Loading Events

« All Events

  • This event has passed.

Colloquium: Stanislaw Spiez

June 22, 2018 @ 2:00 pm - 3:30 pm

Borsuk-Ulam Type Theorems and Existence of Equilibria in Games with Incomplete Information

Stanislaw Spiez
Institute of Mathematics of the Polish Academy of Sciences

“In 1968, R. Aumann, M. Mashler and R. Sterns proved that any undiscounted infinitely repeated two-person zero-sum game of incomplete information on one side has a Nash equilibrium. They posed a problem whether such equilibrium exists in the more general, nonzero-sum, case. A brief description of these games is as follows. A game between two players A and B proceeds in infinitely many successive stages. In the 0-stage a k is chosen from a finite set K of “states of nature” according to a probability distribution known to both players. In any subsequent stage each of the players selects a “pure action” from a finite sets I (for A) and J (for B), gaining a stage-payoff A_k(i;j) (for A) or B_k(i;j) (for B), which depends only on the pure actions i in I and j in J selected in this stage and the “true state of nature” k , chosen at stage 0. At any stage the players also know the pure actions both of them took on proceeding stages and A (but not B) knows the outcome k in K of the 0-stage. We settle in the positive the problem stated above and extend this result to more general games of this type.
Several classical proofs in game theory depend on various fixed point and related theorems. Our proofs depend on new topological results. One of them in its simplest form, states that if x_0 is a point of a compact subset C of R^n and f : C → Y is a mapping such that dimension of f(intC) is less then n , then in the boundary of C there exists a set C_0 mapped by f into a singleton and containing x_0 in its convex hull. The resemblance with the Borsuk-Ulam theorem is that if C is an n-ball and Y = R^n-1, then the later says that C_0 may be taken so as to consist of two points only. We also prove a parametric version of the Borsuk-Ulam theorem which solves a problem related to a conjecture that is relevant for the construction of equilibrium strategies in a very general class of repeated two-player games with incomplete information.
This research is joint with T. Schick, R. S. Simon and H. Torunczyk.”

Light refreshments will be served @ 1:30PM in EMS E424A.

Details

Date:
June 22, 2018
Time:
2:00 pm - 3:30 pm
Event Category:

Venue

EMS Building, Room E424A
E424A; 3200 N Cramer St.
Milwaukee, WI 53211 United States
+ Google Map
Phone
414-229-4836
View Venue Website