Applied Mathematics Colloquium by Jianfeng Zhang: Set Value and Efficiency of Nonzero Sum Games
Speaker: , professor of mathematics, University of Southern California
Title: Set Value and Efficiency of Nonzero Sum Games
Abstract: A game problem is typically ill-posed in various senses. In this talk we investigate two issues concerning Nash equilibria. First, a game may have multiple equilibria with multiple values. We propose to study the set value of games, which roughly speaking is the set of values over all equilibria and thus is by definition unique. We show that the set value function for a dynamic game enjoys many properties of the standard value function for a control problem, most notably the dynamic programming principle, and we shall establish a set-valued Ito formula and develop a notion of set-valued PDE. Somewhat surprisingly, for a dynamic game, the set value is typically convex.
Next, Nash equilibria may not be efficient, in the sense that their aggregate value could be much worse than the social optimum. On the other hand, games are typically unstable. This seemingly bad property can actually be a good news: a small perturbation of the game mechanism may improve the efficiency significantly. Such a feature could be important in practice. For example, assume a government has some limited resources to support two projects. Instead of supporting the more "important" project as the conventional wisdom would suggest, our study suggests it is wiser to invest in the project whose efficiency is more sensitive to small extra support.
Applied Mathematics Colloquium