Question

When playing card games like poker, the strategies available to players can sometimes be summarized as:

• Use Randomness (play unpredictably so that your opponents can't understand your moves).
• Use Math (calculate probabilities based on the cards you can see, and use this information to make your decisions).
• Use Psychology (watch your opponents for signs that they have especially good or bad hands).

A particular pair of players have different levels of skill in each of these strategies, so that their payoffs from any strategy profile are given by the following game table. The strategies here are abbreviated as R, M, and P, respectively.

		Player 2
		R	M	P
Player 1	R	0, 1	2, 0	1, 1
	M	2, 1	0, 2	1, 0
	P	1, 2	1, 1	0, 1

This game has no strictly dominated strategies.

In the space below, answer the following questions:

• Does this game contain any non-rationalizable strategies? (You do not need to list them right now.)
• If so, perform IENBR until none are left. List the order in which you eliminated strategies.
• Does elimination alone reveal the Nash equilibrium of this game? If so, what is it?

Answer 1

*	*	Player 2	Player 2	Player 2
*	Payoff Matrix	R	M	P
Player 1	R	(0,1)	(2,0)	(1,1)
Player 1	M	(2,1)	(0,2)	(1,0)
Player 1	P	(1,2)	(1,1)	(0,1)

Yes, this game does contain non-rationalizable strategies.

We can see that for Player 1, playing P is not a best response to Player 2 playing anything. Hence P1's strategy P can be eliminated by IENBR.

Hence now the payoff matrix looks like:

*	*	Player 2	Player 2	Player 2
*	Payoff Matrix	R	M	P
Player 1	R	(0,1)	(2,0)	(1,1)
Player 1	M	(2,1)	(0,2)	(1,0)

Now we can see that playing R is weakly dominant over playing P for Player 2.

Hence by IENBR, we can remove P from P2.

Hence now the payoff matrix looks like:

*	*	Player 2	Player 2
*	Payoff Matrix	R	M
Player 1	R	(0,1)	(2,0)
Player 1	M	(2,1)	(0,2)

There is no Nash equilibrium in the above game after IENBR.