Economics of Contracts and Information
Dezsö Szalay
University of Bonn
2014
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
Economics of Contracts and Information
University of Bonn
Professor Dezsö Szalay
2014
4. Re…nements of Perfect Bayesian Equilibrium with an Application
to Jobmarket Signalling
This material is based on Gibbons, R. (1992) A primer in Game Theory,
chapter 4
Perfect Bayesian Equilibrium prevents player i from playing a strategy that
is strictly dominated at the beginning of any of i’s information sets. So
player j should not believe that i would play such a strategy!
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
Consider this game:
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
1
There are two PBEs: (L, L0 , p = 1) and R, R 0 , p
2
Both are PBE’s; In the second equilibrium, player 2’s information set is o¤
path.
However:
Strategy M is dominated for Player 1. The highest payo¤ from playing M
(1) is smaller than 2 (Payo¤ from playing R).
That would dictate that 1 p = 0 () p = 1, and this contradicts
1
p
2.
This example does not precisely illustrate the idea!
Strategy M is not only dominated at player 1’s information set, but simply
dominated.
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
De…nition: Consider an information set at which player i has the move.
The strategy si0 is strictly dominated beginning at this information set if
there exists another stategy si such that, for every belief that i could hold
at the given information set, and for each possible combination of the
other players’subsequent strategies, i’s expected payo¤ from taking the
action speci…ed by si and playing the subsequent strategy speci…ed by si is
strictly greater than the expected payo¤ from playing si0 .
Requirement 5: If possible, each player’s beliefs o¤ equilibrium path should
place zero probability on nodes that are reached only if another player plays
a strategy that is strictly dominated beginning at some information set.
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
What is meant by "If possible"?
If we change payo¤ at the node following L, L0 for player 1 to 3/2, then
both L and M are strictly dominated.
But that would imply that p = 0 and p = 1, which is impossible.
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
Example 2 to illustrate Requirement 5:
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
1
Consider the pooling equilibrium f(L, L), (u, d ), p = 0.5, q g for q
2.
The node where t1 played R can be reached only if type t1 plays a
dominated strategy. Formally, the Sender strategies (R, L) and (R, R ) are
dominated.
Moreover, the node where t2 played R can be reached if t2 plays a
strategy that is not dominated (namely (L, R )).
in particular, if the Receiver plays u if R (which is optimal for q < 12 ),
then the Sender of type t2 indeed prefers to play R rather than L.
Hence, requirement 5 would dictate that the belief be q = 0.
Hence requirement 5 destroys the equilibrium f(L, L), (u, d ), p = 0.5, q g
1
for q
2.
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
For Signalling games, the following de…nition is equivalent to the one given
above:
De…nition: In a signaling game, the message mj from M is dominated for
type ti from T if there exists another message mj 0 from M such that
ti ’s lowest possible payo¤ from mj 0 is greater than ti ’s highest possible
payo¤ from mj :
min US (ti , mj 0 , ak ) > maxUS (ti , mj , ak )
a k 2A
a k 2A
Signaling Requirement 5: If the information set following mj is o¤ the
equilibrium path and mj is dominated for type ti then (if possible) the
Receiver’s belief µ(ti j mj ) should place zero probability on type ti . (This
is possible provided mj is not dominated for all the types in T .)
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
In some games, equilibria that satisfy Requirement 5 are nevertheless
unreasonable.
What further requirement can be added to rule out an "unreasonable"
equilibrium?
The intuitive criterion (following Cho and Kreps (1987) QJE))
Consider the following Beer and Quiche Game:
Sender
t1
wimpy
(with prob. 0, 1)
t2
surly
(with prob. 0, 9)
Sender’s message: quiche or beer for breakfast.
Receiver’s action: duel or not with sender.
The wimpy type prefers quiche for breakfast.
The surly type prefers beer for breakfast.
Both types prefer not to duel.
The Receiver would like to duel with the wimpy but not with the surly.
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
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The game tree:
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
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f(Quiche, Quiche ), (not, duel ), p = 0, 1, q g is a PBE for any q
1
2
.
It satis…es Signaling requirement 5, because beer is not dominated for
either sender type.
The minimum payo¤ from having Quiche for the wimpy type is 1, which is
not larger than the maximum payo¤ from having beer, 2.
However the belief o¤ equilibrium path is odd.
1
q
2 means receiver believes the deviation is more likely to come from the
wimpy type.
However,
(a) the wimpy type cannot possibly improve on his equilibrium path (3) by
having beer, while
(b) the surly type could improve on the equilibrium payo¤ (2) by having
beer (as long as q < 12 )
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
The Sender could make the following speech:
Seeing me choose beer should convince you that I am the surly type:
choosing beer could not possibly have improved the lot of the wimpy type,
by (a); and if choosing beer will convince you that I am the surly type then
doing so will improve my lot, by (b).
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
De…nition:
Given a perfect Bayesian equilibrium in a signaling game,
the message mj from M is equilibrium-dominated for type ti from T if ti ’s
equilibrium payo¤, denoted U (ti ), is greater than ti ’s highest possible
payo¤ from mj :
U (ti ) > maxUS (ti , mj , ak ).
a k 2A
Signaling Requirement 6 ("The intuitive criterion):
If the information
set following mj is o¤ the equilibrium path and mj is
equilibrium-dominated for type ti then (if possible) the receiver’s belief
µ(ti j mj ) should place zero probability on type ti . (This is possible
provided mj is not equilibrium-dominated for all the types in T .)
"Beer and Quiche" shows that mj can be equilibrium-dominated but not
dominated for ti . However the reverse is not true. So, if a message is
dominated then it must be equilibrum-dominated.
So, Requirement 6 makes Requirement 5 redundant.
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
Application to the Job market signaling game:
1) Requirement 5 rules out all seperating equilibria but the one where the
low type chooses e (L) and the high type e = es where e (L) and es are
“de…ned” in the following graph:
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
Since
w (e ) = µ(H j e ) y (H, e ) + [1
µ(H j e )] y (L, e ),
if the low type chooses e (L), he can get at least
y [L, e (L)] c [L, e (L)], which is more than he can get by choosing any
e > es .
So e > es is dominated for type L.
But, then, since e¤ort levels e > es are not dominated for type H,
requirement 5 dictates that µ(H j e ) = 1 for e > es .
But then, e > es cannot be an equilibrium, because it would require that
…rms beliefs satisfy µ(H j e ) < 1 for some e > es .
Hence, only one seperating equilibrium satisfying R 5.
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1
2) Again using R 5, in any equilibrium, the high type must get at least
y (H, es ) c (H, es )
This rules out some hybrid and pooling equilibria.
We discuss two cases in sequence; q low and q high.
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
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Case 1: q low
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
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No pooling equilibrium survives the re…nement, because the high type
would get too little utility.
No hybrid equilibrium where low type does the mixing survives the
re…nement, because of the low type’s indi¤erence curve is below the
minimally necessary level of utility for the high type.
No hybrid equilibrium where high type does the mixing survives, because
the expected productivity line is below the average one.
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
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Case 2: q high
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
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Some pooling and hybrid equilibria satisfy signaling requirement 5.
But Requirement 6 rules these equilibrium out.
e > e 0 are equilibrium-dominated for the low type.
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
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e 2 (e 0 , e 00 ) are not equilibrium-dominated for the high type.
Hence, requirement 6 dictates that µ(H j e ) = 1 for e 2 (e 0 , e 00 ).
Dezsö Szalay (University of Bonn)
Economics of Contracts and Information
2014
1/1