- Email: [email protected]
Protocols Forcing Consensus
Journal of Economic Theory ET2125 journal of economic theory 70, 266272 (1996) article no. 0086
Protocols Forcing Consensus Paul Krasucki 1 , 2 Department of Computer Science, Rutgers University, Camden College of Arts and Sciences, Camden, New Jersey 08102 Received July 8, 1991; revised January 24, 1995
We analyze n agents communicating in order to reach consensus. We show how some conditions on the topology of the communication graph (the order in which individuals communicate) are sufficient to guarantee consensus on the value of a function satisfying the ``sure-thing'' condition. Journal of Economic Literature Classification Numbers: C62, C78. 1996 Academic Press, Inc.
Introduction Aumann [1] was the first economist to define the technical notions of knowledge and common knowledge (see also Hintikka [9], Lewis [10], and Schiffer [21]). He proved that if two individuals share their priors, and there is common knowledge among them of their information partitions and their posteriors concerning a certain fixed event, then these posteriors must be the same. Geanakoplos and Polemarchakis [8] extended Aumann's results to a dynamic framework. Instead of assuming common knowledge of posteriors, they investigated a protocol in which agents announce their current posteriors to each other. In this process agents learn and modify their posteriors, and ultimately they agree (the announcements become the same). Cave [6] and Bacharach [3] generalized Aumann's and Geanakoplos Polemarchakis's agreement results to situations in which agents are interested in values of any union-consistent function f, when the values of f after every revision are publicly announced. 1
I thank Rohit Parikh for getting me interested in the problems of consensus, many helpful discussions, reading this manuscript, and valuable comments. I also thank Don Beaver for helpful suggestions and Joseph Gerver, who was patient enough to correct my English. Remarks of an anonymous referee and an editor helped in revisions of the paper. 2 E-mail: Krasuckicrab-rutgers.edu.
266 0022-053196 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
File: AAAAAA 212501 . By:CV . Date:18:07:96 . Time:14:24 LOP8M. V8.0. Page 01:01 Codes: 3896 Signs: 2158 . Length: 50 pic 3 pts, 212 mm
PROTOCOLS FORCING CONSENSUS
267
Here we consider protocols in which values of functions are communicated privately through messages; when agent A communicates with agent B, other agents are not informed about the contents of the message. In this model (first considered by Parikh and Krasucki [18]) union-consistency does not guarantee agreement even with reasonable protocols. In the present paper, we investigate what restrictions on the protocol will guarantee that agreement is reached on the value of any union-consistent function, regardless of the information partitions. Two common protocols in which consensus will be guaranteed by our theorem 1 are the channel protocol and the star protocol. In the channel protocol, information is passed along the ``channel of command'': agent 1 communicates with agent 2, agent 2 with agent 3,. . ., and agent n&1 with agent n, who reverses the flow of information by communicating with agent n&1, and so on, until communication reaches agent 1, who starts the process over again. The star protocol is a centralized protocol in which every agent communicates privately with a distinguished central agent, who immediately replies to every message.
Basic Notions and Results Let W be the space of states (or of possible worlds). There are n participants numbered from 1 to n. Every agent i for 1 i n has a finite information partition P i of W. P i (x) will denote an element of a partition P i containing the world x # W. Agents communicate by sending messages. To simplify notation we will assume that messages are delivered instantaneously (this assumption is not essential in any of the results). A protocol Pr is a function Pr : N + [1, ..., n] 2. If Pr (t)=(s(t), r(t)), then we interpret s(t) as the sender and r(t) as the recipient of the communication which takes place at time t. A protocol is fair if and only if every participant in this protocol communicates (directly or indirectly) with every other participant infinitely often (see [18]). The channel protocol (Fig. 1) and the star protocol (Fig. 2) are both fair protocols.
Fig. 1. Channel protocol. For t>2n&2, s(t)=s(t&2n&2).
File: 642J 212502 . By:MC . Date:04:06:96 . Time:13:00 LOP8M. V8.0. Page 01:01 Codes: 2644 Signs: 2118 . Length: 45 pic 0 pts, 190 mm
268
PAUL KRASUCKI
Fig. 2. Star protocol for n=5. For t>2n&2, s(t)=s((t&1) mod(2n&2)+1).
Let us assume that individuals communicate values of a function f: 2 W D according to a protocol Pr (t)=(s(t), r(t)). We require that the function whose values are communicated satisfies the following condition, called union consistency: 3 for X, Y disjoint, if f (X)=f (Y), then f (X)= f(X _ Y) (the same property was also called the sure thing principle; see also [3] and [6]). The updating procedure is a non-recursive version of the fully rational procedure originally defined in Parikh and Krasucki [18]). The non-recursive description of the updating procedure was motivated by Weyers [23]; the notion of the working partition was also first proposed by Weyers. Let P ti denote an information partition of W for agent i at time t. P ti(x) denotes the set of possible states for i at time t, given that the real state is x; P 0i =P i . Let m(x, t)=f (P ts(t)(x)) be a message sent by s(t) at time t in world x. We will say that two worlds x, x$ are indistinguishable by a message at t (x#t x$) if and only if m(x, t)=m(x$, t). Since #t is an equivalence relations it generates a partition of W (we will denote it by W t and call it the working partition at time t). =P ti . If i was not a recipient of a message at time t, P t+1 i t+1 t t If i=r(t), then P i =P i 6 W . Initially everybody knows only which element of hisher partition contains the real world. At time t, if a person doesn't receive any messages, hisher state of knowledge remains unchanged. Upon receiving a message, a person excludes 3 If a function is not union consistent, then if we take two agents such that one's information partition is a refinement of the other's, they may disagree.
File: 642J 212503 . By:MC . Date:04:06:96 . Time:13:00 LOP8M. V8.0. Page 01:01 Codes: 2402 Signs: 1702 . Length: 45 pic 0 pts, 190 mm
PROTOCOLS FORCING CONSENSUS
269
from hisher set of possible worlds, worlds incompatible with the received message. In [18], Parikh and Krasucki showed that union consistency is not strong enough to guarantee consensus when the number of agents is greater than 2. All the examples in which communicating values of a union-consistent function fails to bring about consensus were round-robin protocols (individual 1 to 2 to 3 to 1. . .). We will show that all such examples must contain a cycle; if there are no cycles in the communication graph, consensus on the value of any unionconsistent function must be reached. We will need two lemmas which correspond to theorems proved in [18]: Lemma 1. There is a t 0 such that for all x, i, and all t, t$>t 0 , P ti(x)= P (x). Hence for all x, i, P ti(x) has a limiting value P i (x). t$ i
Proof. Since all partitions P i are finite, their join P + is also finite. For every i, x, t, P ti(x) is a union of elements of P +. P ti(x) as a function of t is monotone non-increasing; therefore it must converge in a finite number of steps. K Lemma 2. If there is a W, finite partitions P 1 , ..., P n of W and a union consistent function f such that for some fair protocol Pr, the limiting partitions reached at time T are Q 1 , ..., Q n , then there is a protocol Pr$ with the same graph 4 as Pr such that if we use Q 1 , ..., Q n as initial partitions of W in Pr$, no one will gain any knowledge during the execution of Pr$. That is, Q i (x)=Q ti(x) for all i, x, t. Proof. By Lemma 1, T exists. There is no learning after T. Let Pr$=(s$(t), r$(t)) and s$(t)=s(T+t), r$(t)=r(T+t). Clearly Pr and Pr$ have the same graphs. K In particular, if no consensus was reached in the first case, then in the second case also there will be no consensus, and no learning will ever take place. Moreover, the protocol Pr$ is fair if and only if the original protocol Pr was fair. Protocols in which no participant gains any knowledge we will call zero-learning protocols. In zero-learning protocols P ti and W t both remain constant as functions of t. We will characterize protocols in which consensus among any n participants on the value of a union-consistent f is always reached. Using Lemmas 1 4
A graph of the protocol has agents P i for i n as nodes. There is an edge between P i and P j if and only if there are infinitely many communications from P i to P j . See (18).
File: AAAAAA 212504 . By:CV . Date:18:07:96 . Time:14:24 LOP8M. V8.0. Page 01:01 Codes: 3090 Signs: 2327 . Length: 45 pic 0 pts, 190 mm
270
PAUL KRASUCKI
and 2 we can, without loss of generality, consider only situations in which a fair protocol is zero-learning. Lemma 3. In a zero-learning protocol, if i communicates the value of a union consistent f to j, j sends the value of f back to i then i and j were sending the same value. Proof. Let s(t)=i, r(t)=j, s(t+1)=j, r(t+1)=i. If for any y ( y){P ti( y), then the working partitions W t and W t+2 would be difP t+2 i ferent, which would contradict the assumption of zero-learning. Therefore, \y, t,
P ti( y)=P t+2 ( y) i
and
P tj( y)=P t+1 ( y). j
Let E(x, i, t)=[P ti( y) | P ti( y) (P ti 7 P tj )(x)]. Similarly we can define E(x, j, t)=[P tj( y) | P ti( y) (P ti 7 P tj )(x)]. If there is no learning, then all sets in E(x, i, t) are compatible with P ti(x); i.e., they give the same value of f as P ti(x). By the union-consistency of f: f (P tj(x))=f (E(x, i, t))=f ((P ti 7 P tj )(x))=f (E(x, j, t))=f (P ti(x)). K Often when agents communicate, they do not just send messages to each other, but instead they exchange information, they hand each other their bids, then after learning from them they reevaluate their positions (an example of such a process could be reaching an agreement between a buyer and a seller of a house). That would correspond in our model to the situation where sender and receiver in the same instant of time reverse their roles; formally this protocol may be specified as a pair of protocols: P 1 =(s 1(t), r 1(t)), P 2 =(s 2(t), r 2(t)), where \t, s 1(t)=r 2(t), s 2(t)=r 1(t). If P 1 and P 2 are executed concurrently then we will call it a protocol with information exchange. Corollary 1. In any fair protocol with information exchange, consensus on the value of a union-consistent function must be reached. Proof.
Follows from Lemma 3. K
Let us call a protocol cyclic if and only if there are agents i 1 , ..., i k , k 3, such that for all j
Addition and subtraction are mod k.
File: AAAAAA 212505 . By:CV . Date:18:07:96 . Time:14:24 LOP8M. V8.0. Page 01:01 Codes: 3228 Signs: 2170 . Length: 45 pic 0 pts, 190 mm
PROTOCOLS FORCING CONSENSUS
271
Fig. 3. Cyclic protocol (2, 3, 4 form a cycle).
There is an example of a cyclic protocol in Fig. 3. To simplify notation, we will assume that for every n, s(n+1)=r(n), and that communications proceed in rounds, i.e., there is some N (number of communications in a round) such that for all t, s(t)=s(t+N). Theorem 1. Consensus on the value of every union-consistent f, for every set of initial information partitions P 1 , ..., P n is guaranteed if the values of f are communicated according to a non-cyclic, fair protocol Pr . 6 Proof. We assume that the number of agents is at least three. If there are only two agents, consensus is always achieved (Lemma 3, earlier results by Bacharach [3] and Cave [6]). Suppose that in a fair, non-cyclic protocol Pr consensus on the value of f is not reached. By Lemma 2 we can assume without loss of generality that Pr is zero-learning. Furthermore we can assume that our example is minimal; i.e., if we remove any of the agents or communications from the protocol, consensus will be reached. Pr is fair, so for every pair of agents i j , i k there is a chain of communications from i j to i k and from i k to i j . There must be agents (nodes in the graph of Pr ) i $j and i $k such that in every round, if at time t a message m t is sent in Pr by i$j to i$k then m t+1 is sent by i $k to i $j (if there are no such agents, there would be a cycle). By Lemma 3, m t and m t+1 must be the same. We can modify our example by removing communications m t and m t+1 (if an agent i $j is not involved in any other communications, we can remove her as well). Since the protocol is zero-learning, and the messages m t and m t+1 are the same, the consensus is not affected by the change: if there were no consensus between some agents in the original protocol, there would be no consensus in the modified one. That contradicts the assumption of minimality. K The protocol Pr is cyclic, we can always create an example in which communications according to Pr do not lead to consensus. 6 After reading the manuscript of this paper, Professor Rohit Parikh suggested a different formulation of the Theorem 1: consensus on the value of every union-consistent f, for every set of initial partitions P 1 , ..., P n is guaranteed in a protocol Pr with a graph G iff G & G I is connected (G I is the inverse of G).
File: 642J 212506 . By:MC . Date:04:06:96 . Time:13:01 LOP8M. V8.0. Page 01:01 Codes: 3066 Signs: 2311 . Length: 45 pic 0 pts, 190 mm
272
PAUL KRASUCKI
As consequences of Theorem 1 we get consensus results for the channel protocol and the star protocol (the channel protocol and the star protocol are non-cyclic).
References 1. R. Aumann, Agreeing to disagree, Ann. Statist. 4 (1976), 12361239. 2. R. Aumann, Notes on interactive epistemology, unpublished manuscript, 1989. 3. M. Bacharach, Some extensions of a claim of Aumann in an axiomatic model of knowledge, J. Econ. Theory 37 (1985), 167190. 4. A. Brandenburger and E. Dekel, Common knowledge with probability 1, J. Math. Econ. 16 (1987), 237245. 5. L. T. Nielsen, A. Brandenburger, J. Geanakoplos, R. McKelvey, and T. Page, Common knowledge of an aggregate of expectations, Econometrica 58 (1990), 12351239. 6. J. A. K. Cave, Learning to agree, Econ. Lett. 12 (1983), 147152. 7. J. Geanakoplos, Common knowledge, Bayesian learning, and market speculation with bounced rationality, unpublished manuscript, 1988. 8. J. Geanakoplos and H. Polemarchakis, We can't disagree forever, J. Econ. Theory 28 (1982), 192200. 9. J. Hintikka, ``Knowledge and Belief,'' Cornell Univ. Press, Ithaca, NY, 1962. 10. D. Lewis, ``Convention,'' Harvard Univ. Press, Cambridge, MA, 1969. 11. P. Krasucki, Reaching consensus on decisions, in ``Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning About Knowledge,'' Morgan Kaufmann, San Mateo, CA, 1990. 12. R. McKelvey and T. Page, Common knowledge, consensus, and aggregate information, Econometrica 54 (1986), 109127. 13. P. Milgrom, An axiomatic characterization of common knowledge, Econometrica 49 (1981), 219222. 14. P. Milgrom and N. Stokey, Information, trade and common knowledge, J. Econ. Theory 26 (1982), 1727. 15. Y. Moses and G. Nachum, Agreeing to disagree after all, in ``Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning About Knowledge,'' Morgan Kaufman, San Mateo, CA, 1990. 16. R. Parikh, Levels of knowledge in distributed computing, in ``IEEE Symposium on Logic in Computer Science, Boston, 1986,'' pp. 322331. 17. R. Parikh and P. Krasucki, Levels of knowledge in distributed systems, Sadhana, 17 (1992), 167191. 18. R. Parikh and P. Krasucki, Communication, consensus and knowledge, J. Econ. Theory 52 (1990), 178189. 19. A. Rubinstein and A. Wolinsky, On the logic of ``Agreeing to Disagree'' type results, J. Econ. Theory 51 (1990), 184193. 20. D. Samet, Ignoring ignorance and agreeing to disagree, J. Econ. Theory 52 (1990), 190207. 21. S. Schiffer, ``Meaning,'' Oxford U. Press, LondonNew York, 1972. 22. J. Sebenius and J. Geanakoplos, Don't bet on it: contingent agreements with asymmetric information, J. Am. Statist. Assoc. 78 (1983), 424426. 23. S. Weyers, ``Three results on Communication, Information and Common Knowledge,'' CORE Discussion Paper 9228.
File: AAAAAA 212507 . By:CV . Date:18:07:96 . Time:14:25 LOP8M. V8.0. Page 01:01 Codes: 3438 Signs: 2747 . Length: 45 pic 0 pts, 190 mm
Protocols Forcing Consensus Paul Krasucki 1 , 2 Department of Computer Science, Rutgers University, Camden College of Arts and Sciences, Camden, New Jersey 08102 Received July 8, 1991; revised January 24, 1995
We analyze n agents communicating in order to reach consensus. We show how some conditions on the topology of the communication graph (the order in which individuals communicate) are sufficient to guarantee consensus on the value of a function satisfying the ``sure-thing'' condition. Journal of Economic Literature Classification Numbers: C62, C78. 1996 Academic Press, Inc.
Introduction Aumann [1] was the first economist to define the technical notions of knowledge and common knowledge (see also Hintikka [9], Lewis [10], and Schiffer [21]). He proved that if two individuals share their priors, and there is common knowledge among them of their information partitions and their posteriors concerning a certain fixed event, then these posteriors must be the same. Geanakoplos and Polemarchakis [8] extended Aumann's results to a dynamic framework. Instead of assuming common knowledge of posteriors, they investigated a protocol in which agents announce their current posteriors to each other. In this process agents learn and modify their posteriors, and ultimately they agree (the announcements become the same). Cave [6] and Bacharach [3] generalized Aumann's and Geanakoplos Polemarchakis's agreement results to situations in which agents are interested in values of any union-consistent function f, when the values of f after every revision are publicly announced. 1
I thank Rohit Parikh for getting me interested in the problems of consensus, many helpful discussions, reading this manuscript, and valuable comments. I also thank Don Beaver for helpful suggestions and Joseph Gerver, who was patient enough to correct my English. Remarks of an anonymous referee and an editor helped in revisions of the paper. 2 E-mail: Krasuckicrab-rutgers.edu.
266 0022-053196 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
File: AAAAAA 212501 . By:CV . Date:18:07:96 . Time:14:24 LOP8M. V8.0. Page 01:01 Codes: 3896 Signs: 2158 . Length: 50 pic 3 pts, 212 mm
PROTOCOLS FORCING CONSENSUS
267
Here we consider protocols in which values of functions are communicated privately through messages; when agent A communicates with agent B, other agents are not informed about the contents of the message. In this model (first considered by Parikh and Krasucki [18]) union-consistency does not guarantee agreement even with reasonable protocols. In the present paper, we investigate what restrictions on the protocol will guarantee that agreement is reached on the value of any union-consistent function, regardless of the information partitions. Two common protocols in which consensus will be guaranteed by our theorem 1 are the channel protocol and the star protocol. In the channel protocol, information is passed along the ``channel of command'': agent 1 communicates with agent 2, agent 2 with agent 3,. . ., and agent n&1 with agent n, who reverses the flow of information by communicating with agent n&1, and so on, until communication reaches agent 1, who starts the process over again. The star protocol is a centralized protocol in which every agent communicates privately with a distinguished central agent, who immediately replies to every message.
Basic Notions and Results Let W be the space of states (or of possible worlds). There are n participants numbered from 1 to n. Every agent i for 1 i n has a finite information partition P i of W. P i (x) will denote an element of a partition P i containing the world x # W. Agents communicate by sending messages. To simplify notation we will assume that messages are delivered instantaneously (this assumption is not essential in any of the results). A protocol Pr is a function Pr : N + [1, ..., n] 2. If Pr (t)=(s(t), r(t)), then we interpret s(t) as the sender and r(t) as the recipient of the communication which takes place at time t. A protocol is fair if and only if every participant in this protocol communicates (directly or indirectly) with every other participant infinitely often (see [18]). The channel protocol (Fig. 1) and the star protocol (Fig. 2) are both fair protocols.
Fig. 1. Channel protocol. For t>2n&2, s(t)=s(t&2n&2).
File: 642J 212502 . By:MC . Date:04:06:96 . Time:13:00 LOP8M. V8.0. Page 01:01 Codes: 2644 Signs: 2118 . Length: 45 pic 0 pts, 190 mm
268
PAUL KRASUCKI
Fig. 2. Star protocol for n=5. For t>2n&2, s(t)=s((t&1) mod(2n&2)+1).
Let us assume that individuals communicate values of a function f: 2 W D according to a protocol Pr (t)=(s(t), r(t)). We require that the function whose values are communicated satisfies the following condition, called union consistency: 3 for X, Y disjoint, if f (X)=f (Y), then f (X)= f(X _ Y) (the same property was also called the sure thing principle; see also [3] and [6]). The updating procedure is a non-recursive version of the fully rational procedure originally defined in Parikh and Krasucki [18]). The non-recursive description of the updating procedure was motivated by Weyers [23]; the notion of the working partition was also first proposed by Weyers. Let P ti denote an information partition of W for agent i at time t. P ti(x) denotes the set of possible states for i at time t, given that the real state is x; P 0i =P i . Let m(x, t)=f (P ts(t)(x)) be a message sent by s(t) at time t in world x. We will say that two worlds x, x$ are indistinguishable by a message at t (x#t x$) if and only if m(x, t)=m(x$, t). Since #t is an equivalence relations it generates a partition of W (we will denote it by W t and call it the working partition at time t). =P ti . If i was not a recipient of a message at time t, P t+1 i t+1 t t If i=r(t), then P i =P i 6 W . Initially everybody knows only which element of hisher partition contains the real world. At time t, if a person doesn't receive any messages, hisher state of knowledge remains unchanged. Upon receiving a message, a person excludes 3 If a function is not union consistent, then if we take two agents such that one's information partition is a refinement of the other's, they may disagree.
File: 642J 212503 . By:MC . Date:04:06:96 . Time:13:00 LOP8M. V8.0. Page 01:01 Codes: 2402 Signs: 1702 . Length: 45 pic 0 pts, 190 mm
PROTOCOLS FORCING CONSENSUS
269
from hisher set of possible worlds, worlds incompatible with the received message. In [18], Parikh and Krasucki showed that union consistency is not strong enough to guarantee consensus when the number of agents is greater than 2. All the examples in which communicating values of a union-consistent function fails to bring about consensus were round-robin protocols (individual 1 to 2 to 3 to 1. . .). We will show that all such examples must contain a cycle; if there are no cycles in the communication graph, consensus on the value of any unionconsistent function must be reached. We will need two lemmas which correspond to theorems proved in [18]: Lemma 1. There is a t 0 such that for all x, i, and all t, t$>t 0 , P ti(x)= P (x). Hence for all x, i, P ti(x) has a limiting value P i (x). t$ i
Proof. Since all partitions P i are finite, their join P + is also finite. For every i, x, t, P ti(x) is a union of elements of P +. P ti(x) as a function of t is monotone non-increasing; therefore it must converge in a finite number of steps. K Lemma 2. If there is a W, finite partitions P 1 , ..., P n of W and a union consistent function f such that for some fair protocol Pr, the limiting partitions reached at time T are Q 1 , ..., Q n , then there is a protocol Pr$ with the same graph 4 as Pr such that if we use Q 1 , ..., Q n as initial partitions of W in Pr$, no one will gain any knowledge during the execution of Pr$. That is, Q i (x)=Q ti(x) for all i, x, t. Proof. By Lemma 1, T exists. There is no learning after T. Let Pr$=(s$(t), r$(t)) and s$(t)=s(T+t), r$(t)=r(T+t). Clearly Pr and Pr$ have the same graphs. K In particular, if no consensus was reached in the first case, then in the second case also there will be no consensus, and no learning will ever take place. Moreover, the protocol Pr$ is fair if and only if the original protocol Pr was fair. Protocols in which no participant gains any knowledge we will call zero-learning protocols. In zero-learning protocols P ti and W t both remain constant as functions of t. We will characterize protocols in which consensus among any n participants on the value of a union-consistent f is always reached. Using Lemmas 1 4
A graph of the protocol has agents P i for i n as nodes. There is an edge between P i and P j if and only if there are infinitely many communications from P i to P j . See (18).
File: AAAAAA 212504 . By:CV . Date:18:07:96 . Time:14:24 LOP8M. V8.0. Page 01:01 Codes: 3090 Signs: 2327 . Length: 45 pic 0 pts, 190 mm
270
PAUL KRASUCKI
and 2 we can, without loss of generality, consider only situations in which a fair protocol is zero-learning. Lemma 3. In a zero-learning protocol, if i communicates the value of a union consistent f to j, j sends the value of f back to i then i and j were sending the same value. Proof. Let s(t)=i, r(t)=j, s(t+1)=j, r(t+1)=i. If for any y ( y){P ti( y), then the working partitions W t and W t+2 would be difP t+2 i ferent, which would contradict the assumption of zero-learning. Therefore, \y, t,
P ti( y)=P t+2 ( y) i
and
P tj( y)=P t+1 ( y). j
Let E(x, i, t)=[P ti( y) | P ti( y) (P ti 7 P tj )(x)]. Similarly we can define E(x, j, t)=[P tj( y) | P ti( y) (P ti 7 P tj )(x)]. If there is no learning, then all sets in E(x, i, t) are compatible with P ti(x); i.e., they give the same value of f as P ti(x). By the union-consistency of f: f (P tj(x))=f (E(x, i, t))=f ((P ti 7 P tj )(x))=f (E(x, j, t))=f (P ti(x)). K Often when agents communicate, they do not just send messages to each other, but instead they exchange information, they hand each other their bids, then after learning from them they reevaluate their positions (an example of such a process could be reaching an agreement between a buyer and a seller of a house). That would correspond in our model to the situation where sender and receiver in the same instant of time reverse their roles; formally this protocol may be specified as a pair of protocols: P 1 =(s 1(t), r 1(t)), P 2 =(s 2(t), r 2(t)), where \t, s 1(t)=r 2(t), s 2(t)=r 1(t). If P 1 and P 2 are executed concurrently then we will call it a protocol with information exchange. Corollary 1. In any fair protocol with information exchange, consensus on the value of a union-consistent function must be reached. Proof.
Follows from Lemma 3. K
Let us call a protocol cyclic if and only if there are agents i 1 , ..., i k , k 3, such that for all j
Addition and subtraction are mod k.
File: AAAAAA 212505 . By:CV . Date:18:07:96 . Time:14:24 LOP8M. V8.0. Page 01:01 Codes: 3228 Signs: 2170 . Length: 45 pic 0 pts, 190 mm
PROTOCOLS FORCING CONSENSUS
271
Fig. 3. Cyclic protocol (2, 3, 4 form a cycle).
There is an example of a cyclic protocol in Fig. 3. To simplify notation, we will assume that for every n, s(n+1)=r(n), and that communications proceed in rounds, i.e., there is some N (number of communications in a round) such that for all t, s(t)=s(t+N). Theorem 1. Consensus on the value of every union-consistent f, for every set of initial information partitions P 1 , ..., P n is guaranteed if the values of f are communicated according to a non-cyclic, fair protocol Pr . 6 Proof. We assume that the number of agents is at least three. If there are only two agents, consensus is always achieved (Lemma 3, earlier results by Bacharach [3] and Cave [6]). Suppose that in a fair, non-cyclic protocol Pr consensus on the value of f is not reached. By Lemma 2 we can assume without loss of generality that Pr is zero-learning. Furthermore we can assume that our example is minimal; i.e., if we remove any of the agents or communications from the protocol, consensus will be reached. Pr is fair, so for every pair of agents i j , i k there is a chain of communications from i j to i k and from i k to i j . There must be agents (nodes in the graph of Pr ) i $j and i $k such that in every round, if at time t a message m t is sent in Pr by i$j to i$k then m t+1 is sent by i $k to i $j (if there are no such agents, there would be a cycle). By Lemma 3, m t and m t+1 must be the same. We can modify our example by removing communications m t and m t+1 (if an agent i $j is not involved in any other communications, we can remove her as well). Since the protocol is zero-learning, and the messages m t and m t+1 are the same, the consensus is not affected by the change: if there were no consensus between some agents in the original protocol, there would be no consensus in the modified one. That contradicts the assumption of minimality. K The protocol Pr is cyclic, we can always create an example in which communications according to Pr do not lead to consensus. 6 After reading the manuscript of this paper, Professor Rohit Parikh suggested a different formulation of the Theorem 1: consensus on the value of every union-consistent f, for every set of initial partitions P 1 , ..., P n is guaranteed in a protocol Pr with a graph G iff G & G I is connected (G I is the inverse of G).
File: 642J 212506 . By:MC . Date:04:06:96 . Time:13:01 LOP8M. V8.0. Page 01:01 Codes: 3066 Signs: 2311 . Length: 45 pic 0 pts, 190 mm
272
PAUL KRASUCKI
As consequences of Theorem 1 we get consensus results for the channel protocol and the star protocol (the channel protocol and the star protocol are non-cyclic).
References 1. R. Aumann, Agreeing to disagree, Ann. Statist. 4 (1976), 12361239. 2. R. Aumann, Notes on interactive epistemology, unpublished manuscript, 1989. 3. M. Bacharach, Some extensions of a claim of Aumann in an axiomatic model of knowledge, J. Econ. Theory 37 (1985), 167190. 4. A. Brandenburger and E. Dekel, Common knowledge with probability 1, J. Math. Econ. 16 (1987), 237245. 5. L. T. Nielsen, A. Brandenburger, J. Geanakoplos, R. McKelvey, and T. Page, Common knowledge of an aggregate of expectations, Econometrica 58 (1990), 12351239. 6. J. A. K. Cave, Learning to agree, Econ. Lett. 12 (1983), 147152. 7. J. Geanakoplos, Common knowledge, Bayesian learning, and market speculation with bounced rationality, unpublished manuscript, 1988. 8. J. Geanakoplos and H. Polemarchakis, We can't disagree forever, J. Econ. Theory 28 (1982), 192200. 9. J. Hintikka, ``Knowledge and Belief,'' Cornell Univ. Press, Ithaca, NY, 1962. 10. D. Lewis, ``Convention,'' Harvard Univ. Press, Cambridge, MA, 1969. 11. P. Krasucki, Reaching consensus on decisions, in ``Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning About Knowledge,'' Morgan Kaufmann, San Mateo, CA, 1990. 12. R. McKelvey and T. Page, Common knowledge, consensus, and aggregate information, Econometrica 54 (1986), 109127. 13. P. Milgrom, An axiomatic characterization of common knowledge, Econometrica 49 (1981), 219222. 14. P. Milgrom and N. Stokey, Information, trade and common knowledge, J. Econ. Theory 26 (1982), 1727. 15. Y. Moses and G. Nachum, Agreeing to disagree after all, in ``Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning About Knowledge,'' Morgan Kaufman, San Mateo, CA, 1990. 16. R. Parikh, Levels of knowledge in distributed computing, in ``IEEE Symposium on Logic in Computer Science, Boston, 1986,'' pp. 322331. 17. R. Parikh and P. Krasucki, Levels of knowledge in distributed systems, Sadhana, 17 (1992), 167191. 18. R. Parikh and P. Krasucki, Communication, consensus and knowledge, J. Econ. Theory 52 (1990), 178189. 19. A. Rubinstein and A. Wolinsky, On the logic of ``Agreeing to Disagree'' type results, J. Econ. Theory 51 (1990), 184193. 20. D. Samet, Ignoring ignorance and agreeing to disagree, J. Econ. Theory 52 (1990), 190207. 21. S. Schiffer, ``Meaning,'' Oxford U. Press, LondonNew York, 1972. 22. J. Sebenius and J. Geanakoplos, Don't bet on it: contingent agreements with asymmetric information, J. Am. Statist. Assoc. 78 (1983), 424426. 23. S. Weyers, ``Three results on Communication, Information and Common Knowledge,'' CORE Discussion Paper 9228.
File: AAAAAA 212507 . By:CV . Date:18:07:96 . Time:14:25 LOP8M. V8.0. Page 01:01 Codes: 3438 Signs: 2747 . Length: 45 pic 0 pts, 190 mm