods Games

Another canonical example for studying the norms of human cooperation is a public goods game,which can be regarded as a version ofN-person prisoner’s dilemma.([13])

A typical game situation of public goods games can be found in [2,4,7].We shall describe the situation briefly in the follows.A person provides ten dollars to each of ten players,and the players may put their money into a common pool.The person then triples the amount in this pool and divides it equally among these players regardless of the amount of the individual’s contribution.Hence,each player will receive 30 dollars for a maximal return if each player offers a contribution.But each player will be tempted as a free rider,who contributes nothing;that is,being a free rider is the dominating strategy for the players in this game.

More generally,assume that there arenplayers in a group,where players can either contribute some fixed amountcor nothing at all.Suppose again that the return of the public good(i.e.,the payoff to each player of the group)depends on the number of cooperators among the public goods game,denoted asnc.Specifically,the net payoffs for cooperatorsPcand defectorsPdare given by

where 0≤nc ≤n,andrdenotes the interest rate on the common pool.In particular,the parameterris usually assumed in the range 1＜r ＜nin the literature of public goods games.([4])Clearly,C(cooperation)is strictly dominated byD(defection),and the individuals of this group are facing a social dilemma.

Next we shall analyze the social dilemma in public goods games with our framework.The following proposition shows that players have an incentive to chooseCunder Rawls group rationality.

Proposition 11The strategy profile(C,···,C)is the social welfare equilibrium in the strategic game with Rawls group rationality for the public goods game.

ProofUsing our framework,we can construct a game model for modeling public goods gamesGPG=〈N,(Ai)i∈N,(Ui)i∈N,I〉,where

·the set of playersN={1,···,n};

·for each playeri∈N,the set of actionsAi={C,D};

·for each playeri∈N,the utility functionUiis given bywhereA-iis the action profile of players other than playeriandncis the number of cooperators,0≤nc ≤n;

·the degree of idealityI(σ)to a pure strategy profileσis determined by the utility functionsUi.

Note that given a gameGPGand any pure strategy profileσincludingnccooperation actions,its degree of ideality is equal to the payoff of a player choosing the actionC.That is,

We then can find the best response function for each player in the game.In fact,the best response action of any playerito the profile of the other players’actions is“cooperation”.To show this,let us assume that there arenccooperation actions in the profileA-i.If playeriselects actionCgiven the profileA-i,then we have

Again,if playerichooses actionDto response the profileA-i,then we obtain

Hence,we haveI(C,A-i)＞I(D,A-i),which establishesBi(A-i)={C}.As a result,there is a unique social welfare equilibrium(C,···,C)in the public goods gamesGPG. □

Note that the social welfare equilibrium (C,···,C)yields a maximal social welfarenrcand in turn offers a maximal payoff(r-1)cfor each player in the games.In view of this,there is an inspiration for each player to employ Rawls’ maximin criterion as a decision rule when making decisions.

In the public goods games,players have an incentive to deviate from cooperation,by which they consume the public goods without paying any cost.Using Rawls maximin criterion as the decision rule for everyone in a group when facing such social dilemmas provides a valuable approach to solving these problems.A social norm for cooperation,yielding a maximal social welfare by offering contribution,is,after all,a desired one.

4 Conclusion

Traditional game theory assumes that players are self-interested agents who identify their preferences over strategy profiles by maximizing expected utilities.We might face a social dilemma,a conflict between individual and collective interest,when analyzing some games under this assumption.Based on the Rawls’maximin criterion,we have defined the concept of Rawls group rationlity,where we can measure group payoffs (welfare)for strategy profiles that represent the players’ preferences.We have developed a game model under the concept of Rawls group rationality.We have also argued the existence of social welfare equilibria in such games,and established that a strictly dominated action in a traditional strategic game can be used with positive probability in the social welfare equilibria.We have further analyzed two social dilemmas by our model,showing that we have provided an approach to understanding a variety of seemingly puzzling behaviors.

Considering future work,one interesting project is to construct a logic language for capturing the concepts in the game theory established in this paper and exploring the cognitive conditions for social welfare equilibria.