Solving for Nash Equilibria Without Calculus
Solving for Nash Equilibria Without Calculus WITH AN INFINITE NUMBER of strategies, an exhaustive search for Nash equilibria will not work. Considering each strategy profile and asking whether it is a Nash equilibrium will take . . . well, forever! We need to be cleverer than that. Although there is no universal algorithm for solving games with infinite strate- gies—no “plug and chug” method that will always work—there are steps we can take that can, in certain instances, make our task vastly easier.
The trick is to focus on understanding the decision problem faced by a player in a game. We need to get “inside the head” of a player and figure out his incentives. If you were that player, what would you do? How can a player improve his payoff? How does a player best respond to what other players are doing? Once you’ve gained some understanding of a player’s situation, you may have the insight to begin lopping off lots and lots of strategy profiles.
To gain that understanding, one method that often works is to “dive into the problem.” Choose a strategy profile—any strategy profile—and ask whether it
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*The use of the term “rational” here is totally different from how we’ve been using it with respect to behavior.
**For example, equals .25, which terminates, and equals .142857142857142857 . . . , which repeats the sequence 142857.
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6.2 Solving for Nash Equilibria Without Calculus 149
is a Nash equilibrium. Unless you’re extraordinarily lucky, it will not be an equilibrium. What you then need to do is understand why it is not an equilib- rium, for the answer you get may be applicable to many other strategy pro- files. Though this approach may seem rather mystical, you’ll have a better ap- preciation for it when you see it at work in a few examples.