All article in Sports Illustrated some time ago was about how girls were being o

Question

All article in Sports Illustrated some time ago was about how girls were being offered soccer scholarships earlier and earlier by college coaches - sometimes as early as 9^th grade, if not before. Coaches described how they wished no one did that, but given the competitive climate of recruiting they all felt like they needed to. In other words, they described a prisoner's dilemma. Draw out this game in 2 times 2 form with payoffs such that it is a prisoner's dilemma. Clearly label the players and strategies, and circle or underline payoffs corresponding to best responses. Genetically, what are the two things that must be done in order to sustain the cooperative outcome in equilibrium? In addition, explain in words how you might do this in the context of the soccer recruiting example.

Explanation / Answer

Game theory is a situation where two rivals taking optimum decisions from different alternatives. At the time of taking best decision they must consider strategies of rival. Their objective is to select a strategy which will maximize their outcome.

In the problem assume two coaches teaching foorball play to women. They want to attract women of 9th grade or below. In order to attract them to join in the coaching class, two strategies are available. In first strategy some scholaship is paid. In second strategy no such scholarship is paid. Thus both players have same strategies to adopt. They have to select the strategy where their pay off is optimized. Here pay off is the number of students attracted in the soccer game. Consider the following pay off matrix to explain the problem.

The payoff matrix indicates two different pay off in each cell. First one is pay off of coach 1. Second one is for player 2. Select strategies of player 1.

suppose coach 2 has adopted scholarship strategy. Then Caoach 1 will get 6 player if it also offers scholarship. Otherwise it will be ony 2. So he will select strategy to offer scholarship.

Now assume that players 2 has decide not to go for scholarship. Then coach 1 will adopt scholarship. It will maximize its return. He will be able to get 10 players. Thus whatever be the strategies of rival, player 1 will go for scholarship strategy. Here strategy of scholarship is dominating the strategy of no scholarship. So it is optimum one for him.

Similarly consider optimum straegy of coach 2. Suppose coach 1 has decide to go for scholarship. The coach 2 will also go for scholarship. It will give him optimum pay off of 6. If coach 1 select o scholarship strategy, then also coach 2 will go for scholarship. It will enable him to earn highest payoff of 10. Thus scholarship is also best strategy of coach 2.

Hence both coach will offer scholarship. Optimum pay off will be 6,6. It is known as Nash equilibrium. This equilibrium is also stable. ay movement or deviation will make the situation volatile without further gain to both player.

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A special type of eqiuilibrium is sometimes observed when two players inspite of getting a Nash equilbrium select a different strategy. This concept has come from an example of two prisoners. Suppose both are involved in same crime. They have two separate strategies. They are confession of crime and no such confession. Each of them are aware that confession will reduce the period of imprisonment to a large extent. Yet both of them will select the strategy of no confession.

Suppose in this problem, both coaches have attracted some women for Football. But ow they are trying to leave it. So for reducing the quit, they can offer scholarship. Here payoff is indicated by negative figures to indicate number quits. The pay off table is shown below:

Suppose you are selecting optimum strategy of Coach 1. Coach 1 has decided to go for scholarship. Then Coach 1 will also go for scholarship to minimize quit to 4. If Coach 2 is not offering scholarship, the it will offer scholarship. The number of quit of coach one will be only 1. Put * sign against the two pay off receivable by coach 1 from his selected optimum strategy. It is shown above

Now select optimum strategy of Coach 2. If Coach 1 is offering scholarship, then coach 2 will also offer scholarship. Other quit will increase for him. So put a ^ sign in the selected pay off in cell (1,1). If coach 1 is not offering any scholarship, then also coach 2 will offer to get best return of only one quit. Put ^ sign in cell(2,1)

Now compare each cell. You will observe that cell (1,1) has both * and ^ sign. So it is Nash equilibrium. But unfortunately neither of the the two coach will go for this strategy . They will select strategy of ot offering scholarship. Inthat case both coach will be able to reduce quit from 4 to 3.

Thus optimum strategy selected is not to offer scholarship. It is known as prisoners dilema.

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Answer (b):

Cooperative game is observed, whe two players cooperate with each other to better off their position. very often in a oligopoly type of market, it is observed that two firm are ot cooperating. They are reacting with rivals action through price cut. Ultimately price is so low that both firm are suffering huge loss. In order to correct the situation, they will sit together and will form a agreement to cooperate. As per agreement both will sale the product at a fixed price. Also they will operate in the market as agreed in the contract deed.

Success of coopeative game depends upon two points:

1. It will improve existing poor situation.

2. None of them will violate the terms of agreement.

suppose in previous example, two strategies are 1. offer scholarship and 2. Offer some other facilities . Pay off tale is:

In the above pay off matrix, both coach will gain if they select a strategy which its rival has not adopted. If coach1 select scholarship, then coach 2 will select other facility strategy and vice versa. If both of them select samestrategy, then they will suffer. Thus for their own sake, they will sit together and decide on the strategy they will adopt. No one should violate the agreement.

Coach 2 Scholarship No scholarship Coach 1 Scholarship 6,6 10,2 No Scholarship 2,10 4,4

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