Ken Binmore on Game Theory and the Social Contract with a Focus on the Condition of Women in Europe

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DIPARTIMENTO DI ECONOMIA E MANAGEMENT

Corso di Laurea Magistrale Strategia Management e Controllo

TESI DI LAUREA

Ken Binmore on Game Theory and the Social Contract

with a Focus on the Condition of Women in Europe

Candidato Relatore

Ina Koci Prof. Marco Enrico Luigi Guidi

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LIST OF FIGURES

Figure 1. 1: Payoff table ... 9

Figure 1. 2: The Game of Matching Pennies ... 11

Figure 1. 3: The Game of Matching Pennies with numerical payoffs ... 12

Figure 1. 4: Battle of the Sexes ... 13

Figure 1. 5: Prisoners’ Dilemma ... 16

Figure 1. 6: The Trust Minigame ... 19

Figure 1. 7: The equilibrium outcomes of the repeated Trust Minigame ... 19

Figure 2. 1: Hawks and Doves ... 30

Figure 2. 2: Evolutionary dynamics with one population in the ‘Hawks and Doves’ game ... 30

Figure 2. 3: The Chicken Game ... 31

Figure 2. 4: Evolutionary dynamics with one population in the Chicken Game ... 32

Figure 2. 5: Adam and Eve are brother and sister ... 34

Figure 2. 6: Adam and Eve are identical twins ... 34

Figure 2. 7: Extensive form of the Ultimatum Minigame ... 36

Figure 2. 8: Payoff table of the Ultimatum Minigame ... 36

Figure 2. 9: Evolutionary dynamics with several populations in the Ultimatum Minigames ... 37

Figure 3. 1: Kidnapping Game ... 46

Figure 4. 1: The set of feasible social contracts ... 55

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Figure 4. 3: The bargaining solutions of Rawls and Harsanyi when the status quo is

different from the zero point ... 59

Figure 4. 4: The bargaining set ... 66

Figure 4. 5: The Nash bargaining solution ... 71

Figure 4. 6: The Nash bargaining solution in the medium run ... 76

Figure 4. 7: A new bargaining problem ... 79

Figure 5. 1: The payoff table of the Hiring Game ... 102

Figure 5. 2: The extensive form of the Hiring Game ... 102

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INTRODUCTION

This paper aims to conduct an analysis of Ken Binmore’s thinking on human coordination behavior and the evolution of such coordination towards a stable, efficient and, fair social contract.

The social contract is the set of common understandings and conventions that allow the citizens of a society to coordinate their behavior. The validity and purpose of such agreements and conventions do not correspond to strict abstract moral laws. The only iron laws to which men are subject are those codified in their genes, while the conventions and understandings that make up the social contract are relative rules that change in time and space.

Some of these conventions are codified as laws, but the legal system is also changeable in time and space and above all it is relevant for a social contract only to the extent that the laws are actually respected in practice. A social contract is not held up by laws and the constitution, by state officials, by a sense of moral obligation or by God.

As David Hume1_{argued, a social contract is held together like a ‘dry-stone wall’. It does} not need glue or cement because each brick is supported by its neighbors. The rules of a stable social contract manage to coordinate the behavior of men on a balance in the game of life, called by game theorists an equilibrium, without the need for higher moral rules, divine rules or ‘categorical imperatives’2_.

1_{David Hume was a Scottish philosopher considered one of the most radical British Empiricists.} 2_{The categorical imperative is the central philosophical concept in Immanuel Kant’s thought.}

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An evolutionary approach to the theory of social contract identifies three levels of priority:

§ A social contract must be internally stable otherwise it will not be lasting. § It must be efficient in order to compete with alternative social contracts. § It must be fair.

The quest for the first priority leads the analysis in the direction of the search for equilibria in the game of life, through the stylization of the different aspects of reality in simplified models and using game theory as a tool of analysis.

The quest for the second priority is carried out through the identification of Pareto-efficient equilibrium outcomes.

The quest for the third priority involves the coordination of a society on one of the efficient equilibria. The criterion to be used to solve the problem of selection among the efficient equilibrium outcomes in the game of life is equity. The concept of equity is dependent both on cultural and biological evolution. Justice can be conceived in different ways in societies where different conventions have developed; however, a universal principle of justice can be linked to a deep structure that is written in the human genes. Namely this principle is based on the criterion that men use in solving the problem of the selection of a better equilibrium, from the many available in the myriad of coordination games of daily life: the device of the original position.

People who belong to the same culture unconsciously use this device to solve the countless conflicts that arise in their daily lives.

Reading Binmore’s books, I was struck by the fact that almost all the mathematicians, philosophers, politicians to whom the author refers are men and I felt the need to investigate the reasons behind the social disparities between men and women,

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assuming that there are no physiological elements that make men more capable and enterprising than women. As Binmore argues, the status quo depends on the evolution of social conventions and not on the application of absolute laws. In the last chapter of this paper, I tried to investigate this current phenomenon using, precisely, the Binmore paradigm.

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CHAPTER 1 GAME THEORY

Binmore aims to analyze, understand and describe the various aspects of human life and then prescribe those behaviors that would be best for people. Game theory is a fundamental tool of the descriptive part of Binmore’s analysis. In particular it is used as a tool, neutral with respect to moral issues, to predict what will be the behavior of people vis-à-vis different situations that attempts to represent through simplified models. Interactions between human beings can be represented by games. As an example: an elementary school teacher writing up a pupil who behaves badly and giving a grade of merit to those who behave well, plays an educational game with his pupils. Likewise, wife and husband negotiating the division of domestic work and child care play a family coordination game. In a similar fashion, a company and a supplier who negotiate the price of goods are playing a business game, while Salvini and Di Maio who negotiate on coalition policies are playing a political game. Although game theory has a very broad scope of application it fails to provide a solution to all the problems that arise in real life since it only works when people behave rationally. But humans do not always behave in a rational way: as an example, when in love, in panic or find themselves in many other situations in which the lack of rationality hinders this tool.

Game theory can be employed in a vast array of disciplines: evolutionary biology, economics, political science, and even ethics.

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1.1 The models

In order to be able to draw conclusions about real-life phenomena through the tool of game theory, it is important to build a functional game model, i.e. a toy game able to imitate the essential characteristics of the problem it wants to represent and investigate.

In the toy games that will be presented below, only two players are contemplated: Adam and Eve. The payoff table, as shown in figure 1.1, represents the gains for players in relation to the strategies implemented. Eve’s strategies are represented by the rows, while Adam’s strategies are represented by the columns. The choice of a row and a column determines what the players’ payoffs will be. In each cell, bottom left numbers represent Eve’s payoff while those located top right represent Adam’s payoff.

Figure 1. 1: Payoff table

The concept of utility is useful for describing players’ motivations. Indeed, this concept allows to assign each player a numerical value as a possible result of a particular game.

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Specifically, utility is the value of anything the player tries to maximize. Since reason is a tool that people use to avoid inconsistent behavior, and since consistent behavior is considered rational, then a rational individual will always try to maximize his or her own utility. The utility of each player can be fictionally measured in utils. The util is the standard unit of measurement on a player’s scale of utility. Binmore follows the theory of John von Neumann and Oskar Morgenstern3_{which states that a player’s desire for a} specific thing is measured by the size of the risk he or she is willing to take to obtain it (see BINMORE 2007, pp. 8-12). Payoffs therefore represent players’ preferences in a

specified game. These preferences are analyzed through the theory of revealed preference, according to which a consistent player’s preferences are deducted from the observation of the decisions he makes to solve a decision-making problem in which only one person is involved. The comparison between the payoffs of a player links the higher payoff with the strategy that the player would choose to implement in the problem to a single person if he or she knew in advance the choice of his or her opponent.

1.2 Some aspects of human interactions stylized in games

Human interactions can be either conflicting or cooperative.

A toy game that can represent a competitive interaction is ‘The Game of Matching Pennies’.

3 _{John von Neumann (Budapest, 28 December 1903 - Washington, 8 February 1957) and Oskar}

Morgenstern (Görlitz, 24 January 1902 - Princeton, 26 July 1977) are considered to be the co-founders of modern game theory whose birth can be made to coincide with the publication of their book Theory of

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Adam and Eve simultaneously display a coin. Eve wins if both coins show the same face while Adam wins when the coins show different faces. In this case, the revenues of the two players are diametrically opposite.

The strategies available to the players are ‘heads’ and ‘tails’. The following figure shows their payoffs.

Figure 1. 2: The Game of Matching Pennies

In this case, the table shows who wins (shown with the thumb facing up) and who loses (shown with the thumb facing down).

Figure 1.3 shows Adam and Eve’s payoffs as numerical revenue obtained by choosing to assign to the result of defeat the value -1 and to the result of victory the value +1 in the scales of utility of the players.

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Figure 1. 3: The Game of Matching Pennies with numerical payoffs

This is a zero-sum game4_{. Players have antipodal preferences that allow no scope at all} for cooperation. Pure conflict games rarely represent the games that people play in real life.

A toy game of pure coordination that can represent a cooperative interaction is: ‘The Driving Game’.

Adam and Eve have two pure strategies: ‘drive on the left’ or ‘drive on the right’. In this case the players’ payoffs are fully aligned. If the players both drive on the same side, they both win while if they drive on opposite side the outcome will be a car accident. A toy game of impure coordination that can represent an interaction with both cooperative and conflicting characteristics is:

‘Battle of the Sexes’

Adam and Eve are a newlywed couple on their honeymoon in New York. During breakfast they discuss the program of their evening: Adam prefers to go to a boxing

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match while Eve prefers to go to a ballet. Both, however, prefer to be together, and this preference is greater than that of watching their favorite show to the point that the benefits of the satisfaction obtained by attending favorite show are nullified if the players can’t be together. They can’t make a deal because they are separated during the day and don’t have the chance to communicate. Both have two possible strategies: ‘go to the boxing match’ and ‘go to the ballet’.

Players don’t know their opponents’ choices.

§ If Eve knew that Adam was going to the ballet, she would maximize her payoff by choosing to go to the ballet too.

§ If Eve knew that Adam was going to the boxing match, she would maximize her payoff by choosing to go to the boxing match too.

§ If Adam knew that Eve was going to the ballet, he would maximize his payoff by choosing to go to the ballet too.

§ If Adam knew that Eve would go to the boxing match, he would maximize his payoff by choosing to go to the boxing match too.

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A Nash equilibrium5_{is represented by the set of strategies in which each player’s} strategy is the best reply to the strategy choices of the other players’ strategies. The Nash equilibrium is important in the search for analysis of human interactions, through game theory, because:

§ Ideal rational players reason about a game’s solution.

§ People find the solution to the game through an evolutionary process of trial and error.

In the ‘Battle of the Sexes’ there are three Nash equilibria: one in which both players choose to go to the ballet, one in which both players choose to go to the boxing match and a Nash equilibrium in mixed strategies, in which each player chooses with more probability the event each of them prefers.

Rational players will use reason to try to predict their opponents moves by wondering what a rational person would do in a given situation.

In order for game theory to be able to predict people behavior, an evolutionary interpretation must be given to the analysis: mechanisms that favor the more fit strategies at the expense of the less fit ones only stop working when a Nash equilibrium is reached. In equilibrium all the strategies that have survived will be adapted to the given circumstances. Nash equilibrium has a predictive role whenever an adaptation mechanism tends to eliminate strategies that have lower payoffs.

5_{John Forbes Nash (Bluefield, Virginia, 13 June 1928 - Monroe, New Jersey, 23 May 2015) was an}

American mathematician. Studies on game theory led him to win the Nobel Prize for Economics, with J. Harsanyi and R. Selten, in 1994. He developed the principle of Nash equilibrium, which, as part of game theory, is widely used in economic analysis.

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1.3 The problem of human cooperation

This problem is represented through the toy game of the ‘Prisoners’ Dilemma’ (see BINMORE 2007, pp. 5-9).

Adam and Eve are two criminals who have committed a very serious crime. Police is aware of their guilt, but they cannot incriminate them unless one of them confesses. The two criminals are separated from the police and both are faced with the following alternatives:

§ If only one of them confesses, then he or she will go free while the accomplice will be sentenced to the maximum penalty.

§ If they both confess, then they will both be imprisoned but the penalty will not be maximum.

§ If neither confesses, then they will both be charged with tax evasion for which a conviction is certain, but the penalty will be minimal.

The strategies of the two players are: ‘cooperation’ which consists of holding on and not confessing and ‘defection’ which consists in confessing and accusing the opponent. The payoffs shown in figure 1.5 correspond to possible years in jail, so one util represents an extra year of freedom.

Nash equilibrium corresponds to the pair of strategies in which they both choose to defect even if they would have gotten less years in jail if they had cooperated. The defection strategy dominates the cooperation strategy because it is the best reply of each player to any strategy implemented by the other.

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Figure 1. 5: Prisoners’ Dilemma

In such a model, cooperation is not a Nash equilibrium. Rational players do not cooperate in the ‘Prisoners’ Dilemma’ since the conditions for rational cooperation are lacking. This toy game was built in order to represent those situations where cooperation does not arise, but it represents situations that rarely occur in real life. If it were a suitable model to represent the great game of life played by the human species, then human beings would not have evolved as social animals.

1.4 Reciprocity

In order to understand how human society works, it is important to analyze the role of reciprocity (see BINMORE 1994, pp. 114-117). In a single game that takes place only once,

reciprocity will not take place. On the other hand, it arises and develops in situations where participants play the same game several times. However, if the repetition of the game is not infinite, the cooperation will not be rational anyway. If the ‘Prisoners’ Dilemma’ game, as an example, were played by Adam and Eve for n days, then the Nash

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equilibrium would be the same as that of the one-shot version. This can be demonstrated through backward induction: the n-th day the game that will be played by the players will be exactly the one-shot one in which both players will play the defection strategy, thanks to the fact that the game will not be played in the following days. By reasoning in such a way, for each day going back in time, players are aware that no action they decide today will be able to influence what will happen the next day. As a result, rational players will always play the defection strategy.

Since in real life individuals have no definite certainty about the development of their relationships, a model that better represents such real life interactions is a model that accounts for the probability that players will play the same game at least once later, regardless of how many times they have played it in the past: the indefinitely repeated game (see BINMORE 1998, p. 110). The indefinitely repeated version of the ‘Prisoners’

Dilemma’ has many Nash equilibria among which:

The ‘Relentless Strategy’: it requires that Eve always play ‘cooperation’ unless Adam plays ‘defection’. In that case Eve will permanently change her strategy to ‘defection’. If both players play the ‘relentless strategy’, both will be playing ‘cooperation’ the whole time. Such a strategy is a Nash equilibrium because it represents the best response to itself.

Such a game allows to grasp an important aspect of the game of life: mutual cooperation depends on the threat of punishment. People don’t usually offer a service if they don’t get something in return. The punishments that people encounter in real life are identified by different ways of responding to the offenses they have suffered. They may consist of a refusal to deal with the offender or of having to endure the disapproval of those whose respect is necessary or also some more active form of punishment. In this

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last case the punishment might be administrated by the judiciary if the offender has not complied with a legal contract.

In the game of life there are also cases, albeit marginal, in which cooperation is not bound to reciprocity. Such cases occur when people have altruistic personal preferences, where the welfare of other people influences their own utility function, such as among members of a family or among members of a surrogate family where altruistic actions can be explained in terms of collective service; or when they have utilitarian preferences where they will try to maximize the sum of their payoffs and that of other players, assuming that individual welfare has the same weight as the welfare of others. In this case the Nash equilibrium is represented by the strategy (‘cooperation’, ‘cooperation’).

The ‘Trust Minigame’ is a toy game that highlights issues of reputation and trust (see BINMORE 2007, pp. 341-342). In such a model Eve is a shopkeeper who offers her services

in exchange for the corresponding price of the service while Adam is a buyer. Eve’s strategies are ‘offering the service’ and ‘not offering the service’, whereas Adam’s strategies are ‘paying for the service’ and ‘not paying for the service’. Figure 1.6 shows Adam’s and Eve’s payoffs and figure 1.7 shows the sets of payoffs pairs of hypothetical agreements between them under the assumption that previous agreements with legal value can be established: the cooperative payoff region.6

6_{If Adam and Eve use random devices to generate other compromises between the pure outcomes, each}

such randomization generates a convex combination of the payoff pairs in the game’s payoff table. The set of all such convex combination is the cooperative payoff region.

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Figure 1. 6: The Trust Minigame

Figure 1. 7: The equilibrium outcomes of the repeated Trust Minigame

In the one-shot game the subgame-perfect equilibrium7_{is that in which Eve does not} provide the service because she predicts that Adam will not pay. If the game is

7_{A subgame-perfect equilibrium is a strategy profile that isn’t only a Nash equilibrium in the whole game}

but also represents a Nash equilibrium of every subgame of the original game. Not all Nash equilibria are subgame perfect.

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indefinitely repeated, each player can threaten the other that any infringement of the collaboration, which in such case is associated with strategies (‘offer the service’, ‘pay for the service’), will correspond to punitive and uncooperative behavior. According to ‘folk theorem’, if the future matters to the players and the discount factor is close to 1, then any result that offers players a payoff higher than their maximin8_{can represent an} equilibrium outcome in indefinitely repeated games. Folk theorem shows how cooperation can be sustained as an equilibrium in repeated games. Thus, all pairs of this triangle, with the exception of the points located at the top, are mixed equilibria i.e. compromises obtained by throwing coins or taking turns. This area represents the cooperative payoff region of the one-shot of the ‘Trust Minigame’ which is the set of all possible compromises on which Adam and Eve could agree. For the ‘folk theorem’ all pairs of payoffs in the shaded area are Nash equilibria in the infinitely repeated game. In order for the agreements to be maintained and therefore represent the Nash equilibrium, an external agency must exist, enabled to implement any agreement established by the players. If this is not the case, the agreement will be self-regulated and therefore a player will only adhere to the agreement when optimal for him or her if the other player also does so. If the discount factor is close to 1 and the probability of the game being played at least once again is high enough, then all pairs of payoffs in the shaded area are Nash equilibria as long as both players are paid their maximin values, if not more.

8_{The maximin return is the best return that the player can get if the opponent inflicts the worst possible}

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Given any agreement within the cooperative payoff area, for it to be a Nash equilibrium it is necessary for each player to be disincentivized by a hypothetical punishment if they decide to deviate from the established agreement. If the ‘implacable strategy’ is applied as a punishment, in which players punish in the strictest way any deviation, then the worst thing that Eve can do to Adam is to keep him in his payoff maximin. Eve will adopt the strategy that minimizes Adam’s payoff. Adam will respond with the strategy that maximizes his payoff in the worst situation that Eve imposes on him.

The ‘folk theorem’ shows how, in order to successfully achieve cooperation, it is not necessary to have an external agency to implement the contract. If the situation under analysis is repeated over time, then cooperation can be achieved through self-regulatory mechanisms in which individuals behave as policemen. Folk theorem is easily applicable to those realities in which individuals manage to know the deviations of others. However, in large communities, where people do not live close to each other and do not know all the choices of other players, it is more difficult to detect and punish those who deviate.

‘Tit-for-Tat’

In the repeated version of the ‘Prisoners’ Dilemma’ there is another Nash equilibrium if both players use the ‘Tit-for-Tat’ strategy. This strategy advises the player to use ‘cooperation’ the first time and then copy the other player’s in the next time.

Robert Axelrod,9_{following empirical experiments on the indefinitely repeated} ‘Prisoners’ Dilemma’, proposed this strategy as an appropriate paradigm for human cooperation at all levels, describing it as follows:

9_{Robert Axelrod (Chicago, Illinois, 27 May 1943) uses computer simulations to study the mechanisms by}

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What accounts for TIT-FOR-TAT’S robust success is its combination of being nice, retaliatory, forgiving and clear. Its niceness prevents it from getting into unnecessary trouble. Its retaliation discourages the other side from persisting whenever defection is tried. Its forgiveness helps restore mutual cooperation. And its clarity makes it intelligible to the other player, thereby eliciting long-term cooperation. (BINMORE 1998, p.314).

Binmore disagrees with Axerold’s conclusions. He does not share the attribution of the success of such a strategy to its niceness, that is, to the fact that the player never plays ‘defection’ for the first time. According to Binmore, evolution will not necessarily generate gentle behavior. To prove this, he considers the 'Tat-for-Tit' strategy where the player is advised to play ‘defection’ for the first time and then change, if and only if, in the previous time, the other player played 'defection'.

When both players use ‘Tat-for-Tit’, there is a Nash equilibrium and cooperation is achieved (after the first time). For cooperation to evolve it is sufficient for a mutant to be able to recognize a copy of itself.

Another element that Binmore does not share is the assertion that the best strategy is the retaliatory one. Since the retaliatory strategy can only be applied in interactions between couples, it would not explain those cases where cooperation arises in interactions between many people in which it is not necessarily the injured party who punishes the one who deviates. It is not retaliation that is decisive for cooperation but punishment. The latter may also be carried out by people other than the offended person. As an example, a deviation may be punished by persons who are not directly offended but who, as a result, refuse to enter into future alliances with the one that deviated, because of the reputation resulting from the deviation. In order to explain that

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the punishment does not necessarily have to be carried out by the person who has suffered the negative consequences of the deviation, consider a model that represents a fictitious world in which, at all times, only a mother and a daughter are alive. Each player lives for two periods: the period of youth and the period of old age. During the youth period, the daughter produces two forms of bread. Then she gives birth to a daughter and becomes old. Only two players are present at any one time, namely a mother and daughter. It is assumed, therefore, that the mother dies as soon as her daughter gives birth. In old age, the player is unable to produce. Players prefer to eat one form of bread in their youth and one in their old age, but the bread cannot be stored because it dries out after it has been produced. Therefore, this ‘fair’ result can only be achieved if the daughter shares her production with her mother. It is also assumed that the well-being of the mother has no influence on the preferences of the daughter. The strategies of the daughter are: ‘to be conformist’, which means to share the bread with the mother, if and only if, the latter was a conformist during her youth and to punish the mother by not sharing the bread if she was not; ‘to be non-conformist’, which means not to share the bread with the elderly mother even if she was conformist. In this model, the elderly mother is not able to punish her daughter who decides not to share the bread.

From the model it can be deduced that, when the first player to be born is conformist, then it is a subgame-perfect equilibrium when everyone is conformist.

This equilibrium is supported by the punishment that the non-conformist daughter will suffer, not from her mother, but from her daughter who will become a conformist. In real life, there is a convention that the daughter should take care of the elderly mother.

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This duty is an emerging phenomenon that has arisen because people try to give meaning to the equilibria that they find themselves playing in the game of life.

1.5 Bargaining and coalition

Consider now the hypothesis that players can agree before implementing their strategies and that this agreement is binding on both of them (see BINMORE 1998, p. 42).

The basis for such constraints may be in legal systems where players honor their commitments for fear of the consequences of hypothetical complaints, fear of the reputational consequences that certain non-compliant behaviors may have, in conventions arising from education or in a personal innate aversion to immoral behavior. Nash observed that negotiation, which takes place before playing, can be considered, in itself, a kind of game in which the strategies are represented by everything that the players can say or do during the negotiation. The extended game is the game that includes the negotiation phase that precedes it. A strategy of such an extended game shows, first, how to conduct the preliminary negotiations, and then, how to play the starting game depending on what resulted from the negotiations. Such an enlarged game can be assimilated to a non-cooperative game with rules that already include considerations on preliminary negotiations. The analysis must be conducted through search and selection of the Nash equilibrium. If a cooperative solution provides the result of a rational agreement on how to play a given game, then a non-cooperative analysis of the extended game should lead to the same response as cooperative games that offer predictions of easily applicable rational agreements.

In conclusion, this chapter has explained how the game theory can be a useful tool in the search for the first priority of a social contract, namely stability. The latter is

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considered a priority for the identification of social contracts because it is a necessary condition for the feasibility of social contracts themselves. What is not stable cannot be considered a feasible alternative because it lacks the necessary conditions for its implementation. Social contracts are, therefore, stable when they are equilibria in the game of life because:

§ If a game has a rational solution that is common knowledge, then such a solution will certainly be an equilibrium, otherwise, it would mean that players will not believe that it is rational not to implement their best response to the strategies implemented by others. But it can’t be rational to play in a non-optimal way. § If players’ payoffs represent their fitness, then the biological and cultural

evolutionary processes that favor the most fit at the expense of the less fit, will cease to function when an equilibrium is reached in which the survivors are as much as possible fit to the circumstances.

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CHAPTER 2 EVOLUTIONARY BIOLOGY

2.1 Fitness

To try to understand why humans, but also other animal species, behave in a certain way, Binmore relies on Darwinian theories10_{based on the principle of survival of the} fittest. The fitness is decisive in the survival of some behaviors at the expense of others. In order to represent the animals’ behavior through a model, the fitness of a behavioral trait is considered as the average number of children carrying that trait to the next generation, as a consequence of the fact that it has been used in the current generation. The alternative behaviors of the model are referred to one species of animals, consequently the problem focuses on the symmetrical Nash equilibria of symmetrical games. In a symmetrical equilibrium the players all use the same strategy. All finished symmetrical games have at least one symmetrical Nash equilibrium.

Genes are considered to be the hardware of a natural computer. The hardware contains programs that control animals’ behavior. Thus, just as programs can copy themselves

10_{Charles Robert Darwin (Shrewsbury, Shropshire, 12 February 1809 - Downe, London, 19 April 1882) was}

an English naturalist. He claimed that favorable hereditary variations in a population tend to become more and more frequent from one generation to the next, according to a process he called natural selection and that all living beings, including man, are subject, in the succession of generations, to slow but continuous changes, called evolution.

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from one computer to another, so programs imprinted in genes can copy themselves from one guest to another. For this behavior they are called self-replicating.

A replicator is defined as something that replicates itself and determines, in a game, strategic behavior. Genes can be considered replicators that need guests to replicate to survive. Replicators that give their guests ‘high’ fitness will control more guests than those that give low fitness, given that a guest’s fitness is a measure of how often the guest reproduces its genes. Replicators that give the guest a low fitness will die out in favor of those that give a high fitness and, therefore, will determine the strategic behavior of the game. In other words, only those replicators who are able to maximize the fitness of their guests survive. In the model, the players will be the factors that determine which genes survive at the expense of others. As in a model that represents economic aspects of the game of life, players are represented by homo oeconomicus, which will make its strategic decisions aimed at maximizing its enlightened self-interest, likewise in the model that tries to represent evolutionary aspects, players are considered those factors that choose replicators (strategies) that maximize fitness. Evolution occurs when, in a population, variations happen. The sources of such variations can be multiple, among them shedding of genes in the sexual reproduction, the geographical migration, and mutation. The population stabilizes in front of a determined random variation when a population of replicators is stable with respect to each small perturbation. The dynamics of replicators is the simplest model of biological evolution.

A hypothesis of the replicator dynamics states that the proportion of a population hosting a given replicator increases at a rate proportional to two key factors:

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§ The fraction of the population that currently hosts that replicator. It can be seen that the growth rate of a replicator is related to the number of parents who can pass it on to their children.

§ The difference between the current fitness of the guests of the replicator under consideration and the average fitness of all the guests of the population. An evolutionarily stable strategy11_{is a pattern of behaviors adopted by the members of} a population that cannot be improved by the adoption of alternative strategies. They only apply when the players belong to the same population to play a symmetrical game. The properties of the stable strategies are:

§ It must be the best answer to itself.

§ If it is not the only better answer to itself, then it must be a more valid answer to the other alternatives than the alternative answer to itself is.

The latter property requires that an evolutionary stable strategy be better suited to alternative strategies whereby, following an invasion of these (by virtue of the fact that they are also the best response to the evolutionarily stable strategy), the evolutionarily stable strategy will stop the invasion because it represents a better reply to the invader than the invader to himself (see BINMORE 2007, pp. 141-144).

The necessary and sufficient condition for evolutionary stability requires that any evolutionarily stable strategy, in a symmetrical game, is a symmetrical Nash equilibrium. Replicators can be found not only in the biological context, but also in the social and cultural context. Fads, codes of conduct, empirical rules, beliefs, lifestyles and so on can be considered as replicators. They are replicators that spread from one mind to another

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through the mechanisms of education and imitation, namely memes (see BINMORE 1998,

pp. 267-269, 294). One can speak of cultural evolution whenever adaptive dynamics lead to a Nash equilibrium of a game.

The ‘Hawks and Doves’ version of the ‘Prisoners’ Dilemma’ was used by biologist Maynard Smith12_{to show the importance of game theory with reference to some} aspects of evolutionary biology (see BINMORE 1994, pp. 97-101).

In this toy game two birds of the same imaginary species are considered, competing for the opportunity to reproduce. Birds can assume two modes of behavior, that is, two strategies:

§ Behave as a ‘Hawk’, which implies assuming an aggressive and combative behavior to obtain the available resources.

§ Behave like ‘Dove’, which implies assuming a passive behavior and fair sharing of the available resources.

In the toy game the two birds are the players: Adam and Eve.

It is hypothesized that being in possession of a resource increases the fitness of a bird of four utils while fighting of an util only. It is assumed, in fact, that the birds, fighting will hurt each other, so an extra resource obtained by fighting is worth less than an extra resource obtained without fighting.

Figure 2.1 shows the payoff table of the Prisoners’ Dilemma version referring to that model. The payoffs could represent the amount of food or the years of life. As mentioned above, in a biological context, the payoff can be understood as the ‘average incremental fitness’ and that is the amount of extra children which the player can expect

12_{John Maynard Smith (London, 6 January 1920 - Lewes, 19 April 2004) with his book Evolution and the}

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to get on average. It is assumed that players will pass on to their children the instinct that determines whether they will behave like a hawk or a dove. The strategy implemented that will pursue a higher payoff in the game, will expand at the expense of the other.

Figure 2. 1: Hawks and Doves

The only Nash equilibrium in the one-shot game continues to require that both players use ‘hawk’. Since this strategy is strictly dominant, it is also evolutionarily stable. The two birds belong to the same population, so the dynamics of the replicators is one dimensional and is represented in the following figure.

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The only ‘basic of attraction’ is the one in which the population is composed entirely of hawks. If a certain number of mutants were included in the population, they would be eliminated. If the population were all made up of doves, it would be sufficient even the appearance of a very small fraction of hawk-type mutants to lead the doves to extinction.

In the ‘Chicken Game’, Adam and Eve drive on the same street in opposite directions. The street is so narrow that they would not be able to pass both if one of them did not slow down. Players’ strategies are ‘fast’ and ‘slow’. The payoffs are represented in the following figure.

Figure 2. 3: The Chicken Game

This game of conflict is not zero sum because both players have an interest in avoiding a traffic accident that would occur if both drove fast. There are three Nash equilibria: two pure and one mixed. In detail, if we consider that those who drive fast are ‘hawks’

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and those who drive slowly are ‘doves’ the mixed equilibrium requires players to play ‘dove’ for 1/3 of the time and ‘hawk’ for 2/3. Assuming that 1/3 of the population is made up of ‘doves’ and 2/3 of ‘hawks’ and that nature rolls dice to choose two birds from this polymorphic population, it will seem to both players that their opponent is playing the mixed equilibrium strategy. Since the pure strategies they use in a mixed equilibrium are indifferent to the players, both hawks and doves are equally fit. No single bird, therefore, needs to make random moves to support equilibrium.

Maynard Smith observed that the state of mixed population corresponds to a Nash equilibrium and also to an evolutionarily stable strategy. As shown in Figure 2.4, a population composed of twice as many hawks as doves is an asymptotic attractor.

Figure 2. 4: Evolutionary dynamics with one population in the Chicken Game

2.2 Kin selection

It is easier to explain the evolution of cooperation within the family because relatives share certain genes. The degree of kinship corresponds to the probability that a gene, recently changed in the body of an individual, is also present in the body of a relative (see BINMORE 2005, pp. 102-110, 2007, pp. 150-153). The probability that a mutated gene

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comes from the mother instead of the father is 1/2. The probability that it is also present in the body of the aunt is equal to 1/2.

If this mutated gene is present in the aunt’s body, then the probability that the aunt has transmitted it to her child is 1/2.

It follows that the degree of kinship among cousins is equal to: 1/2 ´ 1/2 ´ 1/2 = 1/8.

Inclusive fitness takes account, not only of the effect of individual’s behavior on his own reproductive success, but also of its effect on the reproductive success of his relatives. It depends on one hand on the probability of transmitting their mutated genes to their own children but on the other hand on the probability that these mutated genes, present in other relatives, are transmitted to their respective children.

It follows that, to calculate the fitness of a mutated gene in an individual, it is necessary to add to the direct fitness of that individual (number of children weighted by the degree of relationship between parent and child) the indirect fitness of the relatives (number of children of each relative, weighted by the degree of relationship between parent and child, and subsequently weighed by the degree of relationship between the individual under examination and each relative).

Figure 2.5 considers the ‘Hawk-Dove Game’ played between relatives. The payoffs correspond to the inclusive fitness of each individual. In the case Adam and Eve are brother and sister, whose degree of kinship is 1/2, the payoffs are calculated as follows: 3 = 2 + 2/2

4 = 4 + 0/2 2 = 0 + 4/2 1,5 = 1 + 1/2

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Figure 2. 5: Adam and Eve are brother and sister

In the same way, payoffs are calculated in the case of two identical twins with the degree of kinship equal to 1:

4 = 2 + 2/1 4 = 4 + 0/1 4 = 0 + 4/1 2 = 1 + 1/1

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The Nash equilibrium no longer corresponds to the choice of the strategy ‘hawk’ by both players. Toy games that represent interactions between relatives are very different from those that represent relationships between non-relatives. In particular the individuals’ personal preferences to certain situations are different when relatives interact in such situations. They are influenced by the relatives’ welfare. It can be said that these preferences are influenced by altruism. In this sense, altruism comes to light as an interdependence between the evolutionary success of an individual and that of his relatives. It is therefore much easier for cooperation to arise in relations between relatives.

2.3 Emotions

Emotions play a very useful role in social relationships. Reactions such as anger, envy, pride and others are people’s usual response to certain situations they face in life. These behaviors have survived because they serve to force equilibria in repeated situations. In the ‘Ultimatum Minigame’, a hypothetical philanthropist offers a sum of money that Adam and Eve can get only if they can agree on the mode of division. It is assumed that Eve makes the proposal on the division mode and Adam decides whether to accept or not. If Adam accepts, the sum will be divided as Eve has proposed, while if Adam does not accept, the two players will not receive anything. In the one-shot game, if it is assumed that individuals just want to get as much money as possible, using backward induction, it emerges that, to any positive sum offered by Eve, Adam will accept since little is a better result than nothing. It follows that in the subgame-perfect equilibrium, Eve wins everything.

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In a simplified version of the game, the philanthropist makes a donation of four dollars. Eve's strategies are: make a ‘fair’ offer that consists of dividing the donation into equal parts and make an ‘unfair’ offer that consists of offering Adam only one dollar. Adam’s strategies are: ‘accept’ or ‘not accept’. It is also assumed that the fair strategy will always be accepted. Figure 2.7 shows the extensive form of the game and figure 2.8 the payoff table.

Figure 2. 7: Extensive form of the Ultimatum Minigame

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The subgame-perfect equilibrium is the pair of strategies (unfair, yes). The game has other Nash equilibria as shown in Figure 2.8 by the circled payoffs that represent the best reply of each player to the choices of the other: (fair, no) and others in which Eve chooses ‘fair’ because Adam has in mind to use a mixed strategy in which he chooses to reject the offer with a fairly high probability.

In the indefinitely repeated game the evolutionary process, which shows how players modify over time their strategies in repeated games in order to improve their payoffs, will not necessarily go in the direction of the subgame-perfect equilibrium.

Figure 2.9 shows two different possible evolutionary dynamics of the ‘Ultimatum Minigame’ (see BINMORE 2005, p. 70). It shows the working principles of a toy game in

which players have been extracted from two different populations that evolve separately.

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One represents the best reply dynamic that converges to the subgame-perfect equilibrium and the other is the replicator dynamics where the Nash equilibria in which Eve plays ‘fair offer’ has a wide basin of attraction.

Although ‘accepting the offer’ is always better than ‘rejecting the offer’, the evolutionary pressure against an unfair offer may be so strong that it disappears and the ‘rejecting the offer’ strategy may survive because Adam, at this point, would be indifferent between choosing ‘accept’ or ‘reject’.

Taking up again the considerations on the role of emotions, one can say that, probably, in such a game Adam will be angry at an unfair Eve’s offer. His anger represents evolved behavior in order to force a Nash equilibrium that requires Eve to make a fair offering (see BINMORE 2005, pp. 77-84).

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CHAPTER 3 THE SOCIAL CONTRACT

After analyzing different aspects of the game of life through the game theory, maintaining that this tool aims to know and predict the actions of individuals in hypothetical situations through the identification of Nash equilibria in such situations, it is important to proceed with an analysis that allows to deepen the knowledge of human behavior and evaluating afterwards the consequent results as input for an analysis of the human being as a social being.

3.1 The society

Ken Binmore calls himself a liberal bourgeois, or rather a Whig.13

Whigs do not belong to the right or even to the left (see BINMORE 1998, pp. 152-160,

1994, pp. 1-6). The Left shares Hobbes’s view of society14_{as something wider than the} mere sum of individuals and families that make it up. Just as human beings are made up of organs but transcend the organs from which their bodies are made, in the same way society is a social organism that transcends the human beings that form its constituent parts. But unlike Hobbes, leftists see society as something that is moved by a ‘common

13_{Whigs was a political party in England, active from the end of the 17th to the middle of the 19th century.}

They were considered antithetical to the Tories, a strongly monarchical party that can be considered an ascendant of the modern conservative party.

14_{Thomas Hobbes (Westport, Malmesbury, 5 April 1588 - Hardwick Hall, 4 December 1679) was a British}

philosopher and mathematician who was mainly interested in political philosophy. His most famous and important work is Leviathan.

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will’ or motivated towards a ‘common good’ to which the efforts and aspirations of its human constituent parts must be adequately subordinated.

According to Binmore this vision is not realistic, since the ‘common good’ is a mystified concept. The Left recognizes that men are social animals but makes the mistake of personifying society, endowing it with its own purposes. Deciding to implement reforms of society based on such misunderstandings would lead to their inadequacy.

The Right, on the other hand, recognizes the human’s individual rights and liberties as the basis for the commonality of society. These rights and liberties are determined by absolute standards of justice and therefore immune to change. The Right emphasizes the role of individuals, and focuses on maintaining their individual rights and liberties. But, it loses sight of the fact that men are social animals.

The Whigs agree that society is more than the sum of its parts, in other words the individuals who make it up, because it encompasses all common understandings and conventions. Rights and liberties, which for the extremists of the Right have absolute bases, are in reality nothing more than artificial constructs shaped and developed by social evolution or human ingenuity. It follows that those rights that individuals possess are not something donated and guaranteed by Nature but ground their guarantees on the ideas inherent in the individuals themselves. Such ideas are not permanent, nor are the rights that derive from them.

The Whigs are in favor of the change of the current society because, through the observation of society, it can be seen that the current systems of common conventions that govern life in the main Western societies are not ideal. They feel halfway between those who fix their attention on the wrong problem (Left) and those who fail to recognize the existence of a problem (Right). They can be considered conservative

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reformers, that is to say in favor of reforms, but at the same time very attentive to the considerations of the current social reality, which in part determines the feasibility of the reforms themselves.

3.2 Social contract definition

Individuals are players in the game of life. They have divergent goals and interests that make conflict inevitable. The existence of common understandings and conventions on how the behavior of individuals should be coordinated, allows them to cooperate in order to achieve a balance between different goals and aspirations. The social contract is identified with the set of such a system of common understandings and conventions that allow coordination (see BINMORE 1994, p. 35).

The left-wing socialists propose reforms of the current social contract in the name of the ‘common good’ without any feasibility analysis. They often propose contracts that are not feasible and do not lead to an equilibrium.

The right-wing conservatives aim to maintain the status quo driven by the pursuit of stability but they do not understand that what was stable yesterday is not necessarily stable today. Moreover, they become myopic in the face of opportunities to select a better equilibrium.

The Whigs do not use ideal principles of justice as a standard of comparison in the evaluation of a society, because such standards are determined by abstract contemplation and are not related to the current constitution of the society, and to the historical process through which the current state has been achieved. Accordingly, they argue for the need to recognize feasible social contracts, hence the available equilibria,

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in order to identify what could represent an improvement of the current social contract to which society can be moved by mutual agreement.

Reforms need to be adapted to the system of common understandings currently operating in a society and, given that societies vary in relation to time but also in relation to space, it is not certain that a reform identified as an improvement in the status quo of a given society will also be such with respect to a society located in another continent or in a society of the past.

Any social contract, moral rule, and principle of justice chosen should be feasible in a possible world and should be achievable through a process that is also feasible (see BINMORE 1994, p. 12).

3.3 Individuals

Individuals do not put the interests of the community ahead of their own selfish concerns. As Spinoza15_{argues, talking about its forerunners in moral philosophy,}

… they conceive of men, not as they are, but as they themselves, would like them to be [and hence], instead of ethics, they have generally written satire … and never conceived a theory of … politics, but such as might be taken for a chimera, or might have been formed in Utopia, or in that golden age of the poets … (BINMORE 1994, p.12).

15_{Benedict de Spinoza (Amsterdam, Dutch Republic, 24 November 1632 – The Hague, Dutch Republic, 21}

February 1677) was a Jewish family philosopher. For his openly professed and sustained opinions against religious orthodoxy, he was excommunicated by the Jewish community. His masterpiece, Ethics,

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With regard to social reforms, individuals only consent to reform if it is in their enlightened self-interest to do so. The rules governing their behavior are a mixture of instincts, conventions, customs, and traditions. They are not absolute, nor are they immutable. They are shaped by social, economic, and biological evolutionary forces. Morality is based on these rules and neither absolute nor immutable.

3.3.1 Meme

The hypotheses about human nature are those of neoclassical economics. It is assumed that people act in their own enlightened and widely conceived personal interest (see BINMORE 1994, p.15).

It is the evolutionary, biological, social, and economic forces that are responsible for the maximization of whatever thing. For the purposes of this analysis, it is important to consider not only the evolution of genes but also that of memes. Meme, a term coined by Dawkins16_{, as a parallel of gene, indicates a norm, an empirical rule, a code of conduct} that can be replicated from head to head through imitation and education. Memes are decisive for the behavior of the people in whose heads they are. The environment will be stable when the ‘inferior’ memes are eliminated for the survival of the ‘superior’ ones. The survivors will not necessarily be symbionts but will be in equilibrium with each other in such a way that none of them will gain ground at the expense of others. Therefore, individuals are the carriers of such memes. The notion of personal interest is

16_{In his book The Selfish Gene, considering the comparison with a gene in biological evolution, Richard}

Dawkins suggests using the word ‘’meme’’ when talking about a convention or idea that can spread in a population through imitation or education.

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interpreted as anything that makes the bearer of a meme a place for replicating the meme to other heads.

In the models used to represent different aspects of the game of life, the homo oeconomicus is taken into consideration as a representative model of the homo sapiens. The mechanism of functioning of the meme leads the homo sapiens to behave like homo oeconomicus, without the thoughts of the homo oeconomicus necessarily being attributed to him. The homo oeconomicus represents an incomplete model of the homo sapiens. However, is the best model available (see BINMORE 1998, p. 46).

Personal interest is conceived in a very broad sense, so that an individual can be defined as rational if he or she behaves consistently in achieving whatever he or she sets out to do. Binmore does not see the homo oeconomicus as a species separated from the homo ethicus (see BINMORE 1994. pp. 21-27). The latter is moved by love and duty, which

represent, according to the traditional theory, the cement of society. The love and duty which move the homo ethicus can be compared to ‘sympathy’ and ‘commitment’. Sympathy entails identifying oneself with another person, with a group of people or with a cause so that one no longer distinguishes one’s own goals from those of the entity with which one identifies. Commitment designates completing a course of action even if alternatives leading to better results are available. The homo oeconomicus is not alien to either sympathy or commitment, but he does not share all the virtues of homo ethicus on any occasion. If this were the case, moral problems would evaporate. In most cases people behave altruistically when such behavior is not very expensive. Altruistic behavior appears regular and protracted when individuals are part of a very close-knit group of insiders, such as a family, a tribe, or a group of adolescents. The applicable social contracts in such groups are different from those in force in larger social groups.

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Love and duty are not the cement of modern society: as a matter of fact, it is the reciprocity mechanism that holds sociality together without the need for any cement. Reciprocal behaviors that hold apparently altruistic relationships together do not require nobility of spirit.

3.3.2 Players

In game theory and economics, a player or economic agent is characterized by his personal preferences and beliefs, or as Hobbes would say, by his passions and experiences. Knowing such characteristics of an economic agent, it is possible to predict his behavior in an economic context.

Two other properties characterizing the nature of man can be defined, still following Hobbes, as ‘strength of body’, which entails what an agent can physically do or not do, and the manner in which the agent reasons. In game theory, preferences, beliefs, and the strength of body are the characteristics incorporated into the rules of the game. The fourth characteristic of individuals is reason. The theory of revealed preference deduces a coherent person’s preferences from the decisions he or she makes. People can also behave inconsistently. In such a case they can make mistakes that will put them at a disadvantage compared to people who are consistent.

3.3.3 Commitment and incentives

Players cannot be considered to be able to make commitments that limit their future behavior. Any commitment opportunities must be incorporated into the rules of the game. If, as an example, one player is concerned about the welfare of others, then these

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concerns should already be reflected in the structure of the game’s payoffs. In real life, it is difficult to make sincere commitments and doing so compelling oneself to binding promises.

Figure 3. 1: Kidnapping Game

As an example, in the ‘Kidnapping Game’ Eve kidnapped Adam and demanded a ransom. After the ransom has been paid, Eve has to decide whether to release him or not. She will choose to release him if she is certain that Adam will not reveal her identity to anyone. Adam promises Eve not to reveal her identity once he is free. This toy game shows how the victim, once released by his or her kidnapper, is not bound by any obligation to keep his promise. In fact, in such a model, Adam’s best answer, if Eve releases him, is to reveal the kidnapper’s identity. The only Nash equilibrium require that Eve kills Adam because she thinks that he will not keep his word and will reveal her identity.

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The duties and obligations integrated in a given social contract are honored because it is in the interest of each individual citizen and not because the members of society are committed in some way to honor them on the basis of a previous moral contract that requires members to abide by the contract.

The moral codes of a society are represented by common understandings and conventions which ensure that the constituent parts of it operate uniformly when the society is in good health. Their origin must be sought in the historical theories of biological, social, and political evolution and cannot be sought in the works of abstract thinkers.

Game theorists believe that the levels of cooperation that can be achieved are indeed very high. Cooperation is generated by simple implicit agreements between individuals, which arise to coordinate an equilibrium. Cooperative results can be sustained as equilibria in repeated games. There is no predetermined altruistic element that allows cooperation, in fact even non-altruistic people can achieve it. As Hume argues:

... I learn to do service to another, without bearing him any real kindness: because I foresee, that he will return my service, in expectation of another of the same kind, and in order to maintain the same correspondence of good offices with me or others. And accordingly, after I have serv’d him and he is in possession of the advantage arising from my action, he is induc’d to perform his part, as foreseeing the consequences of his refusal. (BINMORE 2005, pp. 8-9)

Moral laws are only rules for coordinating on equilibria in the game of life which are not supported by the impossibility of breaking the commitment, but by the hypothetical punishment that would follow the inability to keep one’s share of the agreement. The

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punishment could be administered by the judiciary, if the services in question are the object of a legal contract. Punishment may also consist in having to endure the disapproval of those whose respect is necessary, if one wants to maintain the current level of status in the community. Conventional agreements that arise for coordinating on an equilibrium must be self-controlled since the laws of the state are not rules of the game of life but are conventions too. These rules, unlike those of the game of life, can be broken. Even the officials, who have the task of enforcing the laws of the state, are themselves players of the game of life and even they could break their duty and could give in to corrupt behavior. If it is in the enlightened self-interest of individuals to respect the laws and enforce them, then they will be respected and, in a properly defined equilibrium, no one will have an incentive to deviate from the prescribed game. The enlightened self-interest may need the right incentives, as an example in the case of state officials, for them to do their duty. Without such incentives, corrupt behavior could become common practice in their professions (see BINMORE 1998, pp. 271-273).

3.4 The optimal social contract

The optimal social contract must be an improvement of the status quo. For Binmore, the state of nature is identified with the current status quo which derives from the evolution of history, unlike the Lockean state of nature, which identifies the state of nature with an a priori system of ‘natural rights’:

To understand political power aright, and derive it from its original, we must consider, what state all men are naturally in, and this is, a state of perfect freedom to order their actions, and

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dispose of their possessions and persons, as they think fit, within the bounds of the law of Nature, without asking leave or depending upon the will of any other man. A state also of equality, wherein all the power and jurisdiction is reciprocal, no one having more than another; there being nothing more evident, than that creatures of the same species and rank, promiscuously born to all the same advantages of Nature, and the use of the same faculties, should also be equal one amongst another without subordination or subjection, unless the lord and master of them all should, by any manifest declaration of his will, set one above another, and confer on him, by an evident and clear appointment, an undoubted right to dominion and sovereignty.17

To identify the optimal and fair social contract, Binmore uses the device of Rawls’ original position18_{but with differences (see B}_INMORE_{1994, pp. 12-13). Rawls, in order to} identify which should be the ‘just society’ towards which to address reforms, uses the device of the original position, which consists in a position of ignorance of the members of a society with respect to their current role in society. In such acondition, individuals would be able to choose the best social contract. That is to say, the one circumventing the risk of the lottery that assigns the real social roles awarding unfair prizes. The purpose of such a mechanism is to remove everything that makes people different, so that the agreement is taken on a basis of total equality.

17_L_OCKE_,_{J., The Two Treatises of Civil Government, book}_II_{, chap.}_II_{, Thomas Hollis, (London: A. Millar et}

al., 1764).

18_{In his work A Theory of Justice, published in 1971, John Rawls develops the metaphor of the veil of}

ignorance, which allows individuals, who use this artificial device with an effort of thought, to find themselves in the original position.

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Unlike Rawls, Binmore uses the original position device without assuming ignorance of the status quo. Even behind the veil of ignorance, with respect to the role of individuals in society, knowledge of the current social contract is crucial to identifying improvements that can be made.

Another difference is that, according to Rawls, a decision taken behind the veil of ignorance is a decision to be honored a priori, since this mechanism aims to identify those behaviors and decisions that respect predefined moral rules and principles of justice. On the other side, Binmore does not believe that the device of the original position identifies an equilibrium whose rules must be respected by all a priori, as if there existed a previous agreement on what ethical bases apply in the original position. Such a device allows to identify which are the feasible equilibria whose rules will be honored because individuals will be incentivized to do so given the lack of better alternatives.

The hypothetical agreement reached in the original position should not be considered binding because there are moral codes or natural laws a priori. These are just artificial ethical conventions (see BINMORE 1994, p. 38).

3.5 The game of morals

The equilibria that can be achieved in the game of life can be many and have to be compared, in order to select one. The selection is made by playing the ‘game of morals’. It allows to define a fair social contract in the game of life, that is, an equilibrium in the game of life that requires the use of strategies which, if used in the game of morals, would not leave any player with an incentive to appeal to the device of the original