Reaching Consensus on Decisions

We investigate how like-minded agents can reach consensus on their decisions even if they receive different information. The model used here was introduced by Aumann, and subsequently refined by Geanakoplos and Polemarchakis, Bacharach, Cave, Parikh and Krasucki ([Aum76,GP82,Cav83,Bac85,PK]). The main result is that when any number of like-minded agents communicate according to some fair protocol whether they want to trade or not, and their decision is based solely on whether the conditional probability of some fixed event exceeds some threshold value, they must reach consensus in a finite time. We also investigate some necessary conditions which functions communicated have to satisfy in order to guarantee consensus in fair protocols. 1 I N T R O D U C T I O N In order to investigate whether the difference in the kind of information received can justify the speculative trading between rational agents the following model was created ([Aum76,GP82, Cav83,Bac85,PK]). Let W be a set of possible results of an experiment. There are two agents receiving some information about the result. Information is given by choosing one of the elements of the ith partition of W, Pi. It is assumed that Pi's are common knowledge among agents (so type of information available to agents is common knowledge). It is also assumed that agents always receive true information, if the actual state of the world is x, then for all i, x E Pi(x). Both agents are interested in computing a probability of some fixed event, so they could make their decisions based on that. There is given some prior probability distribution on W, and it is shared by both agents. Without any additional information they would both have the same value: p(E). But if an agent 1 learns that the result x is in Pl(X), then he can compute a new probability as p(E[Pl(x)) . This is his posterior probability of E. Similarly, an agent 2 can compute his posterior probability p(EIP2(x)). There is no a priori reason why p(EIPl(x) ) = p(EIP2(x)) , but surprisingly Robert Aumann [Aum76], has shown in 1976 that when the posteriors are common knowledge, then they must indeed be the same. Like-minded agents cannot "agree to disagree". Aumann didn' t address the question how the agents computed their posteriors and how could they become common knowledge. Geanakoplos and Polemarchakis [GP82] first investigated