Solution Set 12 - Zero Sum Games

1. (Increasing payoffs in strictly competitive game)

a. Let ui be player i's payoff function in the game G, let wi be his payoff function in G¢, and let (x*, y*) be a Nash equilibrium of G¢. Then, using part (b) of Proposition 22.2, we have w1(x*,y*) = minymaxxw1(x, y) ³ minymaxxu1(x,y), which is the value of G.

b. This follows from part (b) of Proposition 22.2 and the fact that for any function  f  we have maxxÎX f (x) ³ maxxÎY f (x) if Y ÍX.

c. In the unique equilibrium of the game

 
L R
T
3,3 1,1
1,0 0,1
B
player 1 receives a payoff of 3, while in the unique equilibrium of
 
L R
T
3,3 1,1
4,0 2,1
B
she receives a payoff of 2. If she is prohibited from using her second action in this second game then she obtains an equilibrium payoff of 3, however.

2. (Exercise) Formulate a formal concept which will capture the situation that in a zero-sum game where each player has to choose an action from a set X, player 1 is discriminated in favor. What can you say about the Nash equilibrium in such a game (assuming it exists)?


Let G=({1,2},(A1,A2),(u1,u2)) be a zero sum game with A1=A2=X and u1=-u2. Let us say that G is in favor of player 1 if for any x,y Î X: u1(x,y)³u2(y,x). The intuition for the property is that the only "assymmetries" of the game favor player 1. Let G favor player 1 and (x*,y*) be a NE of G,  then u1(x*,y*)= maxxÎXminyÎXu1(x, y) ³ maxxÎXminyÎXu2(y,x)=maxyÎXminxÎXu2(x,y)=u2(x*,y*). Since u1=-u2, we have that u1(x*,y*)³0³u2(x*,y*), i.e. the player favored by G is also favored in terms of payoffs in any equilibrium of G.