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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 0equilibrium, 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
0equilibrium 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, congestioncompetition
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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 Spring '10
 AsuOzdaglar

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