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==Examples== ===Reputation game=== {| class="wikitable" style="float:right" |- ! {{diagonal split header|Sender | Receiver}} !! Stay !! Exit |- | Sane, prey || P1+P1, D2 || P1+M1, 0 |- | Sane, accommodate || D1+D1, D2 || D1+M1, 0 |- | Crazy, prey || X1, P2 || X1, 0 |} In this game,<ref name=ft91/><ref>, which is a simplified version of a reputation model suggested in 1982 by Kreps, Wilson, Milgrom, and Roberts</ref> the sender and the receiver are firms. The sender is an incumbent firm, and the receiver is an entrant firm. * The sender can be one of two types: ''sane'' or ''crazy''. A sane sender can send one of two messages: ''prey'' and ''accommodate''. A crazy sender can only prey. * The receiver can do one of two actions: ''stay'' or ''exit''. The table gives the payoffs at the right. It is assumed that: * <math>M1>D1>P1</math>, i.e., a sane sender prefers to be a monopoly <math>M1</math>, but if it is not a monopoly, it prefers to accommodate <math>D1</math> than to prey <math>P1</math>. The value of <math>X1</math> is irrelevant since a crazy firm has only one possible action. * <math>D2>0>P2</math>, i.e., the receiver prefers to stay in a market with a sane competitor <math>D2</math> than to exit the market <math>0</math> but prefers to exit than to remain in a market with a crazy competitor <math>P2</math>. * ''A priori'', the sender has probability <math>p</math> to be sane and <math>1-p</math> to be crazy. We now look for perfect Bayesian equilibria. It is convenient to differentiate between separating equilibria and pooling equilibria. * A separating equilibrium, in our case, is one in which the sane sender always accommodates. This separates it from a crazy sender. In the second period, the receiver has complete information: their beliefs are "If accommodated, then the sender is sane, otherwise the sender is crazy". Their best-response is: "If accommodate then stay, if prey then exit". The payoff of the sender when they accommodate is D1+D1, but if they deviate from preying, their payoff changes to P1+M1; therefore, a necessary condition for a separating equilibrium is D1+D1β₯P1+M1 (i.e., the cost of preying overrides the gain from being a monopoly). It is possible to show that this condition is also sufficient. * A pooling equilibrium is one in which the sane sender always preys. In the second period, the receiver has no new information. If the sender preys, then the receiver's beliefs must be equal to the ''apriori'' beliefs, which are the sender is sane with probability ''p'' and crazy with probability 1-''p''. Therefore, the receiver's expected payoff from staying is: [''p'' D2 + (1-''p'') P2]; the receiver stays if-and-only-if this expression is positive. The sender can gain from preying only if the receiver exits. Therefore, a necessary condition for a pooling equilibrium is ''p'' D2 + (1-''p'') P2 β€ 0 (intuitively, the receiver is careful and will not enter the market if there is a risk that the sender is crazy. The sender knows this, and thus hides their true identity by always preying like crazy). But this condition is insufficient: if the receiver exits after accommodating, the sender should accommodate since it is cheaper than Prey. So the receiver must stay after accommodate, and it is necessary that D1+D1<P1+M1 (i.e., the gain from being a monopoly overrides the cost of preying). Finally, we must ensure that staying after accommodate is a best response for the receiver. For this, the receiver's beliefs must be specified after accommodating. This path has probability 0, so Bayes' rule does not apply, and we are free to choose the receiver's beliefs, e.g., "If accommodated, then the sender is sane." Summary: * If preying is costly for a sane sender (D1+D1β₯P1+M1), they will accommodate, and there will be a unique separating PBE: the receiver will stay after accommodating and exit after prey. * If preying is not too costly for a sane sender (D1+D1<P1+M1), and it is harmful to the receiver (''p'' D2 + (1-''p'') P2 β€ 0), the sender will prey. There will be a unique pooling PBE: again, the receiver will stay after accommodate and exit after prey. Here, the sender is willing to lose some value by preying in the first period to build a ''[[reputation]]'' of a predatory firm and convince the receiver to exit. * If preying is neither costly for the sender nor harmful for the receiver, pure strategies will not have a PBE. Mixed strategies will have a unique PBE, as both the sender and the receiver will randomize their actions. ===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=355β374 |doi=10.2307/1882010 |jstor=1882010 }}</ref><ref name=ft91/>{{rp|329β331}}<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. ===Beer-Quiche game=== The Beer-Quiche game of Cho and Kreps<ref>{{Cite journal | last1 = Cho | first1 = In-Koo | last2 = Kreps | first2 = David M. | date = May 1987 | title = Signaling Games and Stable Equilibria | journal = The Quarterly Journal of Economics | volume = 102 | issue = 2 | pages = 179β222 | doi = 10.2307/1885060 | jstor = 1885060| citeseerx = 10.1.1.407.5013 }}</ref> draws on the stereotype of [[Real Men Don't Eat Quiche|quiche eaters being less masculine]]. In this game, an individual B is considering whether to [[duel]] with another individual A. B knows that A is either a ''[[wikt:wimp|wimp]]'' or is ''surly'' but not which. B would prefer a duel if A is a ''wimp'' but not if A is ''surly''. Player A, regardless of type, wants to avoid a duel. Before making the decision, B has the opportunity to see whether A chooses to have [[beer]] or [[quiche]] for breakfast. Both players know that ''wimps'' prefer quiche while ''surlies'' prefer beer. The point of the game is to analyze the choice of breakfast by each kind of A. This has become a standard example of a signaling game. See<ref>{{cite web | url=http://www.econ.ohio-state.edu/jpeck/Econ601/Econ601L15.pdf | title=Perfect Bayesian Equilibrium | author=James Peck | publisher=Ohio State University | access-date=2 September 2016}}</ref>{{rp|14β18}} for more details.
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