Download Perfect Correlated Equilibria

Journal of Economic Theory 2056
journal of economic theory 68, 279302 (1996)
article no. 0018
Perfect Correlated Equilibria*
Amrita Dhillon and Jean Francois Mertens
CORE, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium
Received October 12, 1992; revised June 1, 1994
The (=)-perfect correlated equilibria (PCE) are those induced by a (=)-perfect
equilibrium of some correlation device. The ``revelation principle'' fails for this
conceptthe direct mechanism may not yield a perfect equilibrium. The approximately perfect correlated equilibria (APCE) are the limits of =-PCE, and we obtain
a full characterisation for them. Even the APCE are ``acceptable.'' We argue in an
example that, among those, it is the PCE which seem the ``right'' concept. Journal
of Economic Literature Classification Number: C72. 1996 Academic Press, Inc.
1. Introduction
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We study the analogue of perfection (cf. Selten [6]) for correlated equilibria (Aumann [1]). A standard way of defining a correlated equilibrium
is to specify a class of mechanisms, here correlation devices (i.e., lotteries
that select a private message for each player), to consider the corresponding extended games and to define the correlated equilibria of the
original game as all pairs consisting of such a device and a Nash equilibrium of the corresponding extended game [2].
Every such pair has a ``canonical representation,'' which typically involves
only truth-telling and obedient strategies. For correlated equilibria, this
canonical representation is just the induced distribution on the pure strategy
vectors of the original gamewhere the canonical device just informs each
player of his component of the selected vector, and the corresponding
equilibrium strategies are to follow those recommendations. Thus, such
canonical representations are themselves equilibrium-mechanisms of the
type considered (ibidem)here correlated equilibria. That the direct
mechanism has this property can be viewed as the appropriate generalization of the ``revelation principle.''
* This work was financed, in part, by Contract 26 of the programme ``Po^le d'attraction
interuniversitaire'' of the Belgian government, and in part by National Science Foundation
Grant SES 8922610 at the institute for Decision Sciences, S.U.N.Y., Stony Brook.
279
0022-053196 18.00
Copyright 1996 by Academic Press, Inc.
All rights of reproduction in any form reserved.
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We define perfect correlated equilibria in the same way, just replacing
in the above the Nash equilibrium concept by that of perfect equilibrium. The first motivation is purely methodological and is to
investigate what happens to the revelation principle when the Nash equilibrium concept is replaced by some refinement. Indeed, the need to use
some refinement of Nash equilibrium is felt in a large part of the applications of game theory; similarly the revelation principle is a very widely
used concept, hence in our view the importance of such an investigation.
A specific type of application we have in mind is, for instance, voting
games: in voting situations, the uncertainty voters typically have about
the outcome is much greater than could possibly be explained by a Nash
equilibrium, type of conceptindeed, because of the law of large numbers,
typical uncertainty in such a case would be almost negligible. A correlated
equilibrium-like concept is needed to explain this kind of uncertainty.
On the other hand, a perfection-like refinement is called forotherwise,
even in a situation with two alternatives, A and B, where it is common
knowledge that everybody prefers A to B, one would still have the Nash
equilibrium of everybody (or a large majority) voting B. Clearly a
dominance requirement, like perfection, is needed, to ensure that, at least
in votes between two alternatives, everybody will vote for his preferred
alternative.
Another type of application concerns strategic market games. This too is
a situation where players' typical uncertainty about the outcome is much
greater than could be explained by Nash equilibria (large numbers again).
And here too some perfection-like refinement is needed to avoid the trivial
no-trade equilibria (even in situations with two types of traders and two
goods, where everybody of type 1 owns only-good 1 and cares only about
good 2 and vice versa...).
Conceptually, the reason for looking at (normal form) perfect equilibria is that this is the minimal refinement that yields a complete
system of beliefs of the players which is fully consistent both with the
Harsanyi doctrinewhich underlies the concept of correlation device,
so if the solution concept was not consistent with it, we would no
longer know what we buyand with independence of players' actions.
Indeed, since we allow already arbitrary correlation devices, and (according to the same Harsanyi doctrine) all beliefs are to be traced to private information in such a correlation device, we have to insist on
complete independence at the level of the solution concept of the
extended gameotherwise again, we would no longer know what we are
doing.
Finally, perfect equilibria are conceptually simple, have the right formal
properties (ordinalitycf. [4]), and are mathematically still manageable
perfect correlated equilibria
281
(and, despite this, we obtain in this paper a characterization only in the
two-person case). So before possibly attempting a similar program with
more ambitious refinements (e.g., stable equilibria), it seems a prerequisite
to be able to handle this situation and to get a consistent treatment of the
beliefs, meshing correctly those which are implicit in the correlation
(device) concept with those underlying the solution concept.
Our first finding, in Section 2, is the complete failure of the revelation
principle in this contextthe set of perfect correlated equilibrium distributions includes much more than the set of perfect direct correlated equilibria
(i.e., the direct mechanisms for which obedience is a perfect equilibrium).
Indeed, even a convex combination of two pure strategy perfect equilibria
(thus also perfect direct correlated equilibria) is not necessarily a perfect
direct correlated equilibriumas shown by a two-person example, where,
furthermore, player I 's strategy is the same in both equilibria.
A refinement of correlated equilibrium which purports to be the
analogue of trembling-hand perfection was introduced by Myerson [5]
as ``acceptable correlated equilibrium.'' His definition is based on the
canonical representation of correlated equilibria and introduces ``trembles''
directly into that representation, but without informing the players of their
own trembles. One might therefore suspect that the players would in some
cases assume trembles of their opponents which are impossible given those
opponents' information, and which are such that the opponents, if they had
the required information, would deviate from their recommendations. This
is indeed the case, as shown by a two-person example in Section 4. Before
the example, we show in Section 3 that every perfect correlated equilibrium
distribution is an acceptable correlated equilibriumso by the example,
the inclusion is strictand we obtain characterizations of the two concepts
for two-person games.
Finally, an appendix on ``approximately perfect correlated equilibria''
serves to underscore the necessity of keeping the correlation device fixed in
the definition of perfect correlation equilibrium and sharpens some results
of the paper.
Of course, the major point missing in this paper is to obtain a characterization of perfect correlated equilibria in the N-person case. 1
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1
Our characterization for two-person games is in terms of a canonical correlation device
with as message spaces for each player the set of pure strategy pairs of that playerwhere the
first coordinate of the pair is the actual recommendation and the second is related to the
tremble.
In the Appendix, we characterize also the approximately perfect correlated equilibria in
similar termsthis in the N-player situation.
The associate editor who handled this paper conjectures that the PCED themselves have a
similar characterization even in the N-layer case.
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dhillon and mertens
2. The (Failure of the) Revelation Principle:
Perfect Correlated Equilibria versus
Perfect Direct Correlated Equilibria
We consider a finite game 1, with N as player set (n # N).
Definition. (A) We recall from Selten [6] the following definition of
perfect equilibria (``substitute perfection''): _=(_ n ) n # N is a perfect equilibrium if there exists a sequence { k =({ kn ) n # N of completely mixed strategy
vectors and a sequence = k(0<= k <1) converging to zero such that, for all
k and n, _ n is the best reply of player n when his opponents n~ all use
(1&= k ) _ n~ += k_ kn~ .
(B) Recall also (e.g., Kohlberg and Mertens [3, Appendix D] that,
for two-person games, perfect equilibria are those equilibria where each
player's (mixed) strategy is undominated.
Definition 1. (a) A correlation device is a lottery mechanism
selecting a private message for each player (finite message spaces). (Thus,
formally, it is an (n+1)-tuple d=((M n ) n # N , P), where M n is player n's
finite message space and P is a probability distribution over M=> n # N M n .)
(b) The corresponding extended game 1 d is the game where players
are first informed by the device d of their private messages and next play 1.
(c) The (perfect) correlated equilibria (PCE) of 1 are all pairs
(d, (_ n ) n # N ) where (_ n ) n # N is a (perfect) equilibrium (PE) of 1 d .
(d) The (perfect) correlated equilibrium distributions of 1 (PCED)
are the probability distributions on the product S=> n # N S n of the pure
strategy sets of 1 which are induced by some (P) CE (d, (_ n ) n # N ) of 1.
(e) The (perfect) direct correlated equilibria (PDCE) are the perfect
correlated equilibria where M n =S n and _ n is the identity map from M n to S n .
Remark 1. In the two-person case, point (B) above allows us to verify
by linear programming whether a given direct correlated equilibrium is
perfect: the primal problem will express that no other (behavioural)
strategy dominates obedience, while the dual problem expresses that
obedience is optimal against some completely mixed behavioural strategy
of the opponent.
Remark 2. Clearly, any perfect direct correlated equilibrium is a perfect
correlated equilibrium distribution (being its own distribution).
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Proposition 1. A perfect equilibrium is also a perfect direct correlated
equilibrium.
perfect correlated equilibria
283
Proof. Let _=(_ n ) n # N be a perfect equilibrium, with the associated
sequences = k and { kn .
Define the corresponding direct mechanism in the obvious way: Pr((s n ) n # N )
=> n # N _ n(s n ). In the extended game, let _ n be the identity map, and let
{ kn be the (completely mixed) strategy that plays { kn independently of
the message. For = k == k, we get the required sequence in the extended
game. K
Proposition 2. The set of perfect correlated equilibrium distributions is
convex.
Proof. Given two perfect correlated equilibrium distributions, consider
the corresponding devices and strategies. Construct from the two devices a
single bigger device, where first one of the two devices is selected at random
(with respective probabilities : and 1&:), players are informed of the
device selected, and then that device is operated. The equilibrium strategies
for the single devices yield in the obvious way a strategy for the bigger
device, which is clearly a perfect equilibrium of the extended game. And its
distribution is obviously the convex combination (with weights : and
1&:) of the two distributions we started with. K
With those preliminaries out of the way, we can give our first example,
showing that the inclusion in Remark 2 is stricthence the failure of the
revelation principle, and that the set of perfect direct correlated equilibria is not convex (does not even contain the convex hull of the perfect
equilibria)hence is a very unsatisfactory solution concept.
Example 1.
T
B1
B2
L
M
R
1, 0
0, &2
0, 1
1, 0
0, 1
0, &2
1, 0
0, 0
0, 0
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In this two-person game, any pure strategy of player II is undominated,
so the two equilibria (T, L) and (T, M) are perfect (point(B) above); hence
they are also perfect direct correlated equilibria (Proposition 1). Their
convex combination (T, 12 L+ 12 M) is thus (Proposition 2 and Remark 1)
a perfect correlated equilibrium distribution, but is not a perfect direct
correlated equilibrium. Indeed, in the corresponding direct mechanism,
player I receives message T with probability onehence his pure strategy
set in the extended game is (up to duplication) identical to that in the
original game. Now the ``obedient'' strategy for player II in the extended
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dhillon and mertens
game yields the same expected payoff against each of those three pure
strategies as the mixed strategy ( 12 L+ 12 M) would in the original game. And
since the latter is dominated by the strategy R, the former will therefore be
dominated by the strategy consisting of playing R independently of the
message. Hence, (by point (B) above), the obedient strategy pair is not a
perfect equilibrium of the extended game.
Clearly, there is no simpler conceivable correlation device than that
underlying this convex combination, i.e., to select a public signal, (T, L) or
(T, M), with probability 12 each, and, e.g., by (the proof of) Proposition 2,
our convex combination is indeed a perfect equilibrium of this most simple
and natural device. However, if we wanted to represent this correlated
equilibrium by the direct device, it would be much more complexusing
private signals, etc.and as seen above we would lose the perfection. It is
thus by insisting on representing any correlated equilibrium only by the
direct device (much more complex and less natural in cases like this), and
by insisting on perfection in this representation, that the concept of PDCE
rules even this convex combination out. The point is that player I needs to
have the additional information, not for his action, but for it to be possible
for him to tremble in a way that would justify player II's actionssuch a
way being necessarily correlated with those actions.
3. Perfect Correlated Equilibria and
Acceptable Correlated Equilibria
In this section, we show (Proposition 3) that the perfect correlated equilibrium distributions are a subset of acceptable correlated equilibria [5],
and in the following section a two-person example is given to show that the
inclusion is proper.
To facilitate the analysis of the example, we will give a characterization
of each of the two concepts in the two-person case (Propositions 4 and 5).
We first recall the definition of acceptable correlated equilibria. For
DN, let S D => n # D S D .
Definition 2. An =-correlated equilibrium is a lottery choosing a vector of recommended actions (i.e., a point in S), a coalition D of trembling
players, and a vector of trembles (i.e., a point in S D ) for those players
(hence, formally, it is a probability distribution over S_( DN S D )], such
that:
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(a) Given any vector of recommendations, any coalition of trembling
players not including player n, and any vector of trembles for those players,
the conditional probability of player n also trembling is at most =.
perfect correlated equilibria
285
(b) Given any vector of recommendations, the conditional probability of every coalition of trembling players and every vector of trembles for
these players is strictly positive.
(c) Consider the extended game where each player is first informed
of his recommended action; next the non-trembling players are asked to
movewhile the trembling players are forced to move using the selected
trembles. In this extended game, the obedient strategies form a Nash
equilibrium.
Definition 3. The acceptable correlated equilibria are the limits
(= 0) of distributions (i.e., marginal distributions on S) of =-correlated
equilibria.
Proposition 3. Every perfect correlated equilibrium distribution is an
acceptable correlated equilibrium.
Proof. Consider a perfect correlated equilibriumhence a lottery P on
M=> n M n , (behavioural) strategies _ n and completely mixed trembles { kn
(in the extended game), and a sequence = k converging to zero.
Construct now a probability distribution P k on M =M_S_S N_2 N (with
N
2 =[D | DN]) by selecting, given m # M, s n according to _ n(s n | m n ),
(e n ) n # N # S N according to { kn(e n | m n ), and assigning player n to D with probability = k, all those choices being made independently given m # M. Let ' k be
the distribution of the random variable that maps (m, s, e, D) to (s, D,
(e n ) n # D ): ' k is the required sequence of =-correlated equilibria. K
Proposition 4. In two-person games, the acceptable correlated equilibria
are those correlated equilibria where only undominated strategies are recommended.
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Proof. We first show that, in an =-correlated equilibrium, any recommendation to n that has positive probability is undominated.
From point (b) in the definition of =-correlated equilibrium, given this
recommendation, any combination of actions of the opponents has strictly
positive probability. And from point 2(c), following this recommendation is optimal given this strictly positive probability distributionso the
recommendation is undominated.
Conversely, assume a correlated equilibrium P on S where only
undominated strategies are recommended. Since each such recommendation
is undominated, it is optimal against some completely mixed strategy of the
opponent. This can then be used to select trembles for the opponent (doing
this independently for the two players). The deviating coalition D is chosen
by letting each player tremble independently with probability =. Now, we
have an =-correlated equilibrium, which clearly converges to P. K
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dhillon and mertens
For two-person games, let S and T be the pure strategy sets of players
I and II respectively. In the following characterization of their perfect
correlated equilibria, the probability distribution Q(s 1 , s 2 , t 1 , t 2 ) is to be
interpreted as the joint distribution of the actual recommendations (s 1 , t 1 )
and of trembles (s 2 , t 2 ). S$ and T $ will denote copies of S and T used for
the trembles.
Proposition 5. The perfect correlated equilibrium distributions of a
two-person game (with payoff matrices A and B for players I and II respectively) are the marginal distributions on S_T of the set of all distributions
Q(s 1 , s 2 , t 1 , t 2 ) on S_S$_T_T $ satisfying
_
\s , s : Q(s , s , V, t )(A
_
&
)
&
\s 1 , s : Q(s 1 , s 2 , t 1 , V)(A s1 t1 &A st1 )
(i)
t1
(ii)
1
1
t2
2
2
s1t2
&A st2
# P S$
s2 # S$
# P S$
s2 # S$
and
(iii)
\s 1 , t 1 , t 2 , [Q(s 1 , s 2 , t 1 , t 2 )] s2 # S$ # P S$ ,
where P S$ =[x # R S$ | x=0 or x S >0 \s # S$]together with the dual conditions (i$), (ii$), and (iii$) for player II (involving then P T $ )
Remarks. (1) Equation (iii) together with (iii$) are clearly equivalent
to the positivity condition [Q(s 1 , s 2 , t 1 , t 2 )] (s2, t2 ) # S$_T $ # P S$_T $ i.e.,
Q(s 1 , V, t 1 , V)>0 O Q(s 1 , s 2 , t 1 , t 2 )>0 \s 2 t 2 .
(2) Here above, a summation is assumed over starred indices. for
example, Q(s 1 , s 2 , t 1 , V) = t2 Q(s 1 , s 2 , t 1 , t 2 ) is the probability of the
triplet (s 1 , s 2 , t 1 ).
(3) Only the four three-dimensional marginals of Q appear in conditions (i), (i$), (ii) and (ii$). Those are linked by the requirement that any
pair induces the same marginal on the product of the two common factors.
In addition to those equations; there are a number of facial inequalities
that translate requirements (iii) and (iii$). If one could find those, one
would have reformulated the problem with much fewer variables.
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Proof. Step 1. Given the device d=(M 1 , M 2 , P), use the perfect equilibrium strategies _(s | m 1 ) and {(t | m 2 ) independently to generate s and t
given (m 1 , m 2 ): this defines a bigger correlation device d, with M 1 =M 1 _S
and M 2 =M 2 _T; let also P denote the corresponding distribution.
perfect correlated equilibria
287
We claim that the obedient strategies form a perfect equilibrium of 1 d ,
which has the same distribution as the original correlated equilibrium.
The last point is obvious. Let us first show that, e.g., player I's obedient
strategy is undominated. Otherwise let _~(s | m 1 , s) be a dominating strategy.
Then _(s | m 1 )= s _(s | m 1 ) _~(s | m 1 , s)the corresponding strategy in the
original extended gamewould dominate _ in 1 d . (Indeed, the above shows
also how to construct, for every strategy of player II in 1 d , a corresponding
strategy in 1 d , such that corresponding strategy vectors lead, for every
message (m 1 , m 2 ), to the same distribution of actions, thus the domination
relation between strategy pairs of player II is preserved when going to corresponding strategy pairs in 1 d ). The same argument shows then also that, if
_~ was a profitable deviation, _ would also be one.
Step 2. Therefore, since the obedient strategy is undominated, it is
optimal against some completely mixed behavioural strategy of the opponent. Let _^ and {^ be those strategies, one for each player. Since they are
completely mixed, they can be written as _^ =(1&=) _~ +=_ u and {^ =
(1&=) {~ +={ u for =>0 sufficiently small. Here _ u and { u denote the
uniform strategies (independent of the message), and _~ and {~ are two
behavioural strategies. Let M 1, = =M 1_S_S$, with f 1 as projection to S
and f 2 as projection to S$; let also M 2, = =M 2_T_T $, with g 1 as projection to T and g 2 to T $. Define P = on M 1, =_M 2, = by selecting (m 1 , m 2 , s, t)
according to P, and with P =(s$, t$ | m 1 , m 2 , s, t)=_~(s$ | m 1 , s)_{~(t$ | m 2 , t).
With d = =(M 1, = , M 2, = , P = ), we claim that f 1 is a best reply in 1 d= both
against g 1 and against (1&=) g 2 +={ u (and similarly when inverting the
roles of the players). This amounts to repeating twice (once for each
strategy of II) our proof at the end of Step 1 that the obedient strategy was
a best reply against the obedient strategy.
Step 3. Now we can drop the factors M 1 and M 2 from M 1, = and M 2, =
respectively and remain with a perfect correlated equilibrium with the same
distribution as the original one and with S_S$ and T_T $ as message
spaces. Here ( f 1 , g 1 ) are the equilibrium strategies, and are optimal
against the completely mixed behavioural strategies (1&=) g 2 +={ u and
(1&=) f 2 +=_ u respectively.
Indeed, since all those strategies remain, and less alternatives remain
because less information is given, their best reply properties are preserved.
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Step 4. Denote by P the probability distribution on S_S$_T_T $
thus obtained.
Any P having the properties mentioned in Step 3 will define a perfect
correlated equilibrium. Thus our problem reduces to expressing those conditions on P.
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dhillon and mertens
The optimality of f 1 against g 1 yields, \ s 1 , s 2 , s,
(i)
: P(s 1 , s 2 , t 1 , V)(A s1t1 &A st1 )0.
t1
Similarly, the optimality of f 1 against (1&=) g 2 +={ u yields, \ s1, s2, s ,
(ii)
_
: (1&=) P(s 1 , s 2 , V, t 2 )+=P(s 1 , s 2 , V, V)
t2
1
(A s1t2 &A st2 )0.
*T
&
Together with the analogue inequalities (i$) and (ii$) for player II those are
the conditions on P for the properties stated in Step 3.
Step 5. In P, the coordinates s 2 and t 2 represent signals to the players.
We want to use rather the distribution of their actual actions under the
strategies (1&=) g 2 +={ u and (1&=) f 2 +=_ u in order to get conditions
independent of =. Let thus
Q(s 1 , s 2 , t 1 , t 2 )=(1&=) 2 P(s 1 , s 2 , t 1 , t 2 )+
+(1&=)
=
(1&=) P(s 1 , V, t 1 , t 2 )
*S
=
=2
P(s 1 , s 2 , t 1 , V)+
P(s 1 , V, t 1 , V).
*T
(*S)(*T)
By summing over s 2 and t 2 , we get P(s 1 , V, t 1 , V)=Q(s 1 , V, t 1 , V)the distribution (of ``recommended actions'') is preserved, as it obviously should be.
Summing then over t 2 we get
P(s 1 , s 2 , t 1 , V)=
1
=
Q(s 1 , V, t 1 , V)
Q(s 1 , s 2 , t 1 , V)&
1&=
*S
_
&
(V)
and similarly summing over s 2 yields an expression for P(s 1 , V, t 1 , t 2 ),
hence we can solve our formula for P(s 1 , s 2 , t 1 , t 2 ) in terms of Q:
P(s 1 , s 2 , t 1 , t 2 )=
1
=
Q(s 1 , V, t 1 , t 2 )
Q(s 1 , s 2 , t 1 , t 2 )&
2
(1&=)
*S
_
&
=
=2
Q(s 1 , s 2 , t 1 , V)+
Q(s 1 , V, t 1 , V) .
*T
(*S)(*T )
&
We can compute
(1&=) P(S 1 , s 2 , V, t 2 )+
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=
=
P(s 1 , s 2 , V, V)
*T
1
=
Q(s 1 , V, V, t 2 ) .
Q(s 1 , s 2 , V, t 2 )&
1&=
*S
_
&
(VV)
289
perfect correlated equilibria
Using (V) in inequality (i) yields
: Q(s 1 , s 2 , t 1 , V)(A s1t1 &A st1 )
t1
=
: Q(s 1 , V, t 1 , V)(A s1t1 &A st1 )
*S t1
\s 1 , s 2 , s.
Similarly (VV) in (ii$) yields
: Q(s 1 , s 2 , V, t 2 )(A s1t2 &A st2 )
t2
=
: Q(s 1 , V, V, t 2 )(A s1t2 &A st2 )
*S t2
\s 1 , s, s 2 .
Observe that both right-hand members are the averages over s 2 of the
corresponding left-hand members (s 1 and s fixed).
We have thus a system of inequalities of the type
x s2 =x
\s 2 # S,
where
x =
1
: xs .
*S s2 2
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Taking the average yields x =x, hence x 0 since =<1. Thus x s2 0
\s 2 # S, since =0. Therefore, if x is not identically zero, we have x >0,
and hence x s2 >0 \s 2 # S, since =>0. Conversely, if x is either identically
zero, or strictly positive in each coordinate, we can find =>0 sufficiently
small such that the system of inequalities holds.
Hence the inequalities (i) and (ii) of the statement. And if those
inequalities are satisfied, we obtain, for every s 1 and s, = 1s, s1 >0 for
inequality (i) and = 2s, s1 >0 for (ii) such that the corresponding inequalities
here above are satisfied for all smaller positive =. In particular, let = 0 =
min s, s1 [min(= 1s, s1 , = 2s, s1 )]; all our above inequalities are satisfied for any
smaller =.
By definition of Q, if P(s 1 , V, t 1 , V) (which equals Q(s 1 , V, t 1 , V)) is
positive, then Q(s 1 , s 2 , t 1 , t 2 ) is also positive for every s 2 , t 2 . And otherwise
it is identically zero. Thus inequality (iii) of the statement.
Thus, if all inequalities of the statement hold, we have some = 0 >0 such
that, for the corresponding P computed above, inequalities (i ) and (ii)
hold, and some =$ 0 >0 for inequalities (i$) and (ii$). Thus, to be sure we
have a perfect correlated equilibrium with this distribution (P(s 1 , V, t 1 , V)=
Q(s 1 , V, t 1 , V)), there remains to show that P is nonnegative. Given the formula for P, P(s 1 , s 2 , t 1 , t 2 ) will be nonnegative for = sufficiently small (say
= s1s2t1t2 ) if and only if either Q(s 1 , s 2 , t 1 , t 2 )>0 or Q(s 1 , s 2 , t 1 , V)=
Q(s 1 , V, t 1 , t 2 )=0; this follows from (iii) and (iii$). Hence with = , the minimum of = 0 , =$ 0 and all = s1s2s1t2 , the corresponding P yields a perfect
correlated equilibrium. K
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dhillon and mertens
Proposition 6. The set of perfect correlated equilibrium distributions of
a two-person game remains the same even if one were to allow arbitrary
(nonfinite) correlation devices, thus in the form of a probability space
(0, A, P) together with sub-_ fields M1 and M2 of A for players I and II
respectively. One should then define perfect equilibria of the corresponding
extended game as those Nash equilibria where each player's strategy is
optimal against some completely mixed strategy of his opponent. (Extended
game strategies are behavioural strategies.)
Proof. Step 1 remains the same, except for the proof that the obedient
strategy is undominatednow one has to say that the obedient strategy is
still optimal against the same completely mixed behavioural strategy of the
opponent as the original equilibrium strategy was, and this is the same
argument as for the equilibrium property of the obedient strategies. In Step
2, we can no longer take = independent of the message, but we can choose
= of the form n &1 for some (message-dependent) integer n, and inform the
player of this integer. One obtains then in Step 3 an equivalent correlation
device with M 1 =S_S$_N, M 2 =T_T $_N, and P a probability distribution on M 1_M 2 .
The corresponding probability Q over S_S$_T_T $ is then given by
Q(s 1 , s 2 , t 1 , t 2 )= :
n1 n2
_ (1&n
&1
1
)(1&n &1
2 ) P(s 1 , s 2 , n 1 , t 1 , t 2 , n 2 )
+
n 1&1
(1&n 2&1 ) P(s 1 , V, n 1 , t 1 , t 2 , n 2 )
*S
+
n 2&1
(1&n 1&1 ) P(s 1 , s 2 , n 1 , t 1 , V, n 2 )
*T
+
n 1&1 n 2&1
P(s 1 , V, n 1 , t 1 , V, n 2 ) .
(*S)(*T)
&
It follows first, by summing over s 2 and t 2 , that, if Q(s 1 , V, t 1 , V)>0, then
for some n 1 , n 2 , P(s 1 , V, n 1 , t 1 , V, n 2 )>0, and hence Q(s 1 , s 2 , t 1 , t 2 )>0 for
all s 2 and t 2 . Thus conditions (iii) and (iii$) hold.
Next we obtain, by summing over t 2 , that
_
Q(s 1 , s 2 , t 1 , V)=: (1&n 1&1 ) P(s 1 , s 2 , n 1 , t 1 , V, V)
n1
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+
n 1&1
P(s 1 , V, n 1 , t 1 , V, V) .
*S
&
291
perfect correlated equilibria
And inequalities (i) for player I become
E ns21 =: P(s 1 , s 2 , n 1 , t 1 , V, V)(A s1t1 &A st1 )0 \s 1 , s 2 , n 1 , s.
t1
By the above formula for Q(s 1 , s 2 , t 1 , V), this yields immediately that
: Q(s 1 , s 2 , t 1 , V)(A s1t1 &A st1 )
t1
_
=: (1&n 1&1 ) E ns21 +n 1&1
n1
1
\*S : E +& .
n1
s2
s2
So the left-hand member is nonnegative, and if it is positive for some s 2 ,
this implies that for some n 1 , E n1 is not identically zero; hence s2 E n1s 2 >0
for that n 1 , and hence the left-hand member is positive for all s 2 :
inequalities (i) hold for Q.
Similarly, for inequalities (ii), we obtain by summing the formula for Q
over t 1 that
_
Q(s 1 , s 2 , V, t 2 )=: (1&n 1&1 ) Y(s 1 , s 2 , n 1 , t 2 , )+
n1
n 1&1
Y(s 1 , V, n 1 , t 2 ) .
*S
&
with
_
Y(s 1 , s 2 , n 1 , t 1 )=: (1&n 2&1 ) P(s 1 , s 2 , n 1 , V, t 2 , n 2 )
n2
n &1
+ 2 P(s 1 , s 2 , n 1 , V, V, n 2 ) ,
*T
&
and inequalities (ii) for player I become
F ns21 =: Y(s 1 , s 2 , n 1 , t 2 )(A s1t2 &A st2 0
\s 1 , s 2 , n 1 , s.
t2
Hence
_
: Q(s 1 , s 2 , V, t 2 )(A s1t2 &A st2 )=: (1&n 1&1 ) F ns21 +n &1
1
t2
n1
\
1
: F n1
*S s2 s2
+& .
so that the same argument as above shows that inequalities (ii) are also
satisfied. K
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Remarks. (1) Thus, our restriction to finite correlation devices is
harmless; it does not even cause the loss of boundary points. And it has the
advantage of allowing the use of the standard perfect equilibrium concept.
292
dhillon and mertens
(2) The proof of Proposition 5 also exhibits, for every such Q, a corresponding (finite) canonical correlation devicethe probability distribution
P on S_S$_T_T $.
4. Example 2
Consider the game
}
0, 0
0, 0
0, 0
0, 0
1, &2
0, 0
&3, 1 1, &2
}
&3, 1 .
Observe that the 2_2 game in the bottom right corner is essentially (i.e.,
up to von NeumannMorgenstern transformations) a two-person zero sum
game; the overall game simply consists of first giving each player the
option to refuse to play this 2_2 game.
In this section, we will analyse for this game the different concepts introduced above.
Proposition 7. The correlated equilibrium distributions of example 2 are
}
p 11 p 12 p 13
p 21
0
0
p 31
0
0
}
with p 12 3p 13 , p 13 3p 12 and p 21 2p 31 , p 31 2p 21 (and p 11 0, p ij =1).
(In particular, it is the convex hull of the Nash equilibria.)
Proof. If one had a correlated equilibrium assigning positive probability to one of the strategy pairs where the sum of the payoffs is negative, the
sum of the expected payoffs would be negative, so one of both players'
expected payoff would be negative, while he can guarantee himself zero.
Given now those zeros, there only remains to impose the incentive constraints that each player does not wish to deviate when told to use his first
strategythose yield the specified inequalities. K
Proposition 8. In Example 2, every pure strategy is undominatedhence
(Proposition 4) every correlated equilibrium is acceptable.
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Proof. The second and third strategies are unique best replies to some
pure strategy of the opponent, and for the first strategy, there is clearly no
convex combination of the last two guaranteeing at least zero. K
perfect correlated equilibria
Corollary.
293
Every correlated equilibrium of Example 2 is predominant [5].
Proof. By Proposition 7 and Proposition 8, there is an acceptable
correlated equilibrium where every pure strategy is selected with positive
probability. K
Proposition 9. The perfect correlated equilibrium distributions of
Example 2 satisfy (in addition to Proposition 7) p21 =p 31 =0, and either
3p 12 >p 13 and 3p 13 >p 12 or p 12 =p 13 =0.
Proof. We use Proposition 7 and show first that p 21 =p 31 =0 in every
PCED. Let p t2 = s2 Q 2, s2 , 1, t2 , q t2 = s2 Q 3, s2 , 1, t2 . We first use only weak
inequalitiesi.e., we interpret P S as the nonnegative orthant.
Then inequality (i) of the characterization, when player II is told (1, 2)
and compares with deviating to row 3, yields p 2 2q 2 , and when player II
is told (1, 3) and compares with deviating to row 2, it yields q 3 2p 3 . On
the other hand, inequality (ii) for player I, when told (2, s 2 ) and comparing
with deviating to row 1, yields, when summed over s 2
p 2 3p 3 ,
and similarly (3, s 2 ) yields
q 3 3q 2 .
Hence, one obtains the chain of inequalities
p 2 2q 2 23 q 3 43 p 3 49 p 2 ;
hence p 2 =0, so also q 2 =q 3 =p 3 =0.
Inequalities (iii) and (iii$), i.e., Q(s 1 , t 1 )>0 O Q(s 1 , s 2 , t 1 , t 2 )>0 \s 2 , t 2 ,
imply therefore that p 21 =p 31 =0.
Therefore, in the corresponding canonical device P (Proposition 5), it
becomes superfluous to inform player I of s 1 , since this equals row 1 with
probability one; it is also superfluous to inform player II of t 2 , because
row 1 is optimal against some completely mixed strategy of player II (e.g.,
the uniform strategy), so that player I's strategy in the extended game, to
play row 1 irrespective of what he is told, will still be optimal against some
completely mixed strategy of player II in the extended game (to play
uniformly whatever he is toldi.e., the uniform mixed strategy of the
extended game). In this way, player II receives less information, and his
two optimality conditions will a fortiori be satisfied.
This means we can assume in P, and therefore also in Q, that t 2 is
selected uniformly, independently of any other choices. Thus Q(s 1 , s 2 , t 1 ,
t2 )=0 if s 1 {1 and Q(1, s 2 , t 1 , t 2 )= 13 q s2t1 for some probability matrix q st .
In terms of those, inequalities (i) of player I becomes
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(3q s3 &q s2 ) s # S # P S
and
(3q s2 &q s3 ) s # S # P S .
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dhillon and mertens
Inequalities (ii) of I and (i) of II are automatically satisfied, while inequalities
(ii) of II become q 31 2q 21 , q 21 2q 31 , q 32 2q 22 , and q 23 2q 33 .
Finally, inequalities (iii) and (iii$) become p 1, t >0 O q s, t >0 \s. Hence,
all inequalities will be satisfied if we let
q st =
}
1
3
p 11 p 12 &4=+= 2
p 13 &4=+= 2
1
3
p 11
=
3=&= 2
1
3
p 11
3=&= 2
=
}
provided 0<=<min(3p 13 &p 12 , 3p 12 &p 13 ).
Thus, if p st =0 for s{1 and 3p 13 >p 12 , 3p 12 >p 13 , then p is a PCED.
Since also (Top, Left) is clearly a PCED, (e.g., by Proposition 1 and
Remark 2 that precedes it), it follows that the conditions of Proposition 9
are sufficient.
In the converse direction, it suffices (by symmetry between indices 2 and
3) to show that p 13 >0 O 3p 12 >p 13 .
Otherwise, since 3q s, 2 q s, 3 \s (second inequality (i) for player I), and
since p 1, t = s q s, t , we would get by summing that 3q s, 2 =q s, 3 \shence
also p 12 >0.
Substituting those equations in the last inequality ((ii) for II ) we obtain
q 22 2q 32 which together with the two last inequalities q 32 2q 22 0
yields q 22 =2q 32 =0.
Since p 12 >0, this contradicts our requirement stemming from (iii) that
p 1t >0 O q st >0, \s. K
Corollary. The only perfect direct correlated equilibrium of Example 2
is where both players play their first strategy.
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Proof. From the above proposition, we must show that if a correlation
device selects recommendations L, M, and R for player II with respective
probabilities 1&p&q, p and q, where 3p>q, 3q>p, and p+q1 (and
always recommends T to player I), then in the extended game, player II's
strategy of always following the recommended action is dominated. It suffices for this to show that player II's mixed strategy (1&p&q, p, q) in the
original game is dominated, since then in the extended game, the strategy
consisting of playing the dominating strategy, irrespective of the recommendation will dominate that of following the recommendation. This is
because player II's strategy set remains the same, up to a duplication.
Observe that 3p>q and 3q>p imply both that p>0 and q>0. Let thus
p=, q=, and =>0, and the strategy (1&p&q+2=, p&=, q&=) is the
required dominating strategy. K
perfect correlated equilibria
295
Remark 3. It also follows from the last proposition that the set of
PCED is not an intersection of half-spacesthe intersection of all (open or
closed) halfspaces containing it is the set [( p 11 , p 13 ) | 3p 11 p 13 , 3p 13 p 11 ,
p 11 +p 12 +p 13 =1, | p 11 &p 13 | < 12 ].
Thus, the set of PCED cannot be described by any (even infinite) system
of weak and strict linear inequalities. It also cannot be described as the set
of solutions of a finite system of polynomial inequalities (weak and strict).
This explains the necessity of writing the system of inequalities in
Proposition 5 using the set P S observe that our set of PCED in this example is(isomorphic to) such a set, an open orthant together with its vertex.
Remark 4. To see more clearly the difference between PCE and acceptable correlated equilibria, consider the following (symmetric) variant of the
two-person game constructed in Example 2:
}
0, 0
0, 0
0, 0
0, 0
1, &2
0, 0
&2, 1 1, &2
}
&2, 1 .
Proposition 10. The only PCED is where both the players play their
first strategy.
Proof. Assume that there is a solution, e.g., with p 12 >0. Then the
barycenter of the set of solutions has the full symmetry of the problem;
hence p 12 =p 13 =p 21 =p 31 >0. Let the following 3_3 matrix X 12 represent
the distribution Q (Proposition 5) on S_S$_T_T $ when player I is
recommended strategy 1 and player II is recommended strategy 2:
}
x1 x2
x3
y1
y2
y3 .
z1
z2
z3
}
By symmetry, the corresponding matrix X 13 is obtained from the above
matrix X 12 simply by permuting the last two strategies of each player. Now
consider the inequalities (ii$) when t 1 =2 and t=1. These yield
(z i &2y i ) 3i=1 # P 3hence, use i x i =x, i y i =y, and i z i =z, 2y&z0.
Now consider inequalities (i) where s 1 =1 and s=2: we get (2x&x, 2z&y,
2y&z) # P 3 , and so, z=y=x=0. Thus, p 12 =p 13 =p 21 =p 31 =0. K
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From the above example we see the difference between acceptable and
perfect CE more clearly. Thus, in the first variant (non-symmetric), the
second and third strategy of player I were ruled out because the correlation
required between such a recommendation to I and the recommended
296
dhillon and mertens
trembles to player II in order to induce player I to play these strategies was
too high: it would give player II too much information about the recommended strategy of player I and thus an incentive to deviate from his
recommended action. This was not the case for player I when he is told to
tremble according to the recommendation of player II, since the extra
information that he gets from being told to tremble in this way does not
compensate for the negative consequences of being wrong. Thus when we
change the payoff structure in the second variant we get what is expected:
neither player will play strategy 2 or 3.
Remark 5. One might wonder whether there is (like for acceptable
correlated equilibria) a ``strategy-elimination'' property, in the sense that
the perfect correlated equilibria are just the correlated equilibria satisfying
the additional constraint that every pure strategy which is unused in every
perfect correlated equilibrium has zero probability. Example 2 shows this
is not trueotherwise the set of PCED would always be closed. Even if
one were only interested in the closure of the PCED, we doubt the
property would hold.
Appendix
Approximately Perfect Correlated Equilibria
The approximately perfect correlated equilibria are an intermediate concept between the acceptable correlated equilibria and the perfect correlated
equilibria: a perfect equilibrium is a limit of =-perfect equilibria, but while
for perfect correlated equilibria the correlation device is held fixed while
going to the limit, it is allowed to vary with = for the approximately perfect
correlated equilibria.
This material is relegated to an appendix because it merely serves to
stress the importance of keeping the correlation device fixedotherwise
one may get, in examples like that of Section 4, all the problems of
acceptable correlated equilibria (Proposition 11(b) (Remark 3 below).
Definition 4. (a)
instead of perfect.
Definition 1(c) and (d) apply as well with ``=-perfect''
(b) The approximately perfect correlated equilibrium distributions
APCED of 1 are the limits of the =-perfect correlated equilibrium distributions, with = 0.
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Below, we denote the player set by N, and use G n for n's payoff function,
and S n for his pure strategy set. S (resp. S &n ) is the set of pure strategy
vectors (resp. that of n's opponents).
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perfect correlated equilibria
Proposition 11. (a) The APCED form a compact, convex polyhedron,
depending semi-algebraically on 1.
(b) PCEDAPCEDAcceptable CED, and the first inclusion can
be strict. (In the example of Section 4, CED=APCED.)
(c) The =-perfect correlated equilibrium distributions of 1 are
generated by the distributions Q on S_S that satisfy (with $==(1&=))
{
\n, \s 1n , \s n : [G n(s 1n , s &n )&G n(s n , s &n )]
_
s &n
:
CN "[n]
$ *CQ(s 1n , V, s (N "C )" [n] ; s 2n , s C , V)
=
s2n # Sn
# P Sn
and (Q(s 1, s 2 )) s2 # S # P S \s 1 # S (or [Q(s 1n , s 1&n ; s 2n , s 2&n )] s2n # Sn # P Sn \s 1 # S,
\s 2&n # S &n , \n)
Given a distribution Q on S_S, define Q(s 1, s 2 , t) on S_S_S as having
Q as marginal on s 1, s 2 ), and, given (s 1, s 2 ), each t n independently equals s 1n
with probability 1&= and s 2n with probability =.
Then the constraints can also be written as
{
\n, \s 1n , \s n , : [G n(s 1n , t &n )&G n(s n , t &n )] Q(s 1n , V; s 2n , V; t)
t&n
=
s2n # Sn
# P Sn .
And the distribution on S is generated from Q as the marginal of Q on t # S.
Remark 6. In particular, the APCED are not just the closure of the
PCED. There can even be set of actions of a single player that has probability one under some APCED and probability zero under every PCED.
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Proof. (a) Closedness of the APCED of 1 follows immediately from
the definition.
Observe that, in characterization sub (c) Q is also a linear function of Q;
hence the =-perfect CED on Sthe marginal of Qis also a linear function
. of Q. Denote this set of distributions as D = with C = being the set
of Q's.
Observe the D ='s are decreasing (by part (a) of the definition, since the set
of =-perfect equilibria of any given game decreases with =), and that the set D
of APCED satisfies D= =>0 D = . Hence the semialgebraicity of [(G, P) | P
is an APCED of G]=[(G, P) | \=>0 _Q # C =(G ), d(P, . =(Q))=]and
the convexity of D.
Further, by compactness (and continuity of . = ) we have D = =. =(C = ), to
show that D is a polyhedron; it suffices to show that the D ='s (which
decrease) are polyhedra with a bounded number of faces, and hence that
the C ='s are so.
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dhillon and mertens
To show this, we claim that C = can be obtained in the following way:
For each group of inequalities (given by (n, s 1n , s n ) or by (n, s 1, s 2&n )) check
whether there exists a solution Q of the system which yields strict
inequalities in this group. (The average Q of those Q's is then a solution
of the system which yields strict inequalities wherever possible). If there is
such a solution, replace P sn for this group by R s+n ; otherwise replace the
group of inequalities by equations (with zero right-hand number).
Indeed, this yields a system of linear equations and weak inequalities,
whose set of solutions obviously contains C = and hence C = . And for any
solution Q of this new system, (1&$) Q+$Q is a solution of the original
systemso Q # C = .
Finally, by semi-algebraicity (in =), it follows that there exists = 0 >0 such that
the same groups of inequalities are replaced by equations and the same by weak
inequalities for all == 0 . In particular, our claim follows immediately.
(b) The inclusion PCEDAPCED is obvious: the only difference is
that, in the former, the correlation device is not allowed to vary with =. The
inclusion of the latter into the acceptable correlated equilibria is proved in
Proposition 3 above (the proof did not use the fact that the correlation was
fixed). To show that the first inclusion is strict we show that, in the example of Section 4, all correlated equilibria are approximately perfect (thus
improving Proposition 8). By convexity (point a), it suffices to show that
the extreme points are APCED. By Proposition 9, and since the set of
APCED is closed, we know that it contains all correlated equilibria with
p 21 =p 31 =0. There remains therefore only to show that it contains p 21 = 23,
p 31 = 13 (and the symmetric extreme point).
For this, let, in terms of the characterization subpart (c), Q(s, t; s$, t$)=0 for
t=2 or t=3, Q(1, 1; s$, t$)=' for t$=2 or 3, Q(1, 1; s$, 1)=Q (2, 1; s$, 3)=
Q(3, 1; s$, 2)=' 2, Q(2, 1; s$, 2)=Q(3, 1; s$, 3)=4' 2, Q(2, 1; s$, 1)=2( 19 &;),
Q(3, 1; s$, 1)= 19 &;, with ;=[2'+11' 2 ]3 and '=$6 (and assume
$(==(1&=)) 52 ). Or, to construct directly a correlation device, let s # S 1 =
[1, 2, 3] be the signal to player I and t # S 2 =[1, 2, 3] the signal to player II.
Let
\
'2
'
P= 2# 4'
#
'
2
'2
'2
4' 2
+
with #=(1&2'&11' 2 )3 and '=_6, _1.
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The following strategy pair forms a _-perfect equilibrium of the extended
game: Player I follows the recommendation with probability (1&_) and
plays uniformly with probability _, and player II plays his first pure
strategy with probability 1&_, and with probability _ he plays the recommendation with probability (1&' 2 ) and uniformly with probability ' 2.
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perfect correlated equilibria
(c)
We turn now to the proof of (c).
Let [(M n ) n # N , P, (_ n ) n # N ] be an =-perfect correlated equilibrium. Then
we can write _ n =(1&=) _ n +={ n ; where _ n is optimal against _ &n , and { n
is completely mixed; i.e., { n =(1&$) {^ n +$u n , with {^ n , 1$>0, and u n is
the uniform strategy.
View _ n and {^ n as behavioural strategies, i.e., maps from M n to n and,
given m # M, use them independently (across players, and from each other)
to generate signals s 1n and s 2n respectively, hence a distribution P on
M_S_S.
Clearly the strategies consisting of playing s 1n with probability (1&=) and
with probability =, the mixture s 2n with probability (1&$), and u n with
probability $ are still completely mixed and form an =-perfect equilibrium.
This remains a fortiori true if the original part of the message, m n , is
deleted, leaving thus a distribution P on S_S.
The incentive constraints become then
: [G n(s 1n , t &n )&G n(s n , t &n ] R(s 1n , s 2n , t &n 0 \n, \s 1n , \s 2n , \s n
t&n
with
R(s 1n ; s 2n , t &n )= :
s 1&n , s 2&n
_`
k{n
P(s 1n , s 2n , (s 1k ) k{n , (s 2k ) k{n )
_(1&=) I
tk =s 1k
+=(1&$) I tk =s 2k +
=$
.
*S k
&
Let then
Q(s 1, s 2 )=: P(s 1, s 2 ) ` (1&$) I s 2 =s 2 + $ ;
n
n
*S n
s 2
n
_
&
i.e., Q is the joint distribution of the signals (s 1n , s~ 2n ), when s~ 2n equals (independently for each player) the signal s 2n with probability (1&$) and a
uniform random variable with probability $. Then
P(s 1, s 2 )=
1
$
: Q(s 1, s 2 ) ` I s n2 =s n &
2
(1&$) N s2
*S
n
n
\
+
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as is easily checked by substituting into this formula Q(s 1, s 2 ) by its
definition.
300
dhillon and mertens
It follows, by substituting this formula into the definition of R, that
R(s 1n , s 2n , s &n =
1
$
: Q(s 1, t) I tn =s 2 &
n
1&$ t, s 1
*S n
_
&n
_`
k{n
_(1&=) I
The incentive constraints are now tn X
s 1n , s 2n , s n with
s 1k =sk
n, s 1n , sn
tn
&
&
+=I tk =sk .
[I tn =s 2n &($*S n )]0 \n,
1
s n , sn
X n,
= : [G n(s 1n , s &n )&G n(s n , s &n )]
s 2n
s&n
_ :
s 1&n , s 2&n
Q(s 1, s 2 ) ` [1&=) I s 1k =sk += I s 2 =s ];
k
k
k{n
i.e., the variables X s 2n are all at least $ times their average, for some $>0.
As seen before, this is equivalent to
1
s n , sn
(X n,
)s 2
s2
n
n # Sn
# P Sn
\n, s 1n , s n .
Dividing our expression for X by (1&=) *N&1 yields then the incentive constraints of (c). The only other constraints in our problem are the nonnegativity constraints on P: the definition of Q in terms of P implies then
that \s 1 # S, [Q(s 1, s 2 )] s 2 # S # P s , and conversely, if this holds, one checks
immediately from the formula for P in terms of Q (continuity at zero in $)
that there exists $>0 sufficiently small such that P is nonnegative. K
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Remarks. (1) The set C = can be obtained in a polynomial way as
follows: replace first every group of inequalities by the corresponding weak
inequalities, and obtain, from a standard linear programming algorithm,
which of the inequalities are identically zero on the set of all solutions. For
each of those inequalities, replace the whole group by the corresponding
equations. Repeat now the whole procedure until no more equations are
obtained.
It remains to verify whether, e.g., by an algorithm of this type in the
ordered field of rational fractions in = (with =>0 infinitesimal), one can
similarly obtain in a polynomial way the set of groups of inequalities which
are, for every sufficiently small =>0, identically zero at all solutions.
(2) Is a characterization in terms of less variables feasible (e.g., the
marginal of Q on t # S and for each n, the conditionals of (s 1n , s 2n ) # S n_S n
given t # S ? Even just in the two-person case, such a different characterization might be useful, if only to elucidate the difference (if any) with the
acceptable correlated equilibria (certainly we expect such a difference in the
N-person case).
perfect correlated equilibria
301
(3) The explanation for the misbehaviour of the APCE as seen in the
proof of part (b), is that the correlation in the device, necessary for justifying the recommended actions, is allowed to tend to zerosince only some
conditionals matter (here on columns 2 and 3)and to be swamped by the
``irrational'' part of the players' strategieswhich converges much slower to
zero. For example, in the example discussed, when player II gets his second
signal, the irrational uniform distribution that player I is using changes
completely player II's odds between the last two rows. (And this is possible
because II's conditional on the first row converges to 1although the total
probability of the first row tends to zero).
(4) Since those differences between PCE and APCE appear already
in the two-person case, where perfect equilibria coincide with undominated
equilibria, they seem to suggest that the interpretation of the former as
undominated equilibria (which is an interpretation in the unperturbed
game) is maybe more appropriate than the interpretation as ``trembling
hand'' perfection. Indeed, in the latter it is the corresponding =-equilibria
which are the ``real thing''; the complete system of beliefs is described by a
pair formed of a correlation device and a corresponding =-perfect equilibrium, and it may appear more natural then to pass to the limit on this
complete system of beliefs.
In the N-person case, it was argued in [4] that a perhaps more
appropriate concept of ``admissible best reply'' was that of a best reply
which was still a best reply against a sequence of completely mixed strategy
vectors of the opponents converging to the given strategy vector. The concept of (normal form) perfect equilibrium appears then as the appropriate
extension of that of undominated equilibrium, subject only to the additional common prior requirement (Harsanyi doctrine)the necessity of
which in this context we argued in the introduction.
(5) In the N-person case, we can expect the differences between concepts to become even bigger. In particular, acceptable correlated equilibria
face the additional difficulty there of potentially ascribing highly correlated
trembles to the opponents, even when all players' messages from the device
are completely independent from each other.
References
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1. R. J. Aumann, Subjectivity and correlation in randomized strategies, J. Math. Econ. 1
(1974), 6795.
2. F. Forges, An approach to communication equilibria, Econometrica 54 (6) (1986),
13751385.
302
dhillon and mertens
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3. E. Kohlberg and J. F. Mertens, On the strategic stability of equilibria, Econometrica 54
(1986), 10031039.
4. J. F. Mertens, Ordinality in noncooperative games, Core Discussion Paper 8728, 1987.
5. R. Myerson, Acceptable and predominant correlated equilibria, Int. J. Game Theory 15 (3)
(1986), 133154.
6. R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive
games, Int. J. Game Theory 4 (1975), 2555.