Solution concept

These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game.

Each successive solution concept presented in the following improves on its predecessor by eliminating implausible equilibria in richer games.

In dynamic games, backward induction can be used to eliminate unrealistic Nash equilibria.

Backward induction assumes that players are rational and will make the best decisions based on their future expectations.

For example, consider a dynamic game with an incumbent firm and a potential entrant to the industry.

If the incumbent fights, it will lower its prices, run the entrant out of business (incurring exit costs), and damage its own profits.

Hence, fight can be considered a best response for the incumbent if the entrant does not enter, resulting in another Nash equilibrium.

However, this second Nash equilibrium can be eliminated by backward induction because it relies on a noncredible threat from the incumbent.

By the time the incumbent reaches the decision node where it can choose to fight, it would be irrational to do so because the entrant has already entered.

A perfect Bayesian equilibrium (PBE) is a specification of players' strategies and beliefs about which node in the information set has been reached by the play of the game.

Just as in a Nash equilibrium no player's strategy is strictly dominated, in a PBE, for any information set no player's strategy is strictly dominated beginning at that information set.

Unlike the above solution concepts, no player's strategy is strictly dominated beginning at any information set even if it is off the equilibrium path.

Thus in PBE, players cannot threaten to play strategies that are strictly dominated beginning at any information set off the equilibrium path.

The Bayesian in the name of this solution concept alludes to the fact that players update their beliefs according to Bayes' theorem.

Kohlberg and Mertens (1986) introduced the solution concept of Stable equilibrium, a refinement that satisfies forward induction.

To resolve the problem Jean-François Mertens introduced what game theorists now call Mertens-stable equilibrium concept, probably the first solution concept satisfying both forward and backward induction.

Forward induction yields a unique solution for the burning money game.