Question

Question is based off of Game Theory Mixed strategy and Pure strategy Nash Equilibrium

Each of two players chooses a positive integer. If player i's integer is greater than player j's integer and less than three times this integer, then player j pays $1 to player i. If player i's integer is at least three times greater than player j's integer, then player i pays $1 to player j. If the integers are equal, no payment is made. Each player's preferences are represented by her expected monetary payoff.

(a) Show that the game has no Nash equilibrium in pure strategies.

(b) Show that the pair of strategies in which each player chooses 1, 2, and 5 each with probability 1/3 is a mixed strategy Nash equilibrium.

Answer 1

Let us write the payoff functions first. It will help us understand the proof. (i,j) denotes a pure strategy profile where the first entry is the positive integer chosen by player 'i', and the second entry is the integer chosen by player 'j'.

Since it is a zero-sum game(since one player is always paying the other player), .

(a) We prove by contradiction.

Let be a Nash Equillibrium in pure strategies ( in this game the positive integers are the pure strategies).

Case 1: If  

Fix  ,

Thus, there exists a unilateral deviation for player i. Thus, a contradiction.

Case 2: Without Loss of Generality , if

If then set . Thus again there exists a unilateral deviation for player j. Thus, a contradiction.

If then set . Thus, there exists a unilateral deviation  for player i. Hence a contradiction.

(b) Now, in order to show that the given pair of mixed strategies are indeed Nash Equillibrium, we basically show that they are mutual best responses.

Without loss of generality, fix Player j's mixed strategy. In the table below coloumn 'i' denotes the possible numbers player i can will choose. The triplet corresponding to the values of i represents (outcome of player i when player j chooses 1, outcome of player i when player j chooses 2,outcome of player i when player j chooses 5) . Last coloumn contains the corresponding expected payoffs.

	Outcome triplet	Expected payoff of player i
1	(draw,loss,win)	0
2	(win, draw,loss)	0
3	(loss,win,loss)	-1/3
4	(loss,win,loss)	-1/3
5	(loss,win.draw)	0

It is for this reason player i will choose [1,2,5] each with prob 1/3 only as they maximize the expected payoff given the player j's strategy.

Since the mixed strategies are the same, we can conclude that they are mutual best responses.