outcome and then reviews the beliefs about out-of-equilibrium information sets
that sustain this outcome. The player then applies a refinement criterion that de-
scribes what constitutes a reasonable belief. If, by taking into account the rea-
sonableness of these beliefs and believing that the other players do so too, at
least one player has an incentive to deviate, then this outcome is no longer an
equilibrium in the refined game.
The requirement for belief formation applied in this paper says that it should be
common knowledge among rational players that they never play a strategy pro-
file that a particular player has no incentive to play. We say that a strategy q’ is
weakly dominated by another strategy q’’ for type θ, if, no matter what beliefs
the uninformed player could possibly hold after observing the move of the in-
formed player, the expected payoff of playing q’’ always exceeds the expected
maximum payoff of playing q’ for the informed player.
Definition: Weakly dominated (WD) strategy: A strategy q’ is WD by q’’ for
type θ, if minρNBi(θ,q'',ρ)≥maxρNBi(θ,q',ρ)⇒Pi(θ,q'',1)≥Pi(θ,q',0).
The definition says that even in the case where q’’ is followed by the worst pos-
sible circumstances from the point of view of the informed player, this reduc-
tion level is still preferred to q’, even when q’ is followed by the best possible
circumstances, then q’ is weakly dominated for this type. We want to apply this
requirement to reduce the set of separating equilibria, by invoking the following
requirement.
Requirement on belief formation: If a signal q’ is weakly dominated for one
type θ, but not for the other type, then the uninformed players’ belief should
place zero probability that θ has sent q’, i.e. q’ must be followed by posterior
P(θ∣q ’)=0.
Applying this relatively non-controversial requirement has tremendous cutting
power on the set of separating equilibria, as stated now:
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