How did I figure out that those six strategy profiles are equilibria?
How did I figure out that those six strategy profiles are equilibria? It was not through exhaustive search (I’m far too lazy for that), but from thinking about the companies’ incentives. First, note that it cannot be an equilibrium to have three companies enter and not to have company 1 be one of them. If company 1 is not one of the three entrants, then its payoff is zero, but it can then earn 50 by entering. Hence, if there is a Nash equilibrium with three en- trants, company 1 must be one of them. Next, note that any of the other four companies earns a positive payoff from entry when there are two other
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134 CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
entrants (all of the payoffs are positive in column 2) and that each earns a negative payoff from entry when there are three other entrants (excluding the payoff to company 1, all of the payoffs are negative in column 3). From this state of affairs, we can conclude that any strategy profile in which company 1 enters and two other companies enter is a Nash equilibrium.
■ Is there a Nash equilibrium with more than three entrants? The gross profit for a company when there are four entrants is 150, but only one company has an entry cost that doesn’t exceed 150. Hence, if four com- panies enter, then at least one of them must have a negative payoff, which means that it is better for it to not enter. This logic is also appar- ent from Table 5.6, in which only one company has a nonnegative pay- off under column 3 or column 4. There is, then, no Nash equilibrium with more than three entrants.
An interesting property of equilibrium is that the most efficient companies need not be the ones that enter. The most efficient equilibrium is the one in
which companies 1, 2, and 3 enter, since they have the lowest entry costs. However, there are also equilibria in which companies 4 and 5 enter instead of companies 2 and 3. Given that companies 4 and 5 are anticipated en- tering (along with company 1), entry becomes unprof- itable for companies 2 and 3, even though if they were to change places with companies 4 and 5, they would make more money than 4 and 5.