Case 3:-(t-1)/t>wA>-(t-2)/t and wB>(t-1)/t. By analogous reasoning. A chooses o and B chooses 1 until date t when a switches to 1 The other interesting case to consider is when the signals are similar in trength. For example, suppose that r" where a* is the limit of the sequent [atlee defined by putting r1=2, I2=4, and for t=3, 4,. Notice that if t is even then t <a*<It-I As usual A chooses 0 and B chooses 1 at date 1. At date 2. A observes B's choice in the previous period, realizes that the expected value of b's signal is 1 /2>-x*and switches to 1. By the symmetric argument, B switches to 0. At date 3. A observes B's switch to 0 and realizes that 1/4<wb 1/2, that is, the expected value of wB is greater than T' -WA. So it is optimal for A to choose 0 again. By a symmetric argument B switches back to 1 at date 3 t any even date t, a will choose 1 and B will choose 0 and at any odd date t. A will choose 0 and B will choose 1. b's choice at an even date t reveals that at-2<wb<at-1 and his choice at an odd date rev t-1<WB <Tt-2. By construction, at any odd date t, -WA=I"<at (tt-1+at-2), so it is optimal for A to choose 1 at t+1. Likewise,at any even date t, -WA=I*>It=(at-1+Tt-2), so it is optimal for A to choose 0 at t+1 In fact, we can find a signal w to rationalize any sequence of actions with the properties that for some T, aAt f aBt for t< T and At=TBt =a for t>T. However, the sequences corresponding to T= oo occur with probability 0 and the sequences with T< oo occur with positive probability This example can also be used to illustrate the speed of convergence uniformity of actions. In the first period, the probability that agents choose the same action is 1 /2. In the second period, it is 3/4. In the third period, it is 7/8, and so on. This is a very special example, but simulations of other examples confirm these results Finally, we note that in this simple example, where the signals of the two players are symmetrically distributed, the asymptotic outcome must be Pareto-efficient. This follows from the fact that the agent with the stronger signal, as measured by its absolute value, will ultimately determine the action chosen. However, a simple adjustment to this example shows the possibility of an inefficient outcome. Suppose that A has a signal uniformly distributed on[0, 1] and B has a signal uniformly distributed on[-1. 1]. Then both A and b will choose action 1 at the first date and there will be no learning. However, if WA is close to O and wB is close to -z then action 0 is clearly preferred conditional on the information available to the twoCase 3: −(t−1)/t > ωA > −(t−2)/t and ωB > (t−1)/t. By analogous reasoning, A chooses 0 and B chooses 1 until date t when A switches to 1. The other interesting case to consider is when the signals are similar in strength. For example, suppose that ωA = −ωB = x∗ where x∗ is the limit of the sequent {xt}∞ t=1 defined by putting x1 = 1 2 , x2 = 1 4 , and xt = 1 2 (xt−1 + xt−2) for t = 3, 4, .... Notice that if t is even then xt < x∗ < xt−1. As usual A chooses 0 and B chooses 1, at date 1. At date 2, A observes B’s choice in the previous period, realizes that the expected value of B’s signal is 1/2 > −x∗ and switches to 1. By the symmetric argument, B switches to 0. At date 3, A observes B’s switch to 0 and realizes that 1/4 < ωB < 1/2, that is, the expected value of ωB is greater than x∗ = −ωA. So it is optimal for A to choose 0 again. By a symmetric argument, B switches back to 1 at date 3. At any even date t, A will choose 1 and B will choose 0 and at any odd date t, A will choose 0 and B will choose 1. B’s choice at an even date t reveals that xt−2 < ωB < xt−1 and his choice at an odd date reveals xt−1 < ωB < xt−2. By construction, at any odd date t, −ωA = x∗ < xt = 1 2 (xt−1 + xt−2), so it is optimal for A to choose 1 at t + 1. Likewise, at any even date t, −ωA = x∗ > xt = 1 2 (xt−1 + xt−2), so it is optimal for A to choose 0 at t + 1. In fact, we can find a signal ω to rationalize any sequence of actions with the properties that for some T, xAt 6= xBt for t<T and xAt = xBt = a for t ≥ T. However, the sequences corresponding to T = ∞ occur with probability 0 and the sequences with T < ∞ occur with positive probability. This example can also be used to illustrate the speed of convergence to uniformity of actions. In the first period, the probability that agents choose the same action is 1/2. In the second period, it is 3/4. In the third period, it is 7/8, and so on. This is a very special example, but simulations of other examples confirm these results. Finally, we note that in this simple example, where the signals of the two players are symmetrically distributed, the asymptotic outcome must be Pareto-efficient. This follows from the fact that the agent with the stronger signal, as measured by its absolute value, will ultimately determine the action chosen. However, a simple adjustment to this example shows the possibility of an inefficient outcome. Suppose that A has a signal uniformly distributed on [0, 1] and B has a signal uniformly distributed on £ −1 2 , 1 ¤ . Then both A and B will choose action 1 at the first date and there will be no learning. However, if ωA is close to 0 and ωB is close to −1 2 then action 0 is clearly preferred conditional on the information available to the two agents. 10