Signaling games with pattern recognition

Since the pioneering work of Spence [1], signaling games have been deeply investigated by many researchers around the world [2][3][4][5][6][7][8][9][10]. Generally, there are two players in the model, a Sender (S) and a Receiver (R). The Sender has a certain type t (an element of a finite set T ) which represents his private information. The Sender observes his own type while the Receiver does not know the type of the Sender. There is a strictly positive probability distribution p(t) on T : p(t) represents the prior probability that the Sender is of type t and is the common knowledge.

Based on his knowledge of his own type, the Sender chooses to send a message from a set of possible messages M = {m 1 , • • • , m J }. The Receiver observes the message but not the type of the Sender. Then the Receiver chooses an action from a set of feasible actions A = {a 1 , • • • , a L }. The two players receive payoffs dependent on the Sender’s type, the message chosen by the Sender and the action chosen by the Receiver.

Although the aforementioned model has been widely used by the economic community, people seldom notice clearly that they have made an assumption in the model, i.e., the Receiver is able to discriminate each type of the Sender distinctly. Actually, only by this assumption can the Receiver be able to have a probability distribution on the type set T . However, the justification of this assumption is questionable. Because the type space T is the private information of the Sender, the Sender has no incentive to tell his secret to the Receiver. The Receiver cannot take it for granted that he can discriminate each type of the Sender exactly.

The aim of this paper is to investigate what happens if the aforementioned assumption is canceled. We claim that what the Receiver can do is to recognize the “pattern” of the Sender (The phrase “pattern recognition” comes from computer science). A pattern consists of one type or some types, and one type just belongs to a pattern. The rest of the paper is organized as follows: Section 2 discusses the model of signaling games with pattern recognition. In Section 3, an example is given in detail.

Without loss of generality, consider two players in a signaling game with pattern recognition, i = 1, 2. Player 1 is the Sender who has private information (i.e., the type). Player 2 is the Receiver. Different from classical signaling games (where the Receiver is assumed to know the type space of the sender and discriminate each type exactly), here we relax this assumption and let the Receiver recognize the pattern of the sender (A message can be sent from several patterns. A pattern can send several messages). The pattern space and the belief distribution hold by the Receiver are common knowledge. The dynamic timing sequence of a signaling game with pattern recognition is as follows:

Step 1: Nature (N ) selects the type t of player 1, t ∈ T = {t 1 , • • • , t K }, K ≥ 2. Player 1 knows t, but player 2 doesn’t know it. Furthermore, player 2 cannot discriminate each type in T exactly. The pattern space hold by player 2 is

The belief distribution hold by player 2 is defined on the pattern space

of some types, and one type belongs to a pattern.

Step 2: After observing the type t, player 1 selects a message m from his message space

Step 3: After receiving the message m from player 1, player 2 makes an induction p(t ′ |m) using Bayesian rules, and selects an action a from his feasible action space

Definition 1: The payoffs of player 1 and player 2 are both related to the type t of player 1, the message m and the action a of player 2, i.e., u 1 (t, m, a) and u 2 (t, m, a).

Note: 1) The strategy s 1 of player 1 is a message profile, which is related to the type t of player 1 and the pattern space T ′ hold by player 2, i.e.,

2. The strategy s 2 of player 2 is an action profile, which is related to the message m and the pattern space T ′ hold by player 2, i.e.,

As shown in Fig. 1 Bayesian equilibrium of a signaling game with pattern recognition, we give five basic requirements as follows: R1: The receiver has a belief about which patterns can have sent message m. These beliefs can be described as a probability distribution p(t ′ |m), the probability that the sender is of pattern t ′ if the sender chooses message m. The sum over all patterns t ′ ∈ T ′ of these probabilities has to be 1 conditional on any message m.

The receiver can observe the probability distribution of messages, p(m|t ′ ), when a pattern t ′ is given.

The action that the receiver chooses must maximize the expected utility of the receiver given his beliefs about which patterns can have sent message m. This means that the sum

The action a that maximizes this sum is denoted as a * (m, T ′ ). For simplicity, we still use a * (m) to represent a * (m, T ′ ). R4: For each type t ∈ t ′ that the sender may have, the sender chooses to send the message

that maximizes the sender’s utility u 1 (t, m, a * (m)) given the s

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