Applications of Minmax Regularity - Equilibria in Borel Games with Tail Measurable Payoffs
János Flesch, Arkadi Predtetchinski, Eilon Solan and Galit Ashkenazi-Golan
GAMENET training school on Borel games, 4 August 2021
Repeated Borel games - results
Gale and Stewart (1953): In zero-sum alternating moves games with winning sets, if W is open or closed, the game is determined.
Martin (1975): In zero-sum alternating moves games with winning sets, if W is Borel measurable, then the game is determined. (Ron Peretz)
Mertens – Neyman (1986): In multiplayer nonzero-sum alternating move games an Equilibrium exists. (János Flesch)
Martin (1998): In zero-sum simultaneous move games a value exists. (Ron Peretz)
Talk outline
Our contribution – tail measurable sets
Our contribution – tail measurable payoff functions
Our result : non zero-sum simultaneous move games with tail-measurable payoffs
Equilibrium and epsilon-equilibrium
The minmax
Discontinuity of the minmax
Discontinuity of the minmax – an example
Martin (1998) and Maitra, Sudderth, Purves (1992) prove the regularity of the minmax in case of two players.
Proof method: Like Martin (1998). An auxiliary alternating move game is generated where player I tries to “prove” that the value can be obtained while player II tries to select a play to prove player I wrong.
Regularity of the minmax
Existence of equilibrium for games with tail-measurable winning sets
Existence of equilibrium for games with tail-measurable winning sets
Existence of equilibrium for games with tail-measurable payoff functions
Folk theorem for games with tail-measurable payoff functions
Folk theorem for games with tail-measurable payoff functions – example
Folk theorem for games with tail-measurable payoff functions – example
| L | C | R |
T | | | |
M | | | |
B | | | |
-1,4
4,-1
minmax
minmax
Folk theorem for games with tail-measurable payoff functions – example
| L | C | R |
T | | | |
M | | | |
B | | | |
-1,4
4,-1
minmax
minmax
Folk theorem for games with tail-measurable payoff functions
THANK YOU!
One more regularity result
Period 1
A
B
Payoff: 0
A
B
A
B
A
B
Period 2
Period 3
Period t
Payoff: 1/2
Payoff: 2/3
Payoff: 1 – 1/t
Forever B, Payoff: 0
Why Epsilon Equilibrium?