Dynamic games, beliefs, and sequential rationality

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Dynamic games, beliefs, and sequential
rationality
Consider an altered entry deterrence game played
by an entrant and a monopolist. Suppose there
are two strategies that an entrant can use to
enter in1 and in2 Suppose that the monopolist
is unable to tell which strategy the entrant uses
to enter so the monopolist
observes entry or not but doesnt observe in1 or
in2 Now there are 2 NE 1) (E plays out,
M cuts if E enters) 2) (E plays in1, M
doesnt cut if E enters) Both NE are subgame
perfect - because the only subgame is the entire
game. To eliminate the unreasonable equilibrium,
we will introduce beliefs.
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Monopolist will not cut prices
SPNE is not an effective refinement. - NE1
involves a non-credible threat - both NE are
SPNE here since the only sub-game is the entire
game - both nodes are in the same
information set - we need a different type
of refinement to remove non-credible threats
The basic content of the new equilibrium
refinement - require that players have
beliefs about the likelihood of being at the
various nodes in information sets with
multiple nodes and that players optimize given
their beliefs Require the monopolists strategy
be optimal for some belief about the strategy E
has used. - Cutting prices is not optimal
no matter which entry strategy E used - so for
any probability ? that E used in1
(1
-? ) that E used in2
the monopolist would optimally not cut prices
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System of Beliefs
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Sequentially rational given beliefs
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Strategy profile
Player i has to be optimizing given what
everyone is doing. This is a best response
notion with beliefs that apply for information
sets with multiple nodes Now two factors affect
the optimal choice 1) everyone elses
strategy 2) the players beliefs
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Weak Perfect Bayesian Equilibrium
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2nd Property
Just as an example, this is not an equilibrium
strategy, Suppose E puts probability 5/12 on
staying out 1/3 on in1 1/4 on in2 Then the
information set H is reached with positive
probability, since both in1 and in2 are
played with positive probability.
H
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Es strategies
Consider another strategy, Suppose E stays
out with probability 1, then H is never reached.
In this case, WPBE puts no restrictions on
Ms beliefs about the probability of being at
the node following in1 and in2 It still
requires that M have some beliefs and that Ms
strategy be optimal (sequentially rational)
given those beliefs. There is no system of
beliefs for which cutting price is sequentially
rational for M So the equilibrium where E
stays out can be ruled out using WPBE
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Unique WPBE
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More Generally,
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