Battle Of The Sexes
There are many numerous situations in institutions and industries where they want to achieve a common objective but differ in the mechanics of achieving it. For example, two firms trying to set up some common standards or take some measures but disagree over how to go about it. This type of situation is characterized by a common objective but divergent views or ways of how to attain that objective. Battle of the sexes is one of the coordination games, which explain such situations by drawing a parallel between such situations with the predicament faced by a man and woman going on a date but prefer two different places. As mentioned before, it comes under the category of cooperation games, which have got many practical applications in our day-to-day life.
In the language of Game Theory, a cooperative game is a game in which the players can make binding commitments. These games allow the players to share the benefits from cooperation by making transfers among themselves, which would leave them better off. Because of the existence of interpersonal framework; players come together and through cooperation achieve the most favorable outcome for all the players. Games, which replicate such situations, are called negotiated games because the outcome is reached through negotiations and deliberations. And the outcome, which is the result of such negotiation, is called negotiated settlement. This game is the classic example of how cooperation can be achieved even when people are selfish. It shows how commonality of the objective can resolve conflict.
Battle of the sexes illustrates the conflict between a man who wants to go to a prizefight and a women who wants to go to a ballet. Though selfish, they are deeply in love and would, if need arises, sacrifice their preferences in order to be with each other. Here cooperation, not rivalry, works. This game has two Nash Equilibria, one of which is a strategy combination. Given that the man chooses prizefight, even the women chooses the same. If the woman chooses the ballet so does the man. Hence (Prize Fight, Prize Fight) and (Ballet, Ballet) are the two Nash Equilibria.
Now lets us go into the technicalities of the game, which would aid us in understanding it better.
The payoffs related to different choices are:
Where P1 and B1 are the probabilities of the woman (Sita) attending the prize fight and the ballet respectively. Similarly, P2 and B2 are the probabilities of the man (Rama) attending the Prize fight and the ballet respectively.
- Social Welfare = (P1,P2) (B1,B2)
- Pareto Optimal = (P1,P2) (B1,B2)
- Dominant = none.
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Nash = (P1,P2) (B1,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.
Source:
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.
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.
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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.
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. e.g. 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 labor.
Case Study I -- Selection of Projects for Management Training Segment
Management Traineeship segment, which is an 11-week program, scheduled after IV term in IRMA is designed to provide the students with an opportunity to work closely with experienced managers in an organizational setting.
The Pre-MTS negotiations make a good case for the study of non-zero sum cooperation games, especially “Battle of the Sexes” as rounds of negotiations, manipulations, cooperation, and defections can be witnessed during the selection of the project, the place and the partner.
Rules For Allotment Of MTS Projects
Each participant has to give his/her individual preference regarding the functional area and the states he/she wants to do his/her MTS. The classification of the functional area were marketing, development, systems, finance, human resources and operations.
There are certain weightages for the preferences while deciding the allotment of the projects as shown in the Individual Preference Matrix (IPM).
The weightage for states falling outside this list is 10 points.
Assumptions
- If there is no projects from the states and the functional area given by a participant, there will be no adjustments made in the participants IPM.
- No changes will be allowed in the IPM after submission of the same. All individuals have to give a choice of 6 projects after the list of projects is released. In case more than one participant is interested in a particular project, then draw of lots will be resorted for deciding the projects.
Allotment of the projects
If there is more than one participant who has marked a project as his/her first choice then his/her scores on the IPM will be taken to allot the project. In case of a tie in the scores, draw of lots will be resorted to allot the project. If a participant does not want to take up a project after it is allotted to him/her through the draw of lots then, he will be eligible for allotment of projects only after everybody is allotted. The (IPM) will help in the allotment of projects according to the preferences of the students and minimize the chances of a tie. This will also help in solving cases where more than one individual vies for the same project.
The allotment process will be carried out in the following manner. After the IPM of each student has been collected, all the projects will be displayed under the above-mentioned categories. Each student will be asked to give preferences for four projects. Then the IPM scores of each student will be calculated. The Project will be allocated on the basis of both the IPM score and the project preference
Choosing of the partner
Suppose a person who has the highest score for project "X" has to choose his/her partner. Now he/she will have to choose his/her partner from amongst only those people who have opted for this particular project "X". This group consists of all those people who had filled Project "X" as one of their preferences.
The management traineeship project
If A and B want to be partners, they should fill in the same preference. If they do so then they have a chance to lose, as the project of their choice may not be from their preferred state. So, to maximize their chances of going together they will fill in different choices. This will help them to choose either of the projects if both of them get the project of their choices, either of which will be acceptable to them. For this purpose they would settle for the top preference of either. To attain their objective, both of them need to have total of their individual IPMs the highest for their preferences. Their relative ranking should be such that both of them get the highest ranking for at least one combination of state and functional area each prefers to go and the other should get the next highest total for the one the other has highest ranking for on the IPM. For example if A wants Delhi-Marketing and B fills in Pune-Finance then A’s second preference would be Pune-Finance and B’s second choice would be Delhi-Marketing. Now A’s IPM total for Delhi-Marketing combination would be 100 and similarly B’s total IPM for Pune-Finance would be 100 and their second best options that would be vice versa would have totals of 88 each (Pune-Finance for A and Delhi-Marketing for B).
The payoff matrix
The payoffs
If A chooses Delhi–Marketing then his payoff for Delhi-Marketing would be 100 and B chooses this as his second option then the associated payoff for B will be 88. Similarly if B chooses Pune–Finance as his first option and A chooses this as his second option then the associated payoffs for both will be 100 and 88. The negative payoff for both of them is that they cannot go together and this way their payoff is equivalent to nullity.
In the above matrix we see that in the first row that the candidate A has given his first preference as Delhi-Marketing and the candidate B has his choice as Pune-Finance. Thus they have highest IPM in the given preferences and thus they stand a chance to go to either of the projects. This seems very simple on the face that the candidates get what they want but there may also be other candidates who have the same score in their IPM with the same preferences. This will get them all in a tie thus forcing them to negotiate with one another. Now as they have highest score in two projects, they stand a better chance of getting at least one of them.
Case Study II -- Election in the campus
In the election that was held in the campus for the selection of the Class Representative, there were four nominations for the post, with a hope that they have considerable chances of winning. Let these four candidates be named A, B, C and D. The trends clearly emerged with campaigning reaching an end, which showed that the two candidates A and B were the forerunners.
In the light of emerging trends, C thought that he may be able to divide some of the votes from either side and supported by his core followers, he will be able to win. Candidate D did not want the candidate B to win and felt that his staying in the race can have an impact on the final outcome. So either he wanted candidate A to win or he himself. If he can come to some understanding with candidate, he can fulfill his objective. This situation was also very clear to candidate A and he also opined that either he should win or D should win but not B. So they colluded.
They campaigned by identifying the students who are favorable to both of them for their crucial votes. Candidate B did not have the support of students from all the informal groups across the PRM23, but he had fair chances to win with the help of the support from his own informal group. This will not serve the best interests of the PRM23, which was very well realized by candidates A and D and this was used as the campaigning tool, and the only way to realize this personal and social objective was by forming a cartel. Now it did not matter to them who would win but they did not want the candidate B to win, as they would be satisfied if either of them wins. The problem could have had a very simple solution by making candidate D withdraw from the race and asking all his supporters to vote for candidate A. When the proposal was put forth, few supporters of D, who were willing to cast their vote in favor of C rather than A, did not accept the proposal. As it is detrimental to the interests and objectives of both A and D, candidate D did not withdrew his nomination but advocated his stand to all his supporters voicing that it did not matter whosoever wins as long as either A or D wins. To pacify his reluctant supporters, he had to convey to them that the cooperation is symbiotic only on the face value but in reality he would win and the other candidate would suffer. Both the candidates told this to their reluctant supporters but they had this in their minds and were true to it that if either of them wins the purpose is served.
Election process with respect to the coordination games
Assumptions for the payoff matrix
- Payoffs are in terms of utility maximization; here the satisfaction will be maximized if B does not win which is the overriding consideration for A and D to collude.
- No interpersonal comparisons of utility are permitted that is both A and D are indifferent to the fact that which of them wins as long as B loses
- No side payments are allowed that is there are no external incentives so that either of them defects.
Payoff Matrix
The payoff matrix between candidate A and candidate D
Rules for the election
- Each student will only have one vote
- The vote stands cancelled if it is not marked at the right place
The payoff matrix
The above matrix shows the payoffs associated with the election process in the event of either A or D winning. The first column and row give the possibility of either A or D winning based on the presumption that both cannot win. The payoffs reflect the effect of collusion on maximizing the utility of the players. If they do not collude then the chances of B winning increase thus the penalty being unbearable to both of them.
The payoffs
The payoffs is a win for either of them, the other one is the negative payoffs that the third person will win which is B and this neither of them wants.
Explanation with reference to the battle of the sexes
In the game above we have a game with multiple game equilibria. The game had two players; each of who have the same two possible actions that is each of them can fight the elections individually. The also game has two possible pure strategy Nash- equilibria one of which is strictly preferred by each player that is they both win. The players have some common interest though in that they would prefer to choose the same strategy rather than doing different things as in the case above both of them want either of them to win and for this they have formed a collusion and they are canvassing for each other. Again the need for coordination arises. The payoffs for both of the candidates are more when they cooperate than they do not. In that case the penalty would be very high, here it would mean losing the election.
Case Study III -- Fertilizer Cartel Between India And China
India and China are some of the biggest importers of fertilizers from the west as they are primarily agriculture dependent economies. Huge quantities of fertilizers were imported from US firms, which are subjected to harsh price fluctuations due to their poor bargaining power in the international market. The fertilizer industry in the US is organized and the prices raised by them used to be in collusion as per the demand from importing nations. Due to this factor the Indians and the Chinese could not even purchase from the competitors, as the prices remained the same. This has become a big problem on the demand side for the parties concerned as their need for the inputs rose without a corresponding proportionate rise in the domestic production capabilities. The fertilizer cartel raised the prices arbitrarily without any warnings that entailed heavy outflows of valuable foreign exchange for both India and China.
On assessing the situation, the agro-trade ministers of both the countries who were in US to finalize the purchase transaction on behalf of their respective nations decided to follow some strategy that results in fall of prices. The choices/strategies evaluated are either to collude by not importing or to pressurize the cartel through US government and lobby for lower prices. This to serve the purpose of attaining sufficient clout on the demand side by not making any purchases for as many days as they could hold out for and survive by fall back on their buffer stocks, which were in reserve in their respective countries to bring down the prices considerably as the combined demand of fertilizers from India and China accounted for more than 50% of the cartel’s total sale. If these two did not buy, the cartel would have huge surplus and the prices would crash. This would result in the satisfaction of the objective to purchase at lowest possible prices. Thus, the objective is to satisfy the collective goal and bring the prices of fertilizers down rather than compete with each other for a better deal, as both would lose in the bargain.
Assumptions
Firstly that the buffer stocks with both the countries are around 50,00,000 tonnes and their daily requirements are of around 10,000 tonnes the buffer stocks would last them for around 50 days. Now, if India wants to purchase 15 million tonnes and China wants to purchase 20 million tonnes from the US the price they would have to pay would be US$250 per tonne which was the new raised price quoted by the fertilizer cartel, the old price being US $ 200.
Secondly, it is assumed that the cartel will not offer fertilizers to the other company at cheaper rates to see that one of the players in the game cheats.
Thirdly, The combined pressure by India and china whose share in imports us around 50% can pressurize the US government.
Payoff Matrix
(Figures in millions)
Payoffs
Now, India and China would have to pay excess of (250-200) 15,000,000 US$ and (250-200) 20,000,000 US$ i.e. 750 and 1000 million US$ above the original price of 3000 and 4000 million US$ respectively. India and China have now two choices to attain their objective of bringing prices down either by colluding or by pressurizing. Their combined bargaining power would result in pile up of stocks with the cartel making it difficult for the firms to off load in the market. On the basis of their buffer reserves India and China can decide to hold out on purchases as long as their buffer reached a critical minimum to let the prices would crash by market mechanism. This way they did not have to succumb to arm twisting by the fertilizer cartel. They did not purchase for 3 weeks at a stretch and at the end of the period prices crashed to a level 175 US$ per tonne. Thus their payoffs after collusion came to around 2250 for India and 3000 for China which was 750 and 1000 million US$ below the initial price they would have to pay. This way the collusion resulted in maximization of benefit for both as their overriding objective of standing up to the cartel was met. The different payoff accruing to both i.e. more benefit to China and less to India was of no importance to either. Similarly, another strategy to pressurize the US government to reduce the prices up to US$175 resulting in similar payoffs.
Payoff Matrix:
The matrix gives the payoffs of India and China in case of colluding or pressurizing which should give us multiple Nash Equilibria as they are indifferent between the two out comes.
Application in terms of Battle of Sexes
The outcome of the game is thus the distribution of the total available payoff amongst the two coalition members, which arose as a result of agreements between the players rather than as a predetermined consequence of their strategic choices. Both the players have perfect information about each other’s strategies making the game a success. There are two Nash equilibriums in the game. Both India and China could have achieved same level of payoffs by following the same strategy rather than going by different strategies. Since, the payoffs are assumed to be in monetary terms the coalitions acted by and large to maximize their joint payoff by coordinating strategies. It reflects the fact that in a hostile universe ‘unity is strength’.
Conclusion
Game theory involves the study of “multiperson” decision problems. The multiperson decision problems are not only at the firm level but also at the macro level. In reality the situations are more complex and Game Theory can be applied effectively, if we can draw from other diverse subjects and fields. Game theory comes handy in international trade, tariff decisions and other strategic policy decisions viz. monetary policy.
Coming to Battle of the Sexes, though it 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 more thought is applied, we can apply this game in our dealings and interactions with people, job and other day to day mundane activities
“Assurance games where supplying new rules is considered easier to accomplish than in Prisoner Dilemma games because these are mutually beneficial outcomes that are potential equilibria in the sense that once reached no one has an incentive independently to switch strategies. Equilibria in assurance games, however does not necessarily reward participants equally. Participants prefer a set of rules that will give them the most advantageous outcomes in contrast to continuing “ – Bates
References
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Rasmusen, Eric, Games and Information, 1990, Cambridge, Basic Blackwell
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Ordeshook C, Peter, Game Theory and Political Theory, 1986, Cambridge, Cambridge University Press
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Jones, A.J, Game Theory – Mathematical models of conflict, 1980, England, John Wiley & Sons
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Dixit Avinash, Skeath Susan, Games of Strategy, 1999, London, W.W,Nortan & Company
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Theories of Collective of action and cooperation, 2002, Anand, Institute of Rural Management.
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