Editing Signaling game (section)

===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+D1≥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 ≤ 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+D1≥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 ≤ 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.