Where P1 and B1 are the probabilities of the woman (Alice) attending the prize fight and the ballet respectively. Similarly, P2 and B2 are the probabilities of the man (Bob) attending the Prize fight and the ballet respectively.
- Social Welfare = (P2 ,B1) (P1,B2)
- Pareto Optimal = (P2 ,B1) (P1,B2)
- Dominant = none.
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Nash = (P2 ,B1) (P1,B2)
The worst possible outcome (0,0) is called the “Threat Point”. This payoff is the outcome when both man and the woman act seifishly, by going to the place of their preference. Hence, their primary objective of being together is not fulfilled. Another worst outcome (3,3), which results from what is called the “misplaced altruism”. It happens when man and woman go to different places, not out of selfishness, but out of expectation of finding the other person there. That is, woman goes to the prize fight, expecting the man to be there and vice versa. This outcome is better than the “Threat Point”, even though they are not together, as it atleast gives the satisfaction of considering the other person’s preference.
But the real predicament is the choice of the Nash Equilibrium. To understand how it is done, we need to examine the game in the light of repeated games, first mover advantage, prior communication and focal point. What makes Battle of the sexes different from other games is that its both equilibria are “Pareto-Efficient”. But the feasibility of the achievement of the equilibrium is more a question of psychology than Economics.
The game can be represented graphically for easier understanding. No matter what is the probability of they going to either event together, the result can be represented by the coordinates of a point on the line in red joining the red points (1, 0) and (0,1) in the diagram below. In this diagram the horizontal axis represents Man's payoff; the vertical axis represents Woman's payoff. The blue points are the off-diagonal outcomes of the game.
The red line is called the negotiation set and its equation is the diagonal.
Equation of the red line is Y = f (X) = 1-X,
Where X is the probability of Man going to the prizefight and Y is the probability of woman going to the ballet. The joint payoff of the players is (X Y) and the feasible pay off region is the convex set enclosed by the lines joining the pay off pairs. The coefficient of X is called the pay off trade-off rate, since it is the rate at which man’s pay off is traded-off with the woman’s.
Another dimension to the game can be added, by including a third option or alternative. This alternative should be relevant and should have an effect of changing the outcome of the game. The other alternative can be a food joint, which both of them like. In that case, they would not opt for either of their prior preferences. After negotiating or talking about each other’s preferences and feelings, they may decide to go to their favorite food joint. This new state of affairs can be represented in the matrix given below.
Graphically, it can be explained as following.The line from (0, 1) to (.4, .9) has the equation .25*u + v = 1.The line from (.4, .9) to (1, 0) has the equation 1.5*u + v = 1.5. Without the restaurant option, Man already has a payoff of .45. Therefore, with the restaurant option his new payoff should be greater than .45.
Therefore, only the line from (.4, .9) to (1, 0) seems relevant.
To make the selection of equilibrium easy, Nash introduced few axioms, which made application of cooperative games, here, Battle of the Sexes, to a more fields possible. Few of the axioms are explained below, which are relevant in this particular context.
Nash's Axioms
- Player Symmetry: Swapping the names and payoffs of players doesn't matter.
- Independence of Irrelevant Alternatives: Adding more alternatives do not add weight unless new alternative is the part of the solution.
- Pareto-Optimality: If players can negotiate a settlement that is better than the status quo, they would go for it.
- Individual Rationality: Given a choice, a player will choose the option, which will maximize his benefits. That his he will choose the highest payoff
In the context of understanding Battle of the Sexes better, it would help us to familiarize with some of the terms associated with it.
Repeated Games: Cooperation may not occur in the first instance itself. Man and woman may go to different places, believing the other person is there. But they may end up going to different places. That is, the man may go to ballet believing that woman would be there and vice-versa. Here Nash equilibrium assumes right and constant beliefs, which the players have in each other. With the repetition of the game number of times, they may end up at the same place one night and square for one of the equilibrium.
Therefore in repeated games, the concept of dynamic strategies - that is the change of preferences, play an important role. They replicate the psychological, informational aspect of ongoing relationships
First mover advantage: In many games, the player who commits first or moves first has a “first-mover advantage”. In this game if either of them book tickets for their partner before hand, this situation arises. If this happens, the other person is induced to go with the partner, who committed first. First-mover’s move influences the sequence of the moves.
Mixed Strategy: Availability and the quantum of the information have an important bearing on the course of the game. In this game, if both man and the woman have complete information about each other’s intention, then Pareto-efficient equilibrium can be achieved in the first instance. But on the face of uncertainty due to inadequate, asymmetric information or no information at all, the game has to be repeated to reach the equilibrium. This is where mixed strategy comes in. It can be interpreted in terms of one player’s uncertainty about what the other player would do. It can be viewed as the mapping of the possible information of each player to a probability distribution over actions or outcomes.
Focal points: In real life situations there are numerous equilibria. But out of these only few seem more likely to be feasible. The psychologically compelling Nash equilibria are called focal points. Past experiences and history form the basis of focal points. In Battle of the sexes, the choice of the equilibria, when the game is repeated over a period of time, depends on the on the focal points. Also focal points impart inflexibility to the game. As the number of times the game is repeated, the possibility of the change of Nash Equilibria cannot be ignored, as the payoffs tend to change.
Mediation, communication, negotiation and bargaining also have place in Battle of the Sexes.
Application of Battle of the Sexes
- Setting industry wide standards is a good example of this game. All firms cooperate,
Negotiate and set standards in such a way that it all of them get a favorable pay off.
Eg: computer disk drives, pipe fittings, etc
- Use of common language to formalize a sales agreement between two firms even
They prefer different terms. Common language acts to the advantage of both the firms and help them understand the terms of the agreement better.
- At the macroeconomics level, on such example is the formulation of the tax policies by
The two tax authorities. Though both of them have different ideas and preferences, their
Main objective is that the taxpayers should not benefit from the loopholes in the law. So,
They coordinate on the tax system they wish to devise, in order to avoid the above situation.
- Out of the court settlements also is a good case to explain the working of game theory.
It would save the parties involved from waste of money, time and labour.
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Another important application of battle of the Sexes is in social psychology. John Thibaut and Harold Kelley in their book, The Social Psychology of Groups, use the 'Battle of the Sexes' game to replicate interpersonal interactions of a dyad (two persons in a continuing relationship involving interactions) over time, sourced on either member of the dyad's 'level of satisfaction' with the relationship. Recently, Rusbault and other socio psychologists attempted to model the evolution of abusive relationships based on Battle of the Sexes.
But again the problem with this type of application is the inability to put a figure on players' subjective criterion like how much love does he/she feel for the partner to an objective pay-offs in terms of monetary gains accruing from cooperation.
Though Battle of the Sexes is not as talked about or as widely known as Prisoner’s Dilemma, it has its own range of applications not only in the fields of micro and macro economics but also in social psychology. May be if think about it a bit, we can apply it to our dealings and interactions with people, job and other day to day mundane activities
