In a system of interdependent users, the security of an entity is affected not only by that user’s investment in security measures, but also by the positive externality of the security decisions of (some of) the other users. The provision of security in such system is therefore modeled as a public good provision problem, and is referred to as a security game. In this paper, we compare two well-known incentive mechanisms in this context for incentivizing optimal security investments among users, namely the Pivotal and the Externality mechanisms. The taxes in a Pivotal mechanism are designed to ensure users’ voluntary participation, while those in an Externality mechanism are devised to maintain a balanced budget. We first show the more general result that, due to the nonexcludable nature of security, no mechanism can incentivize the socially optimal investment profile, while at the same time ensuring voluntary participation and maintaining a balanced budget for all instances of security games. To further illustrate, we apply the Pivotal and Externality mechanisms to the special case of weighted total effort interdependence models, and identify some of the effects of varying interdependency between users on the budget deficit in the Pivotal mechanism, as well as on the participation incentives in the Externality mech
@cite_21 introduces a network model of public goods, and studies different features of its Nash equilibria. This model is equivalent to a total effort game with linear investment costs and a general interdependence graph. The authors show that these games always have a Nash equilibrium; i.e., one in which users are either specialists exerting full effort (equivalent to main investors in our terminology), or free-riders. They show that such equilibria correspond to maximal independent sets of the graph, and that specialized equilibira may lead to higher welfare compared to other (distributed) Nash equilibria. Similarly, @cite_22 studies the Nash equilibrium of a linear quadratic interdependence model, and relates the equilibrium effort levels to the nodes' Bonacich centrality in a suitably defined matrix of local complementarities. The work in @cite_11 generalizes these results by studying existence, uniqueness, and closed form of the Nash equilibrium in a broader class of games for which best-responses are linear in other players' actions. All the aforementioned work focuses on the Nash equilibrium in public good provision environments.
@cite_21 @cite_22 @cite_11
2170451139 2166193992 2059431191
Much of Game Theory, including the Nash equilibrium concept, is based on the assumption that players are expectation maximizers. It is known that if players are risk averse, games may no longer have Nash equilibria ([11,6]. We show that 1. Under risk aversion (convex risk valuations), and for almost all games, there are no mixed Nash equilibria, and thus either there is a pure equilibrium or there are no equilibria at all, and, 2. For a variety of important valuations other than expectation, it is NP-complete to determine if games between such players have a Nash equilibrium. We introduce two new natural decision problems, denoted as ? RATIONAL NASH and ? IRRATIONAL NASH, pertinent to the rationality and irrationality, respectively, of Nash equilibria for (finite) strategic games. These problems ask, given a strategic game, whether or not it admits (i) a rational Nash equilibrium where all probabilities are rational numbers, and (ii) an irrational Nash equilibrium where at least one probability is irrational, respectively. We are interested here in the complexities of ? RATIONAL NASH and ? IRRATIONAL NASH. Towards this end, we study two other decision problems, denoted as NASH-EQUIVALENCE and NASH-REDUCTION, pertinent to some mutual properties of the sets of Nash equilibria of two given strategic games with the same number of players. The problem NASH-EQUIVALENCE asks whether or not the two sets of Nash equilibria coincide; we identify a restriction of its complementary problem that witnesses ? RATIONAL NASH. The problem NASH-REDUCTION asks whether or not there is a so called Nash reduction: a suitable map between corresponding strategy sets of players that yields a Nash equilibrium of the former game from a Nash equilibrium of the latter game; we identify a restriction of NASH-REDUCTION that witnesses ? IRRATIONAL NASH. As our main result, we provide two distinct reductions to simultaneously show that (i) NASH-EQUIVALENCE is co- @math -hard and ? RATIONAL NASH is @math -hard, and (ii) NASH-REDUCTION and ? IRRATIONAL NASH are both @math -hard, respectively. The reductions significantly extend techniques previously employed by Conitzer and Sandholm (Proceedings of the 18th Joint Conference on Artificial Intelligence, pp. 765---771, 2003; Games Econ. Behav. 63(2), 621---641, 2008). We consider a finite population simultaneous move game with heterogeneous interaction modes across different pairs of players. We allow for general interaction patterns, but restrict our analysis to games whose pure strategy Nash equilibrium conditions boil down to a set of piece-wise linear conditions, so that an equilibrium is a solution to a linear complementarity problem. We introduce a new class of games for which a suitable linear transformation of the original interaction matrix induces a game with complementarities. We provide general moderation conditions on the interaction matrix such that a game in this class has a unique Nash equilibrium, that we are able to characterize by means of a closed-form expression involving a generalized version of the Katz network measure of node centrality. Finite population non-cooperative games with linear-quadratic utilities, where each player decides how much action she exerts, can be interpreted as a network game with local payoff complementarities, together with a globally uniform payoff substitutability component and an ownconcavity effect. For these games, the Nash equilibrium action of each player is proportional to her Bonacich centrality in the network of local complementarities, thus establishing a bridge with the sociology literature on social networks. This Bonacich-Nash linkage implies that aggregate equilibrium increases with network size and density. We then analyze a policy that consists of targeting the key player, that is, the player who, once removed, leads to the optimal change in aggregate activity. We provide a geometric characterization of the key player identifed with an inter-centrality measure, which takes into account both a player's centrality and her contribution to the centrality of the others. Studying Nash dynamics is an important approach for analyzing the outcome of games with repeated selfish behavior of self-interested agents. Sink equilibria has been introduced by Goemans, Mirrokni, and Vetta for studying social cost on Nash dynamics over pure strategies in games. However, they do not address the complexity of sink equilibria in these games. Recently, Fabrikant and Papadimitriou initiated the study of the complexity of Nash dynamics in two classes of games. In order to completely understand the complexity of Nash dynamics in a variety of games, we study the following three questions for various games: (i) given a state in game, can we verify if this state is in a sink equilibrium or not? (ii) given an instance of a game, can we verify if there exists any sink equilibrium other than pure Nash equilibria? and (iii) given an instance of a game, can we verify if there exists a pure Nash equilibrium (i.e, a sink equilibrium with one state)? In this paper, we almost answer all of the above questions for a variety of classes of games with succinct representation, including anonymous games, player-specific and weighted congestion games, valid-utility games, and two-sided market games. In particular, for most of these problems, we show that (i) it is PSPACE-hard to verify if a given state is in a sink equilibrium, (ii) it is NP-hard to verify if there exists a pure Nash equilibrium in the game or not, (iii) it is PSPACE-hard to verify if there exists any sink equilibrium other than pure Nash equilibria. To solve these problems, we illustrate general techniques that could be used to answer similar questions in other classes of games. Consider an information network with threats called attackers; each attacker uses a probability distribution to choose a node of the network to damage. Opponent to the attackers is a protector entity called defender; the defender scans and cleans from attacks some part of the network (in particular, a link), which it chooses independently using its own probability distribution. Each attacker wishes to maximize the probability of escaping its cleaning by the defender; towards a conflicting objective, the defender aims at maximizing the expected number of attackers it catches. We model this network security scenario as a non-cooperative strategic game on graphs. We are interested in its associated Nash equilibria, where no network entity can unilaterally increase its local objective. We obtain the following results: We obtain an algebraic characterization of (mixed) Nash equilibria. No (non-trivial) instance of the graph-theoretic game has a pure Nash equilibrium. This is an immediate consequence of some covering properties we prove for the supports of the players in all (mixed) Nash equilibria. We coin a natural subclass of mixed Nash equilibria, which we call Matching Nash equilibria, for this graph-theoretic game. Matching Nash equilibria are defined by enriching the necessary covering properties we proved with some additional conditions involving other structural parameters of graphs, such as Independent Sets. We derive a characterization of graphs admitting Matching Nash equilibria. All such graphs have an Expanding Independent Set. The characterization enables a non-deterministic, polynomial time algorithm to compute a Matching Nash equilibrium for any such graph. Bipartite graphs are shown to satisfy the characterization. So, using a polynomial time algorithm to compute a Maximum Matching for a bipartite graph, we obtain, as our main result, a deterministic, polynomial time algorithm to compute a Matching Nash equilibrium for any instance of the game with a bipartite graph. We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by a potential function argument. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. A ρ--approximate pure Nash equilibrium is a state of a (weighted congestion) game from which no player has any incentive to deviate in order to improve her cost by a multiplicative factor higher than ρ. Do such equilibria exist for small values of ρ? And if so, can we compute them efficiently? We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an “approximation” of such games by a new class of potential games that we call Ψ-games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+√5 2 + Oγ for arbitrarily small γ > 0; for latency functions with maximum degree d≥ 2, it is d2d+o(d). The running time is polynomial in the number of bits in the representation of the game and 1 γ. As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a dO(d2)-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is a constant. To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of Ψ-games. The concept of approximating nonpotential games by potential ones is interesting in itself and might have further applications. This paper deals with the complexity of computing Nash and correlated equilibria for a finite game in normal form. We examine the problems of checking the existence of equilibria satisfying a certain condition, such as “Given a game G and a number r, is there a Nash (correlated) equilibrium of G in which all players obtain an expected payoff of at least r?” or “Is there a unique Nash (correlated) equilibrium in G?” etc. We show that such problems are typically “hard” (NP-hard) for Nash equilibria but “easy” (polynomial) for correlated equilibria. We provide a single reduction that demonstrates that in normal-form games: (1) it is -complete to determine whether Nash equilibria with certain natural properties exist (these results are similar to those obtained by Gilboa and Zemel [Gilboa, I., Zemel, E., 1989. Nash and correlated equilibria: Some complexity considerations. Games Econ. Behav. 1, 80-93]), (2) more significantly, the problems of maximizing certain properties of a Nash equilibrium are inapproximable (unless ), and (3) it is -hard to count the Nash equilibria. We also show that determining whether a pure-strategy Bayes-Nash equilibrium exists in a Bayesian game is -complete, and that determining whether a pure-strategy Nash equilibrium exists in a Markov (stochastic) game is -hard even if the game is unobserved (and that this remains -hard if the game has finite length). All of our hardness results hold even if there are only two players and the game is symmetric. The existence of Nash equilibria in noncooperative flow control in a general product-form network shared by K users is investigated. The performance objective of each user is to maximize its average throughput subject to an upper bound on its average time-delay. Previous attempts to study existence of equilibria for this flow control model were not successful, partly because the time-delay constraints couple the strategy spaces of the individual users in a way that does not allow the application of standard equilibrium existence theorems from the game theory literature. To overcome this difficulty, a more general approach to study the existence of Nash equilibria for decentralized control schemes is introduced. This approach is based on directly proving the existence of a fixed point of the best reply correspondence of the underlying game. For the investigated flow control model, the best reply correspondence is shown to be a function, implicitly defined by means of K interdependent linear programs. Employing an appropriate definition for continuity of the set of optimal solutions of parameterized linear programs, it is shown that, under appropriate conditions, the best reply function is continuous. Brouwer's theorem implies, then, that the best reply function has a fixed point. We investigate the existence of optimal tolls for atomic symmetric network congestion games with unsplittable traffic and arbitrary nondecreasing latency functions. We focus on pure Nash equilibria, and consider a natural toll mechanism, which we call cost-balancing tolls. A set of cost-balancing tolls turns every path with positive traffic on its edges into a minimum-cost path. Hence any given configuration is induced as a pure Nash equilibrium of the modified game with the corresponding cost-balancing tolls. We show how to compute in linear time a set of cost-balancing tolls for the optimal solution such that the total amount of tolls paid by any player in any pure Nash equilibrium of the modified game does not exceed the latency on the maximum-latency path in the optimal solution. Our main result is that for congestion games on series-parallel networks with strictly increasing latency functions, the optimal solution is induced as the unique pure Nash equilibrium of the game with the corresponding cost-... In 1951, John F. Nash proved that every game has a Nash equilibrium [Ann. of Math. (2), 54 (1951), pp. 286-295]. His proof is nonconstructive, relying on Brouwer's fixed point theorem, thus leaving open the questions, Is there a polynomial-time algorithm for computing Nash equilibria? And is this reliance on Brouwer inherent? Many algorithms have since been proposed for finding Nash equilibria, but none known to run in polynomial time. In 1991 the complexity class PPAD (polynomial parity arguments on directed graphs), for which Brouwer's problem is complete, was introduced [C. Papadimitriou, J. Comput. System Sci., 48 (1994), pp. 489-532], motivated largely by the classification problem for Nash equilibria; but whether the Nash problem is complete for this class remained open. In this paper we resolve these questions: We show that finding a Nash equilibrium in three-player games is indeed PPAD-complete; and we do so by a reduction from Brouwer's problem, thus establishing that the two problems are computationally equivalent. Our reduction simulates a (stylized) Brouwer function by a graphical game [M. Kearns, M. Littman, and S. Singh, Graphical model for game theory, in 17th Conference in Uncertainty in Artificial Intelligence (UAI), 2001], relying on “gadgets,” graphical games performing various arithmetic and logical operations. We then show how to simulate this graphical game by a three-player game, where each of the three players is essentially a color class in a coloring of the underlying graph. Subsequent work [X. Chen and X. Deng, Setting the complexity of 2-player Nash-equilibrium, in 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006] established, by improving our construction, that even two-player games are PPAD-complete; here we show that this result follows easily from our proof. One reason for wanting to compute an (approximate) Nash equilibrium of a game is to predict how players will play. However, if the game has multiple equilibria that are far apart, or e-equilibria that are far in variation distance from the true Nash equilibrium strategies, then this prediction may not be possible even in principle. Motivated by this consideration, in this paper we define the notion of games that are approximation stable, meaning that all e-approximate equilibria are contained inside a small ball of radius Δ around a true equilibrium, and investigate a number of their properties. Many natural small games such as matching pennies and rock-paper-scissors are indeed approximation stable. We show furthermore there exist 2-player n-by-n approximationstable games in which the Nash equilibrium and all approximate equilibria have support Ω(log n). On the other hand, we show all (e,Δ) approximation-stable games must have an e-equilibrium of support O(Δ2-o(1) e2 log n), yielding an immediate nO(Δ2-o(1) e2 log n) -time algorithm, improving over the bound of [11] for games satisfying this condition. We in addition give a polynomial-time algorithm for the case that Δ and e are sufficiently close together. We also consider an inverse property, namely that all non-approximate equilibria are far from some true equilibrium, and give an efficient algorithm for games satisfying that condition. We consider strategic games in which each player seeks a mixed strategy to minimize her cost evaluated by a concave valuationV (mapping probability distributions to reals); such valuations are used to model risk. In contrast to games with expectation-optimizer players where mixed equilibria always exist (Nash 1950; Nash Ann. Math. 54, 286---295, 1951), a mixed equilibrium for such games, called a V-equilibrium, may fail to exist, even though pure equilibria (if any) transfer over. What is the exact impact of such valuations on the existence, structure and complexity of mixed equilibria? We address this fundamental question in the context of expectation plus variance, a particular concave valuation denoted as RA, which stands for risk-averse; so, variance enters as a measure of risk and it is used as an additive adjustment to expectation. We obtain the following results about RA-equilibria: A collection of general structural properties of RA-equilibria connecting to (i)E-equilibria and Var-equilibria, which correspond to the expectation and variance valuations E and Var, respectively, and to (ii) other weaker or incomparable properties such as Weak Equilibrium and Strong Equilibrium. Some of these structural properties imply quantitative constraints on the existence of mixed RA-equilibria.A second collection of (i) existence, (ii) equivalence and separation (with respect to E-equilibria), and (iii) characterization results for RA-equilibria in the new class of player-specific scheduling games. We provide suitable examples with a mixed RA-equilibrium that is not an E-equilibrium and vice versa.A purification technique to transform a player-specific scheduling game on two identical links into a player-specific scheduling game on two links so that all non-pureRA-equilibria are eliminated while no new pure equilibria are created; so, a particular player-specific scheduling game on two identical links with no pure equilibrium yields a player-specific scheduling game with noRA-equilibrium (whether mixed or pure). As a by-product, the first ?????-completeness result for the computation of RA-equilibria follows. Strong Nash equilibria and Pareto-optimal Nash equilibria are natural and important strengthenings of the Nash equilibrium concept. We study these stronger notions of equilibrium in congestion games, focusing on the relationships between the price of anarchy for these equilibria and that for standard Nash equilibria (which is well understood). For symmetric congestion games with polynomial or exponential latency functions, we show that the price of anarchy for strong and Pareto-optimal equilibria is much smaller than the standard price of anarchy. On the other hand, for asymmetric congestion games with polynomial latencies the strong and Pareto prices of anarchy are essentially as large as the standard price of anarchy; while for asymmetric games with exponential latencies the Pareto and standard prices of anarchy are the same but the strong price of anarchy is substantially smaller. Finally, in the special case of linear latencies, we show that in asymmetric games the strong and Pareto prices of anarchy coincide exactly with the known value @math for standard Nash, but are strictly smaller for symmetric games. Abstract 6 introduced the class of congestion games and proved that they always possess a Nash equilibrium in pure strategies. Here we obtain conditions for the existence of a strong equilibrium in this class of games, as well as for the equivalence of Nash and strong equilibria. We also give conditions for uniqueness and for Pareto optimality of the Nash equilibrium. Except for a natural monotonicity assumption on the utilities, the conditions are expressed only in terms of the underlying congestion game form. It turns out that avoiding a certain type of bad configuration in the strategy spaces is essential to positive results. Journal of Economic Literature Classification Numbers: C71, C72, D62. In this paper we study a stochastic learning model for 2, 2 normal form games that are played repeatedly. The main emphasis is put on the emergence of cycles. We assume that the players have neither information about the payoff matrix of their opponent nor about their own. At every round each player can only observe his or her action and the payoff he or she receives. We prove that the learning algorithm, which is modeled by an urn scheme proposed by Arthur (1993), leads with positive probability to a cycling of strategy profiles if the game has a mixed Nash equilibrium. In case there are strict Nash equilibria, the learning process converges a.s. to the set of Nash equilibria. Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general PLS -complete. We present a surprisingly simple polynomial-time algorithm that computes @math -approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes @math -approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and @math . It also applies to games with polynomial latency functions with constant maximum degree @math ; there, the approximation guarantee is @math . The algorithm essentially identifies a polynomially long sequence of best-response moves that lead to an approximate equilibrium, the existence of such short sequences is interesting in itself. These are the first positive algorithmic results for approximate equilibria in non-symmetric congestion games. We strengthen them further by proving that, for congestion games that deviate from our mild assumptions, computing @math -approximate equilibria is PLS -complete for any polynomial-time computable @math . We study the query complexity of approximate notions of Nash equilibrium in games with a large number of players n. Our main result states that for n-player binary-action games and for constant e, the query complexity of an e-well-supported Nash equilibrium is exponential in n. As a consequence of this result, we get an exponential lower bound on the rate of convergence of adaptive dynamics to approximate Nash equilibria. We study markets of indivisible items in which price-based (Walrasian) equilibria often do not exist due to the discrete non-convex setting. Instead we consider Nash equilibria of the market viewed as a game, where players bid for items, and where the highest bidder on an item wins it and pays his bid. We first observe that pure Nash-equilibria of this game excatly correspond to price-based equilibiria (and thus need not exist), but that mixed-Nash equilibria always do exist, and we analyze their structure in several simple cases where no price-based equilibrium exists. We also undertake an analysis of the welfare properties of these equilibria showing that while pure equilibria are always perfectly efficient (“first welfare theorem”), mixed equilibria need not be, and we provide upper and lower bounds on their amount of inefficiency.