Transforming Monitoring Structures with Resilient Encoders. Application to Repeated Games

An important feature of a dynamic game is its monitoring structure namely, what the players effectively see from the played actions. We consider games with arbitrary monitoring structures. One of the purposes of this paper is to know to what extent an encoder, who perfectly observes the played actions and sends a complementary public signal to the players, can establish perfect monitoring for all the players. To reach this goal, the main technical problem to be solved at the encoder is to design a source encoder which compresses the action profile in the most concise manner possible. A special feature of this encoder is that the multi-dimensional signal (namely, the action profiles) to be encoded is assumed to comprise a component whose probability distribution is not known to the encoder and the decoder has a side information (the private signals received by the players when the encoder is off). This new framework appears to be both of game-theoretical and information-theoretical interest. In particular, it is useful for designing certain types of encoders that are resilient to single deviations and provide an equilibrium utility region in the proposed setting; it provides a new type of constraints to compress an information source (i.e., a random variable). Regarding the first aspect, we apply the derived result to the repeated prisoner’s dilemma.

💡 Research Summary

The paper tackles a fundamental limitation in dynamic and repeated games: the monitoring structure, i.e., what each player actually observes about the actions taken. In many realistic settings the monitoring is imperfect, noisy, or even deliberately obfuscated, which hampers the ability to sustain cooperative equilibria. To overcome this, the authors introduce a public encoder that perfectly observes the entire action profile at every stage and broadcasts a compressed public signal to all players. The central technical challenge is to design a source‑coding scheme that compresses a multidimensional signal whose probability distribution is partially unknown, while each player already possesses side information in the form of private signals received when the encoder is inactive.

The authors formulate this as a novel joint source‑coding problem that blends elements of Slepian‑Wolf (distributed coding) and Wyner‑Ziv (coding with side information) but adds the twist of an unknown component in the source distribution. They propose a “resilient” encoder that remains effective even if a single player deviates from the prescribed strategy (a single deviation). Resilience is achieved by ensuring that the public signal carries enough redundancy—quantified by a correction term Δi for each player—to allow the reconstruction of the true action profile when combined with the player’s private observation. Consequently, the required transmission rate R satisfies

 R ≥ max_i