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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
,θ
−i
)
,θ
i
):=
u
i
((
f, t
)(
θ
i
,θ
−i
)
,θ
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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