Game Theory

Exercise 7:

If the voter believes that the majority of the others votes for Arnold, it is rational to vote for Bush, since he is then indifferent from voting for Arnold or Bush. All prefer Arnold to win the election. So it is also rational to think that the majority will vote for Arnold.

Combining this two arguments for each voter will lead to the problem that it can be the case that the majority will vote for Bush and therefore voting for Bush will not be rational anymore. Therefore voting for Bush is not rationalizable.

Exercise 8:

Lets define our game:

* players: N = {1, 2}

* each player has two states: {w, s} weak or strong

* each player has two actions: {f,y}fight or yield

* player 1 does not know the state of player 2 and assigns probability α to the state strong and 1 − α to the state weak. Player 2 knows its state.

The players Bernoulli payoffs are as follows:

Player 2(weak)

FY

1 , -1

1,0

0,1

0,0

f player 1 y

The best response of player 2 if he is weak is Y if player one plays f and F if player 2 plays y!

Player 2(strong)

FY

-1 , 1

1,0

0,1

0,0

f player 1 y

2

The best response of player 2 if he is strong is always F no matter what player 1 plays!

* Suppose now α < 0.5 (This means it is more likely that player 2 is weak): Player 2(strong)

Fs,Fw Fs,Yw Ys,Fw Ys,Yw

f player 1 y

This implies that it is always better for player 1 to play f. So we get a uniqueNE:s1 =f,s2(w)=Y,s2(s)=F

* Suppose now α > 0.5 (This means it is more likely that player 2 is strong): Player 2(strong)

Fs,Fw Fs,Yw Ys,Fw Ys,Yw

f player 1 y

This implies that it is always better for player 1 to play y. We get again a unique NE:s1 = y, s2(w) = s2(s) = F,