Signaling Games

In game theory, a signaling game is a type of a dynamic Bayesian game.Subsection 8.2.2 in Fudenberg Trole 1991, pp. 326–331

The essence of a signaling game is that one player takes action, the signal, to convey information to another player. Sending the signal is more costly if the information is false. A manufacturer, for example, might provide a warranty for its product to signal to consumers that it is unlikely to break down. A traditional example is a worker who acquires a college degree not because it increases their skill but because it conveys their ability to employers.

A simple signaling game would have two players: the sender and the receiver. The sender has one of two types, which might be called "desirable" and "undesirable," with different payoff functions. The receiver knows the probability of each type but not which one this particular sender has. The receiver has just one possible type.

The sender moves first, choosing an action called the "signal" or "message" (though the term "message" is more often used in non-signaling "cheap talk" games where sending messages is costless). The receiver moves second, after observing the signal.

The two players receive payoffs dependent on the sender's type, the message chosen by the sender, and the action chosen by the receiver.

The tension in the game is that the sender wants to persuade the receiver that they have the desirable type, so they try to choose a signal. Whether this succeeds depends on whether the undesirable type would send the same signal and how the receiver interprets the signal.

Perfect Bayesian equilibrium

The equilibrium concept relevant to signaling games is the "perfect Bayesian equilibrium," a refinement of the Bayesian Nash equilibrium.

Nature chooses the sender to have type $t$ with probability $p$. The sender then chooses the probability with which to take signaling action $m$, which can be written as $Prob(m, t)$ for each possible $t.$ The receiver observes the signal $m$ but not $t$, and chooses the probability with which to take response action $a$, which can be written as $Prob(a, m)$ for each possible $m.$ The sender's payoff is $u(a, m, t)$ and the receiver's is $v(a,t).$

A perfect Bayesian equilibrium combines beliefs and strategies for each player. Both players believe that the other will follow the strategies specified in the equilibrium, as in simple Nash equilibrium, unless they observe something with probability zero in the equilibrium. The receiver's beliefs also include a probability distribution $b(t, m)$ representing the probability put on the sender having type $t$ if the receiver observes signal $m$. The receiver's strategy is a choice of $Prob(a, m).$ The sender's strategy is a choice of $Prob(m, t)$. These beliefs and strategies must satisfy certain conditions:

*Sequential rationality: each strategy should maximize a player's expected utility, given their beliefs.
*Consistency: each belief should be updated according to the equilibrium strategies, the observed actions, and Bayes' rule on every path reached in equilibrium with positive probability. On paths of zero probability, known as "off-equilibrium paths," the beliefs must be specified but can be arbitrary.

The kinds of perfect Bayesian equilibria that may arise can be divided into three categories: pooling equilibria, separating equilibria, and semi-separating. A given game may or may not have more than one equilibrium.
* In a ''pooling equilibrium'', senders of different types all choose the same signal. This means that the signal does not give any information to the receiver, so the receiver's beliefs are not updated after seeing the signal.
* In a ''separating equilibrium'', senders of different types always choose different signals. This means the signal always reveals the sender's type, so the receiver's beliefs become deterministic after seeing the signal.
* In a ''semi-separating equilibrium'' (also called ''partial-pooling''), some types of senders choose the same message, and others choose different messages.
If there are more types of senders than messages, the equilibrium can never be a separating equilibrium (but maybe semi-separating).
There are also ''hybrid equilibria'', in which the sender randomizes between pooling and separating.

Examples

Reputation game

In this game, 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:
* $M1>D1>P1$, i.e., a sane sender prefers to be a monopoly $M1$, but if it is not a monopoly, it prefers to accommodate $D1$ than to prey $P1$. The value of $X1$ is irrelevant since a crazy firm has only one possible action.
* $D2>0>P2$, i.e., the receiver prefers to stay in a market with a sane competitor $D2$ than to exit the market $0$ but prefers to exit than to remain in a market with a crazy competitor $P2$.
* ''A priori'', the sender has probability $p$ to be sane and $1-p$ 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+D1reputation