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21
Approximate Equilibria and Ball Fusion
 Theory of Computing Systems
, 2002
"... We consider sel sh routing over a network consisting of m parallel links through which n sel sh users route their tra c trying to minimize their own expected latency. Westudy the class of mixed strategies in which the expected latency through each link is at most a constant multiple of the optimum m ..."
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Cited by 62 (26 self)
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We consider sel sh routing over a network consisting of m parallel links through which n sel sh users route their tra c trying to minimize their own expected latency. Westudy the class of mixed strategies in which the expected latency through each link is at most a constant multiple of the optimum maximum latency had global regulation been available. For the case of uniform links it is known that all Nash equilibria belong to this class of strategies. We areinterested in bounding the coordination ratio (or price of anarchy) of these strategies de ned as the worstcase ratio of the maximum (over all links) expected latency over the optimum maximum latency. The load balancing aspect of the problem immediately implies a lower bound; lnm ln lnm of the coordination ratio. We give a tight (uptoamultiplicative constant) upper bound. To show the upper bound, we analyze a variant ofthe classical balls and bins problem, in which balls with arbitrary weights are placed into bins according to arbitrary probability distributions. At the heart of our approach is a new probabilistic tool that we call
Exact penalization and necessary optimality conditions for generalized bilevel programming problems
 SIAM J. Optim
, 1997
"... Abstract. The generalized bilevel programming problem (GBLP) is a bilevel mathematical program where the lower level is a variational inequality. In this paper we prove that if the objective function of a GBLP is uniformly Lipschitz continuous in the lower level decision variable with respect to the ..."
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Cited by 24 (15 self)
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Abstract. The generalized bilevel programming problem (GBLP) is a bilevel mathematical program where the lower level is a variational inequality. In this paper we prove that if the objective function of a GBLP is uniformly Lipschitz continuous in the lower level decision variable with respect to the upper level decision variable, then using certain uniform parametric error bounds as penalty functions gives single level problems equivalent to the GBLP. Several local and global uniform parametric error bounds are presented, and assumptions guaranteeing that they apply are discussed. We then derive Kuhn–Tuckertype necessary optimality conditions by using exact penalty formulations and nonsmooth analysis. Key words. generalized bilevel programming problems, variational inequalities, exact penalty formulations, uniform parametric error bounds, necessary optimality conditions, nonsmooth analysis
Analyzing the vulnerability of critical infrastructure to attack and planning defenses
 Tutorials in Operations Research. INFORMS
, 2005
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Circumventing the Price of Anarchy: Leading Dynamics to Good Behavior
"... Many natural games can have a dramatic difference between the quality of their best and worst Nash equilibria, even in pure strategies. Yet, nearly all work to date on dynamics shows only convergence to some equilibrium, especially within a polynomial number of steps. In this work we study how age ..."
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Cited by 10 (4 self)
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Many natural games can have a dramatic difference between the quality of their best and worst Nash equilibria, even in pure strategies. Yet, nearly all work to date on dynamics shows only convergence to some equilibrium, especially within a polynomial number of steps. In this work we study how agents with some knowledge of the game might be able to quickly (within a polynomial number of steps) find their way to states of quality close to the best equilibrium. We consider two natural learning models in which players choose between greedy behavior and following a proposed good but untrusted strategy and analyze two important classes of games in this context, fair costsharing and consensus games. Both games have extremely high Price of Anarchy and yet we show that behavior in these models can efficiently reach lowcost states.
New necessary optimality conditions for bilevel programs by combining MPEC and the value function approach
 SIAM J. Optim
"... Abstract. The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solving such a problem is to replace the lower level problem by its Karus ..."
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Cited by 7 (3 self)
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Abstract. The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solving such a problem is to replace the lower level problem by its Karush–Kuhn–Tucker (KKT) condition and solve the resulting mathematical programming problem with equilibrium constraints (MPEC). In general the classical approach is not valid for nonconvex bilevel programming problems. The value function approach uses the value function of the lower level problem to define an equivalent single level problem. But the resulting problem requires a strong assumption, such as the partial calmness condition, for the KKT condition to hold. In this paper we combine the classical and the value function approaches to derive new necessary optimality conditions under rather weak conditions. The required conditions are even weaker in the case where the classical approach or the value function approach alone is applicable.
Ordinal Game and Generalized Nash and Stackelberg Solutions
 Journal of Optimization Theory and Applications
, 2000
"... Abstract. The traditional theory of cardinal games deals with problems where the players are able to assess the relative performance of their decisions (or controls) by evaluating a payoff (or utility function) that maps the decision space into the set of real numbers. In that theory, the objective ..."
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Cited by 4 (3 self)
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Abstract. The traditional theory of cardinal games deals with problems where the players are able to assess the relative performance of their decisions (or controls) by evaluating a payoff (or utility function) that maps the decision space into the set of real numbers. In that theory, the objective of each player is to determine a decision that minimizes its payoff function taking into account the decisions of all other players. While that theory has been very useful in modeling simple problems in economics and engineering, it has not been able to address adequately problems in fields such as social and political sciences as well as a large segment of complex problems in economics and engineering. The main reason for this is the difficulty inherent in defining an adequate payoff function for each player in these types of problems. In this paper, we develop a theory of games where, instead of a payoff function, the players are able to rankorder their decision choices against choices by the other players. Such a rankordering could be the result of personal subjective preferences derived from qualitative analysis, as is the case in many social or political science problems. In many complex engineering problems, a heuristic knowledgebased rank ordering of control choices in a finite control space can be viewed as a first step in the process of modeling large complex enterprises for which a mathematical description is usually extremely difficult, if not impossible, to obtain. In order to distinguish between these two types of games, we will refer to traditional payoffbased games as cardinal games and to these new types of rank orderingbased games as ordinal games.
Noncooperative Differential Games. A Tutorial
, 2010
"... These notes provide a brief introduction to the theory of noncooperative differential games. After the Introduction, Section 2 reviews the theory of static games. Different concepts of solution are discussed, including Pareto optima, Nash and Stackelberg equilibria, and the coco (cooperativecompet ..."
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Cited by 1 (0 self)
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These notes provide a brief introduction to the theory of noncooperative differential games. After the Introduction, Section 2 reviews the theory of static games. Different concepts of solution are discussed, including Pareto optima, Nash and Stackelberg equilibria, and the coco (cooperativecompetitive) solutions. Section 3 introduces the basic framework of differential games for two players. Openloop solutions, where the controls implemented by the players depend only on time, are considered in Section 4. It is shown
Interference Relay Channels – Part II: 1
, 904
"... In the first part of this paper we have derived achievable transmission rates for the (singleband) interference relay channel (IRC) when the relay implements either the amplifyandforward, decodeandforward or estimateandforward protocol. Here, we consider wireless networks that can be modeled b ..."
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In the first part of this paper we have derived achievable transmission rates for the (singleband) interference relay channel (IRC) when the relay implements either the amplifyandforward, decodeandforward or estimateandforward protocol. Here, we consider wireless networks that can be modeled by a multiband IRC. We tackle the existence issue of Nash equilibria (NE) in these networks where each information source is assumed to selfishly allocate its power between the available bands in order to maximize its individual transmission rate. Interestingly, it is possible to show that the three power allocation (PA) games (corresponding to the three protocols assumed) under investigation are concave, which guarantees the existence of a pure NE after Rosen [3]. Then, as the relay can also optimize several parameters e.g., its position and transmit power, it is further considered as the leader of a Stackelberg game where the information sources are the followers. Our theoretical analysis is illustrated by simulations giving more insights on the addressed issues.
Power Allocation Games in Interference Relay 1 Channels: Existence Analysis of Nash Equilibria
"... We consider a network composed of two interfering pointtopoint 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 optimiz ..."
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We consider a network composed of two interfering pointtopoint 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 noncooperative game) has some predictable outcomes, namely Nash equilibria. This result is proved for three different types of relaying protocols: decodeandforward, estimateandforward, and amplifyandforward. We provide additional results on the problems of uniqueness, efficiency of the equilibrium, and convergence of a bestresponse based dynamics to the equilibrium. This issues are analyzed in a special case of the amplifyandforward protocol and illustrated by numerical simulations in general. Index Terms