Editing Signaling game (section)

===Education game===
[[Michael Spence]]'s 1973 paper on education as a signal of ability starts the economic analysis of signaling.<ref>{{cite journal |author-link=Michael Spence |last=Spence |first=A. M. |year=1973 |title=Job Market Signaling |journal=[[Quarterly Journal of Economics]] |volume=87 |issue=3 |pages=355–374 |doi=10.2307/1882010 |jstor=1882010 }}</ref><ref name=ft91/>{{rp|329–331}}<ref>For a survey of empirical evidence on how important signaling is in education see   Andrew Weiss. 1995. "Human Capital vs. Signalling Explanations of Wages." ''Journal of Economic Perspectives'', 9 (4): 133-154.
DOI: 10.1257/jep.9.4.133.</ref> In this game, the senders are workers, and the receivers are employers. The example below has two types of workers and a continuous signal level.<ref>This is a simplified version of the model in Johannes Horner, "Signalling and Screening," ''The New Palgrave Dictionary of Economics'', 2nd edition, 2008, edited by Steven N. Durlauf and Lawrence E. Blume, http://najecon.com/econ504/signallingb.pdf.</ref> 

The players are a worker and two firms. The worker chooses an education level <math>s,</math> the signal, after which the firms simultaneously offer a wage <math>w_1</math> and <math>w_2</math>, and the worker accepts one or the other. The worker's type, which is privately known, is either "high ability," with <math>a=10</math>, or "low ability," with  <math>a = 0,</math> each type having probability 1/2. The high-ability worker's payoff is <math>U_H= w - s</math>, and the low-ability's is <math>U_{L}= w - 2s.</math> A firm that hires the worker at wage <math>w</math> has payoff <math>a-w</math> and the other firm has payoff 0.  

In this game, the firms compete for the wage down to where it equals the expected ability, so if there is no signal possible, the result would be  <math>w_1=w_2 = .5(10) + .5 (0) =5.</math> This will also be the wage in a pooling equilibrium where both types of workers choose the same signal, so the firms are left using their prior belief of .5 for the probability the worker has high ability. In a separating equilibrium, the wage will be 0 for the signal level the Low type chooses and 10 for the high type's signal. There are many equilibria, both pooling and separating, depending on expectations. 

In a separating equilibrium, the low type chooses <math>s=0.</math> The wages will be <math>w(s=0)=0</math> and <math>w(s=s^*) =10</math> for some critical level <math>s^*</math> that signals high ability. For the low type to choose <math>s = 0</math> requires that   <math>U_L (s = 0) \geq U_L(s=s^*),</math> so <math> 0 \geq 10-2s^*</math>  and we can conclude that <math>s^* \geq 5.</math> For the high type to choose <math>s = s^*</math> requires that  <math>U_H (s = s^*) \geq U_H(s=0),</math> so  <math>10-s \geq 0</math> and we can conclude that <math>s^* \leq 10.</math> Thus, any value of <math>s^*</math> between 5 and 10 can support an equilibrium. Perfect Bayesian equilibrium requires an out-of-equilibrium belief to be specified, too, for all the other possible levels of <math>s</math> besides 0 and <math>s^*,</math> levels which are "impossible" in equilibrium since neither type plays them. These beliefs must be such that neither player would want to deviate from his equilibrium strategy 0 or <math>s^*</math> to a different <math>s.</math> A convenient belief is that <math>Prob(a = High) =0</math> if <math>s \neq s^*;</math> another, more realistic, belief that would support an equilibrium is  <math>Prob(a = High) = 0</math> if <math>s < s^*</math> and <math>Prob(a = High) = 1</math> if <math>s \geq s^*</math>. There is a continuum of equilibria, for each possible level of <math>s^*.</math> One equilibrium, for example, is 
:  <math>s|Low = 0, s|High= 7, w|(s=7)  = 10, w|(s \neq 7)  = 0, Prob(a=High|s=7) = 1, Prob(a=High|s \neq 7) =0. </math>

In a pooling equilibrium, both types choose the same <math>s.</math> One pooling equilibrium is for both types to choose <math>s=0,</math> no education, with the out-of-equilibrium belief <math>Prob(a=High|s>0) = .5.</math> In that case, the wage will be the expected ability of 5, and neither type of worker will deviate to a higher education level because the firms would not think that told them anything about the worker's type. 

The most surprising result is that there are also pooling equilibria with  <math>s = s'>0.</math> Suppose we specify the out-of-equilibrium belief to be <math>Prob(a=High|s< s') = 0.</math> Then the wage will be 5 for a worker with <math>s= s',</math> but 0 for a worker with wage <math>s = 0.</math> The low type compares the payoffs <math>U_L(s=s') = 5 - 2s'</math> to <math>U_L(s=0) =0,</math> and if <math>s'\leq 2.5,</math> the worker is willing to follow his equilibrium strategy of <math>s=s'.</math> The high  type will choose <math>s=s'</math> a fortiori. Thus, there is another continuum of equilibria, with values of <math>s'</math> in [0, 2.5]. 

In the signaling model of education, expectations are crucial. If, as in the separating equilibrium,  employers expect that high-ability people will acquire a certain level of education and low-ability ones will not, we get the main insight: that if people cannot communicate their ability directly, they will acquire education even if it does not increase productivity, to demonstrate ability. Or, in the pooling equilibrium with <math>s=0,</math> if employers do not think education signals anything, we can get the outcome that nobody becomes educated. Or, in the pooling equilibrium with <math>s>0,</math> everyone acquires education they do not require, not even showing who has high ability, out of concern that if they deviate and do not acquire education, employers will think they have low ability.