Question

Game Theory

3) Consider the following game: Person B P X Y e X 313 010 r S O A\B n Y 4\1 112 A If both players choose simultaneously and without cooperation but with full information, what will be the solution to this game? Is there a better outcome for both players? Suppose instead we play the game twice under the following rules: Person A makes a choice that cannot be changed (their choice is for both plays.) Person B makes a simultaneous choice in round 1 but can change their choice for round 2 (after getting the first payout and seeing what Person A chose.) What would the final outcome be? (Technical hint: show the game in extensive form with A at the initial node (going first.) A can't change in round 2 but B can so you should get 8 outcomes. Now, eliminate the 4 outcomes that B will never let happen because of their ability to change their choice in round 2.)

Answer 1

1) nash equilibrium will be (1,2). Yes there's a better outcome which is (3,3). Here the payout will be more than the intial equilibrium.

2) game starts with A making a choice. Both x and y are available choices. He chooses one and then B makes a choice. Again A will choose the same option but now B will have a choice to deviate from the choice he has made initially. In the diagram payouts marked cross will not be available. Because B can increase its payout by deviating from the strategy. Again in the remaining four options- person B needs to choose between x and y when A has chosen x in two cases and y in other two cases. Choosing x will beneficial of person B when person A chooses x because thereafter the outcome is same in the two strategies (x,x,x,x) and( x,y,x,x). Similarly in case when A chooses y , person B should choose y to maximize payment in the first half of the game.

Now with two strategies remaining marked tick in the diagram. We can easily tell the nash equilibrium will be (3,3,3,3) because here the payout is higher for both A and B.

A у Х у A A А А y. x y X B B Х У X х y У X y payout - as per strategy X (X, X, X, X) = (3,3,3,3) (x,x,x,y) (3,3,0,0) ( x, y, x,x) = (0,0,3,3) (x, y, x, y) = (0,ooo) (y, x, y, x) (4,1,4,1) (4, 1, 42) (y, y, y,x) (1,2,4,1) (9 y, y, y) = 112, 1, 2) X Х ( لارلارلارلا) х

available options- (333) XX, X, X) (3ر5,03 لار لار لارلا ) لالا لارو) اراد ا