Question

. (9 points) For the following payoff matrix, X Y S -10,-10 2,-1 TI -1,2 | 1,1 (o) (e pointa) Create a story for this game. Explain briefly all compenents of a strategic game tb) (e pointa) Write the best response functions of each player and identify the pure strategy Nash equilitbria. (e) (3 points) Find the mixed strategy equilibrium. (a) (e points) What are the expected payoffs for both players?

Answer 1

a)Consider a one -shot game between two players. which yields a  payoff matrix as given above in the question.Assume that Player 1 corresponds to the rows while Player 2 corresponds to columns. Each player have two strategies to make .For example,Player 1 can either choose between S & T .Similarly ,Player 2 can either choose X or Y depending on the move which the other player has made. Hence this is an example of a simultaneous games where both players chooses simultaneously having no idea what the other player will choose .They just know the payoffs associated with each move.

If Player 1 chooses 'S' and player 2 chooses' X' then both of them will get negative payoffs. And if player 1 chooses 'T' while player 2 chooses ' Y ' then each would be getting an equal and positive payoff. But if Player 1 chooses 'S' and player 2 chooses 'Y' then player would be getting a positive payoff while the other player would be getting a negative one. Similarly if player 1 goes for 'T' while player 2 go for 'X' then in that case player 1 would get a negative payoff while player 2 would get a positive payoff.

b)Consider the following argument:

If the Player 2 chooses a strategy X , then the best choice of Player 1 is to go for strategy 'T' as the payoff from strategy T (-1) is higher than the payoff from strategy S (-10).

And If the player 2 chooses a strategy Y ,then the best choice of Player 1 is to go for strategy 'S' as it is fetching higher payoff then the payoff received from strategy T.

Similarly The response function in case of Player 1 making the move first will be of following argument v:

If the Player 1 chooses a strategy 'S' first then Player 2 would go for strategy 'Y' because it will yield him a more payoff.

And if the Player 1 chooses a strategy 'T' first Player 2 would go for strategy 'X'

Hence there is no consistency here .

We should remember that in a pure strategy, whichever cell corresponds to the best response for all players is denoted as a Nash Equilibrium.Hence, it is that equilibrium where no player would benefit from changing his/her strategy given the strategy of the other player Hence the cells (S,Y) & (T,X) are the two nash equilibria in this case.For example, consider the case( T,X) ,Here player 1 has decided to go for 'T' while Player 2 goes for 'X' .Would either of them be benefited by doing opposite ,given that the other player doesnt change his initial strategy? not really, as it would yield a lower payoff to them .

Hence these are the two equilibria where each player would prefer one of these two scenarios . Player 1 prefers (S,Y) while Player 2 prefers (T,X) .

Also, (T,Y) is not a nash equlibrium as Player1 could still be better off in case he changes his strategy to S yielding him a higher payoff(2>1)

(c) & (d) let us say player 2 plays X with probability 'q' and 'Y' with probability (1-q) .And player 1 plays S with probability p and T with probability (1-p) . Now when player 1 plays S ,then his payoff depends on whether player 2 is playing X or Y .Similarly when player 1 plays T then his payoff from playing T depends in whether player 2 chooses X or Y . And similarly when player 2 chooses any of the strategy then his payoff depends upon the move of player 1 .Hence there is no consistency here.For example, player 1 will be indifferent between S and T only if Expected payoff from S = Expected payoff from T . These expected payoffs are in turn the functions of X & Y.

So player 1 expectation from S is either -10 or -1 . He'll get -10 if player 2 plays X with probability q while he'll get 2 if player 2 plays Y with probability (1-q). Hence his expected payoff from playing S is : E(S) = -10(q) + 2(1-q). Similarly his expected payoff from playing T is : E(T) = -1(q)+1(1-q). Hence ,it can be said that both E(S) & E(T) are the functions of q & (1-q). Now player 1 would mix these two strategies if E(S) and E(T) are equal to each other . So putting E(S) equal to E(T) , we'll get q=1/10.

Similarly we can calculate expected payoff for player 2 as well .

E(X) = -10p+2(1-p)

E(Y)=-1p+1(1-p)

So putting E(X)=E(Y) gives us p=1\10

Hence mixed strategy nash equilibrium is player 1 plays S with probability p (=1\10) and strategy T with probability (1-p = 1-1\10 = 9\10). And player 2 is going to play X with probability q (=1\10) and Strategy Y with probability 1-q= 1-1\10 = 9\10

Thanks