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Bayesian incentive compatible beliefs
Journal
of Mathematical
Economics
BAYESIAN
10 (1982) 83-103.
INCENTIVE
North-Holland
Publishing
COMPATIBLE
Company
BELIEFS
C. d’ASPREMONT UniversitP Catholique de Louvain,
1348 Louvain-la-Neuve,
Belgium
L.-A. GERARD-VARET* Universite
Louis Pasteur, 67084 Strasbourg,
UniversitP Aix-Marseille
France
III, 13621 Aix-en-Provence,
France
The problem of incentives for correct revelation is studied as a game with incomplete information where players have individual beliefs concerning others’ types. General conditions on the beliefs are given which are shown to be sufficient for the existence of a Pareto-eflicient mechanism for which truth-telling is a Bayesian equilibrium.
1. Introduction Much attention has been devoted recently to a game-theoretic analysis of the strategic use of private information by individual agents in a decentralized mechanism, designed to reach a Pareto optimal outcome.’ In a previous paper’ we have distinguished two approaches to such games of incomplete information. The first approach, the ‘complete ignorance’ approach, relies on strong equilibrium concepts, such as the ‘uniform equilibrium’ or the ‘dominant strategy equilibrium’, which are independent of any particular representation of the players’ expectations. This is not the case for the second approach which is based on Harsanyi (1967/8) concept of ‘Bayesian equilibrium’. One basic result for this approach3 is that under a transferability assumption on the utility functions and an independence assumption on the beliefs (or probability distributions) of the players, there exist Pareto-efficient mechanisms (with feasible transfer schemes) for which *The materials presented in this paper were worked out when the author was visiting the Institut fur Gesellschaftsund Wirtschaftswissenschaften, Universitlt Bonn, Germany. Support from the Institut is gratefully acknowledged. The authors want to thank J.-P. Florens, G. Laffond, J.-F. Mertens, H. Moulin, S. Sorin and all participants at the 1980 Stanford I.M.S.S.S. Summer Workshop for helpful comments. ‘For a recent bibliography, see Green and Laffont (1979). *d’Aspremont and Gerard-Varet (1979). %ee Arrow (1979) and d’Aspremont and Gerard-Varet (1975,1979).
03044068/82/000&0000/$02.75
0
1982 North-Holland
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C. d’dspremont and L.-A. Gerard-Varet, BIC beliefs
‘truth-telling’ is a Bayesian equilibrium. However, it has also been shown that the same result could be obtained under a weaker condition - called the ‘compatibility condition’ - but for discrete probability distributions. In the present work, we shall show how to extend to a more general framework the result obtained under the assumption of discrete beliefs. We shall also provide a detailed discussion4 of the ‘compatibility condition’, implying in particular that it is not a necessary condition for the existence of a Pareto-efficient mechanism for which ‘truth-telling’ is a Bayesian equilibrium. Finally we shall show that a much less restrictive condition is required if Pareto-efficiency is weakened to a property in expected value.
2. The communication
game of incomplete
information
In its simplest form a mechanism based on preference revelation may be described as follows. Suppose a finite set N = {1,. . ., i,. . .,n} of players and a set X of outcomes. To give examples ’ X can be a set of possible public projects or a set of quantity levels of a public good (or bad) or, more generally, the set of joint strategies in some n-person game. Assume that the set of admissible preference orderings of X for any player i in N can be represented by the family of payofffunctions U,(x;a,), where cli is a general parameter varying in some space Ai and summarizing all the private information of player i. We also say that Ai is the space of possible types of player i E N. Furthermore, we shall suppose that all payoff transfers among players are permitted and that every Ui(.;ai) is so calibrated that, for any x in X and any transfer y in R, the payoff of player i E N of parameter ai E Ai is equal to Ui(x; cli)+ y. In other words we assume unrestricted side-payments with full-transferability. The sets X and (A,; iE N} and the functions {Ui; iE N} are common knowledge, but player i E N is privately informed only about his own type in Ai. Let us assume that any player iE N has to independently announce to the others some type as being his own type and that Ai is the space of all his possible messages. A mechanism is then defined as a selection rule s, which is a function from the message n-tuple a = (a,, u2,. . ., a,) in A = x ;= 1 Ai to the set of outcomes, and a transfer scheme t, which associates to each n-tuple of messages UE A a certain vector of transfers (tl(a), . ., t,(a)) in R” made to the individual players. So, for any n-tuple of messages SEA, the payoff of player i E N of type rxiE Ai under the mechanism m = (s, t) can be defined as w~(U; ai) = Ui(S(a); ai) + ti(U). ‘In particular we shall give stronger but more interpretable conditions. ‘See Green and Laffont (1977) or Groves and Loeb (1975).
C. d’Aspremont and L.-A. Gerard-Varet, BIC beliefs
85
Actually, we shall restrict our attention to transfer schemes t which may be called budget-balancing in the sense that
Vae-4 i~tA4=0. Also, in the following, we shall be concerned efficient, ndmely such that V U E A,
by selection rules which are
1 Ui(S(U);ai) = max C Ui(x; a,). XOXisN
ieN
A mechanism m=(s, t) is called efficient if its selection rule is efficient and budget-balancing if its transfer scheme is budget-balancing. These two properties ensure that a mechanism has to be feasible and Pareto-optimal under ‘full information’.6 Hence we shall call Pareto-eficient a mechanism which is both efficient and budget-balancing. In the approach we take here, every player in N is supposed to have only partial information about the value of all players’ types. This generates a game of incomplete information. More specifically we shall associate to each player i E N a measurable space (Ai, pi) and a transition probability Pi from this space (A, di) towards the n - 1 product space denoted (A-i, d-i), where A-i=
X Aj
jsN jPi
and
d_i=odj
JEN j+i
This transition probability, also denoted for simplicity Pi = {P,( ( ai);cliE Ai}, represents the beliefs of player i about the other players’ types conditional to his own type.’ All transitions Pi, taken as functions, are of common knowledge. However, player i beliefs are fully known only when his type CliE Ai is also known. For the sequel, we shall maintain all the following assumptions: l
For every i E N, Ai is a compact matric space and ~2~ denotes its Bore1 Oalgebra.
.
X is a compact metric space.
l
For every iE N, Ui is a continuous function from X x Ai to R.
‘See Harris and Townsend (1981) for a discussion of the notion of ‘full-information’ optimality, also sometimes called ‘ex-post’ optimality. ‘The notion of a transition probability is defined in Neveu (1970). Notice that Pi may be considered as a regular version of the conditional probability given di arising from a basic probability measure over (A, d) =( x isNAi, &,, &J.
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C. d’dspremont and L.-A. GLrard-Varet,BIC beliefs
.
For every i EN, Pi is a continuous function form Ai to the space probability measures over (A _i, JL~), endowed with the topology pointwise convergence. Every P,(. ) cq) is of full support.
l
The space X and all functions efficient-selection rule.
of of
Ui are such that there exists a continuous
We shall call continuous efjcient mechanisms which are continuous and where s is efficient.
these
mechanisms
m=(s,
t)
One particular case, to be called the discrete case, arises when for every player ieN, Ai is a finite set. Then, we define pi as the set of all subsets of Ai and continuity is considered, trivially, with respect to the discrete topology. In the discrete case, every Pi( 1ai) will simply denote a function from ki to R, such that Ca_ioA_iPi(cr~iIai)=l. We may now introduce the game of incomplete information which can be associated to a mechanism m. We first define, for every player i E N, a set of (normalized) strategies as the set A: of measurable functions (decision rules) aF:A,+A, associating to each possible type aie Ai a message (or revealed type) ai=a,*(a,). We then define, for every player i e N, for every type cli E Ai, and for any choice aTi at) of strategies by the other players, the conditional ai*_,,a,*,,,..., =(a?,..., expected payoff which player i is supposed to maximize on A, I?‘: (a, a? i; EJ = .j
where, for notational
Wy(a, a? ,(a _ i); cci)Pi(da _ i 1C(i),
-1 convenience,
a* i(c( i) = (a:(a,),
. . ., a,“_
ltai
- A
a,*,
lCc(i+
I),..
A Bayesian equilibrium in such a game is a n-tuple = x isN AT such that, for every iEN and every CAKE Ai, VaiE Ai,
WT(ai, Z i; ai) 2 W~($(CfJ,
., 434J. of strategies
a* E A*
E* i; c(i).
We are interested here by those mechanisms which are such that every player has ‘interest to reveal’ the information he is privately controlling. Consider, for that matter, the truth-telling strategy denoted ri,* for ieN and function). Thus, a defined by V tii6 Ai, @(ai)=czi (i.e., ci: is the identity mechanism m is said to be Bayesian Incentive Compatible (for short BIC) if of truth-telling strategies is a Bayesian and only if the n-tuple circa* equilibrium in the game of incomplete information associated to m.
3. Pareto-efficiency
and the BIC-property
The difficulty associated to the determination of efficient BIC-mechanisms arises essentially when one requires, in addition, that the transfers cancel
C. d’Aspremont and L.-A. Gerard-bet,
BIG heli@Ys
87
each other (budget-balance). In a previous paper [d’Aspremont and GCrardVaret (197911, we have given a ‘compatibility condition’ imposed on the players’ beliefs which, in the discrete case (every Ai is finite), is sufficient for the existence of a Pareto-efficient BIC-mechanism. The present section is devoted to the more general case. In that framework we first give a condition, involving both individual beliefs and payoffs, which is necessary and sufficient for the existence of a Pareto-efficient mechanism having a slightly weaker BIC-property. We finally obtain a sufficient condition, concerning the beliefs alone which coincides, for the discrete case, with the ‘compatibility condition’. We say that, for every e>O, a mechanism m satisfies the E-BIG-property iff
i.e., in words, the n-tuple 8* EA* of truth-telling strategies is a Bayesian cequilibrium of the game associated to m. In order to investigate the possibility of finding Pareto-efficient E-BICmechanisms let us introduce some more notation. Let C(A) be the space of real-valued continuous function on A. Let T denote the space C(A)” of continuous functions from A to R” and ?= {TV T; xisNti(.) ~0). Lcl. for every in N, Gi be the linear VZEC(A),
application
from C(A) to RAf defined by
G,(z)(u~,c~~)~~~ ,S. [z(cr)-z(Ui,a~i)]Pi(drx~i(cri).
-I To interpret this definition, suppose ZE C(A) is a ‘payoff function’ of some player i, defined on the space A of all types. Then the quantity Gi(z)(ai,ai) appears as an expected gain (or loss) obtained by player i ex ante for deviating from the situation where all players use their truth-telling strategies: this expected gain results from his announcing of a,~ Ai, while being of type ai. Thus it may be called player i’s expected gain from (unilateral) deviation at (a, ai) w.r.t. z. Now, for any selection rule s and i E N, we denote by ~(u,
Gus ,S [Vi(s(u, ai); ai)- Ui(s(a); Ui)]Pi(da-i -L
With this notation the problem of finding BIC-mechanism amounts to find a solution linear inequalities: V~EN, where s is assumed
VaiEAi,
VU~EA~,
1C(i).
a continuous Pareto-efficient t E T to the following system
u~#N,,
to be efficient and continuous.
Eof
Gi(ti) (a, ai) > G(u, q) - E, (I)
C. d’dspremont and L.-A. Gerard-Varet, BIC beliefs
88
Let ,4 denote the set of vectors of finite Borel-measures ;1 = (4,k *f -3&)#O, where each li is a measure on (Ai x Ai, di@ai). Then we can state: Theorem 1. A necessary and sufficient condition for the existence continuous Pareto-efficient &-BIC-mechanism is: Vi E A, if VZEC(A),
of a
Vi,jEN,
then & 6 Cq(aiy 4 - ~1Udai,dd < 0. Proof:
(2)
We proceed in two steps:
Step I. First we can show that if, for any z E C(A) and i EN, Gi(z) is a continuous function defined on A? then the result derives directly from the necessary and sufficient condition given in Theorem 15 in Ky Fan (1956). Indeed letting Y denote the product space x ;..C(Af) with llyll =maxNxAf 1yi(ai,Bi)l, YE Y? and lltll =maxNxA Iti(~ t E IT; the application G=(G, ,..., Gi,..., G,) from T to Y is continuous and the set Int Q dzfInt(y E Y:y lo} #a. So Ky Fan’s condition may be written: t/g # 0 to the conjugate convex cone Q* of Q (i.e., Q* belonging ={gE Y*:VyEQ,g(y)zO)) if
(4
V te r
g(G(t))=O,
then (b)
g(oS-_E?
for
(oS-E)d~f(P, +E ,..., O;--E ,..., Os,-E).
NOW VYE I: VgE Q*, g(y)=~isNgi(yi), where every gi is the bounded linear functional defined on C(A?) by g,(y,) =g(O, . . ., yi, . . ., 0). Clearly g,(y,) 2 0 if y,~O. Hence, by Riesz representation theorem, to every such gi there corresponds a unique finite Bore1 measure Ai on A? such that
(4
vYi E Ai,
gi(YJ
= f Yi(ai,Wddai, W. A?
If follows immediately that (b) is equivalent to (2). To get the equivalence of (a) and (l), suppose that (1) holds for some 1 E A. Then for any t E T and any ieN, k G,(tJ(a, 4Udai,daJ=~2
G,(tJ(al, a,)A, (da,, da,), 1
C. d’dspremont
and L.-A. Gtrard-Varet,
89
BIC beliefs
and hence we may associate some gE Q*, g#O, such that g satisfies (c) and Vt E
g(G(t))=
~
C S Gi(ti)(U, ai)Ai(da, dai) iEN AT
To prove the converse take any g E Q*, g# 0, such that, Vt E ?; g(G(t)) = 0 and take any z E C(A). If (a) holds we may then define, for any pair {i,j} cN, some FE ;i; such that 6 = z, 4 = - z and & = 0, for k $ {i, j), and thus get
0= dG(fJ)= gAGi - g_iGj(z))
where 1 E A is defined according to (c). Therefore the two conditions are equivalent remains only to show the following. Step 2.
V1’EN,
VZ~ C(A),
and to prove the result it
Gi(Z)E C(A:).
Take any z E C(A). By the definition of G, it is enough to show that both SA_i~(~i.or_i)Pi(da_i)ai) and ~A_iz(ai~G(-i)Pi(d~-i~~i) belong to C(Af). Since Af
is compact z is the limit of sums of the form CfE’=,fi(aJgkK(a_i), where every fi E C(Ai) and every tiKE C(A _ i). Take any sequence (a?, c$‘) in A? which is convergent to (a:,$‘). Clearly, by assumption or definition, we have that lim P,(* 1BY) = P,(. ) a:), wl-*m Hence, by a general convergence Royden (1968, p. 232)]
lim f”, (a:) = f”, (a:), m-m argument
lim f”, (al) = f”, (a:). m-tao
[see e.g. Proposition
lim J ~(r-i)Pi(dr-i/a~)=A~,~(a_i)Pi(da-iJcxp), m+OIA
-i
-,
and, by Lebesgue convergence theorem, Vai
E
A,
1.[~_~~~~~(~3R:(n-i)]Pi(da-ilod -I
1
Pi(da_i\ai).
18 in
90
Finally,
C. d’Aspremont and L.-A. Gtrard-Varet, BIC beliej?
using properties lim
of limits, we get
J ~(a:, CL _ ,)P,(da _ i 1 a;)
f?l~a,
A-,
= lim
5
lim
fi(ay)
f g~(a_i)Pi(da-iIa~)
=,S .z(ap,a-i)Pi(da_i 1a:). -8 Similarly, lim 1 m-m A_i
z(u:,
ct _
JPi(da _i ( ay) = dirnmkilfk(al) -t
,S $K(a_ i)Pi(da _ i / LX:) -I
An unsatisfactory feature of the necessary and sufficient this theorem is that it involves not only the given transition also the given payoff functions. It is a joint condition on the utilities of the players. Our purpose now is to state which will concern only the beliefs of the players. This conform to the classical problem of preference revelation. that this condition is equivalent, in the discrete case, to condition’ stated in d’Aspremont and Gerard-Varet (1979). Condition C.
conditon stated in probabilities but the beliefs and on another condition seems to be more We shall see later the ‘compatibility
V~EA, if
VZEC(A),
‘diEN,
VjEN,
,S2 Gi(z)(Ui, GUd~t~da,)= ,S2 Gj(z)(aj, ajP,Xduj, daj),
(1)
Vz
(2)
then E
C(A),
V i E N,
S Gi(z)(Ui,aJii(dai, A;
dai) =O.
To enunciate Condition C verbally, interpret i = (A,, . . ., &,. . ., i,) as being a system of individual scaling rules where Ai gives weights which are attributed
91
C. d’Aspremont and L.-A. Gerard-Varet, BIC beliefs
ex ante to player i for every value of his private information and declaration. Thus JA: Gi(z)(ai,cr,)li(dai,dai) is player i weighted sum, under the system A, of all possible expected gains from deviation w.r.t. ZEC(A). Condition C says that if for some system AE/~ of individual scaling rules the weighted sums of all possible expected gains from deviation are made comparable and identical, across payoff functions ZE C(A) and players, then they should be identically zero. Assume that all players met before knowing of any mechanism and before having been privately informed about their own type. They plan to cooperate against the mechanism. Condition C says that there is no room for such cooperation in the sense that there does not exist a system ;l~:n of individual scaling rules making the weighted sum of conditional expected gains from deviation of every player both comparable and equal to some non-zeroamount. In order to show that Condition C is suficient to give the existence of a solution t E 7 to system (I), we need the following lemma where a continuous efficient BIC-mechanism is constructed: Lemma 1.
Let s be a continuous eficient selection rule, 3 t* E T such that V i E N,
Proof.
V ai E Ai,
V cliE Ai,
Gi(ti)(ai,q) 2 Q(ae a,).
Define ViEN,
VaEA,
Since every Ui and efficiency of s,
s are
t:(a)=,&iU,(.S(a);aj).
continuous
functions,
t* E 7: Moreover
by the
ui(s(@);c(J + t;(a)>=Ui(s(ai, ff _i); ai)+ @(ai, M-i). Integrating
on both sides the result follows.
With this lemma
0
we can state and prove the following
theorem:
Theorem 2. If the given beliefs {P,iE N} satisfy Condition exists a continuous, Pareto-efficient .+BIG-mechanism. Proof We show that, and sufficient condition
with Lemma of Theorem
C, then there
1, Condition C implies the necessary 1. First notice that this last condition
C. d’dspremont and L.-A. Gerard-Varet, BIC beliefs
92
has the same antecedent any 1~ A, if (3)
V t E T,
as Condition
C. Also, by Lemma 1, we have, for
C S Gi(ti)(a, ai)li(dUi, da3 = 0, ioN
A:
then (4)
C S ~(Ueai)li(dUedcli)~O. isN
At
Indeed, for t* E Tas defined in Lemma 1, we get O= C ieN
J 1 Gi(t,*)(Q, ai)~i(dU, dai)~ 1 J O;(U, UJli(daip dai). At
isN
A_i
AZ
Clearly (3) above is equivalent to the consequent (2) of Condition C. Therefore (4) must hold, which implies the consequent (2) of the condition in Theorem 1, since the vector measure 1 is different from zero and s>O. The result follows. 0 We shall now show that Condition C is equivalent to a statement about individual beliefs. For that let d” = (EE&; E= X ixNEi and V~EN, Ei E &J be the semi-algebra of rectangles in d. Then we show: Theorem 3. VLGA, if (1)
A family of beliefs {Pi, i E N} satisfies Condition C if and only if
VEEJZZO, ViEN,
VjEN,
.,i E,Pi(E- i 1@i(dai~ dad I
_f Pi(Ei ( @ddai, dd E,xA,
I
then (4
VEEZP,
ViEN,
A C, P,(E_i) cr,)il,(dU,dai) -,,i,, 1 i
I
PJE-i I dU%~d4=Q I
Let, for every i E N and 1 E A, F: denote the following bounded linear functional on C(A):
ProoJ:
V z E C(A),
C. d’Aspremont and L.-A. GLrard-Varet, BIC beliefs
93
By Riesz representation theorem there corresponds a unique finite signed Bore1 meaure Qi, on A such that, Vz E C(A), F:(z) = IAZ(U)Q:(da). Now take, for every iE N and 1~ /1, the signed measure P” on (A, J@‘) defined by the property
We see that
SZ(Cr)P:(da)= S S z(a)Pi(da _ i 1ai)li(dU, deli)
V Z E C(A),
A
i.e., Pf =Qf.The theorem follows.
A_iAf
0
In the discrete case, Theorem 3 shows that Condition C is equivalent to the ‘compatibility condition’ already mentioned.g Therefore by Theorem 7 in d’Aspremont and Gerard-Varet (1979), Theorem 2 can be strenghtened in the discrete case by putting E=O. 4. On compatible beliefs Our purpose in the present section is to provide a detailed analysis of the compatibility condition - Condition C - determining families of beliefs which are admissible for the existence of Pareto-efficient mechanisms satisfying the BIG-property. We first consider two conditions which are stronger than Condition C, but such that each one of them has a more specific interpretation. Second, we study connections between Condition C and another restriction upon beliefs known in the literature as the ‘consistency’ condition. In that case a weaker budget-balancing property can 90ne may provide for the statement of Theorem 3 an interpretation similar to the one given for Condition C. Indeed, in a Bayesian set-up, the beliefs Pi give the bets P,@_i ).) of player in N on events E_,E& -i concerning the others’ information type, conditional to the player’s own information .di. JMathE-
D
94
C. d’dspremont and L.-A. GLrard-Varet, BlC beliefs
be defined and a stronger positive result can be obtained. Last, we show that Condition C is not necessary by exhibiting a family of discrete beliefs which do not fullil the requirement and however is such that, whatever may be the players’ payoffs (under the full-transferability assumption), there exist efficient budget-balancing BIC-mechanisms. Take the beliefs of a particular player, say player ie N. We say that player i beliefs are free of any dependency with respect to his own types or, for short, that he has free beliefs iff” Vtli E Ai,
Vai E Ai,
Pi(’ ) pi) = Pi(. / $).
In terms of information, free beliefs mean that the true beliefs of player i E N are common knowledge. Consider now the whole family {Pi; i E N} of beliefs. We state: Condition F.
There exists at least one player who has free beliefs.
We shall now see that Condition F is sufficient for having Condition and, as a consequence, the result of Theorem 2. We first prove: Lemma 2.
C,
Given a family of beliefs (Pi; i E N}, for every 1 E A, if ViEN,
VjEN,
V i E N,
V Ei E di,
VEE&‘O,
then li(Ei X AJ = Iq(Ai X Ei).
Proof. Take any i E N and consider We get, for j # i, &(A, x Ei) - li(Ei x Ai) =
E =
s “1
Ei x A_i in SP in the antecedent.
P,(E -j 1a$Xdaj, dmj)
“j ‘. *j
- A l A. I
“‘Thus, d-i and .di are a-algebras which Over (A, sJ) generating Pi.
Pj(E- j 1aj)A,{daj,duj)= 0. Cl
J
are independent
with respect
to the proh;~hili~y
C. d’Aspremont and L.-A. GPrard-Varet, BIC beliefs 7‘11co1wn
95
Given a family of beliefs {Pi; iE N), Condition F implies Condition
4.
C‘. Proof. Suppose w.1.o.g. that player 1 has free beliefs, i.e., V cr,,V Cc,,Pl( ( x,) = Pl(. / El)= xl(.) and that Condition C does not hold. Then, by Theorem 3, there exist 2 EA and E E d”, such that Vj E N - (11,
=A.!~,‘XE-jlaj)~~,~daj,daj!-E [, P,(E-jIaj)~j(daj,daj)#O. J
I
J
This is a contradiction
to Lemma
2.
J
0
Since Condition F implies Condition C, its only advantage is that it provides an interpretation for an important subcase, in which the beliefs of at least one player are fully known. An even stronger condition is to require that every player has free beliefs. We shall call it Condition F* [it is called the independence condition in d’Aspremont and Gerard-Varet (1979)]. In this case we obtain a stronger result, namely that there exists a Pareto-efficient BIC-mechanism. However other interpretable subcases may be considered. An n-person normal form game over the spaces {A,; i E N), taken as stategy spaces, is a function o from A to R” where wi is the payoff function of player iE N. The game is zero-sum if xi.N~izO. Moreover we say that the family (Pi; i E N) provides a strict correlated equilibrium for the game iff ViEN,
VcrieAi,
~~,wi(U,a-i)Pi(da_ilai)
Vai~Ai,
ai #
ai,
,wi(ai,a~i)Pi(da~ilcri). -I
This non-cooperative equilibrium notion has to be considered as referring to an underlying random mechanism designed by the players for selecting their strategies. Thus the Pi’s represent the beliefs of the players about the outcome of this mechanism.” We now consider the following new condition beliefs {Pi; iE N}. In order to match Condition version of this condition:
relative to the family of C, we give a continuous
Condition B. There exists a zero-sum game, {(Ai, 0,); i E N), with continuous payoff functions and for which the family of beliefs {Pi; i E N} provides a strict correlated equilibrium. “See Aumann
(1974).
C. d’dspremont and L.-A. Gh-ard-Varet, BIC beliefs
96
The statement of Condition B may be written as follows:
3WE7; such that V1’EN, Vai~Ai,
VUiEAi,
Gi(oJ(a,a;)>O.
Consider on the other hand any family of payoff functions {Ui; iEN} and take an efficient selection rule s. Define, as in the beginning of section 3, the numbers Of(a, CQ),ieN, ai~Ai, cciEAi. From Condition B, each oi being multiplied by a constant correctly chosen, one easily deduce the solvability of system (I) with E= 0, i.e., 3tQ
such that V i E N,
V EliE A,
VU,E A,
Gi(ti)(a, xi) 2 oS(ai, EJ.
Thus, we prove: Theorem 5. Assuming Condition B, there exists a (continuous) eficient and budget-balancing BIC-mechanism. One interpretation of Theorem 5 is that the individual beliefs characterized by Condition B must introduce sufficient ‘disagreements’ between the players in order to make ‘correlation’ worthwhile in a game having a strong competitive character.” Comparisons of Condition B with Conditions the following characterization: Lemma 3. such that
F, F* and C, are based on
Given {Pi; i E N), Condition B holds $ and only if there is no A E A,
VZEC(A),
ViEN,
VjEN,
Proof Using, as in Theorem 1, Ky Fan (1956), a necessary and sufficient condition for solving system (I), in which we take q()-E identically null, is “See Aumann (1974 p. 67, and example 2.1,~. 68). This property captures features those which are noticed in an example given in Laffont and Maskin (1979, p. 307).
similar
to
97
C. d’Aspremont and L.-A. GPrard-Varet, BIG belief7
simply: there is no g #O in Q* such that, Vt E =i; g(G(t))=O. By the same reasoning as in Theorem 1 we get the result. 0 Theorem (i)
6.
Given a family
Condition
of beliefs {Pi; i E N}, if Condition
B holds, then
C holds.
(ii) Condition F does not hold, Proofi (i) is immediate from Lemma 3. To prove (ii), suppose w.1.o.g. that player 1 has free beliefs represented by nl. Then Condition B, implies
such that Vai~Ai,
Vai~Ai,
,wi(ai,m_Jx(da-i), -1
,S mi(a,u-Jn(da-J>,S
--I which forms a set of contradictory
inequalities.
0
Theorem 6 shows that the beliefs satisfying Condition F and those satisfying Condition B are two distinct subclasses of those satisfying Condition C. However, the size of the class of beliefs satisfying C but neither B nor F, remains an open question. Our next objective is to compare Condition C with another condition known as ‘consistency’.’ 3 Let, for that matter, pi be the set of all probability measures over (Ae~i) and A be the set of all probability measures over (A, d). We say that {Pi; i E N} is a consistent family of beliefs if 3~~
x~i,
ARES,
RfO,
ioN
such that QEEEO,
VisN,
g,Pi(E_,lai)vi(daJ=R(E).
Thus the Pi’s are, for the respective players, the different conditionalizations of the same joint probability. Our first result is for the two player case: Theorem 7. Assuming N = (1,2} and given (PI, P2), beliefs, Condition C is equivalent to Condition F*.
‘%ee, for example,
Harsanyi
(1967/g) for a detailed
discussion
a pair of consistent
of this condition.
C. d’Aspremont and L.-A. Gerard-Varet, BIC beliefs
98
Proof
Given
{P1,P2},
3v,~A?,,
by consistency ~v,EJ%?~,
we have R#O,
UREA?,
such that VEEAP,
Also, the negation
W4L2f,
of Condition
i#L
d, Pi(Ej ) ~i)vi(dai) = R(E).
C is, by Theorem
3, equivalent
to
IAEA, such that
is equal for both i and all E E ~2’ and different Define ,? E A to be V i E { 1,2},
V E, Fi E d,
from zero for some E in ~2”.
li(Ei x Fi) = v,(E,). v,(F,).
Thus, VEEJP,
=
ViE{1,2},
= R(E)-R(Ai = R(E)-
If
ILd, 1
1,Pi(Ej(aJvi(daJ -
Condition
-v,(E,).v,(E,)#O
either. Theorem
vi(daJ
X Ej) ’ Vt(EJ
V,~Ej). Vi(Ei).
F*
does not hold then it is well and so Condition for some EEJP, 4 implies the other direction. 0
However, this negative result following example demonstrates:
known that C does not
only holds in the case of two players,
Theorem 8. For INI > 2, there exist consistent families Condition C and not Condition F*.
R(E) hold
as the
qf beliefs which satisfy
C. d’Aspremont and L.-A. Gerard-Varet, BIC beliefs
99
According to Theorem 2, it is sufficient to find a consistent family of beliefs for which only one player has free beliefs (i.e., satisfying Condition F). Consider ~=u,2,3), A,={a,,cr,},A,=(81,Bz), A,={y,,yJ, the following beliefs satisfy the requirements: Proof:
For player 1, P,
a1 a2
Pl Yl
B2Yl
blY2
B2Y2
4115 119
l/15 219
8115 219
2115 419
For player 2, P,
4115
l/l5
8/15
2115
119
219
219
419
3124 3124
3124 3124
6124 6124
For player 3, P,
Yl “92
12124 12124
0
As a last example, we shall take the simplest case of two players, each having only two types, i.e., N={1,2),
A I = {@I>a,),
J42={Bl?BzI.
We shall, for this case, consider the following class of beliefs, choosing 6 #$ 02651:
100
C. d’dspremont and L.-A. Gkrard-Varet, BIG beliefs
Notice that none of these beliefs are free and that it is a consistent family of beliefs. Consider now any set X of outcomes and any family of two payoff functions {U1, U,} satisfying the side-payment full-transferability assumption. Take any selection rule s which is efficient. As in the proof of Theorem 1, finding a balanced transfer scheme t E RN xA,such that m=(s, t) is a BICmechanism, consists in showing that, V(i,(al, a,), Arta,, c(i), &(fi,, pz), &(Bz, Pi)) E R4, - {O},if v t E =i;
then
Now, assuming (i), we get
and
But, for the given family of beliefs, we have
Pl(Pl I%)-P1(P1 Ic4=P,h implying (iii)
A, =&=&.
)B1)-P2@1I B2)=26- 17
C. d’dspremont and L.-A. Gerard-Varet, BIG beliefs
Now, given (iii) and the assumed
By applying
twice the efficiency
101
family of beliefs, we have in (ii)
property,
this is smaller
than or equal to
Thus, there exists, for the given family of beliefs, an efficient budget-balancing mechanism which is BIC. Also, since S#$ Condition C does not hold (see Theorem 3). Therefore, we may state: Theorem 9. Condition C is not necessary budget-balancing mechanism which is BZC.
for
the existence
of an efficient
The last situation which we shall consider using the same approach is the one where we keep the consistency condition but without requiring Condition C. In this case we can only find efficient-BIC-mechanisms satisfying a weak budget-balancing property, namely,
R(da) = 0.
For the sake of simplicity we shall consider this new property, expected-budget-balancing property, only in the disrete case:
called the
C. d’Aspremont and L.-A. Ghard-Varet,
102
BIC beliefs
Theorem 10. In the discrete case and for any consistent family of beliefs, there exists an efficient-BIC-mechanism which satisfies the expected-budgetbalancing property. Proof of Theorem 9. We have following system of inequalities: V i E IV,
to show
V ai E Ai,
V CliE Ai,
the existence
of t E T solving
the
Gi(ti)(ai, gi) 2 q(ai, NJ,
and
where s is some efficient selection rule. A well-known [see e.g. Ky Fan (1956)] necessary and sufficient for the existence of a solution of (II) is, VLE A, VOLER, if
condition
(1) then
(2) But
=,?A
&
tiara,
Cpitcl-i
( @i(ai,
ai)-Pi(M-i
L 1 So that (1) is equivalent
(1’)
ViEN,
14~i(%~ai)l~
to the following:
Vcci~Ai,
aTA. [Pi(a-i1@i)ii(aiT cci)-pi(a-i/ 4U%ai)l =PRta). I
1
Since, for some CX-i E A Pi, En, R(cc_ i, ui) # 0, we get in (1’) that ,u = 0. Therefore the necessary and sufficient condition becomes, V ;1E A, if V t E 7:
1 C itNa,tA,a,tA!
1 Gi(tJ(ai, aJLi(ai, tli)=O,
(1)
C. d’Aspremont
103
and L.-A. GCrard-Varet, BIG belief7
then iL
oTA,
I
JA,
II
This holds by Lemma
UsCai,
aiVi(ai~
ai) 5
O.
I 1 (see the argument
in the proof of Theorem
2).
0
In conclusion, the first contribution of this section is to provide some interpretable conditions on the beliefs implying Condition C, and thus representing sufficient conditions for the existence of a Pareto-efficient BICmechanism [whatever the (transferable) payoff functions]. They require that the beliefs of at least one player be fully known or that the beliefs of all players be sufficiently in ‘disagreement’ to make correlation, in some sense, profitable. However, the Compatibility Condition C is not necessary. There are families of beliefs which do not satisfy Condition C but have such strong symmetry properties that it remains possible to design a Pareto-optimal feasible mechanism which is Bayesian incentive compatible. The second contribution concerns the much more general ‘consistency’ condition. Under this condition we prove (for the discrete case) that it is always possible to find an efficient BIC-mechanism, provided the budget is only balanced in expected value.
References Arrow, K.J., 1979, The property rights doctrine and demand revelation under incomplete information. in: Economics and human welfare (Academic Press, New York). d’Aspremont, C. and L.-A. Gerard-Varet, 1975, Individual incentives and collective efficiency for an externality game with incomplete information, CORE DP 7519 (Universite Catholique de Louvain, Louvain-la-Neuve). d’Aspremont, C. and L.-A. Gerard-Varef 1979, Incentives and incomplete information, Journal of Public Economics 11, 2545. R.J., 1974, Subjectivity and correlation in randomized strategies, Journal of Aumann. Mathematical Economics 1, 67-96. Green, J. and J.-J. Lalfont, 1977, Characterization of satisfactory mechanisms for the revelation of preference for public goods, Econometrica 45, 783-809. Green, J. and J.-J. Laffont, 1979, Incentives in public decision making (North-Holland, Amsterdam). Groves, T. and M. Loeb, 1975, Incentives and public inputs, Journal of Public Economics 4, 21 l-226. Harris, M. and R.M. Townsend, 1981, Resource allocation under asymmetric information, Econometrica 49, 33-64. Harsanyi, J.C., 1967/g, Games with incomplete information played by ‘Bayesian Plavers’, Part IIII, Management Science 14, 158-182, 320-334,486-502. Ky Fan, 1956, On systems of inequalities, in: H.W. Kuhn and A.W. Tucker, eds., Linear inequalities and related systems (Princeton University Press, Princeton, NJ). Laffont, J.-J. and E.S. Maskin, 1979, A differential approach to expected utility maximizing mechanisms, J.-J. Laffont, ed., Aggregation and revelation of preferences (North-Holland, Amsterdam). Neveu, J.. 1970, Bases mathematiques du calcul des probabilites (Masson. Paris). Royden, J.L., 1968, Real analysis (Macmillan, New York).
of Mathematical
Economics
BAYESIAN
10 (1982) 83-103.
INCENTIVE
North-Holland
Publishing
COMPATIBLE
Company
BELIEFS
C. d’ASPREMONT UniversitP Catholique de Louvain,
1348 Louvain-la-Neuve,
Belgium
L.-A. GERARD-VARET* Universite
Louis Pasteur, 67084 Strasbourg,
UniversitP Aix-Marseille
France
III, 13621 Aix-en-Provence,
France
The problem of incentives for correct revelation is studied as a game with incomplete information where players have individual beliefs concerning others’ types. General conditions on the beliefs are given which are shown to be sufficient for the existence of a Pareto-eflicient mechanism for which truth-telling is a Bayesian equilibrium.
1. Introduction Much attention has been devoted recently to a game-theoretic analysis of the strategic use of private information by individual agents in a decentralized mechanism, designed to reach a Pareto optimal outcome.’ In a previous paper’ we have distinguished two approaches to such games of incomplete information. The first approach, the ‘complete ignorance’ approach, relies on strong equilibrium concepts, such as the ‘uniform equilibrium’ or the ‘dominant strategy equilibrium’, which are independent of any particular representation of the players’ expectations. This is not the case for the second approach which is based on Harsanyi (1967/8) concept of ‘Bayesian equilibrium’. One basic result for this approach3 is that under a transferability assumption on the utility functions and an independence assumption on the beliefs (or probability distributions) of the players, there exist Pareto-efficient mechanisms (with feasible transfer schemes) for which *The materials presented in this paper were worked out when the author was visiting the Institut fur Gesellschaftsund Wirtschaftswissenschaften, Universitlt Bonn, Germany. Support from the Institut is gratefully acknowledged. The authors want to thank J.-P. Florens, G. Laffond, J.-F. Mertens, H. Moulin, S. Sorin and all participants at the 1980 Stanford I.M.S.S.S. Summer Workshop for helpful comments. ‘For a recent bibliography, see Green and Laffont (1979). *d’Aspremont and Gerard-Varet (1979). %ee Arrow (1979) and d’Aspremont and Gerard-Varet (1975,1979).
03044068/82/000&0000/$02.75
0
1982 North-Holland
84
C. d’dspremont and L.-A. Gerard-Varet, BIC beliefs
‘truth-telling’ is a Bayesian equilibrium. However, it has also been shown that the same result could be obtained under a weaker condition - called the ‘compatibility condition’ - but for discrete probability distributions. In the present work, we shall show how to extend to a more general framework the result obtained under the assumption of discrete beliefs. We shall also provide a detailed discussion4 of the ‘compatibility condition’, implying in particular that it is not a necessary condition for the existence of a Pareto-efficient mechanism for which ‘truth-telling’ is a Bayesian equilibrium. Finally we shall show that a much less restrictive condition is required if Pareto-efficiency is weakened to a property in expected value.
2. The communication
game of incomplete
information
In its simplest form a mechanism based on preference revelation may be described as follows. Suppose a finite set N = {1,. . ., i,. . .,n} of players and a set X of outcomes. To give examples ’ X can be a set of possible public projects or a set of quantity levels of a public good (or bad) or, more generally, the set of joint strategies in some n-person game. Assume that the set of admissible preference orderings of X for any player i in N can be represented by the family of payofffunctions U,(x;a,), where cli is a general parameter varying in some space Ai and summarizing all the private information of player i. We also say that Ai is the space of possible types of player i E N. Furthermore, we shall suppose that all payoff transfers among players are permitted and that every Ui(.;ai) is so calibrated that, for any x in X and any transfer y in R, the payoff of player i E N of parameter ai E Ai is equal to Ui(x; cli)+ y. In other words we assume unrestricted side-payments with full-transferability. The sets X and (A,; iE N} and the functions {Ui; iE N} are common knowledge, but player i E N is privately informed only about his own type in Ai. Let us assume that any player iE N has to independently announce to the others some type as being his own type and that Ai is the space of all his possible messages. A mechanism is then defined as a selection rule s, which is a function from the message n-tuple a = (a,, u2,. . ., a,) in A = x ;= 1 Ai to the set of outcomes, and a transfer scheme t, which associates to each n-tuple of messages UE A a certain vector of transfers (tl(a), . ., t,(a)) in R” made to the individual players. So, for any n-tuple of messages SEA, the payoff of player i E N of type rxiE Ai under the mechanism m = (s, t) can be defined as w~(U; ai) = Ui(S(a); ai) + ti(U). ‘In particular we shall give stronger but more interpretable conditions. ‘See Green and Laffont (1977) or Groves and Loeb (1975).
C. d’Aspremont and L.-A. Gerard-Varet, BIC beliefs
85
Actually, we shall restrict our attention to transfer schemes t which may be called budget-balancing in the sense that
Vae-4 i~tA4=0. Also, in the following, we shall be concerned efficient, ndmely such that V U E A,
by selection rules which are
1 Ui(S(U);ai) = max C Ui(x; a,). XOXisN
ieN
A mechanism m=(s, t) is called efficient if its selection rule is efficient and budget-balancing if its transfer scheme is budget-balancing. These two properties ensure that a mechanism has to be feasible and Pareto-optimal under ‘full information’.6 Hence we shall call Pareto-eficient a mechanism which is both efficient and budget-balancing. In the approach we take here, every player in N is supposed to have only partial information about the value of all players’ types. This generates a game of incomplete information. More specifically we shall associate to each player i E N a measurable space (Ai, pi) and a transition probability Pi from this space (A, di) towards the n - 1 product space denoted (A-i, d-i), where A-i=
X Aj
jsN jPi
and
d_i=odj
JEN j+i
This transition probability, also denoted for simplicity Pi = {P,( ( ai);cliE Ai}, represents the beliefs of player i about the other players’ types conditional to his own type.’ All transitions Pi, taken as functions, are of common knowledge. However, player i beliefs are fully known only when his type CliE Ai is also known. For the sequel, we shall maintain all the following assumptions: l
For every i E N, Ai is a compact matric space and ~2~ denotes its Bore1 Oalgebra.
.
X is a compact metric space.
l
For every iE N, Ui is a continuous function from X x Ai to R.
‘See Harris and Townsend (1981) for a discussion of the notion of ‘full-information’ optimality, also sometimes called ‘ex-post’ optimality. ‘The notion of a transition probability is defined in Neveu (1970). Notice that Pi may be considered as a regular version of the conditional probability given di arising from a basic probability measure over (A, d) =( x isNAi, &,, &J.
86
C. d’dspremont and L.-A. GLrard-Varet,BIC beliefs
.
For every i EN, Pi is a continuous function form Ai to the space probability measures over (A _i, JL~), endowed with the topology pointwise convergence. Every P,(. ) cq) is of full support.
l
The space X and all functions efficient-selection rule.
of of
Ui are such that there exists a continuous
We shall call continuous efjcient mechanisms which are continuous and where s is efficient.
these
mechanisms
m=(s,
t)
One particular case, to be called the discrete case, arises when for every player ieN, Ai is a finite set. Then, we define pi as the set of all subsets of Ai and continuity is considered, trivially, with respect to the discrete topology. In the discrete case, every Pi( 1ai) will simply denote a function from ki to R, such that Ca_ioA_iPi(cr~iIai)=l. We may now introduce the game of incomplete information which can be associated to a mechanism m. We first define, for every player i E N, a set of (normalized) strategies as the set A: of measurable functions (decision rules) aF:A,+A, associating to each possible type aie Ai a message (or revealed type) ai=a,*(a,). We then define, for every player i e N, for every type cli E Ai, and for any choice aTi at) of strategies by the other players, the conditional ai*_,,a,*,,,..., =(a?,..., expected payoff which player i is supposed to maximize on A, I?‘: (a, a? i; EJ = .j
where, for notational
Wy(a, a? ,(a _ i); cci)Pi(da _ i 1C(i),
-1 convenience,
a* i(c( i) = (a:(a,),
. . ., a,“_
ltai
- A
a,*,
lCc(i+
I),..
A Bayesian equilibrium in such a game is a n-tuple = x isN AT such that, for every iEN and every CAKE Ai, VaiE Ai,
WT(ai, Z i; ai) 2 W~($(CfJ,
., 434J. of strategies
a* E A*
E* i; c(i).
We are interested here by those mechanisms which are such that every player has ‘interest to reveal’ the information he is privately controlling. Consider, for that matter, the truth-telling strategy denoted ri,* for ieN and function). Thus, a defined by V tii6 Ai, @(ai)=czi (i.e., ci: is the identity mechanism m is said to be Bayesian Incentive Compatible (for short BIC) if of truth-telling strategies is a Bayesian and only if the n-tuple circa* equilibrium in the game of incomplete information associated to m.
3. Pareto-efficiency
and the BIC-property
The difficulty associated to the determination of efficient BIC-mechanisms arises essentially when one requires, in addition, that the transfers cancel
C. d’Aspremont and L.-A. Gerard-bet,
BIG heli@Ys
87
each other (budget-balance). In a previous paper [d’Aspremont and GCrardVaret (197911, we have given a ‘compatibility condition’ imposed on the players’ beliefs which, in the discrete case (every Ai is finite), is sufficient for the existence of a Pareto-efficient BIC-mechanism. The present section is devoted to the more general case. In that framework we first give a condition, involving both individual beliefs and payoffs, which is necessary and sufficient for the existence of a Pareto-efficient mechanism having a slightly weaker BIC-property. We finally obtain a sufficient condition, concerning the beliefs alone which coincides, for the discrete case, with the ‘compatibility condition’. We say that, for every e>O, a mechanism m satisfies the E-BIG-property iff
i.e., in words, the n-tuple 8* EA* of truth-telling strategies is a Bayesian cequilibrium of the game associated to m. In order to investigate the possibility of finding Pareto-efficient E-BICmechanisms let us introduce some more notation. Let C(A) be the space of real-valued continuous function on A. Let T denote the space C(A)” of continuous functions from A to R” and ?= {TV T; xisNti(.) ~0). Lcl. for every in N, Gi be the linear VZEC(A),
application
from C(A) to RAf defined by
G,(z)(u~,c~~)~~~ ,S. [z(cr)-z(Ui,a~i)]Pi(drx~i(cri).
-I To interpret this definition, suppose ZE C(A) is a ‘payoff function’ of some player i, defined on the space A of all types. Then the quantity Gi(z)(ai,ai) appears as an expected gain (or loss) obtained by player i ex ante for deviating from the situation where all players use their truth-telling strategies: this expected gain results from his announcing of a,~ Ai, while being of type ai. Thus it may be called player i’s expected gain from (unilateral) deviation at (a, ai) w.r.t. z. Now, for any selection rule s and i E N, we denote by ~(u,
Gus ,S [Vi(s(u, ai); ai)- Ui(s(a); Ui)]Pi(da-i -L
With this notation the problem of finding BIC-mechanism amounts to find a solution linear inequalities: V~EN, where s is assumed
VaiEAi,
VU~EA~,
1C(i).
a continuous Pareto-efficient t E T to the following system
u~#N,,
to be efficient and continuous.
Eof
Gi(ti) (a, ai) > G(u, q) - E, (I)
C. d’dspremont and L.-A. Gerard-Varet, BIC beliefs
88
Let ,4 denote the set of vectors of finite Borel-measures ;1 = (4,k *f -3&)#O, where each li is a measure on (Ai x Ai, di@ai). Then we can state: Theorem 1. A necessary and sufficient condition for the existence continuous Pareto-efficient &-BIC-mechanism is: Vi E A, if VZEC(A),
of a
Vi,jEN,
then & 6 Cq(aiy 4 - ~1Udai,dd < 0. Proof:
(2)
We proceed in two steps:
Step I. First we can show that if, for any z E C(A) and i EN, Gi(z) is a continuous function defined on A? then the result derives directly from the necessary and sufficient condition given in Theorem 15 in Ky Fan (1956). Indeed letting Y denote the product space x ;..C(Af) with llyll =maxNxAf 1yi(ai,Bi)l, YE Y? and lltll =maxNxA Iti(~ t E IT; the application G=(G, ,..., Gi,..., G,) from T to Y is continuous and the set Int Q dzfInt(y E Y:y lo} #a. So Ky Fan’s condition may be written: t/g # 0 to the conjugate convex cone Q* of Q (i.e., Q* belonging ={gE Y*:VyEQ,g(y)zO)) if
(4
V te r
g(G(t))=O,
then (b)
g(oS-_E?
for
(oS-E)d~f(P, +E ,..., O;--E ,..., Os,-E).
NOW VYE I: VgE Q*, g(y)=~isNgi(yi), where every gi is the bounded linear functional defined on C(A?) by g,(y,) =g(O, . . ., yi, . . ., 0). Clearly g,(y,) 2 0 if y,~O. Hence, by Riesz representation theorem, to every such gi there corresponds a unique finite Bore1 measure Ai on A? such that
(4
vYi E Ai,
gi(YJ
= f Yi(ai,Wddai, W. A?
If follows immediately that (b) is equivalent to (2). To get the equivalence of (a) and (l), suppose that (1) holds for some 1 E A. Then for any t E T and any ieN, k G,(tJ(a, 4Udai,daJ=~2
G,(tJ(al, a,)A, (da,, da,), 1
C. d’dspremont
and L.-A. Gtrard-Varet,
89
BIC beliefs
and hence we may associate some gE Q*, g#O, such that g satisfies (c) and Vt E
g(G(t))=
~
C S Gi(ti)(U, ai)Ai(da, dai) iEN AT
To prove the converse take any g E Q*, g# 0, such that, Vt E ?; g(G(t)) = 0 and take any z E C(A). If (a) holds we may then define, for any pair {i,j} cN, some FE ;i; such that 6 = z, 4 = - z and & = 0, for k $ {i, j), and thus get
0= dG(fJ)= gAGi - g_iGj(z))
where 1 E A is defined according to (c). Therefore the two conditions are equivalent remains only to show the following. Step 2.
V1’EN,
VZ~ C(A),
and to prove the result it
Gi(Z)E C(A:).
Take any z E C(A). By the definition of G, it is enough to show that both SA_i~(~i.or_i)Pi(da_i)ai) and ~A_iz(ai~G(-i)Pi(d~-i~~i) belong to C(Af). Since Af
is compact z is the limit of sums of the form CfE’=,fi(aJgkK(a_i), where every fi E C(Ai) and every tiKE C(A _ i). Take any sequence (a?, c$‘) in A? which is convergent to (a:,$‘). Clearly, by assumption or definition, we have that lim P,(* 1BY) = P,(. ) a:), wl-*m Hence, by a general convergence Royden (1968, p. 232)]
lim f”, (a:) = f”, (a:), m-m argument
lim f”, (al) = f”, (a:). m-tao
[see e.g. Proposition
lim J ~(r-i)Pi(dr-i/a~)=A~,~(a_i)Pi(da-iJcxp), m+OIA
-i
-,
and, by Lebesgue convergence theorem, Vai
E
A,
1.[~_~~~~~(~3R:(n-i)]Pi(da-ilod -I
1
Pi(da_i\ai).
18 in
90
Finally,
C. d’Aspremont and L.-A. Gtrard-Varet, BIC beliej?
using properties lim
of limits, we get
J ~(a:, CL _ ,)P,(da _ i 1 a;)
f?l~a,
A-,
= lim
5
lim
fi(ay)
f g~(a_i)Pi(da-iIa~)
=,S .z(ap,a-i)Pi(da_i 1a:). -8 Similarly, lim 1 m-m A_i
z(u:,
ct _
JPi(da _i ( ay) = dirnmkilfk(al) -t
,S $K(a_ i)Pi(da _ i / LX:) -I
An unsatisfactory feature of the necessary and sufficient this theorem is that it involves not only the given transition also the given payoff functions. It is a joint condition on the utilities of the players. Our purpose now is to state which will concern only the beliefs of the players. This conform to the classical problem of preference revelation. that this condition is equivalent, in the discrete case, to condition’ stated in d’Aspremont and Gerard-Varet (1979). Condition C.
conditon stated in probabilities but the beliefs and on another condition seems to be more We shall see later the ‘compatibility
V~EA, if
VZEC(A),
‘diEN,
VjEN,
,S2 Gi(z)(Ui, GUd~t~da,)= ,S2 Gj(z)(aj, ajP,Xduj, daj),
(1)
Vz
(2)
then E
C(A),
V i E N,
S Gi(z)(Ui,aJii(dai, A;
dai) =O.
To enunciate Condition C verbally, interpret i = (A,, . . ., &,. . ., i,) as being a system of individual scaling rules where Ai gives weights which are attributed
91
C. d’Aspremont and L.-A. Gerard-Varet, BIC beliefs
ex ante to player i for every value of his private information and declaration. Thus JA: Gi(z)(ai,cr,)li(dai,dai) is player i weighted sum, under the system A, of all possible expected gains from deviation w.r.t. ZEC(A). Condition C says that if for some system AE/~ of individual scaling rules the weighted sums of all possible expected gains from deviation are made comparable and identical, across payoff functions ZE C(A) and players, then they should be identically zero. Assume that all players met before knowing of any mechanism and before having been privately informed about their own type. They plan to cooperate against the mechanism. Condition C says that there is no room for such cooperation in the sense that there does not exist a system ;l~:n of individual scaling rules making the weighted sum of conditional expected gains from deviation of every player both comparable and equal to some non-zeroamount. In order to show that Condition C is suficient to give the existence of a solution t E 7 to system (I), we need the following lemma where a continuous efficient BIC-mechanism is constructed: Lemma 1.
Let s be a continuous eficient selection rule, 3 t* E T such that V i E N,
Proof.
V ai E Ai,
V cliE Ai,
Gi(ti)(ai,q) 2 Q(ae a,).
Define ViEN,
VaEA,
Since every Ui and efficiency of s,
s are
t:(a)=,&iU,(.S(a);aj).
continuous
functions,
t* E 7: Moreover
by the
ui(s(@);c(J + t;(a)>=Ui(s(ai, ff _i); ai)+ @(ai, M-i). Integrating
on both sides the result follows.
With this lemma
0
we can state and prove the following
theorem:
Theorem 2. If the given beliefs {P,iE N} satisfy Condition exists a continuous, Pareto-efficient .+BIG-mechanism. Proof We show that, and sufficient condition
with Lemma of Theorem
C, then there
1, Condition C implies the necessary 1. First notice that this last condition
C. d’dspremont and L.-A. Gerard-Varet, BIC beliefs
92
has the same antecedent any 1~ A, if (3)
V t E T,
as Condition
C. Also, by Lemma 1, we have, for
C S Gi(ti)(a, ai)li(dUi, da3 = 0, ioN
A:
then (4)
C S ~(Ueai)li(dUedcli)~O. isN
At
Indeed, for t* E Tas defined in Lemma 1, we get O= C ieN
J 1 Gi(t,*)(Q, ai)~i(dU, dai)~ 1 J O;(U, UJli(daip dai). At
isN
A_i
AZ
Clearly (3) above is equivalent to the consequent (2) of Condition C. Therefore (4) must hold, which implies the consequent (2) of the condition in Theorem 1, since the vector measure 1 is different from zero and s>O. The result follows. 0 We shall now show that Condition C is equivalent to a statement about individual beliefs. For that let d” = (EE&; E= X ixNEi and V~EN, Ei E &J be the semi-algebra of rectangles in d. Then we show: Theorem 3. VLGA, if (1)
A family of beliefs {Pi, i E N} satisfies Condition C if and only if
VEEJZZO, ViEN,
VjEN,
.,i E,Pi(E- i 1@i(dai~ dad I
_f Pi(Ei ( @ddai, dd E,xA,
I
then (4
VEEZP,
ViEN,
A C, P,(E_i) cr,)il,(dU,dai) -,,i,, 1 i
I
PJE-i I dU%~d4=Q I
Let, for every i E N and 1 E A, F: denote the following bounded linear functional on C(A):
ProoJ:
V z E C(A),
C. d’Aspremont and L.-A. GLrard-Varet, BIC beliefs
93
By Riesz representation theorem there corresponds a unique finite signed Bore1 meaure Qi, on A such that, Vz E C(A), F:(z) = IAZ(U)Q:(da). Now take, for every iE N and 1~ /1, the signed measure P” on (A, J@‘) defined by the property
We see that
SZ(Cr)P:(da)= S S z(a)Pi(da _ i 1ai)li(dU, deli)
V Z E C(A),
A
i.e., Pf =Qf.The theorem follows.
A_iAf
0
In the discrete case, Theorem 3 shows that Condition C is equivalent to the ‘compatibility condition’ already mentioned.g Therefore by Theorem 7 in d’Aspremont and Gerard-Varet (1979), Theorem 2 can be strenghtened in the discrete case by putting E=O. 4. On compatible beliefs Our purpose in the present section is to provide a detailed analysis of the compatibility condition - Condition C - determining families of beliefs which are admissible for the existence of Pareto-efficient mechanisms satisfying the BIG-property. We first consider two conditions which are stronger than Condition C, but such that each one of them has a more specific interpretation. Second, we study connections between Condition C and another restriction upon beliefs known in the literature as the ‘consistency’ condition. In that case a weaker budget-balancing property can 90ne may provide for the statement of Theorem 3 an interpretation similar to the one given for Condition C. Indeed, in a Bayesian set-up, the beliefs Pi give the bets P,@_i ).) of player in N on events E_,E& -i concerning the others’ information type, conditional to the player’s own information .di. JMathE-
D
94
C. d’dspremont and L.-A. GLrard-Varet, BlC beliefs
be defined and a stronger positive result can be obtained. Last, we show that Condition C is not necessary by exhibiting a family of discrete beliefs which do not fullil the requirement and however is such that, whatever may be the players’ payoffs (under the full-transferability assumption), there exist efficient budget-balancing BIC-mechanisms. Take the beliefs of a particular player, say player ie N. We say that player i beliefs are free of any dependency with respect to his own types or, for short, that he has free beliefs iff” Vtli E Ai,
Vai E Ai,
Pi(’ ) pi) = Pi(. / $).
In terms of information, free beliefs mean that the true beliefs of player i E N are common knowledge. Consider now the whole family {Pi; i E N} of beliefs. We state: Condition F.
There exists at least one player who has free beliefs.
We shall now see that Condition F is sufficient for having Condition and, as a consequence, the result of Theorem 2. We first prove: Lemma 2.
C,
Given a family of beliefs (Pi; i E N}, for every 1 E A, if ViEN,
VjEN,
V i E N,
V Ei E di,
VEE&‘O,
then li(Ei X AJ = Iq(Ai X Ei).
Proof. Take any i E N and consider We get, for j # i, &(A, x Ei) - li(Ei x Ai) =
E =
s “1
Ei x A_i in SP in the antecedent.
P,(E -j 1a$Xdaj, dmj)
“j ‘. *j
- A l A. I
“‘Thus, d-i and .di are a-algebras which Over (A, sJ) generating Pi.
Pj(E- j 1aj)A,{daj,duj)= 0. Cl
J
are independent
with respect
to the proh;~hili~y
C. d’Aspremont and L.-A. GPrard-Varet, BIC beliefs 7‘11co1wn
95
Given a family of beliefs {Pi; iE N), Condition F implies Condition
4.
C‘. Proof. Suppose w.1.o.g. that player 1 has free beliefs, i.e., V cr,,V Cc,,Pl( ( x,) = Pl(. / El)= xl(.) and that Condition C does not hold. Then, by Theorem 3, there exist 2 EA and E E d”, such that Vj E N - (11,
=A.!~,‘XE-jlaj)~~,~daj,daj!-E [, P,(E-jIaj)~j(daj,daj)#O. J
I
J
This is a contradiction
to Lemma
2.
J
0
Since Condition F implies Condition C, its only advantage is that it provides an interpretation for an important subcase, in which the beliefs of at least one player are fully known. An even stronger condition is to require that every player has free beliefs. We shall call it Condition F* [it is called the independence condition in d’Aspremont and Gerard-Varet (1979)]. In this case we obtain a stronger result, namely that there exists a Pareto-efficient BIC-mechanism. However other interpretable subcases may be considered. An n-person normal form game over the spaces {A,; i E N), taken as stategy spaces, is a function o from A to R” where wi is the payoff function of player iE N. The game is zero-sum if xi.N~izO. Moreover we say that the family (Pi; i E N) provides a strict correlated equilibrium for the game iff ViEN,
VcrieAi,
~~,wi(U,a-i)Pi(da_ilai)
Vai~Ai,
ai #
ai,
,wi(ai,a~i)Pi(da~ilcri). -I
This non-cooperative equilibrium notion has to be considered as referring to an underlying random mechanism designed by the players for selecting their strategies. Thus the Pi’s represent the beliefs of the players about the outcome of this mechanism.” We now consider the following new condition beliefs {Pi; iE N}. In order to match Condition version of this condition:
relative to the family of C, we give a continuous
Condition B. There exists a zero-sum game, {(Ai, 0,); i E N), with continuous payoff functions and for which the family of beliefs {Pi; i E N} provides a strict correlated equilibrium. “See Aumann
(1974).
C. d’dspremont and L.-A. Gh-ard-Varet, BIC beliefs
96
The statement of Condition B may be written as follows:
3WE7; such that V1’EN, Vai~Ai,
VUiEAi,
Gi(oJ(a,a;)>O.
Consider on the other hand any family of payoff functions {Ui; iEN} and take an efficient selection rule s. Define, as in the beginning of section 3, the numbers Of(a, CQ),ieN, ai~Ai, cciEAi. From Condition B, each oi being multiplied by a constant correctly chosen, one easily deduce the solvability of system (I) with E= 0, i.e., 3tQ
such that V i E N,
V EliE A,
VU,E A,
Gi(ti)(a, xi) 2 oS(ai, EJ.
Thus, we prove: Theorem 5. Assuming Condition B, there exists a (continuous) eficient and budget-balancing BIC-mechanism. One interpretation of Theorem 5 is that the individual beliefs characterized by Condition B must introduce sufficient ‘disagreements’ between the players in order to make ‘correlation’ worthwhile in a game having a strong competitive character.” Comparisons of Condition B with Conditions the following characterization: Lemma 3. such that
F, F* and C, are based on
Given {Pi; i E N), Condition B holds $ and only if there is no A E A,
VZEC(A),
ViEN,
VjEN,
Proof Using, as in Theorem 1, Ky Fan (1956), a necessary and sufficient condition for solving system (I), in which we take q()-E identically null, is “See Aumann (1974 p. 67, and example 2.1,~. 68). This property captures features those which are noticed in an example given in Laffont and Maskin (1979, p. 307).
similar
to
97
C. d’Aspremont and L.-A. GPrard-Varet, BIG belief7
simply: there is no g #O in Q* such that, Vt E =i; g(G(t))=O. By the same reasoning as in Theorem 1 we get the result. 0 Theorem (i)
6.
Given a family
Condition
of beliefs {Pi; i E N}, if Condition
B holds, then
C holds.
(ii) Condition F does not hold, Proofi (i) is immediate from Lemma 3. To prove (ii), suppose w.1.o.g. that player 1 has free beliefs represented by nl. Then Condition B, implies
such that Vai~Ai,
Vai~Ai,
,wi(ai,m_Jx(da-i), -1
,S mi(a,u-Jn(da-J>,S
--I which forms a set of contradictory
inequalities.
0
Theorem 6 shows that the beliefs satisfying Condition F and those satisfying Condition B are two distinct subclasses of those satisfying Condition C. However, the size of the class of beliefs satisfying C but neither B nor F, remains an open question. Our next objective is to compare Condition C with another condition known as ‘consistency’.’ 3 Let, for that matter, pi be the set of all probability measures over (Ae~i) and A be the set of all probability measures over (A, d). We say that {Pi; i E N} is a consistent family of beliefs if 3~~
x~i,
ARES,
RfO,
ioN
such that QEEEO,
VisN,
g,Pi(E_,lai)vi(daJ=R(E).
Thus the Pi’s are, for the respective players, the different conditionalizations of the same joint probability. Our first result is for the two player case: Theorem 7. Assuming N = (1,2} and given (PI, P2), beliefs, Condition C is equivalent to Condition F*.
‘%ee, for example,
Harsanyi
(1967/g) for a detailed
discussion
a pair of consistent
of this condition.
C. d’Aspremont and L.-A. Gerard-Varet, BIC beliefs
98
Proof
Given
{P1,P2},
3v,~A?,,
by consistency ~v,EJ%?~,
we have R#O,
UREA?,
such that VEEAP,
Also, the negation
W4L2f,
of Condition
i#L
d, Pi(Ej ) ~i)vi(dai) = R(E).
C is, by Theorem
3, equivalent
to
IAEA, such that
is equal for both i and all E E ~2’ and different Define ,? E A to be V i E { 1,2},
V E, Fi E d,
from zero for some E in ~2”.
li(Ei x Fi) = v,(E,). v,(F,).
Thus, VEEJP,
=
ViE{1,2},
= R(E)-R(Ai = R(E)-
If
ILd, 1
1,Pi(Ej(aJvi(daJ -
Condition
-v,(E,).v,(E,)#O
either. Theorem
vi(daJ
X Ej) ’ Vt(EJ
V,~Ej). Vi(Ei).
F*
does not hold then it is well and so Condition for some EEJP, 4 implies the other direction. 0
However, this negative result following example demonstrates:
known that C does not
only holds in the case of two players,
Theorem 8. For INI > 2, there exist consistent families Condition C and not Condition F*.
R(E) hold
as the
qf beliefs which satisfy
C. d’Aspremont and L.-A. Gerard-Varet, BIC beliefs
99
According to Theorem 2, it is sufficient to find a consistent family of beliefs for which only one player has free beliefs (i.e., satisfying Condition F). Consider ~=u,2,3), A,={a,,cr,},A,=(81,Bz), A,={y,,yJ, the following beliefs satisfy the requirements: Proof:
For player 1, P,
a1 a2
Pl Yl
B2Yl
blY2
B2Y2
4115 119
l/15 219
8115 219
2115 419
For player 2, P,
4115
l/l5
8/15
2115
119
219
219
419
3124 3124
3124 3124
6124 6124
For player 3, P,
Yl “92
12124 12124
0
As a last example, we shall take the simplest case of two players, each having only two types, i.e., N={1,2),
A I = {@I>a,),
J42={Bl?BzI.
We shall, for this case, consider the following class of beliefs, choosing 6 #$ 02651:
100
C. d’dspremont and L.-A. Gkrard-Varet, BIG beliefs
Notice that none of these beliefs are free and that it is a consistent family of beliefs. Consider now any set X of outcomes and any family of two payoff functions {U1, U,} satisfying the side-payment full-transferability assumption. Take any selection rule s which is efficient. As in the proof of Theorem 1, finding a balanced transfer scheme t E RN xA,such that m=(s, t) is a BICmechanism, consists in showing that, V(i,(al, a,), Arta,, c(i), &(fi,, pz), &(Bz, Pi)) E R4, - {O},if v t E =i;
then
Now, assuming (i), we get
and
But, for the given family of beliefs, we have
Pl(Pl I%)-P1(P1 Ic4=P,h implying (iii)
A, =&=&.
)B1)-P2@1I B2)=26- 17
C. d’dspremont and L.-A. Gerard-Varet, BIG beliefs
Now, given (iii) and the assumed
By applying
twice the efficiency
101
family of beliefs, we have in (ii)
property,
this is smaller
than or equal to
Thus, there exists, for the given family of beliefs, an efficient budget-balancing mechanism which is BIC. Also, since S#$ Condition C does not hold (see Theorem 3). Therefore, we may state: Theorem 9. Condition C is not necessary budget-balancing mechanism which is BZC.
for
the existence
of an efficient
The last situation which we shall consider using the same approach is the one where we keep the consistency condition but without requiring Condition C. In this case we can only find efficient-BIC-mechanisms satisfying a weak budget-balancing property, namely,
R(da) = 0.
For the sake of simplicity we shall consider this new property, expected-budget-balancing property, only in the disrete case:
called the
C. d’Aspremont and L.-A. Ghard-Varet,
102
BIC beliefs
Theorem 10. In the discrete case and for any consistent family of beliefs, there exists an efficient-BIC-mechanism which satisfies the expected-budgetbalancing property. Proof of Theorem 9. We have following system of inequalities: V i E IV,
to show
V ai E Ai,
V CliE Ai,
the existence
of t E T solving
the
Gi(ti)(ai, gi) 2 q(ai, NJ,
and
where s is some efficient selection rule. A well-known [see e.g. Ky Fan (1956)] necessary and sufficient for the existence of a solution of (II) is, VLE A, VOLER, if
condition
(1) then
(2) But
=,?A
&
tiara,
Cpitcl-i
( @i(ai,
ai)-Pi(M-i
L 1 So that (1) is equivalent
(1’)
ViEN,
14~i(%~ai)l~
to the following:
Vcci~Ai,
aTA. [Pi(a-i1@i)ii(aiT cci)-pi(a-i/ 4U%ai)l =PRta). I
1
Since, for some CX-i E A Pi, En, R(cc_ i, ui) # 0, we get in (1’) that ,u = 0. Therefore the necessary and sufficient condition becomes, V ;1E A, if V t E 7:
1 C itNa,tA,a,tA!
1 Gi(tJ(ai, aJLi(ai, tli)=O,
(1)
C. d’Aspremont
103
and L.-A. GCrard-Varet, BIG belief7
then iL
oTA,
I
JA,
II
This holds by Lemma
UsCai,
aiVi(ai~
ai) 5
O.
I 1 (see the argument
in the proof of Theorem
2).
0
In conclusion, the first contribution of this section is to provide some interpretable conditions on the beliefs implying Condition C, and thus representing sufficient conditions for the existence of a Pareto-efficient BICmechanism [whatever the (transferable) payoff functions]. They require that the beliefs of at least one player be fully known or that the beliefs of all players be sufficiently in ‘disagreement’ to make correlation, in some sense, profitable. However, the Compatibility Condition C is not necessary. There are families of beliefs which do not satisfy Condition C but have such strong symmetry properties that it remains possible to design a Pareto-optimal feasible mechanism which is Bayesian incentive compatible. The second contribution concerns the much more general ‘consistency’ condition. Under this condition we prove (for the discrete case) that it is always possible to find an efficient BIC-mechanism, provided the budget is only balanced in expected value.
References Arrow, K.J., 1979, The property rights doctrine and demand revelation under incomplete information. in: Economics and human welfare (Academic Press, New York). d’Aspremont, C. and L.-A. Gerard-Varet, 1975, Individual incentives and collective efficiency for an externality game with incomplete information, CORE DP 7519 (Universite Catholique de Louvain, Louvain-la-Neuve). d’Aspremont, C. and L.-A. Gerard-Varef 1979, Incentives and incomplete information, Journal of Public Economics 11, 2545. R.J., 1974, Subjectivity and correlation in randomized strategies, Journal of Aumann. Mathematical Economics 1, 67-96. Green, J. and J.-J. Lalfont, 1977, Characterization of satisfactory mechanisms for the revelation of preference for public goods, Econometrica 45, 783-809. Green, J. and J.-J. Laffont, 1979, Incentives in public decision making (North-Holland, Amsterdam). Groves, T. and M. Loeb, 1975, Incentives and public inputs, Journal of Public Economics 4, 21 l-226. Harris, M. and R.M. Townsend, 1981, Resource allocation under asymmetric information, Econometrica 49, 33-64. Harsanyi, J.C., 1967/g, Games with incomplete information played by ‘Bayesian Plavers’, Part IIII, Management Science 14, 158-182, 320-334,486-502. Ky Fan, 1956, On systems of inequalities, in: H.W. Kuhn and A.W. Tucker, eds., Linear inequalities and related systems (Princeton University Press, Princeton, NJ). Laffont, J.-J. and E.S. Maskin, 1979, A differential approach to expected utility maximizing mechanisms, J.-J. Laffont, ed., Aggregation and revelation of preferences (North-Holland, Amsterdam). Neveu, J.. 1970, Bases mathematiques du calcul des probabilites (Masson. Paris). Royden, J.L., 1968, Real analysis (Macmillan, New York).