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lecture6 notes

G price competition models congestion competition

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Unformatted text preview: osition 3. Since G k is “essentially finite” for each k , it follows using Nash’s Theorem that it has a 0-equilibrium, which we denote by σk . Then, by Proposition 2, σk is a 2αk -equilibrium of G . Since Σ is compact, {σk } has a convergent subsequence. Without loss of generality, we assume that σk → σ. Since 2αk → 0, σk → σ, by Proposition 1, it follows that σ is a 0-equilibrium of G . 12 Game Theory: Lecture 6 Continuous Games Discontinuous Games There are many games in which the utility functions are not continuous (e.g. price competition models, congestion-competition models). We next show that for discontinuous games, under some mild semicontinuity conditions on the utility functions, it is possible to establish the existence of a mixed Nash equilibrium (see [Dasgupta and Maskin 86]). The key assumption is to allow discontinuities in the utility function to occur only on a subset of measure zero, in which a player’s strategy is “related” to another player’s strategy. To formalize this notion, we introduce the following set: for any two players i and j , let D be a finite index set and for d ∈ D , let fijd : Si → Sj be a bijective and continuous function. Then, for each i , we define S ∗ (i ) = {s ∈ S | ∃ j �= i such that sj = fijd (si ).} (1) 13 Game Theory: Lecture 6 Continuous Games Discontinuous Games Before stating the theorem, we first introduce some weak continuity conditions. Definition Let X be a subset of Rn , Xi be a subset of R, and X−i be a subset of Rn−1 . (i) A function f : X → R is called upper semicontinuous (respectively, lower semicontinuous) at a vector x ∈ X if f (x ) ≥ lim supk →∞ f (xk ) (respectively, f (x ) ≤ lim inf k →∞ f (xk )) for every sequence {xk } ⊂ X that converges to x . If f is upper semicontinuous (lower semicontinuous) at every x ∈ X , we say that f is upper semicontinuous (lower semicontinuous). (ii) A function f : Xi × X−i → R is called weakly lower semicontinuous...
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