We now know that the CPa is equivalent to the assumption that, at any state w EQ, players are not willing to engage in bets over the realization of w-contingent random variable Thus, it entails a very strong notion of agreement For instance, in our elaboration of the Cournot example, denote by pt the prior proba- bility of cell ti of Ti, and by q the prior probability of cell t2 of T2. Then the conditional probabilities indicated above imply that the priors pi and p2 must satisfy the following State p1()p2(a Po P 0 2P02 q W012 0 0 (1-q) P2H(1-q)3 Table 1: The priors in the Elaborated Cournot Model Disregarding the last column for the time being, the table yields a set of necessary conditions for the existence of a common prior. First, from the line corresponding to w121, P12 Hence, from the line corresponding to w122, we must have 5q=4(1-q), or However, from the line corresponding to wo22, Po2=23=1. This is impossible, because the per's must add up to one. Thus, there can be no common prior The rightmost column of table 1 illustrates a bet that in accordance with the results cited above, at each state w yields a positive expected payoff to Player 1 conditional on t1(w) and a negative expected payoff to Player 2 conditional on t2(w). Hence, at each state w of the model, Player 1 would want to offer a bet stating that, at each state w, Player 2 would pay her X(w) dollars, and Player 2 would readily accept it To see this, note that E[]=1, E[Xt]=1, E(X+02]=5 and E(Xt]=5 the other hand, E[X t2=-1 and E[Xt2=- If players' conditional beliefs were consistent with a common prior, no such mutually acceptable bet would exist. This statement is also known as the no-trade theorem. Indeed the results cited above state that the converse is also true: if no such mutually acceptable bet exists, then players' conditional beliefs are consistent with a common prior. For instance, in the simpler"textbook"treatment of the Cournot model (in which the set of states is the setWe now know that the CPA is equivalent to the assumption that, at any state ω ∈ Ω, players are not willing to engage in bets over the realization of ω-contingent random variables. Thus, it entails a very strong notion of agreement. For instance, in our elaboration of the Cournot example, denote by p cx the prior proba￾bility of cell t cx 1 of T1, and by q the prior probability of cell t 1 2 of T2. Then the conditional probabilities indicated above imply that the priors p1 and p2 must satisfy the following equalities: State ω p1(ω) p2(ω) X(ω) ω011 p01 1 2 q 1 ω 1 2 11 p 1 2 1 0 1 ω021 1 2 p02 0 3 ω 1 2 21 1 2 p 1 2 2 1 2 q -2 ω012 0 0 0 ω 1 2 12 0 0 0 ω022 1 2 p02 3 4 (1 − q) -2 ω 1 2 22 1 2 p 1 2 2 1 4 (1 − q) 3 Table 1: The priors in the Elaborated Cournot Model Disregarding the last column for the time being, the table yields a set of necessary conditions for the existence of a common prior. First, from the line corresponding to ω 1 2 21, p 1 2 2 = q. Hence, from the line corresponding to ω 1 2 22, we must have 1 2 q = 1 4 (1 − q), or 2q = 1 − q, so q = 1 3 = p 1 2 2 . However, from the line corresponding to ω022, p02 = 3 2 2 3 = 1. This is impossible, because the pcx’s must add up to one. Thus, there can be no common prior. The rightmost column of Table 1 illustrates a bet that, in accordance with the results cited above, at each state ω yields a positive expected payoff to Player 1 conditional on t1(ω), and a negative expected payoff to Player 2 conditional on t2(ω). Hence, at each state ω of the model, Player 1 would want to offer a bet stating that, at each state ω 0 , Player 2 would pay her X(ω 0 ) dollars, and Player 2 would readily accept it. To see this, note that E[X|t 01 1 ] = 1, E[X|t 1 2 1 1 ] = 1, E[X|t 02 1 ] = 1 2 and E[X|t 1 2 2 1 ] = 1 2 ; on the other hand, E[X|t 1 2 ] = −1 and E[X|t 2 2 ] = − 3 4 . If players’ conditional beliefs were consistent with a common prior, no such mutually acceptable bet would exist. This statement is also known as the no-trade theorem. Indeed, the results cited above state that the converse is also true: if no such mutually acceptable bet exists, then players’ conditional beliefs are consistent with a common prior. For instance, in the simpler “textbook” treatment of the Cournot model (in which the set of states is the set 6