On games without approximate equilibria
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On games without approximate equilibria Yehuda John Levy1 Accepted: 26 August 2020 © The Author(s) 2020
Abstract This note shows that the work by Simon and Tomkowicz (Israel J Math 227(1):215– 231, 2018) answers another outstanding open question in game theory in addition to the non-existence of approximate Harsányi equilibrium in Bayesian games: it shows that strategic form games with bounded and separately continuous payoffs need not possess approximate equilibria. Keywords Approximate equilibrium · Bayesian games · Discontinuous games A strategic form game, with bounded and separately continuous payoffs, is a triple N , (X i )i∈N , (u i )i∈N , where N a finite collection of players, and for each i ∈ N , X i isa compact convex set in some locally convex topological vector space, and u i : i∈N X i → R is a bounded function which is affine in X i and which is, for each j ∈ N , continuous in X j when other components are held constant. A particular case of such games is when X i is the set of probability measures Δ(Ai ) over a compact set Ai in some metric space endowed with the topology of weak convergence, and u i is the multi-linear extension of some bounded separatelycontinuous payoff function known as the mixed extension (to this extent, i Ai embeds naturally into on i Ai , X = i i i Δ(Ai ).) In the two-player zero-sum case, Sion’s minmax theorem, Sion (1958), shows that the separate continuity is sufficient to guarantee existence of the value; the existence of optimal strategies follows. (Indeed, if u : X 1 × X 2 → R is separately continuous, x → min y∈Y u(x, y) is upper-semicontinuous and hence obtains a maximum at an optimal strategy of the maximizer, i.e. of Player 1; and similarly for Player 2.) In the non-zero-sum case, however, an equilibrium need not exist; Vieille (1996) presents an example of a two-player non-zero-sum game ‘on the square’, i.e. in which each agent has action space [0, 1], the mixed extension of which possesses no equilibrium.
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Yehuda John Levy [email protected] Adam Smith Business School, University of Glasgow, Glasgow, UK
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Y. J. Levy
It has remained open until now, however, the question of whether such games must possess ε-equilibria for each ε > 0; a profile x = (xi )i∈N is a ε-equilibrium iff sup u i (yi , x−i ) ≤ u i (x) + ε, ∀i ∈ N
(1)
yi ∈X i
(The case ε = 0 is simply an equilibrium.) This note shows that the example Simon and Tomkowicz (2018) gives a negative answer to this question. The Bayesian game model of Simon and Tomkowicz (2018) fits into the following general model of Bayesian games: An underlying state space Ω with σ -algebra F and probability measure μ is given on F. There are N players, each (for simplicity) with a finite action set Ai , a bounded measurable payoff function ri : Ω × i∈N Ai → R, and a knowledge σ -algebra Fi ⊆ F. (This model is similar to the model in Stinchcombe (2011), except there the state space is a product of types spaces for each player, endowed with product σ -algebra; the general framework recalled here transl
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