Lunch In Theory This Thursday (12:00, 9/4, GCS 502c)
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Grayson York
Sep 3, 2025, 2:11:51 PM (6 days ago) Sep 3
to usc-theo...@googlegroups.com, usc-t...@googlegroups.com, Jianfeng Zhang
Hi all,
Please join us for Lunch in Theory this Thursday 9/4 at 12:00 in GCS 502c.
Reminder: please bring your own lunch, as lunch will not be provided. However, we will have an excellent speaker this week. Jianfeng Zhang will be visiting from the math department to give the following talk. Looking forward to seeing everyone!
Grayson
Title: Set Value and Efficiency of Nonzero Sum Games
Abstract: In this talk we study Nash equilibria and more importantly the values of general nonzero sum games. Unlike control problems, a game problem is typically ill-posed in various sense. In this talk we investigate two issues.
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 pertubation of the game mechanism may improve the efficiency significantly. We shall study two mechanism designs and their effects on efficiency improvement. Such understanding 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 can be improved more by the limited extra support.
Abstract: In this talk we study Nash equilibria and more importantly the values of general nonzero sum games. Unlike control problems, a game problem is typically ill-posed in various sense. In this talk we investigate two issues.
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 pertubation of the game mechanism may improve the efficiency significantly. We shall study two mechanism designs and their effects on efficiency improvement. Such understanding 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 can be improved more by the limited extra support.
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