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**11 - 16**of**16**### Statistical prediction of the outcome of a game

"... Many machine learning problems involve predicting the joint strategy choice of some goaldirected “players ” engaged in a noncooperative game. Conventional game theory predicts that that joint strategy satisfies an “equilibrium concept”. The relative probabilities of the joint strategies satisfying t ..."

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Many machine learning problems involve predicting the joint strategy choice of some goaldirected “players ” engaged in a noncooperative game. Conventional game theory predicts that that joint strategy satisfies an “equilibrium concept”. The relative probabilities of the joint strategies satisfying that concept are not given, and all other joint strategies are given probability zero. As an alternative, I view this prediction problem as one of statistical inference, where the “data ” includes the game specification. This replaces the game theory issue of how to specify a set of equilibrium joint strategies with the issue of how to specify a density function over joint strategies. I explore a Bayesian version of such a Predictive Game Theory (PGT) using the entropic prior and a likelihood that quantifies the rationalities of the players. A popular game theory equilibrium concept parameterized by player rationalities is the Quantal Response Equilibrium concept (QRE). I show that for some games the local peaks of the posterior density over joint strategies approximate the associated QRE’s, and derive the associated correction terms. I also discuss how to estimate parameters of the likelihood from observational data. I end by discussing how PGT can be used to define an equilibrium concept, thereby solving a long-standing problem of conventional game theory.

### Majid Nili-Ahmadabadi, Member, IEEE

"... algorithms usually work on repeated extended, or stochastic games. Generally RL is developed for discrete systems both in terms of states and actions. In this paper, a hierarchical method to learn equilibrium strategy in continuous games is developed. Hierarchy is used to break the continuous domain ..."

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algorithms usually work on repeated extended, or stochastic games. Generally RL is developed for discrete systems both in terms of states and actions. In this paper, a hierarchical method to learn equilibrium strategy in continuous games is developed. Hierarchy is used to break the continuous domain of strategies into discrete sets of hierarchical strategies. The algorithm is proved to converge to Nash-Equilibrium in a specific class of games with dominant strategies. Then, it is applied to some other games and the convergence in shown. This approach is common in RL algorithms that they are applied to problem where no proof of convergence exits. I.

### Algorithms, Theory

"... Many applications in multiagent learning are essentially convex optimization problems in which agents have only limited communication and partial information about the function being minimized (examples of such applications include, among others, coordinated source localization, distributed adaptive ..."

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Many applications in multiagent learning are essentially convex optimization problems in which agents have only limited communication and partial information about the function being minimized (examples of such applications include, among others, coordinated source localization, distributed adaptive filtering, control, and coordination). Given this observation, we propose a new non-hierarchical decentralized algorithm for the asymptotic minimization of possibly time-varying convex functions. In our method each agent has knowledge of a time-varying local cost function, and the objective is to minimize asymptotically a global cost function defined by the sum of the local functions. At each iteration of our algorithm, agents improve their estimates of a minimizer of the global function by applying a particular version of the adaptive projected subgradient method to their local functions. Then the agents exchange and mix their improved estimates using a probabilistic model based on recent results in weighted average consensus algorithms. The resulting algorithm is provably optimal and reproduces as particular cases many existing algorithms (such as consensus algorithms and recent methods based on the adaptive projected subgradient method). To illustrate one possible application, we show how our algorithm can be applied to coordinated acoustic source localization in sensor networks.

### Game Theory and Distributed Control

, 2012

"... Game theory has been employed traditionally as a modeling tool for describing and influencing behavior in societal systems. Recently, game theory has emerged as a valuable tool for controlling or prescribing behavior in distributed engineered systems. The rationale for this new perspective stems fro ..."

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Game theory has been employed traditionally as a modeling tool for describing and influencing behavior in societal systems. Recently, game theory has emerged as a valuable tool for controlling or prescribing behavior in distributed engineered systems. The rationale for this new perspective stems from the parallels between the underlying decision making architectures in both societal systems and distributed engineered systems. In particular, both settings involve an interconnection of decision making elements whose collective behavior depends on a compilation of local decisions that are based on partial information about each other and the state of the world. Accordingly, there is extensive work in game theory that is relevant to the engineering agenda. Similarities notwithstanding, there remain important differences between the constraints and objectives in societal and engineered systems that require looking at game theoretic methods from a new perspective. This chapter provides an overview of selected recent developments of game theoretic methods in this role as a framework for distributed control in engineered systems.