Separable Quitting Games with Perfect Information and Differential Equations

Eilon Solan

We introduce a new approach to study subgame-perfect equilibrium payoffs in stochastic games: the differential equations approach.

We apply our approach to quitting games with perfect information. Those are sequential game in which at every stage one of n players is chosen; each player is chosen with probability 1/n

)the extension to the case where the choice is not uniform is also discussed(. The chosen player i decides whether he quits, in which case the game terminates and the terminal payoff is some vector a_i in Rⁿ, or whether he continues, in which case the game continues to the next stage. If no player ever quits, the payoff is some vector a₀ in Rⁿ.

We define a certain differential inclusion, prove that it has at least one solution, and prove that every vector on a solution of this differential inclusion is a subgame-perfect equilibrium payoff.