Download Extensive Games with Imperfect Information

Extensive Games with Imperfect
Information
Jeff Rowley
Abstract
This document will draw heavily on definitions presented in Osborne; An
Introduction to Game Theory and, for later definitions, Jehle & Reny;
Advanced Microeconomic Theory. This is done mainly for accuracy of
definition rather than clarity of presentation. Examples will reference
Jehle & Reny by a ⋆ if they are taken from that textbook, but if no
reference is made, assume that the example is from Osborne.
The document will first define an extensive game with imperfect information. A more intuitive explanation of what each feature included in the
definition means will then be given, and an example included. The reader
will then be given formal definitions for Nash equilibrium and subgame
perfect equilibrium. An example will be used to demonstrate why subgame perfect equilibrium loses much of its bite when an extensive game
is modified such that a player has imperfect information, and why it is
necessary to introduce a new equilibrium concept. Up to this point, most
of the definitions introduced will involve, or could be extended to involve,
mixed strategies. For games with imperfect information it is useful to
introduce a new strategy profile: a behavioural strategy. It will be stated
that, where every player exhibits perfect recall, the two concepts are a
dual approach to the same problem but that behavioural strategies are
relatively easier to work with. Sequential rationality and consistency will
be defined and imposed as necessary conditions for the existence of a weak
sequential equilibrium. The final part of the document will work through
some illustrative examples.
1
Extensive Games
A significant problem in economics is that of imperfect information, and of
particular note, asymmetric information. In the language of games, information
is imperfect if some player i is unable to distinguish between two histories h, h′
of the game. This definition does not exclude a player from possessing imperfect
information over two states ω, ω ′ of the world; rather, consider ω, ω ′ to be two
1
1.1 Terminal Histories
1 EXTENSIVE GAMES
histories of the game which follow a move by nature1 . Now, if h, h′ cannot
be distinguished by i, then they lie in the same information partition Ii 2 . A
consequence of this is that the actions available to i after the histories h, h′ must
be the same. Formally, if h, h′ ∈ Ii then Ai (h) = Ai (h′ ) = Ai (Ii )3 .
Extensive Games with Imperfect Information. An extensive game with
imperfect information is formally defined as consisting of
X A set of players I = {1, ..., H}.
X A set of sequences (terminal histories) having the property that no sequence
is a proper subhistory of any other sequence.
X A function (the player function) that assigns either a player or nature
to every sequence that is a proper subhistory of some terminal history.
Standard convention dictates that the player function is denoted P (·).
X A function that assigns to each history h such that P (h) = Υ a probability
distribution over the actions available after that history, where each such
distribution is independent of every other such distribution. Less formally,
there exists a state space Ω with an accompanying probability distribution.
X For each player, a partition (the player’s information partition) of the set
of histories assigned to that player by the player function such that for
every history h in any given member of the partition, the set Ai (h) of
actions available is the same.
X For each player, preferences over the set of lotteries over terminal histories. The payoff function is Bernoulli, ui : ×ωk ∈Ω S(ωk ) → R.
For completeness, an alternative description of the game replaces the notion of
information partitions with a signal function τi : Ω → Ii and any signals ti (ω)
which i receives. The analogous interpretation arises when ω, ω ′ ∈ Ii if and only
if ti (ω) = ti (ω ′ ). Replacing ω with h and Ω with H allows for the possibility of
this system being used more generally to describe extensive games with imperfect
information. The notion of information sets may be easier to understand and
formulate.
This definition is complete but demanding. To benefit the understanding
of the reader it will be necessary to decompose this definition into each of its
required elements and express them in less formal language. Presentation of an
extensive game with imperfect information will be illustrative of how the formal
description is applied in practice.
1.1
Terminal Histories
A terminal history is simply a possible outcome of the game. It is a description
of what actions are played at each node of the whole game Γ(∅) such that a
terminal node is reached, including actions made by nature. For this reason, a
terminal history is often referred to as a sequence.
1I
shall denote nature as Υ. This is my own convention and not standard notation.
word partition is used rather than set to highlight the possibility of player i being
unable to distinguish between multiple sets of histories which are partitions of the information
set
3 Otherwise player i could distinguish between h, h′ as the set of actions available to her
would be different which clearly violates the assumption of h, h′ ∈ Ii .
2 The
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1.2 Player Function
1.2
1 EXTENSIVE GAMES
Player Function
Consider some history h ⊂ ht where ht is any terminal history of Γ(∅). Then
the player function assigns either nature Υ or a player i ∈ I to move at the
node Γ(h). That is, the player function prescribes the order of play. Formal
notation of the player function P (h) = i when player i is required to move (to
be replaced by Υ when nature is required to move).
1.3
States
Wherever nature is required to move, it assigns some probability distribution
p(Ωh ) over its available actions at that history of the game. States are randomly
selected according to this distribution. This simplification is sufficient to grasp
an understanding the concept of how states are selected with any elaboration
merely confusing.
1.4
Information Partition
The defining characteristic of games of imperfect information is that a player
doesn’t know exactly where they are in the game. That is, they do not know
which sequence of actions has been played to date. The history h consisting
of a sequence of actions a1 , a2 might not be differentiable from the history
h′ = (ã1 , a2 ). In contrast, the history h′′ = (a1 , ã2 ) might be differentiable from
h and h′ . For this example, h, h′ ∈ I while h′′ ∈ J ; h, h′ are a partitioned set
of the entire information set while h′′ is a singleton contained within the entire
information set.
1.5
Preferences
Each player has preferences over each outcome of the game. There will exist
a utility function which maps these preferences to R such that there is ordinal
comparability between any two outcomes. The condition that preferences are
Bernoulli is simply a technical assumption on the form of the utility function
over different states.
Example 315.1
A card game in which nature deals a high or a low card to a gambler who then
decides whether to raise (R) or see (S). Her opponent then decides whether to
pass (P ) or meet (M ).
Players I = {1, 2}.
Terminal Histories (H, S), (H, R, P ), and (H, R, M ); (L, S), (L, R, P ),
and (L, R, M ).
Player Function P (∅) = Υ, P (H) = P (L) = 1, and P (R) = 2.
States Ω = (H, L); p∅ (H) =
1
2
and p∅ (L) = 12 .
Information Partitions Player 1 observes nature’s move: she can distinguish
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2 EQUILIBRIUM STRATEGIES
between the state H and the state L. Player 2 cannot distinguish between the
history hH = (H, R) and the history hL = (L, R) as hR , hL belong to the same
information partition.
Preferences Preferences are according to the following table.
P
1, −1
0, 0
R, S
0, 0
S, R
1, −1
1
1
2, −2
− 21 , 12
S, S
0, 0
0, 0
R, R
2
M
Equilibrium Strategies
Let the set of all actions available to some player i following the history h be
denoted by Ai (h). Recall, that for any histories h, h′ belonging to the same
information partition Ii , it must be the case that Ai (h) = Ai (h′ ) = Ai (Ii ).
A consequence is that, upon reaching the information set Ii , player i can only
specify a single action (or mixture of actions) to be played, regardless of whether
the history is h, h′ . This last fact is hopefully intuitive to the reader and yet
places some caveats on what constitutes a strategy.
Strategy in an Extensive Game. A (pure) strategy of player i in an extensive
game is a function that assigns to each of i’s information set Ii an action in
Ai (Ii ).
That is, a strategy for an extensive game is a prescription of what actions
player i will take wherever she thinks she is in the game every time she is
required to move, regardless of whether that information set is reached or not.
Convention is to write a pure strategy as s and a mixed strategy as α.
Nash Equilibrium of an Extensive Game. The mixed strategy profile α∗
in an extensive game is a (mixed strategy) Nash equilibrium if, for each player
i and every mixed strategy αi of player i, player i’s expected payoff to α∗ is at
least as large as her expected payoff to (αi , α∗−i ) according to a payoff function
whose expected value represents player i’s preferences over lotteries.
ui (α∗ ) ≥ ui (αi , α∗−i )
Generally, to solve for the Nash equilibria of an extensive game, one can
reduce the game to normal form and solve for the Nash strategies. However,
Nash equilibria need not be credible.
Example 317.1
Imagine an entry game in which a challenger can choose to enter a market or
to stay out of the market (Out). Prior to entering, the challenger can either
make preparations (Ready) or rush in (U nready). The incumbent then has the
choice of whether to acquiesce (A) or to fight (F ). Fighting is costly for both
firms, and preparing is costly for an entering firm. For now, assume that there
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2 EQUILIBRIUM STRATEGIES
is perfect information.
Players I = {Challenger, Incumbent}.
Terminal Histories (Ready, A), (Ready, F ), (U nready, A), (U nready, F ),
(Out, A), and (Out, F ).
Player Function P (∅) = Challenger, and P (Ready) = P (U nready) =
Incumbent.
Preferences Preferences are according to the following table.
A
F
Ready
3, 3
1, 1
U nready
4, 3
0, 2
Out
2, 4
2, 4
There are two Nash equilibrium strategies for this game: (U nready, A) and
(Out, F ). Clearly, the latter is not credible in so much as, if the challenger
enters, the optimal strategy of the incumbent is to acquiesce. For games with
perfect information, the notion of subgame perfect equilibrium was introduced.
Subgame Perfect Nash Equilibrium. The strategy profile s∗ in an extensive
game with perfect information is a subgame perfect Nash equilibrium if, for every
player i, every history h after which it is player i’s turn to move (P (h) = i),
and for every strategy si of player i, the terminal history Oh (s∗ ) generated by s∗
after the history h is at least as good according to player i′ s preferences as the
terminal history Oh (si , s∗−i ) generated by the strategy profile (si , s∗−i ) in which
player i chooses si while every other player j chooses s∗j . Equivalently, for every
player i and every history h after which it is player i’s turn to move,
ui (Oh (s∗ ) ≥ ui (Oh (si , s∗−i ))
for every strategy si of player i, where ui is the payoff function that represents
player i’s preferences and Oh (s) is the terminal history consisting of h followed
by the sequence of actions generated by s after h.
This is a rather involved definition. A concise summary of subgame perfect
equilibrium is that the strategy profile s∗ be a Nash equilibrium, and also that
the sequence of actions s∗ (h) following the history h be a Nash equilibrium of
the proper subgame Γ(h).
A necessary condition for subgame perfect equilibrium is that the candidate
equilibrium is a Nash equilibrium. This is a useful condition since the set of
subgame equilibria must be a subset of the Nash equilibria which have already
been found. For Example 317.1, only the Nash equilibrium (U nready, A) is
subgame perfect.
Unfortunately, when a game is modified such that there is imperfect information, subgame perfect equilibrium can lose much of its bite. The reason is
that Γ(h) is a proper subgame of the whole game Γ(∅) if and only if the information partition to which h belongs is a singleton set. Formally, I(x) = {x}
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3
BEHAVIOURAL STRATEGIES
for some node x. Where a history h belongs to a nonsingleton information partition, that history does not constitute the initial node of a proper subgame.
In some cases the number of subgames is restricted such that only the whole
game remains as an improper subgame of itself. In this extreme case, subgame
perfect equilibrium converges to Nash equilibrium.
Example 317.1 (Again)
Let the game be modified such that the incumbent cannot distinguish between
a prepared challenger and an unprepared challenger.
States There is a single state which occurs with certainty.
Information Partition The challenger has a single information set and is perfectly informed. The incumbent cannot distinguish between the history Ready
and the history U nready as these histories belong to the same information partition.
Both Nash equilibrium strategies for the game, (U nready, A) and (Out, F ),
are also subgame perfect equilibrium strategies as the only subgame of Γ(∅) is
the game itself.
3
Behavioural Strategies
It is here that the notion of beliefs is introduced. For games with perfect information, such a concept was implicit in so much as a player’s belief about her
opponent’s strategy was simply equal to the strategy itself. When dealing with
extensive games with imperfect information a player’s beliefs need not be determined solely by her opponent’s strategy. For example, in a variant of the entry
game considered in Example 317.1 where the incumbent optimally prefers to
acquiesce to a prepared challenger but fight an unprepared one, the Nash equilibrium of the game is (Out, F ). However, for F to be the optimal action for the
incumbent when the information set containing the histories Ready, U nready
is reached, her belief must be that an entering challenger is more likely to be
unprepared. As the optimal strategy for the challenger is Out, the challenger
has no basis on which to form this belief. The optimal strategy of the challenger
is now explicitly dependent upon her beliefs.
Belief System. A belief system in an extensive game is a function that assigns to each information set a probability distribution over the histories in that
information set.
With this definition to hand, it is now that a distinction can be drawn between mixed strategies α and behavioural strategies β. The principal difference
between the two strategy profiles is that mixed strategies are defined ex-ante
whilst behavioural strategies are determined interim. This is a slightly incorrect
statement but one that effectively captures the distinction between the concepts.
A more rigorous description is as follows. Let Si be the set of all pure strategies
available to player i. A mixed strategy profile assigns a probability distribution
over all si ∈ Si where αi (si ) is the probability that any one strategy si is played.
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4 WEAK SEQUENTIAL EQUILIBRIUM
P
It must be that αi (si ) ∈ [0, 1] for all si , and si ∈Si αi (si ) = 1. In contrast, a
behavioural strategy specifies a probability distribution over the actions available to some player at one
P of their information sets. Formally, βi (ai , Ii ) ∈ [0, 1]
for all ai ∈ Ai (Ii ), and ai ∈Ai (Ii ) βi (ai , Ii ) = 1.
Behavioural Strategy in an Extensive Game. A behavioural strategy of
player i in an extensive game is a function that assigns to each of i’s information
sets Ii a probability distribution over the actions A(Ii ), with the property that
each probability distribution is independent of every other distribution.
In the above definition, if the reader is more comfortable with the terminology information partition then they may replace information set with information partition. Technically, set is the more correct terminology to use; a
partitioned set is a collection of proper subsets (or partitions) that forms the
original set.
In most cases, mixed strategy profiles and behavioural strategy profiles
amount to a primal and dual approach to the same problem; that is, they
are equivalent. The necessary condition for this to be true is that all players
exhibit perfect recall.
Perfect Recall. An extensive form game has perfect recall if whenever two
nodes x and y = (x, a, a1 , ..., ak ) belong to a single player, then every node in
the same information set as y is of the form w = (z, a, a′1 , ..., a′k ) for some node
z in the same information set as x.
The reader will notice that y and w contain the common member a. Perfect
recall demands that each player always remembers what he knew in the past
about the history of play. In particular, any two information sets which a
player’s information set do not allow her to distinguish between can differ only
in the actions taken by other players4. In other words, perfect recall fails when
the player takes some action a at x and some action a′ at z and yet y, w lie in
the same information set.
4
Weak Sequential Equilibrium
The dependency between a player’s strategy and her beliefs means that listing
the sequence of actions that she will choose at any point in the game is no longer
sufficient to completely describe her play; a complete description of her strategy
will necessarily include her beliefs.
Assessment. An assessment in an extensive game is a pair consisting of a
profile of behavioural strategies and a belief system.
Prior to defining the conditions of sequential rationality and consistency
under which an assessment is an equilibrium, it will be a useful exercise for the
reader to think of an appropriate equilibrium concept.
For extensive games with perfect information, subgame perfect equilibrium
imposed additional caveats on candidate equilibria by requiring that strategy
profiles were Nash equilibria for every subgame of the game. The problem when
subgame perfection was applied to games with imperfect information was that
4 Jehle
& Reny.
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4.1 Sequential Rationality
4 WEAK SEQUENTIAL EQUILIBRIUM
1
L
R
M
[µ]
0, 5
l
4, 0
[1 − µ]
2
m
r
−1, 1
l
0, 4
0, 4
m
−1, 1
r
4, 0
Figure 1: The game presented in example 7.27 of Jehle & Reny. There is one
Nash equilibrium (L, m). This is also the unique subgame perfect equilibrium.
However, this equilibrium does not satisfy sequential rationality.
nodes belonging to nonsingleton information sets could not be considered subgames. To summarise, subgame perfection places no restriction on the strategy
following nonsingleton information sets as these cannot be considered subgames.
A natural extension would therefore be to impose optimal behaviour following
every information set. To incorporate the importance of beliefs in determining behavioural strategies, some notion of optimality given beliefs must also be
incorporated.
4.1
Sequential Rationality
Sequential Rationality. Each player’s strategy is optimal in the part of the
game that follows each of her information sets, given the strategy profile and
her belief about the history in the information set that has occurred. Precisely,
for each player i and each information set Ii of player i, player i’s expected
payoff to the probability distribution OIi (β, µ) over terminal histories generated
by her belief µi at Ii and the behaviour prescribed subsequently by the strategy
profile β is at least as large as her expected payoff to the probability distribution
OIi ((γi , β−i ), µ) generated by her belief µi at Ii and the behaviour prescribed
subsequently by the strategy profile (γi , β−1 ), for each of her behavioural strategies γi .
Simply put, sequential rationality imposes that for any arbitrary belief system µ, each player is maximising their expected payoff given the other players’
strategies, after every information set.
Example 7.27 ⋆
The game tree is presented in Figure 1. There is one Nash equilibrium (L, m).
This is also the unique subgame perfect equilibrium. However, this equilibrium
does not satisfy sequential rationality. Suppose that player 2’s information set is
reached with positive probability. Sequential rationality states that the action m
should be optimal given her belief about how the information set was reached5 .
Clearly, regardless of which value is assigned to µ, such that µ ∈ [0, 1], either
5 The
belief µ corresponds to the probability with which player 2 thinks the history is M .
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4.2 Consistency
4 WEAK SEQUENTIAL EQUILIBRIUM
1
(1)
(0)
2
( 13 )
[µ]
( 23 )
[1 − µ]
3
Figure 2: The game presented in example 7.29 of Jehle & Reny. Given the
behavioural strategy of player 1, player 3’s information set is never reached.
Nonetheless, consistency still places some restrictions on her beliefs.
the action l or the action r is optimal. Hence, the subgame perfect equilibrium
is not sequentially rational.
4.2
Consistency
Beliefs are obviously pivotal to the optimality of behavioural strategies; if beliefs are altered slightly, it might be that a Nash profile that was previously
sequentially rational is no longer. Yet no restriction has been placed upon how
beliefs are formed, or their accuracy.
Consistency. For every information set Ii reached with positive probability
given the strategy profile β, the probability assigned by the belief system to each
history h⋆ in Ii is given by
P r(h∗ |β)
h∈Ii P r(h|β)
P r(h⋆ ) = P
(Bayes’ rule).
Consistency imposes quite strong restrictions on beliefs for those information
sets that are reached with positive probability. To see this, the reader should
look at Bayes’ rule and think what it means. The numerator is the probability
of some history h⋆ being reached given the behavioural strategy profile β. What
should immediately be apparent is that consistency imposes that beliefs be realistic in the sense that they reflect play. The denominator is the probability that
the information set that contains h⋆ is reached given the behavioural strategy
profile β.
4.2.1
Trembling Hand
The question then is how beliefs should be derived such that they satisfy consistency when an information set is not reached given the behavioural strategy
profile β. It is here that the idea of a trembling hand is introduced.
Suppose that, in Figure 2, the behavioural strategy of player 1 was instead
to play left with probability 1 − ǫ and right with probability ǫ. Then, the
information set would be reached with positive probability. µ, 1 − µ are then
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4.2 Consistency
4 WEAK SEQUENTIAL EQUILIBRIUM
1
(0)
[µ]
(1)
(0)
[1 − µ]
2
Figure 3: A game in which Bayes’ rule does not uniquely define the beliefs
µ, 1 − µ. A trembling hand can be used but does not place strict constraints on
the formation of the beliefs.
uniquely defined by Bayes’ rule.
µ=
ǫ · 13
ǫ · 31 + ǫ ·
2
3
1−µ=
ǫ · 23
ǫ· +ǫ·
2
3
1
3
Bayes’ rule thus uniquely defines µ = 31 , 1−µ = 23 . Using the idea of a trembling
hand which assigns positive probability to every information set being reached
is useful in determining consistent beliefs if an information set is never reached
under an arbitrary behavioural strategy profile β.
This is not to say that the notion of a trembling hand can be used in every circumstance; in many games, the trembling hand demonstrates that some
arbitrary beliefs can be assigned to some histories contained in an information
set that is never reached. Consider the game in Figure 3. If probabilities ǫ1 , ǫ2
are assigned to left and middle, then the beliefs µ, 1 − µ depend on the relative
magnitudes of ǫ1 , ǫ2 . To proceed further, the exact constraints placed upon the
beliefs by consistency must be formalised.
Consistent Assessments. An assessment (β, µ) for a finite extensive form
game Γ is consistent if there is a sequence of completely mixed behavioural
strategies, β n , converging to β, such that the associated sequence of Bayes’ rule
induced systems of beliefs, µn , converges to µ.
According to this definition, whenever a trembling hand is applied to some
behavioural strategy profile β and the trembles ǫ → 0 such that β n → β, then
the beliefs µn derived using Bayes’ rule for the behavioural strategy profile β n
will be consistent for µ.
To illustrate, consider again Figure 3. Suppose that the ǫ1 , ǫ2 are the probabilities assigned by player 1 to the actions left and middle as part of some
behavioural strategy profile β n .
ǫ1
ǫ1 + ǫ2
ǫ2
1−µ=
ǫ1 + ǫ2
µ=
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4.3 Weak Sequential Equilibrium
5
INTUITIVE CRITERION
In this case, any arbitrary beliefs can be assigned to the events left and middle.
To see this, suppose that ǫ1 = ǫ2 . Then as ǫ1 , ǫ2 → 0, µn , 1 − µn → 21 , 12 .
Alternatively, if ǫ1 = ǫ22 then µn , 1 − µn → 0, 1. The reader is invited to try
other combinations of ǫ1 , ǫ2 such that any belief system can be generated.
To summarise, consistency requires that beliefs be derived using Bayes’ rule
wherever possible. Wherever an information set is not reached, Bayes’ rule is
undefined. A trembling hand can be applied to a behavioural strategy profile β
such that Bayes’ rule is defined as the trembles ǫ → 0. In some cases, the trembling hand does indeed define consistent beliefs but in others merely illustrates
the point that beliefs can be arbitrarily assigned whenever an information set
is not reached according to the behavioural strategy profile β.
4.3
Weak Sequential Equilibrium
As sequential rationality and consistency have been defined, it is now possible
to define a weak sequential equilibrium of an extensive game with imperfect
information.
Weak Sequential Equilibrium. An assessment (β, µ) (consisting of a behavioural strategy profile β and a belief system µ) is a weak sequential equilibrium if it satisfies sequential rationality and weak consistency of beliefs with
strategies.
5
Intuitive Criterion
Consider the Beer-Quiche game set out in Figure 4. This type of game is a
signalling game as the receiver forms her beliefs over which type of the sender
she based upon what type of breakfast she has eaten (the message she receives).
An assessment for the receiver will thus consist of a strategy following the
signal Beer and a strategy following the signal Quiche, and a belief system over
the probability that the sender is Strong given the signal she receives. This
is consistent with the definition previously given that a player must specify a
strategy at every information set, and the beliefs she forms over histories in each
information set.
There are two broad types of equilibria that could exist for this game: pooling equilibria in which both types of sender consume the same breakfast, and
separating equilibria in which each type of sender consumes a different breakfast. In the latter, the receiver has the advantage of knowing exactly which type
of her opponent she faces based on the breakfast they consume.
It is claimed that there exist no separating equilibria for this game. The
receiver would always choose to F ight a weak opponent and to Acquiesce to a
strong opponent. For any separating equilibria, the W eak type would always
prefer to consume whatever breakfast the Strong type is consuming as she
would derive a strictly higher payoff, regardless of the suggested separating
equilibrium.
Let p > 0.5 so that the probability of asender being strong is greater than
the probability of an opponent being weak. There are two candidate pooling
equilibria. The first is where both types of sender drink Beer and the second
is where both types of sender eat Quiche. It should be obvious that Bayes’
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5
−1, 1
F
F
B
0, 3
N
2
1, 0
F
B
0, 2
INTUITIVE CRITERION
−1, 0
Q
S
p
N
Υ
2
1−p
F
W
N
0, 2
1, 1
Q
N
0, 3
Figure 4: A Beer-Quiche game in which the receiver prefers to F ight a weak
opponent but Acquiesce to a strong one. Each type of her opponent has a
favourite breakfast but she would forgo that breakfast if it meant not fighting.
The receiver’s payoffs are listed first.
rule uniquely determines that, for any information set reached with positive
probability, the belief of the receiver that the sender is Strong is simply equal
to p. It is claimed that there exist two pooling equilibria for this game:
((B, B; N, F ), (p, 1 − p; λ, 1 − λ)),
((Q, Q; F, N ), (λ, 1 − λ; p, 1 − p)).
Here, the first pair list the strategies of the two types of sender, the second pair
list the strategies of the receiver following the signals Beer and Quiche, the
third pair list the belief of the receiver that the sender is Strong or W eak given
the signal Beer, and the fourth pair list the belief of the receiver that the sender
is Strong or W eak given the signal Quiche. Consistency demands that λ ≤ 21 6 .
It is left up to the reader to show that these assessments do indeed constitute
weak sequential equilibria.
Careful examination of these pooling equilibria will reveal that one of them is
not entirely credible; the pooling equilibrium in which both types of sender eats
Quiche demands that µ be such that the receiver thinks it more likely a sender
drinking Beer is W eak. A further mechanism with which to impose credibility
on equilibria is required. The notion of an intuitive criterion is introduced.
Prior to defining the intuitive criterion, some discussion must be entered
into to determine what constitutes an incredible equilibrium. Consider again
the Quiche pooling equilibrium. What makes this equilibrium incredible? Put
simply, given the equilibrium behavioural strategy profile β ⋆ , the belief that a
6 Note
that a trembling hand does nothing to tighten the constraint on these beliefs.
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5
INTUITIVE CRITERION
W eak type will be more likely to play Beer seems unrealistic. In equilibrium,
the W eak type gets a payoff of 3. If a W eak type were ever to deviate to sending
the message Beer, the most she could ever hope to get would be a payoff of
2. The belief system which supports this equilibrium is thus incredible. Any
intuitive criterion must incorporate some means of ruling out such equilibria.
Equilibrium Dominance. A message m can be eliminated for some sender
type θ if
U ⋆ (θ) > max U (m, a, θ)
a
⋆
where U (θ) is the equilibrium payoff to θ from playing m⋆ .
This definition formalises the argument presented in the paragraph above. A
message m′ is equilibrium dominated for a sender type θ if the payoff from sending m′ always yields less utility than the payoff from sending the equilibrium
message m∗ . Applying this to the game in Figure 4: the possible messages are
Beer, Quiche, the possible sender types are Strong, W eak, and the possible actions for the receiver are F ight, Acquiesce. Fixing θ = W eak, the message Beer
is equilibrium dominated by the message Quiche as, regardless of whether the
receiver chooses to F ight or to Acquiesce, Quiche always yields a higher payoff
given that the receiver opts to Acquiesce when she sees the sender consumes
Quiche.
Intuitive Criterion. No message m which is equilibrium dominated will ever
be sent by a type θ. The belief system µ must be consistent with the probability
that the message m being sent is zero.
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Jeff Rowley, 2012