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

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 Repeated Game Formulation of Energy-Efficient Decentralized Power Control

Decentralized multiple access channels where each transmitter wants to selfishly maximize his transmission energy-efficiency are considered. Transmitters are assumed to choose freely their power control policy and interact (through multiuser interference) several times. It is shown that the corresponding conflict of interest can have a predictable outcome, namely a finitely or discounted repeated game equilibrium. Remarkably, it is shown that this equilibrium is Pareto-efficient under reasonable sufficient conditions and the corresponding decentralized power control policies can be implemented under realistic information assumptions: only individual channel state information and a public signal are required to implement the equilibrium strategies. Explicit equilibrium conditions are derived in terms of minimum number of game stages or maximum discount factor. Both analytical and simulation results are provided to compare the performance of the proposed power control policies with those already existing and exploiting the same information assumptions namely, those derived for the one-shot and Stackelberg games.Comment: 25 pages, 5 figures, accepted for publication in IEEE Transaction on Wireless Communicatio

Resilient Source Coding

This paper provides a source coding theorem for multi-dimensional information signals when, at a given instant, the distribution associated with one arbitrary component of the signal to be compressed is not known and a side information is available at the destination. This new framework appears to be both of information-theoretical and game-theoretical interest: it provides a new type of constraints to compress an information source; it is useful for designing certain types of mediators in games and characterize utility regions for games with signals. Regarding the latter aspect, we apply the derived source coding theorem to the prisoner's dilemma and the battle of the sexes

Coordination in State-Dependent Distributed Networks: The Two-Agent Case

This paper addresses a coordination problem between two agents (Agents $1$ and $2$) in the presence of a noisy communication channel which depends on an external system state $\{x_{0,t}\}$. The channel takes as inputs both agents' actions, $\{x_{1,t}\}$ and $\{x_{2,t}\}$ and produces outputs that are observed strictly causally at Agent $2$ but not at Agent $1$. The system state is available either causally or non-causally at Agent $1$ but unknown at Agent $2$. Necessary and sufficient conditions on a joint distribution $\bar{Q}(x_0,x_1,x_2)$ to be implementable asymptotically (i.e, when the number of taken actions grows large) are provided for both causal and non-causal state information at Agent $1$. Since the coordination degree between the agents' actions, $x_{1,t}$ and $x_{2,t}$, and the system state $x_{0,t}$ is measured in terms of an average payoff function, feasible payoffs are fully characterized by implementable joint distributions. In this sense, our results allow us to derive the performance of optimal power control policies on an interference channel and to assess the gain provided by non-causal knowledge of the system state at Agent $1$. The derived proofs readily yield new results also for the problem of state-amplification under a causality constraint at the decoder.Comment: Published in 2015 IEEE International Symposium on Information Theor

Combining coded signals with arbitrary modulations in orthogonal relay channels

We consider a relay channel for which the following assumptions are made. (1) The source-destination and relay-destination channels are orthogonal (frequency division relay channel). (2) The relay implements the decode-and-forward protocol. (3) The source and relay implement the same channel encoder, namely, a onvolutional encoder. (4) They can use arbitrary and possibly different modulations. In this framework, we derive the best combiner in the sense of the maximum likelihood (ML) at the destination and the branch metrics of the trellis associated with its channel decoder for the ML combiner and also for the maximum ratio combiner (MRC), cooperative-MRC (C-MRC), and the minimum mean-square error (MMSE) combiner

Power Allocation Games in Interference Relay Channels: Existence Analysis of Nash Equilibria

International audienceWe consider a network composed of two interfering point-to-point links where the two transmitters can exploit one common relay node to improve their individual transmission rate. Communications are assumed to be multiband, and transmitters are assumed to selfishly allocate their resources to optimize their individual transmission rate. The main objective of this paper is to show that this conflicting situation (modeled by a non-cooperative game) has some stable outcomes, namely, Nash equilibria. This result is proved for three different types of relaying protocols: decode and-forward, estimate-and-forward, and amplify-and-forward. We provide additional results on the problems of uniqueness, efficiency of the equilibrium, and convergence of a best-response-based dynamics to the equilibrium. These issues are analyzed in a special case of the amplify-and-forward protocol and illustrated by simulations in general

Charging Games in Networks of Electrical Vehicles

In this paper, a static non-cooperative game formulation of the problem of distributed charging in electrical vehicle (EV) networks is proposed. This formulation allows one to model the interaction between several EV which are connected to a common residential distribution transformer. Each EV aims at choosing the time at which it starts charging its battery in order to minimize an individual cost which is mainly related to the total power delivered by the transformer, the location of the time interval over which the charging operation is performed, and the charging duration needed for the considered EV to have its battery fully recharged. As individual cost functions are assumed to be memoryless, it is possible to show that the game of interest is always an ordinal potential game. More precisely, both an atomic and nonatomic versions of the charging game are considered. In both cases, equilibrium analysis is conducted. In particular, important issues such as equilibrium uniqueness and efficiency are tackled. Interestingly, both analytical and numerical results show that the efficiency loss due to decentralization (e.g., when cost functions such as distribution network Joule losses or life of residential distribution transformers when no thermal inertia is assumed) induced by charging is small and the corresponding "efficiency", a notion close to the Price of Anarchy, tends to one when the number of EV increases.Comment: 8 pages, 4 figures, keywords: Charging games - electrical vehicle - distribution networks - potential games - Nash equilibrium - price of anarch

Mean Field Energy Games in Wireless Networks

This work tackles the problem of energy-efficient distributed power control in wireless networks with a large number of transmitters. The problem is modeled by a dynamic game. Each transmitter-receiver communication is characterized by a state given by the available energy and/or the individual channel state and whose evolution is governed by certain dynamics. Since equilibrium analysis in such a (stochastic) game is generally difficult and even impossible, the problem is approximated by exploiting the large system assumption. Under an appropriate exchangeability assumption, the corresponding mean field game is well defined and studied in detail for special cases. The main contribution of this work is to show how mean field games can be applied to the problem under investigation and provide illustrative numerical results. Our results indicate that this approach can lead to significant gains in terms of energy-efficiency at the resulting equilibrium.Comment: IEEE Proc. of Asilomar Conf. on Signals, Systems, and Computers, Nov. 2012, Pacific Grove, CA, US

Stochastic Differential Games and Energy-Efficient Power Control

One of the contributions of this work is to formulate the problem of energy-efficient power control in multiple access channels (namely, channels which comprise several transmitters and one receiver) as a stochastic differential game. The players are the transmitters who adapt their power level to the quality of their time-varying link with the receiver, their battery level, and the strategy updates of the others. The proposed model not only allows one to take into account long-term strategic interactions but also long-term energy constraints. A simple sufficient condition for the existence of a Nash equilibrium in this game is provided and shown to be verified in a typical scenario. As the uniqueness and determination of equilibria are difficult issues in general, especially when the number of players goes large, we move to two special cases: the single player case which gives us some useful insights of practical interest and allows one to make connections with the case of large number of players. The latter case is treated with a mean-field game approach for which reasonable sufficient conditions for convergence and uniqueness are provided. Remarkably, this recent approach for large system analysis shows how scalability can be dealt with in large games and only relies on the individual state information assumption.Comment: The final publication is available at http://www.springerlink.com/openurl.asp?genre=article\&id=doi:10.1007/s13235-012-0068-