Imperfect-Recall Abstractions with Bounds in Games

Imperfect-recall abstraction has emerged as the leading paradigm for practical large-scale equilibrium computation in incomplete-information games. However, imperfect-recall abstractions are poorly understood, and only weak algorithm-specific guarantees on solution quality are known. In this paper, we show the first general, algorithm-agnostic, solution quality guarantees for Nash equilibria and approximate self-trembling equilibria computed in imperfect-recall abstractions, when implemented in the original (perfect-recall) game. Our results are for a class of games that generalizes the only previously known class of imperfect-recall abstractions where any results had been obtained. Further, our analysis is tighter in two ways, each of which can lead to an exponential reduction in the solution quality error bound. We then show that for extensive-form games that satisfy certain properties, the problem of computing a bound-minimizing abstraction for a single level of the game reduces to a clustering problem, where the increase in our bound is the distance function. This reduction leads to the first imperfect-recall abstraction algorithm with solution quality bounds. We proceed to show a divide in the class of abstraction problems. If payoffs are at the same scale at all information sets considered for abstraction, the input forms a metric space. Conversely, if this condition is not satisfied, we show that the input does not form a metric space. Finally, we use these results to experimentally investigate the quality of our bound for single-level abstraction

Evolutionary dynamics in heterogeneous populations: a general framework for an arbitrary type distribution

A general framework of evolutionary dynamics under heterogeneous populations is presented. The framework allows continuously many types of heterogeneous agents, heterogeneity both in payoff functions and in revision protocols and the entire joint distribution of strategies and types to influence the payoffs of agents. We clarify regularity conditions for the unique existence of a solution trajectory and for the existence of equilibrium. We confirm that equilibrium stationarity in general and equilibrium stability in potential games are extended from the homogeneous setting to the heterogeneous setting. In particular, a wide class of admissible dynamics share the same set of locally stable equilibria in a potential game through local maximization of the potential

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

Learning from Neighbors about a Changing State

Agents learn about a changing state using private signals and past actions of neighbors in a network. We characterize equilibrium learning and social influence in this setting. We then examine when agents can aggregate information well, responding quickly to recent changes. A key sufficient condition for good aggregation is that each individual's neighbors have sufficiently different types of private information. In contrast, when signals are homogeneous, aggregation is suboptimal on any network. We also examine behavioral versions of the model, and show that achieving good aggregation requires a sophisticated understanding of correlations in neighbors' actions. The model provides a Bayesian foundation for a tractable learning dynamic in networks, closely related to the DeGroot model, and offers new tools for counterfactual and welfare analyses.Comment: minor revision tweaking exposition relative to v5 - which added new Section 3.2.2, new Theorem 2, new Section 7.1, many local revision

Utility indifference pricing and hedging for structured contracts in energy markets

In this paper we study the pricing and hedging of structured products in energy markets, such as swing and virtual gas storage, using the exponential utility indifference pricing approach in a general incomplete multivariate market model driven by finitely many stochastic factors. The buyer of such contracts is allowed to trade in the forward market in order to hedge the risk of his position. We fully characterize the buyer's utility indifference price of a given product in terms of continuous viscosity solutions of suitable nonlinear PDEs. This gives a way to identify reasonable candidates for the optimal exercise strategy for the structured product as well as for the corresponding hedging strategy. Moreover, in a model with two correlated assets, one traded and one nontraded, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Finally, numerical results are provided.Comment: 32 pages, 5 figure

Optimal investment under multiple defaults risk: A BSDE-decomposition approach

We study an optimal investment problem under contagion risk in a financial model subject to multiple jumps and defaults. The global market information is formulated as a progressive enlargement of a default-free Brownian filtration, and the dependence of default times is modeled by a conditional density hypothesis. In this Ito-jump process model, we give a decomposition of the corresponding stochastic control problem into stochastic control problems in the default-free filtration, which are determined in a backward induction. The dynamic programming method leads to a backward recursive system of quadratic backward stochastic differential equations (BSDEs) in Brownian filtration, and our main result proves, under fairly general conditions, the existence and uniqueness of a solution to this system, which characterizes explicitly the value function and optimal strategies to the optimal investment problem. We illustrate our solutions approach with some numerical tests emphasizing the impact of default intensities, loss or gain at defaults and correlation between assets. Beyond the financial problem, our decomposition approach provides a new perspective for solving quadratic BSDEs with a finite number of jumps.Comment: Published in at http://dx.doi.org/10.1214/11-AAP829 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org