First impressions matterRabin and Schrag

Model

Two states of the world A and B; x 2 fA;Bg exhaustive and mutually exclusive. The agent has prior beliefsabout x by P[x = A] = P[x = B] = 1=2: In evey period t 2 f1; 2; 3; :::g the agent receives a signal st 2 fa; bgthat is correlated with the true state of the world. Signals received in dierent times are independent andidentically distributed.

P[st = ajx = A] = P[st = bjx = B] = for some 2 (0:5; 1) : This implies that

P[st = bjx = A] = P[st = ajx = B] = 1 We suppose that the agent may misintrepret signals. In every t 2 f1; 2; 3; :::g the agent perceives a signalt 2 f; g : If the agent perceives a signal t = he believes that he actually received a signal st = a andif the perceives t = he believes he has received st = b: In probabilistic notation these mean that

P[t = jst = a;P[x = A] 1=2] = P[t = jst = b;P[x = A] 1=2] = 1:But he has some bias

P[t = jst = b;P[x = A] > 1=2] = P[t = jst = a;P[x = A] < 1=2] = q;with q 2 [0; 1] :

If q = 0 the conrmation bias is absent and s = a and s = b are always converted to = and = respectively.

If q = 1 he always misreads signals. The higher is q the more extreme is the conrmatory bias. If agents are not aware of their conrmation bias, then

Suppose that the agent has received n signals and n signals, where n > n : His updated posteriorbelies are given by:

P[t = jst = a] = P [t = jst = b] = 1:and

P[t = jst = b] = P [t = jst = a] = 0:Now

P[x = Ajs1 = a; s2 = a; s3 = b] = P[s1 = a; s2 = a; s3 = bjx = A]P[x = A]P[s1 = a; s2 = a; s3 = bjx = A]P[x = A] +P[s1 = a; s2 = a; s3 = bjx = B]P[x = B]

=P[s1 = ajx = A]P[x = A] +P[s2 = ajx = A]P[x = A] +P[s3 = bjx = A]

P[s1 = ajx = A]P[x = A] +P[s2 = ajx = A]P[x = A] +P[s3 = bjx = A]P[x = A] +P[s1 = ajx = B]P[x = B] +P[s2 = ajx = B]P[x = B] +P[s3 = bjx = B]P[x = B]]

=2(1 )P[x = A]

2(1 )P[x = A] + (1 )2 P[x = B]Suppose that the agent has received n signals and n signals, where n > n : His updated posteriorbelies are given by:

P[x = Ajn; n ] = n(1 )nP[x = A]

n(1 )nP[x = A] + (1 )n nP[x = B]

1

If P[x = A] = P[x = B] = 1=2; then it reduces to

P[x = Ajn; n ] = n(1 )n

n(1 )n + (1 )n n

=nn

nn + (1 )nn :

2