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Bayesian Equilibria and Shifting Power Bargaining Games
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Definition An outcome ω ∈ Ω is incentive compatible if and only if for all r, r ∈ [r, r],
U
R
(r; r) = max
r
U
R
(r ; r),
(5)
or
U
R
(r; r) ≥ U
R
(r ; r).
(6)
While incentive compatibility is necessary and sufficient to characterize the entire set of Bayesian
equilibria for a any game, it is sometimes useful to restrict this set by examining only those equilib-
rium that meet additional constraints. For example, we may want to add some sort of participation
or individual rationality constraints on the set of incentive compatible equilibria. We discuss such
possibilities in the conclusion.
Our first result, which also holds in discrete time, establishes a “compound” monotonicity
condition that must hold for all incentive compatible outcomes.
Lemma 1 Incentive compatibility implies that for all r ≥ r ,
p(r)e
−ρt(r)
≤ p(r )e
−ρt(r )
.
(7)
Proof By incentive compatibility, U
R
(r;r) ≥ U
R
(r ;r) and U
R
(r ;r ) ≥ U
R
(r;r ). Substituting in
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| | Authors: Gochal, Joseph. and Levy, Jack. |
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Definition An outcome ω ∈ Ω is incentive compatible if and only if for all r, r ∈ [r, r],
U
R
(r; r) = max
r
U
R
(r ; r),
(5)
or
U
R
(r; r) ≥ U
R
(r ; r).
(6)
While incentive compatibility is necessary and sufficient to characterize the entire set of Bayesian
equilibria for a any game, it is sometimes useful to restrict this set by examining only those equilib-
rium that meet additional constraints. For example, we may want to add some sort of participation
or individual rationality constraints on the set of incentive compatible equilibria. We discuss such
possibilities in the conclusion.
Our first result, which also holds in discrete time, establishes a “compound” monotonicity
condition that must hold for all incentive compatible outcomes.
Lemma 1 Incentive compatibility implies that for all r ≥ r ,
p(r)e
−ρt(r)
≤ p(r )e
−ρt(r )
.
(7)
Proof By incentive compatibility, U
R
(r;r) ≥ U
R
(r ;r) and U
R
(r ;r ) ≥ U
R
(r;r ). Substituting in
9
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