This game has two pure strategy equilibria, namely one of the two firms enters the market niche and the other stays out. But, unlike the games we have encountered thus far, neither player has a dominant strategy. When a player has no dominant strategy, she should consider playing a mixed strategy. In a mixed strategy, each of the various pure strategies is played with some probability, say for Strategy 1, for Strategy 2, etc with . What would be the best mixed strategies for Firms A and B? Denote by the probability that Firm A enters the market niche. Therefore is the probability that Firm A stays out. Similarly, Firm B enters the niche with probability and stays out with probability . The key insight to a mixed strategy equilibrium is the following. Every pure strategy that is played as part of a mixed strategy equilibrium has the same expected value. If one pure strategy is expected to pay less than another, then it should not be played at all. The pure strategies that are not excluded should be expected to pay the same. We now apply this principle. The expected value of the ``Enter'' strategy for Firm A, when Firm B plays its mixed strategy, is
The expected value of the ``Stay out'' strategy for Firm A is . Setting we get
Using , we obtain
As you can see, the payoffs of this mixed strategy equilibrium, namely (0,0), are inefficient. One of these firms could make a lot of money by entering the market niche, if it was sure that the other would not enter the same niche. This assurance is precisely what is missing. Each firm has exactly the same right to enter the market niche. The only way for both firms to exercise this right is to play the inefficient, but symmetrical, mixed strategy equilibrium. In many industrial markets, there is only room for a few firms - a situation known as natural oligopoly. Chance plays a major role in the identity of the firms that ultimately enter such markets. If too many firms enter, there are losses all around and eventually some firms must exit. From the mixed strategy equilibrium, we can actually predict how often two firms enter a market niche when there is only room for one: with the above data, the probability of entry by either firm is 2/3, so the probability that both firms enter is . That is a little over 44% of the time! This is the source of the inefficiency. The efficient solution has total payoff of 100, but is not symmetrical. The fair solution pays each player the same but is inefficient. These two principles, efficiency and fairness, cannot be reconciled in a game like Market Niche. Once firms recognize this, they can try to find mecanisms to reach the efficient solution. For example, they may consider side payments. Or firms might simply attempt to scare off competitors by announcing their intention of moving into the market niche before they actually do so.
An important case where mixed strategies at equilibrium are always efficient is the case of constant-sum games. Interestingly, the optimal mixed strategies can be computed using linear programming, one linear program for each of the two players. We illustrate these ideas on an example.
The payoff matrix is as follows.
Suppose Player 1 chooses the mixed strategy , where
The expected payoff of Player 1 is when Player 2 plays Stone, when Player 2 plays Paper and when Player 2 plays Scissors. The best Player 2 can do is to achieve the minimum of these three values, namely . Player 1, on the other hand, should try to make the quantity v as large as possible. So, Player 1 should use probabilities , and that maximize . This can be written as the following linear program.
Similarly, we obtain Player 2's optimal mixed strategy , where
by solving the linear program
For the Stone, Paper, Scissors game, the optimal solutions are and with .
In general, for any 2-person zero-sum game, let and be the optima of the linear programs for Players 1 and 2 respectively. Because of the special form of these linear programs, it can be shown that . So, the payoff at equilibrium is . The fact that, for each player, the equilibrium solutions are the optimal solutions of a linear has another consequence: All equilibrium solutions have the same payoff.