Question

#1. (30 points) Consider the following normal-form game. (a) (10 points) Find all pure strategy Nash equilibria. (b) (20 points) Find all mixed strategy Nash equilibria. EFG | A 0,0 3, 4, 1 B5,5 0,01,-1 C 2.0 1,0 2,6 D 1,0 1,4 6,3

Answer 1

A) Pure Nash equilibrium is a specification of a strategy for each player such that no player would benefit by changing his strategy, provided the other players don't change their strategies. This concept, as simple as it sounds, often leads to counterintuitive ”solutions” (bolded in above figures).

In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

In terms of game theory, if each player has chosen a strategy, and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and their corresponding payoffs constitutes a Nash equilibrium.

B)

A mixed strategy Nash equilibrium involves at least one player playing a randomized strategy and no player being able to increase his or her expected payoff by playing an alternate strategy. A Nash equilibrium in which no player randomizes is called a pure strategy Nash equilibrium.

Let us consider the matching pennies game again, "Matching pennies again".

	Heads	Tails
Heads	1,-1	-1,1
Tails	-1,1	1,-1

Suppose that Row believes Column plays Heads with probability p. Then if Row plays Heads, Row gets 1 with probability p and –1 with probability (1 – p), for an expected value of 2p – 1. Similarly, if Row plays Tails, Row gets –1 with probability p (when Column plays Heads), and 1 with probability (1 – p), for an expected value of 1 – 2p. This is summarized in Figure 16.14 "Mixed strategy in matching pennies".

If 2p – 1 > 1 – 2p, then Row is better off, on average, playing Heads than Tails. Similarly, if 2p – 1 < 1 – 2p, then Row is better off playing Tails than Heads. If, on the other hand, 2p – 1 = 1 – 2p, then Row gets the same payoff no matter what Row does. In this case, Row could play Heads, could play Tails, or could flip a coin and randomize Row’s play.