that R knows that t is between ti-1 and ti, and in the special case quadratic utility where R wants action a to be as close to t as possible, we can show that quite intuitively the optimal action is the middle of the interval: Indifference condition What happens at t = ti? The sender has to be indifferent between sending either message mi-1 or mi. 1 ≤ i≤ N-1 This gives information about N and the ti. → Practically: We consider a partition of size N. One can show that N must be small enough so that the numerator is positive. This determines the maximum allowed value where is the ceiling of , i.e. the smallest positive integer greater or equal to . Example: We assume that b = 1/20. Then N* = 3. We now describe all the equilibria for N=1, 2, or 3 (see Figure 2). N = 1: This is the babbling equilibrium. t0 = 0, t1 = 1; a1 = 1/2 = 0.5. N = 2: t0 = 0, t1 = 2/5 = 0.4, t2 = 1; a1 = 1/5 = 0.2, a2 = 7/10 = 0.7. N = N* = 3: t0 = 0, t1 = 2/15, t2 = 7/15, t3 = 1; a1 = 1/15, a2 = 3/10 = 0.3, a3 = 11/15. With N = 1, we get the coarsest possible message, which does not give any information. So everything is red on the top left panel. With N = 3, the message is finer. However, it remains quite coarse compared to full revelation, which would be the 45° line, but which is not a Nash equilibrium. With a higher N, and a finer message, the blue area is more important. This implies higher utility. Disclosing more information benefits both parties. See also Game theory Handicap principle Screening game Signaling game Small talk Trash-talk Notes References Game theory Asymmetric information