Chapter 6. Nash Equilibria

10.

Identify all the Nash equilibria in the games in Figures 59, 60, and 61.

Figure 59

Figure 60

Figure 61

Answers

10.

In Figure 59 A,X is the only Nash equilibrium. A is a dominant strategy for Player One, and X is a dominant strategy for Player Two. Since a dominant strategy gives a player the highest payoff no matter what, a player would always regret not playing a dominant strategy. Thus, if a dominant strategy exists, a player must play it in any Nash equilibrium.

In Figure 60 A,Y and B,W are the Nash equilibria. If Player One plays A, Player Two cannot do any better than playing Y, so Player Two would be happy playing Y in response to A. Similarly, if Player Two chooses Y, Player One cannot do any better than playing A. Thus A,Y is a Nash equilibrium. By similar logic one can show that B,W is also a Nash equilibrium. X is a strictly stupid strategy for Player Two, so it will never be part of any Nash equilibrium since Player Two will always be unhappy playing X. A,W is not a Nash equilibrium because if Player Two Plays W, Player One would be unhappy playing A because he would get a higher payoff picking B. B,Y is not a Nash equilibrium because if Player One picks B, Player Two would regret playing Y since he would have gotten a higher payoff playing W in response to B.

Figure 61 is the matching pennies game we have previously seen, and it is also an outguessing game. Recall that in an outguessing game the players always randomize. In this game there is no Nash equilibrium where either player always plays one strategy. For example, if Player One played A and Player Two played X, then Player Two would regret not playing Y in response to Player One choosing A. For any of the four possible combinations (A,X; A,Y; B,X; and B,Y) one of the players would always regret his choice. Although this book hasn’t fully motivated this result, there is a Nash equilibrium where each player randomizes and picks each strategy half the time.

11.

Two firms can produce widgets. It costs firm one $10 to produce each widget, and it costs firm two $13 to produce each widget. Both firms choose the selling price of their widgets. The customers will buy their widgets from whomever charges the least. If the firms set the same price, all customers, for some reason, go to firm one. What is the Nash equilibrium of this game?

Answers

11.

There is a Nash equilibrium where both firms charge $13. This outcome is a Nash equilibrium because if firm two charges $13, firm one gets no customers if it charges more than $13, and all the customers if it charges $13 or less. Thus, firm one’s optimal response to firm two’s charging $13 is for it to charge $13, too. If firm one charges $13, then there is no possible way for firm two ever to make a profit. If it charges less than $13 (which is firm two’s cost of production), then firm two would actually lose money on every sale. If firm two charges $13 or more, then it would get no customers. Thus, if firm one charges $13, firm two would not regret charging $13 because it could not possibly do better.

There does not exist a Nash equilibrium where either firm charges more than $13. If this occurred, each firm would want to slightly undercut the other and take all the customers. There can’t be a Nash equilibrium where a firm charges an amount below its cost and loses money, because such a firm would rather charge a very high amount and get no customers.

There are, however, many other Nash equilibria besides the one where both firms charge $13. It’s a Nash equilibrium for both firms to charge the same price if that price is between $10 and $13. In these equilibria all the customers will, by assumption, go to firm one, so firm two will make zero profit. Recall, however, that if firm one charges less than $13, it’s impossible for firm two to make money anyway. Thus, if firm one charges less than $13, firm two does not regret a strategy that gives it zero (as opposed to a negative) profit. Furthermore, if firm two is charging between $10 and $13, firm one can still make a profit on each customer, so it wants to charge the highest possible price and still take all the customers. Thus, if firm two charges between $10 and $13, firm one’s optimal response is to charge the same price.