Answer to Question #91923 in Microeconomics for John Manda

Question

Players A and B are engaged in a coin-matching game. Each shows a coin as either heads or tails. If the coins match, B pays A $1. If they differ, A pays B $1. a. Write down the payoff matrix for this game, and show that it does not contain a Nash equilibrium. b. How might the players choose their strategies in this case?

Answer 1

Let's write this game in normal form! If the pennies match, A wins one penny and B loses one, so their payoffs are (1,-1). If they pennies don't match, A loses one and B wins one, so their payoffs are (-1,1). If we say

• choice 1 is heads • choice 2 is tails

then the normal form looks like this:

payoff for A:

$$A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$$

payoff for B:

$$B = \begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix}$$

If you examine this game, it's easy to see no pure strategy is a Nash equilibrium. If A chooses heads, B will want to choose tails. But if B chooses tails, A will want to choose tails. And if A chooses tails, B will want to choose heads. And if B chooses heads, A will want to choose heads! There's always someone who would do better by changing their choice.

Given a 2-player normal form game, a pair of mixed strategies (p,q), one for player A and one for player B, is a Nash equilibrium if:

(p, q) p represents A and q represents B

1) For all mixed strategies p′ for player A, p′ ⋅Aq ≤ p ⋅ Aq . p′ ⋅ Aq ≤ p ⋅ Aq.

2) For all mixed strategies q′ for player B, p ⋅ Bq′ ≤ p ⋅ Bq.