Some Results on Consensus 245 Some Results on Consensus Extended Abstract

We will show how reaching consensus among n individuals communicating in pairs depends on the topology of the communications graph. In particular we show that the consensus on the value of union consistent function is guaranteed in any non-cyclic fair protocol. We also anylyze protocols where individuals exchange (simultaneously) information in pairs. Finally we prove a surprising result that a certain non-trivial level of common knowledge of some formula which was not initially common knowledge is a necessary condition for a disagreement. 1 I N T R O D U C T I O N There are many situations in which rational agents, trying to accomplish the same objective, act differently. Typical example are two people who trade stocks. Both want to maximize their profits and one is selling the stocks while the other one is buying the stocks of the same company. If we assume that they are like-minded (given the same information they would have acted in the same way) the only explanation is that they have received different information. What if they are aware of the others actions so they have reason to believe that the information received by others leads to different conclusions? Can they 'agree to disagree' ? What if they can observe others actions and change their own actions accordingly? Will they start to act the same way? We will continue the work started by Aumann [Aum76] and subsequently expanded by Geanakoplos and Polemarchakis [GP82] and others. Specifically we will use the model of one-toone communications among n agents as introduced by Parikh and Krasucki [PK90]. 2 D E S C R I P T I O N O F T H E M O D E L W is the space of states (or of possible worlds). There is n agents (individuals) numbered from 1 to n. Every individual i has an accessibility relation ,~,i. If w is the real world, i's information allows him to exclude all the worlds w ~ which are not accessible from w, but he cannot distinguish between the worlds w" s.t. w ~i w". We will assume that for every i, ~ is an equivalence relation so its equivalence classes form a partition of W. We will further assume that all the partitions are finite. We will use P~ to denote a partition of an individual i given by ~i.