On the Impossibility of an Exact Imperfect
Monitoring Folk Theorem
Eduardo Azevedo
yAbstract
It is shown that, for almost every two-player game with imperfect monitoring, the conclusions of the classical folk theorem are false. So, even though these games admit a well-known approximate folk theorem, an exact folk theorem may only be obtained for a measure zero set of games.
A complete characterization of the e¢cient equilibria of almost every such game is also given, along with an ine¢ciency result on the imperfect monitoring prisoner’s dilemma.
1
Introduction
This thesis deals with extensions of the folk theorem to games of incomplete in-formation. Section1brie‡y reviews some of the related theory and applications of repeated games, as described by the excellent surveys of Kandori (2006), Mailath and Samuelson (2006) and Aumann (1985). The original results are presented in subsection1:4. Researchers in the …eld will probably prefer to skip it, as a technical introduction is given in section2.
1.1
Repeated games and e¢ciency
1According to Kandori (2006), economics recognizes three general ways to achieve e¢cient social outcomes: competition, contracts and long term relationships. The …rst way, competition, has been recognized since classical economists. It is the notion that interactions between a great number of participants, in a free market, dealing in undi¤erentiated commodities, will bring about e¢ciency. As put by Friedman (1962):
I would like to thank Aloisio Araujo, Vinicius Carrasco, Vijay Krishna, Paulo Maduro, Andreu Mas-Colell, Stephen Morris, and seminar participants at IMPA and Getulio Var-gas Foundation Graduate School of Economics for valuable comments and suggestions; and, specially, Humberto Moreira, for great teaching and superb advising work. Any remaining mistakes are my own.
yGetulio Vargas Foundation Graduate School of Economics.
1This subsection follows closely Kandori (2006), to whom all opinions and interpretations
So long as the e¤ective freedom of exchange is maintained, the central feature of the market organization of economic activity is that is prevents one person from interfering with another in respect to most of his activities. The consumer is protected by the seller because of the presence of other sellers with whom he can deal. The seller is protected from coercion by the consumer because of other consumers to whom he can sell.
In economic theory, this notion is represented by the …rst and second welfare theorems.
Yet, in many practical situations, commodities and services are highly dif-ferentiated, and perfect competition is unfeasible. These situations may require some degree of coordination, to align social and private interests. Contracts and long term relationships are the two basic ways in which this coordination is disciplined. Through contracts, parties may compromise themselves to an e¢cient course of action, assured that any deviation is subject to the prescribed penalties. Also, in relationships, parties may commit themselves to an e¢-cient plan, knowing that incompliance will be met by retaliation from the other party. In economic theory, contractual enforcement is the subject of the theory of contracts and mechanism design, and relationships are treated by repeated games.
Just as competition is often unavailable, contracting is also impossible in many cases. In many complex interactions, even de…ning cooperation may be di¢cult. For example, in a joint research project, how could one de…ne whether a participant is contributing enough? Also, even in cases where cooperation may be de…ned, it may be impossible, or very costly, for a court to actually verify compliance to the contract. There are also instances where a court of power su¢cient to punish defections does not even exist. This is the case of a cartel (given its illegal nature), medieval trade or international policy coordination, as measures against global warming.
1.2
The classical folk theorem
2Both the results and the mechanics of the previous example hold in a very general setting, a result known as the folk theorem. This theorem has been known since the …rst years of modern noncooperative game theory, in the late 1950s, but was not published, and its authorship is obscure - hence its name. An early version of the theorem appears in Shubik (1959) (the Nash equilibrium was introduced in Nash (1950)). The theorem is stated bellow, after some basic de…nitions.
In this thesis, an in…nitely repeated game is de…ned as the in…nitely repeated play of a stage game in periods t = 0;1;2; with players payo¤s being the discounted sum (1 )Pt 0
tgt
i, where gti is player i’s payo¤ on period t.
The factor1 is used to normalize the supergame payo¤s to average stage game payo¤s. A payo¤ vector is (strictly) individually rational if it (strictly) dominates the stage game’s minimax point. A payo¤ vector is feasible if it is the in the convex hull of the set of pure strategy payo¤s. More formal de…nitions are given in section3. Intuitively, these are very fundamental restrictions on possible equilibrium values, namely that payo¤s are phisically possible, and that each player gets more than he would get by behaving with minimum rationality. The folk theorem may be stated as follows.
Theorem 1 (folk theorem) For almost every in…nitely repeated game of per-fect information, every feasible strictly individually rational payo¤ is an equilib-rium, for a su¢ciently large discount factor .
This result tells us that the set of equilibria of repeated games of complete information is, essentially, unrestricted. This is a remarkable …nding. One thing it says is that, even though repeated games and equilibrium strategies may be very complex, a complete description of equilibrium payo¤s is available. Indeed, it is not a trivial theorem, and a large literature is concerned with its many proofs and generalizations, including Aumann (1959), Friedman (1971), Rubinstein (1979), Fudenberg and Maskin (1986) and Abreu, Dutta and Smith (1994).
The folk theorem has, nevertheless, attracted criticism to the lack of pre-dictive power of the theory (Kandori 2006). Yet, Kandori (2006) iterprets this result as showing that parties in a relationship may agree on any payo¤ as an equilibrium, and commit to this equilibrium as a relational contract. Whichever the interpretation, this thesis will address the extension of this theorem to games of incomplete information. It will be shown that, on the imperfect monitoring case, the theory does, quite generally, have some predictive power.
1.3
Repeated games with imperfect information
So far we have discussed games of perfect information. But this assumption often does not hold. Firms in a cartel may not be able to directly observe each other’s outputs (Green and Porter 1984), or their costs (Athey and Bagwell 2001). Workers in a joint project also may not observe each other’s e¤ort (Radner, Myerson and Maskin 1986). One of the pioneering works on repeated games with imperfect information was Aumann and Maschler (1966), dealing with gradual disarmament.
Yet, despite many potential economic applications, the theory of repeated games of incomplete information advanced much slower than the complete in-formation case. While the complete inin-formation folk theorem was known in the late 1950s, Aumann (1985) wrote on a survey (…rst published in 1981, and reprinted verbatim in 1985) that
In the …rst instance one simply seeks a characterization of equi-librium payo¤s, in the spirit of the folk theorem for the complete information case. Obviously the incentive-compatibility of revela-tions ia a central issue here, and one could perhaps have hoped for a solution related to the IIC set of Roger Myerson. Unfortunately the issue seems much more complex.
It was only in the works of Fudenberg and Levine (1994) and Fudenberg, Levine and Maskin (1994) that such results were obtained. These authors worked on an speci…c type of incomplete information - games with imperfect public monitoring. In these games, while players’ actions are private informa-tion, they receive a public signal of the actions taken by others. Still, this class includes most applications, and is much better understood than the pri-vate monitoring case. The Fudenberg, Levine and Maskin folk theorem may be stated as:
Theorem 2 (FLM folk theorem) For almost every game of incomplete in-formation with a su¢ciently large signal space, any feasible strictly individually rational payo¤ may be approximated by equilibria, as the discount factor ap-proaches1.
It must be noted that the FLM folk theorem is anapproximate folk theorem. It only says that every feasible individually rational payo¤ may be approximated by equilibria. In contrast, the classical folk theorem states that every feasible individually rational payo¤ is exactly an equilibrium value. In this sense, the FLM folk theorem is usually viewed as a result for in…nitely patient players.
Athey and Bagwell (2001) discovered an example of such a game with e¢cient equilibria, pointing to this possibility. Yet, this thesis answers the previous question in the negative.
1.4
Original results
This thesis shows that, for almost every game with imperfect public monitoring, the conclusions of the classic folk theorem are false. That is, there are feasible, strictly individually rational payo¤s which are not equilibria. This implies that it is impossible to …nd an exact imperfect monitoring folk theorem - any such result would have to be restricted to a measure0 class of games (see Theorem 4).
It will also be shown that, on the repeated prisoner’s dilemma with imperfect monitoring, generically, no e¢cient equilibria exist. This result depends only on the combinatorial structure of the game’s payo¤s, and applies to similar games, such as models of collusion with moral hazard, as in Green and Porter (1984) (see Theorem 3).
Also, a complete description of the e¢cient equilibria of almost every game with imperfect public monitoring is given. This description is similar to the one presented in Fudenberg, Levine and Takahashi (forthcoming), though it is not contained nor contains their characterization. The former is restricted to two-player games, but players may be impatient. As for the latter, it applies for
n-player games, but requires in…nite patience. The two previous results follow directly from this characterization (see Lemma 3), and are also restricted to two-player games.
2
Technical overview
It is well known that repeated games with imperfect monitoring admit an ap-proximate folk theorem (Fudenberg, Levine and Maskin 1994). For a large class of such games, any feasible individually rational payo¤ may be approximated by perfect public equilibria (PPE), as players become in…nitely patient. However, the question of whether such payo¤s can be achieved exactly in equilibrium is still open.
In this thesis it will be shown that the answer for this question is that, generically, no. Formally, for almost every game with imperfect monitoring, there are feasible, strictly individually rational payo¤s which are not PPEs. That is, the conclusions of the classical folk theorem are false for almost every game with imperfect monitoring. Therefore an exact folk theorem may only be obtained for a zero measure class of such games.
Figure 1: The feasible payo¤s in the prisoner’s dilemma.
a very hard problem, for almost every two-player game they may be found by solving a static mechanism design problem.
The …rst description of an e¢cient PPE for a repeated game with imper-fect monitoring was given by Athey and Bagwell (2001) in a repeated duopoly game. Lemma 3 extends their techniques to get a complete description of the set of e¢cient PPE for almost every two-player game. Another related work is Fudenberg, Levine and Takahashi (forthcoming). They present an algorithm for determining the limit set of e¢cient PPEs. Although their techniques ap-ply to then-player case, they only characterize the limit set as players become in…nitely patient. In contrast, our characterization, while being limited to the
2-player case, gives the set of e¢cient PPE for …xed values of impatience factor less than one. The characterization will also be used to obtain an ine¢ciency result for the repeated imperfect monitoring prisoner’s dilemma.
These points can be illustrated by the following classical example which will be revisited along the exposition of the thesis.
2.1
The prisoner’s dilemma
Consider a repeated prisoner’s dilemma with imperfect monitoring, usually de-scribed as a partnership problem. Two players own a …rm and operate it. Each period, they may either cooperate (c) or defect (d), i.e., their e¤ort levels are
si2 fc; dg. Although the …rm’s pro…ty2Y is observed, it is randomly
distrib-uted according to (js). Players’ payo¤s are ri =y=2 si. Therefore, while
The average payo¤ to playeriis de…ned as
gi(s) =E(rijs) =
Z
ri(y; s)d (yjs):
For the game to be interesting, they should be distributed as in a prisoners’ dilemma, as indicated on Figure 1. That is,
cd1 < dd1< cc1< dc1
dc2 < dd2< cc2< cd2
where we have abused notation by using pairs such ascdto denote both strategy pro…les and average payo¤s asg(cd).
This stage game is repeated every period and players discount future utility by some impatience factor0< <1.
It is well known that, in this example, the approximate folk theorem holds generically if Y has at least three elements (Fudenberg, Levine and Maskin, 1994). That is, that any feasible individually rational payo¤ may be approx-imated by PPEs, as approaches to 1. Yet, we will show that generically the prisoner’s dilemma has no e¢cient equilibria,3 for any discount factor (see Theorem 3).
Also, this means that the game has feasible strictly individually rational payo¤s which are not equilibrium values. So, the conclusions of the classic folk theorem fail. We will show that this phenomenon is very general. Except for some trivial cases, an exact folk theorem is false for almost every game with imperfect monitoring (see Theorem 4).
However, the example is somewhat misleading, as the anti-folk theorem is not simply an ine¢ciency result. Indeed, there are open sets of games which have e¢cient equilibria. Thus, the general case will be somewhat more complicated. About the prisoner’s dilemma ine¢ciency, Radner, Myerson and Maskin (1986) showed that in some cases where the approximate folk theorem fails, the equilibria are bounded away from the e¢cient frontier. However, we are not aware of ine¢ciency results in cases where the approximate folk theorem holds in the literature.4
We now lay down the model, give the characterization of the e¢cient equi-libria and prove the results.
3
The Framework
3.1
The stage game
Two players play a stage game att= 0;1;2; . Playeritakes actions si in a …nite set Si. Actions are not directly observable, but they induce probabilities
3Throughout the paper, we will use equilibria interchanging with PPEs.
4In the recent book on this topic, Mailath and Samuelson (2006) have not mentioned any
(js)on a…nite set of public outcomesY. The actual payo¤ri(si; y)to a player
depends on his own action and on the public outcome. But not directly on the other player’s action (although it does a¤ect the distribution ofy). We also allow for mixed actions iin Si, the space of probabilities onSi. Probabilities (j )
and payo¤sri( i; y)are de…ned as usual, for any mixed strategy pro…le .5
We may collect these elements into the following:
De…nition 1 A stage game is a list(S1; S2; Y; :S ! Y; r1: (S1; Y) ! R; r2: (S2; Y) !R); whereS1,S2 andY are …nite sets.
Let gi( ) = E(ri( i; y)j ) be the average gain of player i from playing a
pro…le .
3.2
The supergame
The public history at timet is de…ned as ht = (y1
; y2
; yt 1
). Players also observe their own actions, and playeri’s private history ishi
t= (a
1
i; a
2
i; at
1
i ).
A strategy for playeriis a sequence of mapsf t
i :ht hti ! Sig1t=0. In the repeated game players maximize
vi = (1 )
X t
Egti
for0 <1. The factor(1 )normalizes supergame payo¤s as average stage game payo¤s.
Our solution concept is the perfect public equilibrium, henceforth PPE. That is, a subgame perfect equilibria in which players condition their actions only on the public history. Abreu Pearce and Stachetti (1990) show that, as long as other players play public strategies, there is no gain in conditioning on private history.6 LetP P E( )denote the set of perfect public equilibrium values for a given discount factor .
A typical stage game has a strategic form such as
l r
u m d
0 @AC BD
E F
1 A
Throughout this thesis we will use lettersA; B; C; to denote both action pro…les, such asll;and their payo¤ vectors,g(ll).
We admit correlated equilibria, as in Aumann (1987): every period players can condition their action on a public randomization device. This makes the set of feasible payo¤s a convex polygon - the convex hull of the pure strategy payo¤s. Figure 2 shows the set of feasible payo¤s of a typical game.
5In the prisoner’s dilemma game,S
i=fcooperate;defectg,yis the …rm’s pro…t andri= y=2 si.
6Yet, Kandori and Obara (2006) show that e¢ciency may be improved if all players use
Figure 2: The set of feasible payo¤s for a generic game.
For simplicity, we also make the following assumption, which holds for almost every game.
A1. The set of feasible payo¤s of the supergame, V;is a polygon such that each side contains only two pure action pro…les.7
3.3
The recursive method
We rely heavily on recursive methods, so we now state some de…nitions and known results which will be useful. Abreu, Pearce and Stachetti (1990), Fuden-berg and Tirole (1991), Mailath and Samuelson (2006) and FudenFuden-berg, Levine and Maskin (1994) are good references.
The key idea of the recursive approach is to factor equilibria into an action pro…le to be played on the current period and continuation values to be played conditional on the public outcomey. The continuation values for each players are expressed by reward functionsu:Y !R2
.
De…nition 2 A reward function u -enforces if for i = 1;2 and every s in Si:
vi = (1 )gi( ) + E(ui(y)j ) (1)
(1 )gi(si; i) + E(ui(y)jsi; i)
that is,
E(ui(y)j ) E(ui(y)jsi; i)
1
(gi(si; i) gi( )):
A key element of the recursive approach is the Bellman mapT. The Bellman map of a setW R2
is the set of values that may be achieved using promises onW.
De…nition 3 The Bellman map T is de…ned for compact subsets ofR2
by
T(W) = cofg( ) +E(uj (j )) :utakes values in W and -enforces g;
wherecois the convex hull. A setW inR2
is self-generating ifW T(W).
The key facts we will use are summarized in the next:
Proposition 1 (Known results) The set P P E( ) is compact and convex, and is the largest …xed point of T. All self-generating sets are contained in P P E( ). T takes compact sets on compact sets.
4
The e¢cient PPEs
4.1
Mixing property
We will now give, for almost every game, a complete description of the e¢cient equilibria. Since the set of feasible payo¤s is a polygon, we just need a method to determine which are the equilibria on each side of the polygon. Then, by applying this method to each side of the polygon, we may …nd all e¢cient equilibria on a …nite number of steps.
Consider, from now on, a sideABof the set of feasible values. For simplic-ity, assume AB to be negatively inclined with slope m, as in Figure 2. Also, suppose AB contains no static equilibria. Our strategy will be to somehow restrict the di¢cult problem of determining the game’s e¢cient PPEs to the one-dimensional segmentAB.
In general this cannot be done, because for the equilibrium payo¤s in AB
one may use equilibrium values outside AB as punishments. So, we need a generic condition which we call mixing. A strategy pro…le is locally mixing if deviations from it cannot be detected with certainty. The stage game is said to be mixing if every pro…le is locally mixing.
De…nition 4 The strategy pro…le s is locally mixing if (jsei; s i) (js)8 for i = 1;2 and every esi in Si. The stage game is mixing if all (js) are equivalent, for every s in S. Or equivalently, if every pro…le s in S is locally mixing.
Mixing is the key property that allows us to work out the equilibria on any given sideABof feasible values polygon without having to know all the equilibria of the game. Because mixing implies that any public history may happen with positive probability, any punishments prescribed in equilibrium may actually be carried out with positive probability. Thus, equilibria with values onAB may only use continuation values and strategy pro…les onAB.
LetI( ) =AB\P P E( ). In terms of the Bellman map these facts translate into the following:
8That is, (jes
Lemma 1 IfA andB are locally mixing, the closed intervalI( )is the largest self-generating closed interval inAB.
Proof. I( )is a closed interval, once it is the intersection ofABwithP P E( ), which is know to be closed and convex. By Proposition 1 every self generating interval inABis contained in I( ). Because of mixing, T(W)\AB T(W\
AB). But sinceT(P P E( )) =P P E( ), we have
I( ) = P P E( )\AB=T(P P E( ))\AB T(P P E( )\AB);
that is,I( )is self-generating.
4.2
The characterization
In this subsection we will work out what the set of equilibria in AB is. This result is recorded as Lemma 3.
Lemma 1 shows that to have the characterization, all that we have to do is to …nd the largest self-generating interval onAB. This will be done now. Take some interval [a; b] in AB. We check what are the conditions for [a; b] to be self-generating. Let[a0; b0]beT([a; b])\AB. By the de…nition ofT,
a0= (1 )A+ E(u) = (1 )A+ E(u a); (2)
where the expectation is conditional onAanduis the promise function enforcing
A with minimal Eu1. Let " be the horizontal distance betweenE(u) and a. By rearranging equation (2), we see thata0 aif and only if
1 " a1 A1; (3)
that is,[a; b]is self-generating if and only if (3) holds. We now calculate" . By its de…nition, we know that
" = min
u Eu1 a1 (4)
s.t. u -enforcesA
u(y)2[a; b], for ally2Y
and that uis the promise function that solves program (4). Since -enforcing is invariant by translation, we will show that, at least for large enough, it is possible to …nd" by a much simpler program.
Consider the program
e" = min
u maxy u1 (5)
s.t. u -enforcesA
Eu= 0
in whichuis normalized to have0 average andmis the slope of segment AB. This program is in fact very simple, because ifusatis…es its constraints for some
, then 1 1 0
0 usatis…es the constraints for
0
. Therefore,e" =1 e"1=2. The next lemma shows formally that the programs are equivalent for large enough.
Lemma 2 If is large enough, then programs (4) and (5) are equivalent.
Proof. First let us prove thate" " . If u satis…es the constraints in (4), thenu Eusatis…es the constraints in (5). Now, takeue1=2solving program (5) for = 1=2 and letu = 1 u1=2 be the solutions for other values of . Then,
u=ue +a+" (1; m)satis…es all restrictions of program (4), except for having allu1(y) b1. But if we take high enough, this restriction is also satis…ed, so
" e" .9
By (3),[a; b]is self-generating if and only if
e"1=2 a1 A1: (6)
Condition (6) allows us to describe the conditions for a segment to be self-generating in very simple terms. Since the set of equilibria onABis the largest self-generating interval, this also allows us to describe this set. We now sum up the previous discussion in a lemma that describes all the e¢cient equilibria on
AB.
De…nePi be de…ned as
min
u maxy ui (7)
s.t. u1=2-enforces
Eu= 0 (balanced expected payment)
u2=mu1 (budget balance).
which is the amount of punishment that has to be in‡icted on playerifor him to play his unfavorable action . Therefore,P1
A is exactly what we callede"1=2. From (6) we have the following:
Lemma 3 Suppose thatA andB are locally mixing. Let aand b be points in AB with a1 = A1+PA1 and b2 =B2+PB2 (see Figure 3). Then, for large
enough the set of equilibria inAB is
[a; b] if a1< b1;
empty, otherwise.
Hence, the only restrictions on the equilibria are that players have to get at least what they would get playing their least favored action plus the amount of punishment necessary to enforce it.
9This proof and some computer simulations show that, in practice, does not have to be
Figure 3: The set of equilibria onAB.
Note that, to enforce pro…le A (resp. B), player1 (resp. 2) must receive punishment strictly greater than his gain from deviating to his best response to
A(resp. B). We have the following useful:
Lemma 4 Suppose that the stage game is mixing. At any equilibrium payo¤ inAB, player1 (resp. 2) receives strictly more than on his best response toA (resp. B).
Proof. Let Ab be the pro…le where player 2 plays A2 and player 1 plays his best response to A2. For program (7) applied to A; since u enforces A and
E(ujA) = 0, we must have
E( u1jAb) Ab1 A1:
Therefore,max
y ui>Ab1 A1 andA1+P
1
A>Ab1.
5
The anti-folk theorem
The anti-folk theorem and the ine¢ciency result for the prisoner’s dilemma follow easily from the previous lemmata. We start with the ine¢ciency in the prisoner’s dilemma:
Theorem 3 Let be a mixing stage game and AB be a segment negatively inclined contained in the Pareto frontier (as in Figure 3). If any player plays
the same action on pro…lesAandB; then there is no PPE payo¤ in AB.
Proof. For simplicity suppose again thatABis negatively inclined and thatA
action on pro…lesAandB. Therefore, player1’s best response toA2dominates
B1. By Lemma 4 there is no e¢cient equilibrium inAB.
For the prisoner’s dilemma example of section 2.1, since the Pareto frontier is made up of segmentscd; ccandcc; dc, this theorem applies immediately. Hence, for almost every prisoner’s dilemma there are no e¢cient equilibria.
We now turn to the anti-folk theorem. The additional assumptions used are necessary because there are pathological examples in which the folk theorem is true for the stage game, so these cases have to be excluded.
Theorem 4 (Anti-folk) The folk theorem is false generically for games with feasible payo¤s strictly dominating the minimax point and without e¢cient static equilibria. That is, almost every such game has feasible individually rational payo¤s which are not equilibrium payo¤s.
Proof. Consider a game with mixing and only two pure strategy pro…les on each edge of the feasible set (this is the generic situation). Take a segment
AB with a point strictly dominating the minimax value. Suppose that it is negatively inclined, and vertices labeled as in the Lemma 3.
In any equilibrium payo¤s inABplayer1gets strictly more than by playing his best response toA. But the best response toAyields at least as much utility as his minimax value. The positively inclined case is similar. This completes the proof.
6
Conclusions
Our main contribution is to show that, even though games with imperfect moni-toring admit an approximate folk theorem (e.g., Fudenberg, Levine and Maskin, 1994), an exact folk theorem is valid only for a zero measure set of such games. This …nding also shows that the classical folk theorem is extremely unstable with respect to imperfect monitoring. Given a game with perfect monitoring, if its informational structure su¤ers a random perturbation, the classical folk theorem will be false with probability one.
We must point out that our anti-folk theorem is not simply an ine¢ciency result, since there are open sets of games with e¢cient PPEs. Yet, the mes-sage that the prisoner’s dilemma and other similar examples gives is that exact e¢ciency is di¢cult to achieve for repeated games.
Finally, we believe that the characterization and the method presented here may be also useful to studying particular games as well.
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