Communication, consensus, and knowledge

JOURNAL

OF ECONOMIC

THEORY

52, 178-189 (1990)

Communication,

Consensus,

and Knowledge

ROHIT PARIKH C.I.S. Department, Brooklyn College, City University of New York, Brooklyn, New York 11210 and C.I.S. and Mathematics Departments, Graduate Center, City University of New York, New York, New York 10036

AND PAUL KRASUCKI * C.I.S. Department, Brooklyn College, City University of New York, Brooklyn, New York 11210 and C.I.S. Department, Graduate Center, City University of New York, New York, New York 10036 Received August 24, 1987; revised October 26, 1989

We continue the work of Aumann (Ann. Statist. 4 (1976), 1236-1239), Geanakoplos and Polemarchakis, (J. Econ. Theory 28 (1982) 192-200), Cave (Econ. Lett. 12 (1983), 147-152), and Bacharach (J. Econ. Theory 37 (1985), 167-190) on common knowledge and consensus, extending the results of Geanakoplos and Polemarchakis, Cave, and Bacharach to the case where the number n of communicants is greater than two, but communication is in pairs. When n > 2, then communication is pairs does not lead to common knowledge for the group. We show that, nevertheless, conditional probabilities will converge in any fair protocol for communication, i.e., when none of the participants is blocked from communication. We show that the Cave-Bacharach generalisation of the results of Geanakoplos and Polemarchakis and of Aumann does not hold for pairwise communication when n > 2, and hence any proof for this case must use methods different from theirs. In particular, our proof uses a convexity condition that is obeyed by conditional probabilities, but is not implied by Cave’s union consistency principle. Journal of Economic Literature Classification Numbers: 026, 213. 0 1990 Academic Press, Inc.

INTRODUCTION

In 1976 Robert Aumann [l] proved the following result. Suppose that a group of n individuals share a common prior distribution, but have The research of both authors was supported by NSF grant DCR 8504825. * Current address: Department of Computer Science, Rutgers University, Camden College of Arts and Sciences, Camden, New Jersey 08102.

178 0022-0531/90 $3.00 Copyright 0 1990 by Academic Press, Inc. AI1 rights of reproduction in any form reserved.

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different private information partitions. If there is common knowledge in the group of the posterior probabilities of some proposition A, then these probabilities must all be equal. Aumann did not address the question of why t&se posteri common knowledge at all. This question was addressed by and Polemarchakis [S] who proved that for a group two of even if these posteriors are not common knowledge, but if the the group tell each other their respective probabilities, then ~ornrn~~ knowledge will eventually transpire and the probabilities will all become equal. Subsequently, Cave [3] and Bacharach [2] proved that the results of Aumann and of Geanakoplos and Polemarchakis will hold not only for conditional probabilities, but also for any function f from sets of worlds to some domain D, provided only that f has the property that if X, Y are disjoint and f(X) =f( Y), then f(X) = f(Xu Y). This property is called ~~~~~ consistency by Cave and the sure thing princijde by Savage, ES]. All the authors mentioned above deal with the case n = 2 except for Cave considers arbitrary finite IZ but in the context where all members of the p hear all the statements that any of them makes. one of these authors raises the following general questions: (1) are the states of knowledge created in a group of individuals communication takes place? and (2) what happens in the ca communication is not to the whole group, as in an auction, as happens commonly in commercial transactions? We illustrate the first point by an example. Suppose I telephone you an give you some information; then the fact I have told you is common knowledge between us. If, however, I write you a letter e~~tai~in~ information, then when you receive the letter, you know the fact you have received the letter) but I do not know that you know, not know that you have received my letter. If you acknowledge the letter and I receive your acknowledgement then I know that you know, but, again, you do not know that I know that you know. lin fact, ~ornrno~n knowledge cannot be created by letters provided that they are undated and there is some uncertainty in the time of arrival. See [6]. less, the fact remains that the entire correspondence, except very last step, is known to both of us. Moreover, even the last ste known to the one whose turn it is to respond. Thus it can be shown, u the Cave-Bacharach argument, that eventual consensus will necess transpire between two people, regardless of whether they ~ornrn~~i~a~e by phone or by letter. The situation changes if we have more than two people. could communicate in a setting where all of them are auction-the case considered by Cave-achieves just that. 642/52!1-13

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not all commercial transactions have the setting of auctions. The details of many sales are known only to the parties to the transactions, although each party may participate in many transactions and with many different persons. Thus we can ask what happens when transactions in a group of individuals take place in pairs. What happens in this case is that no common knowledge is arrived at by the group: i.e., no new fact becomes common knowledge as a result of communication which was not common knowledge to begin with, see [4, 6, 7, 8 J. The reason here is that an individual knows only that portion of the dialogue in which he or she participates. It can be shown, nonetheless, that when n individuals communicate probabilities to each other, two at a time, and every individual is repeatedly involved in communication, then their probabilities do become equal. We show that this result generalises to arbitrary functions f which satisfy a certain convexity condition. This condition is properly stronger than Cave’s “union consistency” property, and indeed, union consistency is not strong enough for N > 2. For there does exist a function J; with the union consistency property, such that repeated communication of values of f from individual 1 to 2 to 3 to 1 .... fails to bring about consensus. This last fact shows that the proof of Geanakoplos and Polemarchakis, which uses the union consistency property of conditional, probabilities (and only that), does not work for the case where n > 2. We also give a general definition of the formula to be used for updating, which tells us how an individual should revise his set of possible worlds when he receives a communication.

BASIC NOTIONS AND RESULTS

We assume given a space W, the space of states (or of possible worlds), and n participants with finite partitions Pi of W. Let P,(x) be the equivalence class of world x in partition Pi. Let xz y mean that for all i, P,(x) = Pi( y). We will say that a subset X of W is closed if x in X and XE y imply that y is also in A’. If P+ is the coarsest common refinement (join) of the P,, then Xis closed iff it is a union of P+ equivalence classes. Note that since the Pi are finite, so is P+. Similarly, we will use the expression i-closed to describe a union of elements of the partition Pi. Note that an i-closed set is closed, but not necessarily vice versa. By a protocol Pr we mean a pair of functions s(t), r(t) from the natural numbers ( 20) to the set { 1, .... n). Here t stands for time and s(t), r(t) are, respectively, the sender and the recipient of the communication which takes place at time t.

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Dn~~NrrroN. Given a protocol (s(t), r(t)) consi er the directed graph whose vertices are the participants (1, .... n> and su that there is an ed from i to j iff there are infinitely many t such that s(t) = i and r(e) = the protocol is fair if the graph above is strongly comected, i.e., if a path of directed edges which passes through every vertex at least once, returning to its origin. If the participants are observing a fair protocol, then not only will each participant be a recipient and a sender infinitely many times, but each participant will receive information from every other, possibly i~dire~tI~, infinitely many times. Let f be some function from the set of subsets of the state space W to some domain D (f is the function whose values are communicated). If f is intended to be a conditional probability, then we assume given a probability measure n on W and some event A contained in IV. Then f(X) for XE W is just 7c(A1X) = n(A n X)/n(X) where z(X) f 0. More generally, let the domain D be the reals. We say that the J+’is convex if for all subsets X, Y of IV, if X and Y are disjoint a are closed then there are positive reals a: b such that cp f(X u Y) = affix) + bf( Y). In other words, f(Xu Y) lies in th val between f(X) and f(Y). Note that we will get f(X, u X dl~,d + . . . + u&(X,) provided that the Xi are pairwise disjoint c sets and a,, .... ak will be positive reals such that a, + . . + ak = 1. also that if the f(X)) are all equal, then f(X, u . . . u ir,) =f(X, )~ convexi.ty implies the union consistency property, at least fo the snly subsets of W that we will need IS be concerned with Suppose now that Pr = (s(t), r(t)) is some protocol. We define by induction on t the message m(x, t) sent at time t, and C(x, E’,t)l the set of possible states for i at time t, given that the real state is x. C(x, i, 0) = P,(x) then C(x, i, tt l)=C(x, i, t)n (ylm(y, t)=m(x, t)J, w m(x, I) =fl% 4th t)). Otherwise, C(x, i, t + 1) = C(x, i, t). We let U(X, i, t)= f(C(x, i, t)) so that i;rl(x, t) is just u(x, s(t), 1). assume that all the communications are instantaneous, but this assurn~ti~~ is not essential Ifi=r(t),

hadA 1. If x= y then for all i, t, m(x, t) = m(y, t) and C(x, i, t) = C(Y, i, t). Proof. Straightforward, by induction on t. +h%.lREM 1. There is a T such that for all’ x, i, and all t, 1’ > T, C(x, i, t) = C(x, i, t’). Hence for all x, i, v(x, i, 1) has a limiting value

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v(x, i, GO).Moreover, if f is convex, and the protocol is fair, then this limiting value does not depend on i; i.e., v(x, i, 00)) = v(x, j, 00)) for all i, j. COROLLARY. If W is a probability space such that all nonempty elements of Pi have nonzero probability, and A is some event, then the limiting probability of A is the same for all i, j, provided that the protocol is fair,

Proof of Corollary. We note that in this case P(u, i, t), the probability of A for i at time t, where the real world is U, is z(A ( C(u, i, t)) and z(A (X) is a convex function of X. We now turn to the proof of the theorem itself. For all X, i, C(x, i, t) is a decreasing function of t. Also it is a nonempty union of P+ equivalence classes. Since P+ is finite, it is eventually constant. Let t(x, i) be the least t such that for t’ > t, the value of C(x, i, t’) is the same. Then t(x, i) depends only on the P+ equivalence class of x. Since P+ has finitely many equivalence classes, and there are finitely many i, there is a T such that C(x, i, t) is constant for all t’ > T regardless of x, i. T does of course depend on the choice of the protocol (s(t), r(t)). It follows also that m(x, t)=f(C(x, s(t), t)) depends only on x and s(t) if t > T. We write C(x, i, co) to indicate the limiting value of C(x, i, t). To proceed with the proof of the theorem, we assume for simplicity that N = 3, and that for all t, s(t) = t mod 3 and r(t) = t + 1 mod 3. This protocol is evidently fair (we will call this protocol a “round-robin” protocol). The reader should note that our arguments will also apply to any n and to any fair protocol. We will say that p is a possible limiting value for i if there is an x such that p = f((C(x, i, co)). Let the possible limiting values for individual 1 be pl, .... pk with p1 < pz < . . .
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and hence ( y Jf( C( y, 1, t)) = p) must contain C(x, 2, “c). Thus (a) contains C(x, 2, 00). Thus E(p) is a union of sets of the form C(U, 2, 0~)) and hence, by convexity, p must be an average of those values qi which are compatible with it. Consider pl, the smallest of the p”s. The set of vah~es q compatible with it must contain at least one element dp: and hence ql 2, and r2 with pi for i > 2. We can now repeat our argument and show that p2 = q2 = r2 and any of them forces the others. By induction we get, for all i, pi = qi = ri and all force each other. What happens if n > 3 or if we are using a fair protocal other than roun robin ? We note now that our argument used only the fact that we are able to find a chain t, < t, < . . . < t,, all greater than T and such that (a) dt,) (b) the (6) the (1 may occur

=1 sender at t,,, is the recipient at t, and chain passes through all partiticipants, finally returning to 1. in between also).

But this is implied Theorem 1.

by the fact that the protocol

is fair. This proves

IEXAMPLE 1. Let us consider the case of statistical inference by Bayesian individuals. Suppose there are two dice; d, is fair but the other, d,, is fair only between (1,2, 3,4, 5}, never showing 6 d, is worth $1 and dz is worth $100. The two dice are put in a box which is shaken, and then one of the dice is taken out and tossed, showing a 2. Of two gamblers is told that the result was under 4, while B is told that it is even. evaluates the probability that the die is d2 as 6/11, and hence his expected value for the die will be $55. However, B thinks that the probability that the die is d2 is only 4/9 and his expected value for the offered die will be us if the die is offered at $50, clearly A would buy it, and B would not. What will happen if they repeatedly communicate their values of the die to each other? Will these values converge?

To put this problem abstractly, suppose that the random variable X takes values in a set W. X is distributed according to one of a finite number of probability distributions II,, D,, .... Dk, We also have n individuals each

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of whom has his information partiton Pi of the set W. The individuals share a utility function U: (Or, D,, .. .. Dk} -+ R, say u(D,)=q. The result w of an experiment performed (to measure X) is (partially) communicated to each person by specifying P,(w). The individuals communicate values of f: 2w -+ R where for A z W, f(A) = Cf= r ~$0,) A) . ui. In order for f to be defined for all i-closed subsets of IV, we need to assume that for A E P+, p(A) > 0. It is easy to check then that for disjoint, non-empty subsets A 1, A 2 of W, if P(A 1 u A,) # 0, then f(A,uA,)=f(A,).(p(A,)/p(A,uA,))+f(A,).(p(A,)/p(A,uA,)). Hence f is convex so that when values of f are exchanged, then by Theorem 1, consensus will be reached. The fact that there are only two gamblers is of course not essential. EXAMPLE 2. We now give an example to show that Cave’s union consistency property does not imply consensus for IZ> 2. Let the space W equal the set (1, 2, 3, 4, 5, 6, 7, S}. The function f satisfying the union consistency property takes integer values and is defined as follows: let the subsets of W be numbered X,, X,, ... and let num(X,) be i. Now we let

f((l,Z})=f((3,4))=f((l,.Z, 3,4H=l f((5,6})=f((7,8})=f((5,6,7,8))=2 f((1,3})=f(15,7})=f(Il, 3,5,7))=3 8))=f({% 49% 8))=4 f((1,5})=f((2,6))=f((1,2,5,6})=5 f((3,7})=f({4, S))=ff(T 4>7> 8))=6

f((2,4})=f((fi

and for all other subsets X of W, f(X) = num(X) + 6. This ensures that f has the union consistency property. Now the participants are A, B, C and their partitions P,, P,, PC are defined by PA = { (1, 2}, (3, 41, 15, 619 17, 81) Pg= ((1, 31, (2,4),

{5, 71, {6,8>>

Pc= {{I, 5j, (2, 61, 13, 71, (4, 811. The real world is 1. Now note that A sends a 1 to B so B knows that the real world is in (1,2,3,4). But he learns nothing since he knew anyway that it was in { 1, 3). Similarly, B sends a 3 to C who learns noting thereby and C sends a 5 to A who learns nothing. Since the three participants have learned

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nothing in the first round, they will repeat their signals ad ~~~~iturn a consensus will not take place. The following, rather colorful example is due to Aumann. Suppose t are three detectives, Adams, Brown, and Cox. In detective school, learn that adults are more reliable than teenagers (maturity), teenagers than children (accuracy), and children than adults (disingenu Th are three material witnesses in the case: an ad&, a teenager, chi It is common knowledge that Adams interviews the adult rown the child and the adult, and Cox the teenager ctably, Adams believes the adult, Brown the child, and teenager. Adams now tells Brown whom he is going to arrest, whit knew, since he also interviewed the adult, but predictably, Brown prefers the testimony of the child. He, in turn, tells Cox who be is going to and so on. Nobody gains any information from wrhar he is tol nobody’s mind is changed by what he is told. If, however, there were only two detectives, then Cave’s unio consistency theorem would apply, and they could not differ on whom they were going to arrest. We have shown above that the union consistency property does not lea to a consensus in the case of pairwise communication in a group of more then 2 people. Still, it is a natural question whether we can weaken the exity condition so that consensus will still be guaranteed. e will say that f is weakly convex if for disjoint, closed A, lies in the closed interval between f(A) and f(B). The difference now is that even en f(A) and f(B) are different, f(A u p3) may equal one of S(A) and ). With simple convexity, this would be prohibited. Clearly weak convexity implies the union consistency pro erty and is implied by convexity. EXAMPLE 3. We give an example of a weakly convex S such that when values of f are communicated by 4 persons using the “round-robin” protocol then no consensus is reached. Let the space W be the set (a, b, c, d, e}. Let the function f: 2 w -+ (0, 1 > be defined as follows: For convenience, we write a for (a>, b for (b) etc.

fta) = f(b) = 1 ftc)=ftd)=fte)=O f(auc)=f(aud)=f(aue)=f(buc)=/(bwd) =f(bue)=f(aub)=l. f(c u d) =f(c

u e) =f(du

e) = 0

186

PARIKHANDKRASUCKI f(aucud)=f(budue)=f(cudve)=O f(aUdUe)=f(bucud)=f(aucue)=f(bucue) =f(aubuc)=f(aubud)=f(aLJbue)=l ff(aucUdue)=f(bucudue)=O f(aubucud)=f(aubudue)=f(aubucue)=i f(aubucudue)=l.

It can be easily checked that f’ is weakly convex, Intuitively, say that A dominates B if f(A) and f(B) are different, A and B are disjoint, and f(A u B) equals f(A). Then in our example, either of a and b dominates over any one of c, d, e, and even over any two of c, d, e, except that c, d together dominate a, and d, e together dominate b. However a, b together dominate over any combination of c, d, e. We have 4 participants and their partitons are defined as follows: PI = ({e>, ia, c, d), (b)3 Pz = ((4

(a, 4 e), (b})

Pj= i(c),

(b,d, e>, (a)>

p4 = {{e),

{by G 4, (a> >

If the real world is d then 1 sends value 0 to 2, but 2 learns nothing since P,(d)= (a, d, e> s (w 1f(P, (w))=O) = {a, c, d, e) and 2 sends value 1 to 3. Similarly 3 learns nothing upon receiving 1 from 2 and sends value 0 to 4, 4 learns nothing and sends 1 to 1 and 1 also does not learn anything. This will now continue with no consensus ever being reached. We have shown that weak convexity does not work for n = 4 (and in fact for n 34, whereas the union consistency property is strong enough for vr= 2. Thus the question arises what happens if n = 3. THEOREM 2. If f is weakly convex and the protocol Pr is fair then if 3 participants communicate balues off according to Pr then consensus on the value off must be reached.

ProoJ: Without loss of generality we can restrict ourselves to the “round-robin” protocol. As in the proof of Theorem 1 we can define sets of possible limiting values for 1,2, 3. In order to prove our theorem we observe that the following lemmas hold: LEMMA 2. ff there are a space W, finite partitions P,, ,.., P, of W, and a weakly convex function f s.t. for some fair protocol Pr, the sets of possible

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limiting values are PI, .... P”, then there exist finite partitions Pi, .... Pi of @’ aazdQ protocol Pr’ with the same graph as Pr such that by executing Pr’, we get the same set of limiting values, but no one gains any knowledge during the execution of Pr’. I.e., C(w, i, I) = C(w, i, 02) for d i, W. ProojI Simply take Pi(x) to be C(x, i, cc) and let Pr’(t) = (s(t + T), rjt + T)), where Pr(t) = (s(t), r(t)) and T is such that stability (no further change in the C’s) is always reached by time T. In particular, if no consensus was reached in the first case, then in the second case also there will be no consensus, and moreover, no learning will ever take place. LEMIVIA 3. Suppose there are a space W, a world w E convex ,function f such that when w is the actual world, and values c$S are communicated according to a fair protocol, then no consensus takes place. Then there is another weakly convex function s’, taking values in (0, k > such that no consensus is reached when values off’ are Corn~M~icu~e~ according to the same protocol.

To see that this lemma is true, assume by Lemma 2 that tb C(x, if a) are the same as the sets Pi(x). Let x, i, j be such that i, on the limiting value when the real world is x, and say u(x, i, ~0) = .f(Pi(x)) < v(x, j, 00) =f(P,(x)). Let a be such that f(Pi(x)) < c! 4c

f(Pi(xj].

And 01is not a value of $ Now let, for ail closed X9 f’(X) = 0 if f(X) < CI

and

f’(X) = 1 if f(X) > o(.

The reader can easily check that f is weakly emvex and consensus not take place. In fact, since f supplies less information than f, there can learning with f’ either. e note that for 2-valued functions, weak convexity is equivalent to the union consistency property. The reason Theorem 2 does not hold for the union consistency property is that Lemma 2 does not hold, so at the reduction from the generai case to the 2-valued case cannot be car d out. To prove the theorem, assume that the values of a weakly convex function f are communicated between 1, 2, 3 using the ~~round~ro protocol and consensus is not reached. By Lemmas 1 and 2 we can ass that f is (0, 1 )-valued and no knowledge is gamed during t the protocol. Call a closed set X unstable iff for some persons i, j in a world w from X, i and j have different values. Let U = I.J (A’ ( X is unstable). Uf 0 since we assumed that consensus is not reached. We can now express U in two ways: (i) as the union of all C(w, i, a) s.t. S(C(w, i, co)) = 0 and f(C(w, i 0 1, co))= I and

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(ii) as the union of all C(w, i, co) s.t. f( C(w, i, co)) = 1 and f(C(w,iOl,oo))=O(wherexOy=x-ymod3). It is not hard to see that each union is disjoint. But then the first representation forces us to assign f(U) = 0 while the second representation requires f(U) = 1 so we have a contradiction. This proves Theorem 2. COMMON

KNOWLEDGE

Is NOT NECESSARY

If a function f is convex, or f is weakly convex and n < 3 then by Theorems 1 and 2 a consensus on the value of f in a fair protocol will eventually be reached; i.e., the values of S will become the same. We now point out that there need be no common knowledge in this situation. Suppose that individuals a, 6, c, etc. do not have watches, but at time t with t = 1 mod 3, a receives a signal from outside that causes him to send a value to b. Similarly with b and c. In that case, a does not know when b communicates with c, b does not know when c communicates with a etc. The argument for convergence is not affected. However, we can now show that if some proposition P is common knowledge at time t + 1, then it was common knowledge at time t. For if P is common knowledge, so is the fact that P is common knowledge. (This is a property of common knowledge). Suppose that t = 2 mod 3. Then it is not known to a that b communicated with c. If P is common knowledge at t + 1, then letting C(P) denote the fact that P is common knowledge, C(P) is true and so is K,(C(P)), i.e., a knows C(P). But a’s knowledge did not change between t and t + 1 so K,(C(P)) was true at t, and hence so was C(P) since a cannot know a false proposition. Applying backward induction on t, we see that C(P) was true at t = 0, before communication started. However, the consensus value is not known in general before communication starts. Hence it was not common knowledge and never will be common knowledge, though it will be known to everyone 161. This point is of some importance, It is shown in [7] that knowledge can occur at various levels, and while common knowledge is the highest of these levels, it would be wasteful to demand it when a lower level will do, and is attainable. It is important in a society that everyone know (at least approximately) the prices of commodities. However, it does not make sense to demand that everyone know the details of every commercial transaction. ACKNOWLEDGMENTS We thank R. Aumann, A. Chandra, H. Gaifman, J. Halpern, S. Lipschutz, A. Mate and H. Polemarchakis for fruitful discussions. In particular, the example with detectives was given to us by Aumann, example 3 is joint with Chandra, and example 1 is joint with Gaifman.

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Statist. 4 (1976), 1236-1239. a claim of Aumann in an axiomatic model of knowledge, J. &on. Theory 37 (1985), 167-190. 3. J. A. K. CAVE, Learning to agree, Econ. Lett. 12 (19833, 147-152. 4. M. CHANDY AND J. Mrsrtz, How processes learn in “Proceedings of 4th ACM Conference on Principles of Distributed Computing,” pp. 204-214, 1985. 5. J. GEANAKOPL~S AND H. POLEMARCHAKIS, We can’t disagree forever. ,7. Icon. Theory 28 (1982), 192-200. 6. J. HALPERN AND Y. MOSES, Knowledge and common knowledge in a distributed environment, in “Proc. 3rd ACM Conference on Principles of Distributed Computing,” pp. 5tL-51, 1984 [To appear in J. Assoc. Comput. Mach.] 7. R. PASUKH, Levels of knowledge in distributed computing, in “IEEE Symposium on Logic in Computer Science, Boston, 1986,” pp. 322-331. 8. R. PARIKH AND R. RAMANUJAM, Distributed processes and the logic of knowledge, iiz “Logics of Programs” (R. Parikh, Ed.), pp. 256268, Springer Lecture Notes in Computer Science, Vol. 193, Springer-Verlag, Berlin/New York, 1985. 9. L. SAVAGE, “The Foundations of Statistics,” pp. 21-26, Dover pubiications, New York, 1972. 2. M.

AUMANN, Agreeing to disagree, Ann. BACHARACH, Some extensions of