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L 2, 2
M 3, 1
R 0, 0
r
2, 2
1, 0
0, 1
2
2
L
M
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` @r
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3, 1
1, 0
1c
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@ R
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0, 0
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ECON5210 Topics in microeconomics — Spring semester 2009
Example illustrating sequential equilibrium
Consider the above example. An assessment is the following object:
(β, µ) = ((β1 (∅), β2 ({M, R})), µ({M, R})) ,
where
• β1 (∅), player 1’s behavioral strategy, is a probability distribution over
player 1’s strategies, L, M , and R,
• β2 ({M, R}), player 2’s behavioral strategy, is a probability distribution
over player 2’s strategies, ` and r, and
• µ({M, R}) is player 2’s belief conditional on 2 being asked to play over
the two histories, M and R, that lead to her information set.
A sequential equilibrium is an assessment that is sequentially rational and
consistent. Sequential rationality precludes that player 1 assigns positive probability on R, since R is strictly dominated for 1. Hence, in any sequential
equilibrium, β1 (∅)(R) = 0.
There are two types of sequential equilibria.
Sequential equilibrium where β1 (∅)(M ) > 0. In this case, 2’s information set is reached with positive probability, and the consistency of the
assessment entails that µ({M, R} is determined from β1 (∅) using Bayes’ rule:
β1 (∅)(M )
= 1,
β1 (∅)(M ) + β1 (∅)(R)
β1 (∅)(R)
µ({M, R})(R) =
= 0.
β1 (∅)(M ) + β1 (∅)(R)
µ({M, R})(M ) =
1
With these beliefs, β2 ({M, R}) is part of a sequentially rational assessment if
and only if
β2 ({M, R})(`) = 1 ,
β2 ({M, R})(r) = 0 .
This in turn means that β1 (∅) is part of a sequentially rational assessment if
and only if
β1 (∅)(L) = 0 ,
β1 (∅)(M ) = 1 ,
β1 (∅)(R) = 0 .
Hence, there is unique sequential equilibrium where β1 (∅)(M ) > 0:
(β, µ) = ((β1 (∅), β2 ({M, R})), µ({M, R})) = (((0, 1, 0), (1, 0)), (1, 0)) .
Sequential equilibrium where β1 (∅)(M ) = 0. Hence, β1 (∅) is given as
follows:
β1 (∅)(L) = 1 ,
β1 (∅)(M ) = 0 ,
β1 (∅)(R) = 0 .
This behavioral strategy of player 1 is part of a sequentially rational assessment
if and only if β2 ({M, R}) satisfies
β2 ({M, R})(`) = 1 − p ,
β2 ({M, R})(r) = p ,
with p ≥ 21 , because otherwise player 1 would want to have β1 (∅)(M ) = 1. If
1
2
≤ p < 1, then player 2’s behavioral strategy is part of a sequentially rational
assessment if and only if
µ({M, R})(M ) = 12 ,
µ({M, R})(R) = 21 .
2
To check that this is part of a consistent assessment, consider
β = ((β1 (∅), β2 ({M, R}))) ,
where is a “small” positive number and
β1 (∅)(L) = 1 − 2
β2 ({M, R})(`) = β2 ({M, R})(`) ,
β1 (∅)(M ) = β2 ({M, R})(r) = β2 ({M, R})(r) ,
β1 (∅)(R) = .
Then µ({M, R} is determined from β1 (∅) using Bayes’ rule:
β1 (∅)(M )
1
=
=
,
β1 (∅)(M ) + β1 (∅)(R)
+
2
β1 (∅)(R)
1
µ({M, R})(R) = =
= .
β1 (∅)(M ) + β1 (∅)(R)
+
2
µ({M, R})(M ) =
Furthermore, β → β as → 0. This shows consistency and establishes that
(β, µ) = ((β1 (∅), β2 ({M, R})), µ({M, R})) = ((1, 0, 0), (1 − p, p)), ( 12 , 21 ) ,
is a sequential equilibrium if and only if p ≥ 12 .
If β2 ({M, R})(r) = 1, then player 2’s behavioral strategy is part of a sequentially rational assessment if and only if
µ({M, R})(M ) = 1 − q ,
µ({M, R})(R) = q ,
with q ≥ 12 . To check that this is part of a consistent assessment, consider
β = ((β1 (∅), β2 ({M, R}))) ,
where is a “small” positive number and
β1 (∅)(L) = 1 − β2 ({M, R})(`) = ,
β1 (∅)(M ) = (1 − q)
β2 ({M, R})(r) = 1 − ,
β1 (∅)(R) = q
.
3
Then µ({M, R} is determined from β1 (∅) using Bayes’ rule:
1−q
β1 (∅)(M )
=
= 1−q,
β1 (∅)(M ) + β1 (∅)(R)
1−q+q
q
β1 (∅)(R)
=
µ({M, R})(R) = = q.
β1 (∅)(M ) + β1 (∅)(R)
1−q+q
µ({M, R})(M ) =
Furthermore, β → β as → 0. This shows consistency and establishes that
(β, µ) = ((β1 (∅), β2 ({M, R})), µ({M, R})) = (((1, 0, 0), (0, 1)), (1 − q, q)) ,
is a sequential equilibrium if and only if q ≥ 21 .
A final question: Are the sequential equilibria of the second type reasonable?
Geir B. Asheim, 2 February 2009
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