Convergence and Approximation Results for Non-Cooperative Bayesian Games: Learning Theorems

• Published in: Economic theory 1994-01, Vol.4 (6), p.843-857
• Main Authors: Koutsougeras, Leonidas C., Yannelis, Nicholas C.
• Format: Article
• Language: English
• Subjects: Approximation; Banach space; Bayes theorem; Game theory; Learning; Martingales; Mathematical functions; Mathematical sets; Mathematical theorems; Nash equilibrium; Perceptron convergence procedure; Rationality
• ISSN: 0938-2259; 1432-0479
• DOI: 10.1007/BF01213815

Let T denote a continuous time horizon and$\{G^{t}\colon t\in T\}$be a net (generalized sequence) of Bayesian games. We show that: (i) if$\{x^{t}\colon t\in T\}$is a net of Bayesian Nash Equilibrium (BNE) strategies for$G^{t}$, we can extract a subsequence which converges to a limit full information BNE strategy for a one shot limit full information Bayesian game. (ii) If$\{x^{t}\colon t\in T\}$is a net of approximate or$\varepsilon _{t}\text{-}{\rm BNE}$strategies for the game$G^{t}$we can still extract a subsequence which converges to the one shot limit full information equilibrium BNE strategy. (iii) Given a limit full information BNE strategy of a one shot limit full information Bayesian game, we can find a net of$\varepsilon _{t}\text{-}{\rm BNE}$strategies$\{x^{t}\colon t\in T\}$in$\{G^{t}\colon t\in T\}$which converges to the limit full information BNE strategy of the one shot game.