Encompassing anonymity in signaling games

Signaling games are an important class of games in the literature of game theory, finding wide spread application in modelling financial behavior in markets, economic reasoning in job searches, and evolutionary behavior in the emergence of languages. Fundamentally, signaling games consist of multiple senders and a common receiver. Each sender belongs to one of multiple types. Senders transmit their messages to the receiver. The receiver takes an action for each received message, which results in a pair of rewards for the receiver and the corresponding sender respectively. Each sender therefore chooses a message that maximizes his/her reward knowing the optimal response of the receiver. In classical signaling games, the senders reward is a deterministic function of the transmitted message and receiver's action, while the receivers reward is a function of the action and belief about the senders' type. Therefore, in a signaling game, the message received provides information about a sender's type which, in practical commercial contexts, is a violation of the sender's anonymity. In this work, the payoff of a signaling game is adjusted to incorporate the information revealed to the receivers such that this information leakage is minimized from the sender's perspective. The existence of Bayesian-Nash equilibrium is proven in this work for the signaling games even after the incorporation of “type anonymity”. In particular, when the reward is modeled as a weighted sum of the anonymity and direct signaling reward, there exists a threshold on the anonymity weighting coefficient; when the coefficient exceeds the threshold, a pooling equilibrium exists-senders of all types transmit the same message, and when the coefficient is below the threshold, a separating equilibrium exists-senders of each type transmit a distinct message. Furthermore when the cardinality of the message set is 2, the separating equilibrium is shown to be the unique Bayesian Nash equilibrium. The proposed signaling game is suitable to model the problem of routing in the datagram networking where Quality of service (delay or throughput) and the source-destination anonymity are competing requirements.