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associate with it a revelation-type pre-Bayesian game, in which each payoff
function p i is defined by
p i (( θ i −i ) i ):= u i (( f, t )( θ i −i ) i ) .
Then the mechanism is incentive compatible iff in the associated pre-Bayesian
game for each player truth-telling is a dominant strategy.
By Groves's Theorem 1.28 we conclude that in the pre-Bayesian game
associated with a Groves mechanism, ( π 1 ( · ) ,...,π n ( · )) is a dominant strategy
ex-post equilibrium.
1.9 Conclusions
1.9.1 Bibliographic remarks
Historically, the notion of an equilibrium in a strategic game occurred first
in Cournot [1838] in his study of production levels of a homogeneous product
in a duopoly competition. The celebrated von Neumann's Minimax Theorem
proved by von Neumann [1928] establishes an existence of a Nash equilibrium
in mixed strategies in two-player zero-sum games. An alternative proof of
Nash's Theorem, given in Nash [1951], uses Brouwer's Fixed Point Theorem.
Ever since Nash established his celebrated theorem, a search has continued
to generalise his result to a larger class of games. A motivation for this
endeavour has been the existence of natural infinite games that are not
mixed extensions of finite games. As an example of such an early result
let us mention the following theorem due to Debreu [1952], Fan [1952] and
Glicksberg [1952].
Theorem 1.35
Consider a strategic game such that
each strategy set is a non-empty compact convex subset of a complete
metric space,
• each payoff function p i is continuous and quasi-concave in the ith argu-
ment. 5
Then a Nash equilibrium exists.
More recent work in this area focused on the existence of Nash equilibria
in games with non-continuous payoff functions, see in particular Reny [1999]
and Bich [2006].
5 Recall that the function p i : S → R is quasi-concave in the i th argument if the set
{s i ∈ S i
| p i ( s i ,s −i ) ≥ p i ( s ) } is convex for all s ∈ S .
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