In this paper we extend Rabin’s (1993) model of fairness equilibria to groups of individuals and define a new solution concept we name “group-fairness equilibria.” We model two games with two players, where each player in each game belongs to one of two groups. We analyze how the outcome of one game may affect the outcome of the other and how the existence of one individual with a particular grudge or liking towards the player she is playing with can impact the outcome of both games. We analyze some applications of our model.

En este artículo extendemos el modelo de “equilibrio justo” de Rabin (1993) a grupos de individuos y definimos un nuevo concepto de solución, al cual llamamos “equilibrio de justicia de grupos”. Nosotros modelamos dos juegos con dos jugadores, donde cada jugador en cada juego pertenece a alguno de los dos grupos. Analizamos cómo el resultado de un juego puede afectar el resultado del otro juego y como la existencia de un individuo con un odio o aprecio por el jugador con el que está jugando puede impactar el resultado de ambos juegos. También analizamos algunas aplicaciones de nuestro modelo.

Our objective in this article is to analyze how the emotion of fairness between the members of different groups can affect the outcome of apparently independent games. We extend

In his seminal paper, Rabin introduced the emotion of fairness to game theory. Rabin develops a utility function that incorporates the assumption that players want to be “kind” to other players who are kind with them, and the assumption that players want to be “unkind” to other players that are unkind with them. Rabin develops a solution concept: “fairness equilibria” that includes the emotion of fairness. For example, in the Prisoner´s Dilemma the outcome where both players play “cooperate” may be a fairness equilibrium for small payoffs.

However, some aspects of reality are absent from his analysis. First, individuals see themselves as parts of groups and care about the treatment toward the members of their groups. And second, it is easier for individuals to cooperate if they belong to the same group, for example if they are relatives. There is evidence that individuals tend to treat better those individuals that belong to their own group, even if the group was formed randomly (

Our objective is to extend Rabin’s Fairness Equilibria to represent fairness between groups.

In this article we model games where individuals play directly only in pairs. However, we assume that individuals take into consideration the interaction of players in other games when they form their beliefs regarding kindness. While most game theorists have assumed that players only care about what happens in the games they play, our aim is to model how the outcome in one game may affect the outcome of other games.

The closest work to ours is

In section “Model”, we introduce our model by extending Rabin’s Fairness Equilibria to groups of individuals. Rabin’s model is defined over a single game of two players. However, in order to analyze the interaction between groups of individuals, we have to work with more individuals. We work with the easiest case: Two games of two players in each game, where one of the players in each game belongs to one of two groups.

Rabin defines a belief-of-kindness function that represents how kind an individual A thinks another individual B has acted towards her. We extend this function to include not only A’s beliefs of the kindness or cruelty of B’s actions towards A herself, but also A’s beliefs regarding the kindness or unkindness of other members of B’s group toward other members of A’s group. Next, we define a function of how kind A thinks she has acted towards B. Finally we define a utility function that uses these two kindness functions to represent that an individual wants to be kind to an individual that belongs to a group whose members have been kind to members of her own group and wants to be unkind to an individual that belongs to a group whose members have been unkind to the members of her own group. Our equilibrium occurs when each individual plays in such a way as to maximize his or her own utility and when individuals beliefs regarding other individuals actions are correct and individuals beliefs regarding other individuals beliefs are also correct. We refer to this solution concept as a “group-fairness equilibrium.”

Also, we analyze how the outcome of a single game can impact the outcome of the other game. We get the following results:

First, a combination of strict Nash equilibrium in both games will always be a groupfairness equilibrium for large values of the payoffs. Second, for very high payoffs, there are not positive group-fairness equilibria.

Third, in the case where the payoffs of the games are small, the outcomes where individuals are maximizing the other players’ payoffs is a group-fairness equilibrium if any player do not have a specific grudge for her opponent; and the outcomes where individuals are minimizing the other players payoffs is a group-fairness equilibrium if every player have a positive grudge for their opponents.

In section “Group-Fairness over two periods” we extend our model to the case of two periods, when a single game is played first and then the other. We apply our model to the case of a monopoly that gives away a product for free to an individual in need in order to improve its perception among its consumers. If the individual in need belongs to the same group as the consumer, the consumer may be willing to pay a higher price to the monopoly in order to repay its kindness. The kinder the monopoly is to the individual in need and the closer that individual is to the consumer, the higher the price the consumer will be willing to pay for the product.

In section “Facing somebody from our own group”, we analyze the case where one individual plays against another member of her own group. We assume that individuals think better of members of their own groups and treat them better as a result. In this section we analyze an example in which one consumer buys a product from a monopoly and we show that the consumer is willing to pay a higher price if the owner of the firm belongs to the same group as her.

In section fifth we conclude and discuss possible extensions.

For simplicity, we analyze the case of two games with two players each game, where one player
in each game belongs to one of two groups. ,
_{1}_{2}

We assume that there is a partition

_{i} is the set of (possible mixed) strategies for
player _{i}
_{i} is a strategy for individual
_{ij}
_{i} are the beliefs of individual
_{iji}
_{j} are the beliefs of individual
_{i}
_{i},
_{j}_{i}
_{i} and individual _{j}
_{j}.We refer to
_{i} as the "material"
payoffs of player

Individuals sometimes belong to groups whose members are very tightly-knit, such as members of the same family, and sometimes belong to groups whose members are not so tightly-knit. We use the variable

We introduce the variable

We define a kindness function, which is taken from

where _{ij}. As

where _{ij} we are assuming that individuals judge more favorably those individuals they exogenously like.

The first two terms of the numerator and the denominator of

The choice of

Now we define a player’s utility function using our definitions of kindness functions.

This utility function represents a situation where an individual’s utility increases if she is kind to somebody that is kind to her (or if she is kind to the partner of somebody that is kind to her own partner) and unkind to somebody that is unkind (or if she is unkind to the partner of somebody that is unkind to her own partner). We now define a new solution concept that includes the idea that an individual wants to reciprocate when playing against somebody that belongs to a group in which somebody has been kind or unkind to somebody in her own group. We name this solution concept: “Croup-Fairness equilibria”. We follow

This solution concept represents the emotion of fairness where in the presence of acts of kindness or unkindness, individuals may want to reciprocate not only to those individuals who committed the acts, but also to every member of the same group, even those that does not have any relation to those acts. As a result, the outcomes of different games for members of the same groups may be related.

One drawback of using Rabin’s “fairness equilibrium” as a framework for our model is that we cannot prove the existence of the group-fairness equilibrium defined above, as the utility function is not continuous. However, we work with Rabin’s framework given its clarity and the familiarity of his model.

In this section we analyze the relation that exists between the outcome of two games. We pay special attention to analyze the cases where a particular liking or grudge can affect the outcome of both games. We give some general propositions, but before, we write three of Rabin’s definitions that we use in our results. Rabin works with a game of two players and therefore his definitions are for a game of two players. The first two definitions: a mutual- min and mutual-max strategies are useful as they are, but we extend the third one, the definition of positive and negative outcomes, for the case of two games of two players each game.

Rabin defines a mutual-max strategy as a strategy where both players mutually maximize each other’s material payoffs and a mutual-min strategy as a strategy where both players mutually minimize each other’s material payoffs. An example of a mutual-max strategy is where both players, in the battle of the sexes, go to the same event, and an example of a mutual-min is where both players, in the prisoner´s dilemma, play defect.

_{i},
_{j}) ϵ
(_{i},
_{j}) is a mutual-max outcome

_{i},
_{j}) ϵ
(_{i},
_{j}) is a mutual-min outcome

Other definition that we take from Rabin, is that of a positive and negative outcome. Rabin defines the outcome of a game as a positive if the sign of the kindness function for both players is positive and negative if the sign of the kindness function for both players is negative. We extend Rabin’s definition of positive and negative outcomes for the case of two games and four players by defining a positive outcome as an outcome where the four players are kind and a negative outcome for the case of two games as an outcome where the four players are unkind.

_{i} > 0 b) An outcome is weakly
positive if for _{i} ≥ 0. c) An
outcome is strictly negative if for ,. _{i} < 0 d) An outcome is weakly
negative if for _{i} ≤ 0.

Now we use these definitions in the following propositions. In proposition 1 we show the relation that exists between group-fairness equilibria and the Nash equilibria of both games when the material payoffs are high. For this, we analyze a set of games that are exactly the same, except that the size of their material payoffs vary with a variable

Given that the material payoffs for each player only depend on the strategies of herself and
her opposite, we can think of the whole game as composed of two materially
independent games, one formed by players 1 and 2 and the other formed by players 3
and 4. Let g_{1} be the set of games that consists of players 1 and 2 and
the set of strategies _{1} and
_{1} and payoff functions ,
_{1}
_{1},
_{2}_{2}
_{1},
_{2}_{1}
_{1} be the game that corresponds
to _{2} be the
set of games that consists of players 3 and 4, the set of strategies
_{3} and
_{4} and payoff functions ,
_{3}
_{3},
_{4}_{4}
_{3},
_{4}_{2}
_{2} be the game that corresponds
to

Let’s denote the whole game that is composed by games
_{1}_{2}
_{1}_{2}

Proposition 1 is a direct translation of Rabin’s propo.sition 5 to group- fairness. (The proof of proposition 1, as well as all other proofs are in the Appendix.) If the material payoffs increase, the importance of fairness considerations becomes smaller. As the material payoffs increase arbitrarily, eventually the material payoffs dominate the fairness considerations and the group-fairness equilibria become the combination of Nash equilibria for both games. Proposition 2 tell us that as the material payoffs grow arbitrarily large in both games, players cannot have positive emotions.

Proposition 2 tell us that as the material payoffs grow large, the positive group-fairness equilibria are eliminated and only the weakly negative and neutral group-fairness equilibria are left. As the income increases, the material payoffs dominate the fairness considerations. Because individuals are maximizing their own material payoffs, other players would not think they are being kind and the positive emotions are eliminated. Proposition 3 analyzes group-fairness when the material payoffs for both games become small. Part a) of the proposition tell us that any combination of mutual- max and any combination of mutual-min is a group-fairness equilibrium is a group-fairness equilibrium when the material payoffs are very small and individuals have an exogenous liking (grudge) for their opponents. Part b) tells us that when every game has only one mutual-max (mutual-min) outcome and the material payoffs are very small and individuals have an exogenous liking (grudge) for their opponents, then the combination of these outcomes are a group-fairness equilibrium.

_{ij} > 0 for all

Part a) of proposition 3 is a direct translation of Rabin’s proposition 3. As material payoffs approach zero, the game is dominated by the fairness considerations. In the case that an outcome that is strictly mutual-max for both games, every player is playing a strategy that maximizes the material payoffs of the other player and therefore they are being kind to each other (or at least, they are not unkind to each other). If no player has a grudge against each other, nobody wants to change their strategy since they want to be kind to each other in response. In the case that an outcome is a strictly mutual-min for both games, every player is playing a strategy that minimizes the material payoffs of the other player and therefore they are being unkind to each other. In this case, if no player has a special liking for the player they are playing with, nobody wants to change strategy since they want to be unkind to each other in response. Part b) tell us the effect that exogenous grudges or likings of the individuals can have in the outcome of the game. In the case that each game has one mutual-max outcome and the material payoffs are small, then if every player has a liking for her opposite player, the group-fairness of the game has to be the combination of the mutual-max for each game. In the case that each game has one mutual-min outcome and the material payoffs are small, then if every player has a grudge against her opposite player, then the group-fairness of the game has to be the combination of the mutual-min for each game.

The last three propositions refer to the case where the material payoffs increase or decrease
for both games. In the next two propositions we analyze the group-fairness when the
material payoffs of one game change while the material payoffs for the other game
are left constant. We define the playoffs of game one as function of
_{2} represents the set of games that
consists of players 3 and 4 and payoff functions
_{3}(_{3},_{4})
and
_{4}(_{3},_{4}).
_{2}
_{2} is the game that corresponds
to Y. We denote the whole game that is composed by games
_{1} (_{2} (

We analyze the case where

Proposition 4 refers to the case where the material payoffs of one game grow large. As the material payoffs of one game grow, the fairness consideration of this game dominate the fairness consideration of the other game. If the outcome of this game is strictly negative, then both players in this game are unkind to each other and (if nobody has an exogenous liking) both players in the other game do not want to be kind to each other (they will not necessarily will be unkind, as this depends also of the material payoffs, however, they will not be kind).

_{1,} if game 2
has a strictly negative outcome then game 1 has a weakly negative outcome.

Proposition 5 refers to the case where the material payoffs of one game become very small, while the material payoffs for the other game are left constant. As the material payoffs of one game become small, individuals care more about what’s happen in the other game and the emotions of fairness for that game would be dominated by the emotions of fairness of the other game: players would be kind if the players in the other game are kind and players would be unkind if players in the other game are unkind. In this scenario, a player in game 1 with a large enough liking or grudge to determine the outcome of the single game, will also determine the outcome for game 2 and the groupfairness equilibria for the whole game.

_{2}, if game 1
has a strictly positive outcome then game 2 has a weakly positive outcome and if
game 1 has a strictly negative outcome, then game 2 has a weakly negative
outcome.

In

In this section, we extend our model to include the secuential case, where one game is played first and then the other, in order include the situation where players anticipate that the kindness or unkindness of their actions will have an effect on the behavior of other players. For example, firms may donate to charity in order to influence consumers, who may want to buy from kind firms. Similarly, some individuals may commit hate crimes to generate negative emotions among the members of opposing ethnic groups in order to create divisions. In this section, we extend our model of group-fairness to the two period case where single game 1 is played first and then single game 2. Players in single game 1 may want to influence the actions of players in game 2.

for all

Note that for players 3 and 4,

Players in game 1 maximize their utility in the knowledge that their actions will affect the actions of players in game 2. Given that they want their partners to be treated kindly in game 2, they may be kind in the first period in order to make their partner’s opponent be kind in return, even if they are treated badly. ^{1}

We assume that there is a single consumer who wants to buy a single unit of a product from a monopoly. The consumer’s valuation of the product is given by β, while the marginal cost for the monopoly is zero. Simultaneously, the monopoly chooses the price and the consumer chooses a reservation price

The timing of the game is as follows: in the first period the monopoly decides whether or not to help the individual in need and in the second period the monopoly sells its product to the consumer. We solve this problem by backward induction. In the second period, the consumer chooses the reservation price (

If the monopoly helps the individual in need in the first period and if the monopoly prices at

where

If the consumer buys the product from the monopoly, she will not be being kind to the monopoly, given that she is performing an action that improves her own material payoffs (unless the consumer pays a higher price that her valuation of the product). However, if she does not buy the product (by choosing a reservation price higher than the price of the monopoly) she will be unkind, given that she is sacrificing her material payoffs in order to punish the monopoly.

In the second period, the monopoly charges a price that makes the consumer indifferent between consuming and not consuming. The maximum price the monopoly is able to charge is:

and if the monopoly doesn’t help the individual in need, the maximum price that the monopoly is able to charge is:

therefore, the monopoly is able to increase its price by helping the individual in need. If the cost of helping the individual in need is lower than the extra revenue this brings, that is, if

We should note that the price the consumer is willing to pay can be higher than her valuation of the product. If

As mentioned in the introduction, individuals tend to treat those individuals that belong to
their own groups better. In Prisoners Dilemma experiments, cooperation is more
common between members of the same group (Yamag-ishi & Kiyonari, 2000). _{ij} (

_{ij}
(

this is equivalent to the kindness function of

By including ν, our model takes into account that if both players belong to the same group, cooperation may be sustained for higher values of

As family members are less likely to defect in a Prisoner´s Dilemma-type situation, then in some cases institutions will be organized around members of the same family. For example,

There is evidence that individuals in developed countries evaluate the products from their own countries more favorably (

We extend the example in the section “Application: Firms giving to Charity” by assuming that
the consumer has an exogenous liking for the monopoly that is a lineal function of
the closeness of the owner of the monopoly to the consumer:
_{ij} (

An individual is willing to pay a higher price the higher the value of

In equation (2) we can see that for

In this article we extended

Second, we used our utility function to define a new solution concept: “groupfairness equilibrium” and analyzed the relation between the outcome of both games. Finally, we applied our model to explain why firms invest in programs of social responsibility to improve their image, which increases the price that customers are willing to pay for their products.

Our work could be extended in a number of ways. For example, group-fairness could be paired with generalized reciprocity. Generalized reciprocity is when an individual (that is not related at all) is targeted by the offences of third parties. This could explain why minorities sometimes are targeted on hard times, even if the minorities does not have anything to do with the problems.

Proof of proposition 1

Part a) By contradiction: If

Part b) If

Proof of proposition 2

As _{
ij
} = 0 and player _{
ij
} < 0 and player

_{
ij
} > 0 for all

Proof of proposition 3

Part a) If the outcome is strictly mutual-max and σ_{ij} is positive
for _{ij} is negative, then

Part b) For every _{ij} high
enough that _{ij} that makes

_{1}, if
game 2 has a strictly negative outcome then game 1 has a weakly negative
outcome.

Proof of proposition 4

If σ_{ij} is bounded and _{ij} for
_{1} is dominated by the
outcome of game 2. If the outcome in game 2 is negative, the value of _{1} is
negative. Given the utility we defined in definition 3 if _{ij} would not be positive.
Therefore, in the group-fairness equilibria, the fairness functions for both
players of game 1 are weakly negative and are weakly negative outcomes.

_{2}, if
game 1 has a strictly positive outcome then game 2 has a weakly positive outcome
and if game 1 has a strictly negative outcome, then game 2 has a weakly negative
outcome.

Proof of proposition 5

If σ_{ij} and Y become arbitrarily small, the value of _{2} is
dominated by the outcome of game 1. Also, as the material payoffs of game 2
become arbitrarily small, the emotional payoffs for both players of game 2
dominate their material payoffs.

If the outcome in game 1 is positive, the value of _{2}, is
positive. If

If the outcome in game 1 is negative, the value of _{2}, is
negative. If