Games on graphs with a public signal monitoring

We study pure Nash equilibria in games on graphs with an imperfect monitoring based on a public signal. In such games, deviations and players responsible for those deviations can be hard to detect and track. We propose a generic epistemic game abstraction, which conveniently allows to represent the knowledge of the players about these deviations, and give a characterization of Nash equilibria in terms of winning strategies in the abstraction. We then use the abstraction to develop algorithms for some payoff functions.Comment: 28 page

The price of re-establishing perfect, almost perfect or public monitoring in games with arbitrary monitoring

This paper establishes a connection between the notion of observation (or monitoring) structure in game theory and the one of communication channels in Shannon theory. One of the objectives is to know under which conditions an arbitrary monitoring structure can be transformed into a more pertinent monitoring structure. To this end, a mediator is added to the game. The objective of the mediator is to choose a signalling scheme that allows the players to have perfect, almost perfect or public monitoring and all of this, at a minimum cost in terms of signalling. Graph coloring, source coding, and channel coding are exploited to deal with these issues. A wireless power control game is used to illustrate these notions but the applicability of the provided results and, more importantly, the framework of transforming monitoring structures go much beyond this example.Comment: Proc. of the 4th ACM International Workshop on Game Theory in Communication Networks, 201

A minority game with bounded recall.

This paper studies a repeated minority game with public signals, symmetric bounded recall, and pure strategies. We investigate both public and private equilibria of the game with fixed recall size. We first show how public equilibria in such a repeated game can be represented as colored subgraphs of a de Bruijn graph. Then we prove that the set of public equilibrium payoffs with bounded recall converges to the set of uniform equilibrium payoffs as the size of the recall increases. We also show that private equilibria behave badly: A private equilibrium payoff with bounded recall need not be a uniform equilibrium payoff.folk theorem; de Bruijn sequence; imperfect monitoring; uniform equilibrium; public equilibrium; private equilibrium;

An Approximate Subgame-Perfect Equilibrium Computation Technique for Repeated Games

This paper presents a technique for approximating, up to any precision, the set of subgame-perfect equilibria (SPE) in discounted repeated games. The process starts with a single hypercube approximation of the set of SPE. Then the initial hypercube is gradually partitioned on to a set of smaller adjacent hypercubes, while those hypercubes that cannot contain any point belonging to the set of SPE are simultaneously withdrawn. Whether a given hypercube can contain an equilibrium point is verified by an appropriate mathematical program. Three different formulations of the algorithm for both approximately computing the set of SPE payoffs and extracting players' strategies are then proposed: the first two that do not assume the presence of an external coordination between players, and the third one that assumes a certain level of coordination during game play for convexifying the set of continuation payoffs after any repeated game history. A special attention is paid to the question of extracting players' strategies and their representability in form of finite automata, an important feature for artificial agent systems.Comment: 26 pages, 13 figures, 1 tabl

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.Comment: Springer, Dynamic Games and Applications, 201

Enforcing International Trade Agreements with Imperfect Private Monitoring: Private Trigger Strategies and the Possible Role of the WTO

International trade disputes often involve the WTO as a third party that generates impartial opinions on potential violations when countries receive imperfect and private signals of violations. To identify the role that the WTO plays in enforcing trade agreements, this paper first explores what countries can achieve without the WTO by characterizing optimal private trigger strategies (PTS) under which each country triggers a punishment phase by imposing an explicit tariff based on privately-observed imperfect signals of the other country's concealed trade barriers. It identifies the condition under which countries can restrain the use of concealed barriers based on PTS and establishes that countries will not reduce the cooperative protection level to its minimum attainable level under the optimal PTS. This paper then considers third-party trigger strategies (TTS) under which the WTO allows each country to initiate a punishment phase based on the WTO's judgment (i.e., its signals) about potential violations. The WTO thus changes the nature of punishment-triggering signals from private into public, enabling countries to use punishment phases of any length under TTS, which in turn facilitates a better cooperative equilibrium. The optimal TTS will involve an asymmetric and minimum punishment if the probability of a punishment phase being triggered is lower than a critical level, but it will entail punishments involving a permanent Nash tariff war if the probability of a punishment phase is higher than a certain level. A numerical comparison of the optimal TTS and optimal PTS indicates that the contribution of the WTO is likely to be significant when the signals of potential violations are relatively accurate, as this enables countries to use a more efficient punishment, such as an asymmetric and minimum punishment.Concealed Trade Barriers, Imperfect Private Monitoring, International Trade Agreements, Repeated Game, Trade Disputes, Trigger Strategies, WTO

Belief-Based Equilibria in the Repeated Prisoners' Dilemma with Private Monitoring

We analyze the infinitely repeated prisoners' dilemma with imperfect private monitoring and discounting. The main contribution of this paper is to construct ``belief-based'' strategies, where a player's continuation strategy is a function only of his beliefs. This simplifies the analysis considerably, and allows us to explicitly construct sequential equilibria for such games, thus enabling us to invoke the one-step deviation principle of dynamic programming. By doing so, we prove that one can approximate the efficient payoff in any prisoners' dilemma game provided that the monitoring is sufficiently accurate. Furthermore, for a class of prisoners' dilemma games, one can approximate every individually rational feasible payoff. These results require that monitoring be sufficiently accurate, but only require a uniform lower bound on the discount rate.

Enforcing International Trade Agreements with Imperfect Private Monitoring: Private Trigger Strategies and a Possible Role for the WTO

International trade disputes often involve the WTO as a third party that generates impartial opinions of potential violations when countries receive imperfect and private signals of violations. To identify the role that the WTO plays in enforcing trade agreements, this paper first characterizes what countries can achieve alone in a repeated bilateral trade relationship in which they can secretly raise their protection levels through concealed trade barriers. In particular, countries adopt gprivate trigger strategies (PTS)h under which each country triggers a punishment phase by imposing an explicit tariff based on its privately observed imperfect signals of such barriers. This paper identifies the condition under which countries can restrain the use of concealed barriers based on simple PTS, where each country imposes its static optimal tariff in all periods under any punishment phase: The sensitivity of private signals rises in response to an increase in concealed protection. Any equilibrium payoff under almost strongly symmetric PTS will be identical to the one under simple PTS, as long as the initial punishment is triggered by a static optimal tariff, justifying the paper's focus on simple PTS. With countries maximizing their expected payoffs under the optimal PTS, they will not push down the cooperative protection level to its minimum attainable level, thus not setting it to the free trade level even when it is attainable. To analyze a possible role of the WTO, this paper considers gthird-party trigger strategies (TTS)h under which the WTO allows each country to initiate a punishment phase based on the WTO's judgment (signals) about potential violations. The WTO changes the nature of punishment-triggering signals from private into public, enabling countries to use punishment phases of any length under TTS, which in turn facilitates a better cooperative equilibrium. The optimal TTS will involve an asymmetric and minimum punishment if the probability of a punishment phase being triggered is low enough, but it will entail punishments involving a permanent Nash tariff war if the probability of a punishment being triggered is high enough. A numerical comparison of the optimal TTS and optimal PTS indicates that the contribution of the WTO is likely to be significant when the signals of potential violations are relatively accurate. The WTO enables countries to adopt a more efficient punishment, such as the asymmetric and minimum punishment, which in turn enables countries to be less tolerant of potential violations and attain a higher level of cooperation as a result.