We study social cost losses in Facility Location games, where $n$ selfish agents install facilities over a network and connect to them, so as to forward their local demand (expressed by a non-negative weight per agent). Agents using the same facility share fairly its installation cost, but every agent pays individually a (weighted) connection cost to the chosen location. We study the Price of Stability (PoS) of pure Nash equilibria and the Price of Anarchy of strong equilibria (SPoA), that generalize pure equilibria by being resilient to coalitional deviations. A special case of recently studied network design games, Facility Location merits separate study as a classic model with numerous applications and individual characteristics: our analysis for unweighted agents on metric networks reveals constant upper and lower bounds for the PoS, while an $O(\ln n)$ upper bound implied by previous work is tight for non-metric networks. Strong equilibria do not always exist, even for the unweighted metric case. We show that $e$-approximate strong equilibria exist ($e=2.718...$). The SPoA is generally upper bounded by $O(\ln W)$ ($W$ is the sum of agents' weights), which becomes tight $Θ(\ln n)$ for unweighted agents. For the unweighted metric case we prove a constant upper bound. We point out several challenging open questions that arise.
Facility location games have been a topic of major interest in economics, operations research and computer science, starting from the seminal work by Hotelling. In the classical pure location Hotelling game businesses compete for maximizing customer attraction by strategically locating their facilities, assuming no price competition, while customers are attracted to their closest facilities. Surprisingly, very little rigorous work has been presented on multi-unit facility location games, where the classical pure location Hotelling games are extended to contexts where each player is to locate s…
In this paper we present a novel generic mapping between Graphical Games and Markov Random Fields so that pure Nash equilibria in the former can be found by statistical inference on the latter. Thus, the problem of deciding whether a graphical game has a pure Nash equilibrium, a well-known intractable problem, can be attacked by well-established algorithms such as Belief Propagation, Junction Trees, Markov Chain Monte Carlo and Simulated Annealing. Large classes of graphical games become thus tractable, including all classes already known, but also new classes such as the games with O(log n) t…
We introduce Q-Nash, a quantum annealing algorithm for the NP-complete problem of Fnding pure Nash equilibria in graphical games. The algorithm consists of two phases. The first phase determines all combinations of best response strategies for each player using classical computation. The second phase finds pure Nash equilibria using a quantum annealing device by mapping the computed combinations to a quadratic unconstrained binary optimization formulation based on the Set Cover problem. We empirically evaluate Q-Nash on D-Wave's Quantum Annealer 2000Q using different graphical game topologies.…
We study the problem of checking for the existence of constrained pure Nash equilibria in a subclass of polymatrix games defined on weighted directed graphs. The payoff of a player is defined as the sum of nonnegative rational weights on incoming edges from players who picked the same strategy augmented by a fixed integer bonus for picking a given strategy. These games capture the idea of coordination within a local neighbourhood in the absence of globally common strategies. We study the decision problem of checking whether a given set of strategy choices for a subset of the players is consist…
We study the computation of approximate pure Nash equilibria in Shapley value (SV) weighted congestion games, introduced in [19]. This class of games considers weighted congestion games in which Shapley values are used as an alternative (to proportional shares) for distributing the total cost of each resource among its users. We focus on the interesting subclass of such games with polynomial resource cost functions and present an algorithm that computes approximate pure Nash equilibria with a polynomial number of strategy updates. Since computing a single strategy update is hard, we apply samp…
The problem of the existence of Berge equilibria in the sense of Zhukovskii in normal form finite games in pure and in mixed strategies is studied. The example of a three-player game that has Berge equilibrium neither in pure nor in mixed strategies is given.
In recent years, empirical game-theoretic analysis (EGTA) has emerged as a powerful tool for analyzing games in which an exact specification of the utilities is unavailable. Instead, EGTA assumes access to an oracle, i.e., a simulator, which can generate unbiased noisy samples of players' unknown utilities, given a strategy profile. Utilities can thus be empirically estimated by repeatedly querying the simulator. Recently, various progressive sampling (PS) algorithms have been proposed, which aim to produce PAC-style learning guarantees (e.g., approximate Nash equilibria with high probability)…
We study natural improvement dynamics in weighted congestion games with polynomial latencies of maximum degree $d\geq 1$. We focus on two problems regarding the existence and efficiency of approximate pure Nash equilibria, with a reasonable small approximation factor, in these games. By exploiting a simple technique, we firstly show that such a game always admits a $d$-approximate potential function. This implies that every sequence of $d$-approximate improvement moves by the players leads to a $d$-approximate pure Nash equilibrium. As a corollary, we also obtain that, under mild assumptions o…
This paper studies the existence of pure Nash equilibria in resource graph games, which are a general class of strategic games used to succinctly represent the players' private costs. There is a finite set of resources and the strategy set of each player corresponds to a set of subsets of resources. The cost of a resource is an arbitrary function that depends on the load vector of the resources in a specified neighborhood. As our main result, we give complete characterizations of the cost functions guaranteeing the existence of pure Nash equilibria for weighted and unweighted players, respecti…
Lipschitz games, in which there is a limit $λ$ (the Lipschitz value of the game) on how much a player's payoffs may change when some other player deviates, were introduced about 10 years ago by Azrieli and Shmaya. They showed via the probabilistic method that $n$-player Lipschitz games with $m$ strategies per player have pure $ε$-approximate Nash equilibria, for $ε\geqλ\sqrt{8n\log(2mn)}$. Here we provide the first hardness result for the corresponding computational problem, showing that even for a simple class of Lipschitz games (Lipschitz polymatrix games), finding pure $ε$-approximate equil…
In \emph{bandwidth allocation games} (BAGs), the strategy of a player consists of various demands on different resources. The player's utility is at most the sum of these demands, provided they are fully satisfied. Every resource has a limited capacity and if it is exceeded by the total demand, it has to be split between the players. Since these games generally do not have pure Nash equilibria, we consider approximate pure Nash equilibria, in which no player can improve her utility by more than some fixed factor $α$ through unilateral strategy changes. There is a threshold $α_δ$ (where $δ$ is …
We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each players payoff depends on moves of other players. We say that a game has small neighborhood if the utility function for each player depends only on (…
The probabilistic serial (PS) rule is a prominent randomized rule for assigning indivisible goods to agents. Although it is well known for its good fairness and welfare properties, it is not strategyproof. In view of this, we address several fundamental questions regarding equilibria under PS. Firstly, we show that Nash deviations under the PS rule can cycle. Despite the possibilities of cycles, we prove that a pure Nash equilibrium is guaranteed to exist under the PS rule. We then show that verifying whether a given profile is a pure Nash equilibrium is coNP-complete, and computing a pure Nas…
This paper deals with the complexity of the problem of computing a pure Nash equilibrium for discrete preference games and network coordination games beyond $O(\log n)$-treewidth and tree metric spaces. First, we estimate the number of iterations of the best response dynamics for a discrete preference game on a discrete metric space with at least three strategies. Second, we present a sufficient condition that we have a polynomial-time algorithm to find a pure Nash equilibrium for a discrete preference game on a grid graph. Finally, we discuss the complexity of finding a pure Nash equilibrium …
Congestion games constitute an important class of games to model resource allocation by different users. As computing an exact or even an approximate pure Nash equilibrium is in general PLS-complete, Caragiannis et al. (2011) present a polynomial-time algorithm that computes a ($2 + ε$)-approximate pure Nash equilibria for games with linear cost functions and further results for polynomial cost functions. We show that this factor can be improved to $(1.61+ε)$ and further improved results for polynomial cost functions, by a seemingly simple modification to their algorithm by allowing for the co…
We prove that a deterministic n-person shortest path game has a Nash equlibrium in pure and stationary strategies if it is edge-symmetric (that is (u,v) is a move whenever (v,u) is, apart from moves entering terminal vertices) and the length of every move is positive for each player. Both conditions are essential, though it remains an open problem whether there exists a NE-free 2-person non-edge-symmetric game with positive lengths. We provide examples for NE-free 2-person edge-symmetric games that are not positive. We also consider the special case of terminal games (shortest path games in wh…
In this work we study of competitive situations among users of a set of global resources. More precisely we study the effect of cost policies used by these resources in the convergence time to a pure Nash equilibrium. The work is divided in two parts. In the theoretical part we prove lower and upper bounds on the convergence time for various cost policies. We then implement all the models we study and provide some experimental results. These results follows the theoretical with one exception which is the most interesting among the experiments. In the case of coalitional users the theoretical u…
University of Aberdeen（阿伯丁大学）；University of Artois（阿尔图瓦大学）
In this paper we describe an argumentation-based representation of normal form games, and demonstrate how argumentation can be used to compute pure strategy Nash equilibria. Our approach builds on Modgil's Extended Argumentation Frameworks. We demonstrate its correctness, prove several theoretical properties it satisfies, and outline how it can be used to explain why certain strategies are Nash equilibria to a non-expert human user.
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 potential function arguments. Unfortunately, this proof of existence is inefficient and computing equilibria is such games is a {\sf PLS}-hard problem. The situation gets worse when superlinear latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and equilibria may not even exist. Given these obstacles, we consider approximate eq…
Recently, a new model extending the standard replicator equation to a finite set of players connected on an arbitrary graph was developed in evolutionary game dynamics. The players are interpreted as subpopulations of multipopulations dynamical game and represented as vertices of the graph, and an edge constitutes the relation among the subpopulations. At each instant, members of connected vertices of the graph play a 2-player game and collect a payoff that determines if the chosen strategies will vanish or flourish. The model describes the game dynamics of a finite set of players connected by…
We study the problem of finding robust equilibria in multiplayer concurrent games with mean payoff objectives. A $(k,t)$-robust equilibrium is a strategy profile such that no coalition of size $k$ can improve the payoff of one its member by deviating, and no coalition of size $t$ can decrease the payoff of other players. We are interested in pure equilibria, that is, solutions that can be implemented using non-randomized strategies. We suggest a general transformation from multiplayer games to two-player games such that pure equilibria in the first game correspond to winning strategies in the …
This paper concerns computing approximate pure Nash equilibria in weighted congestion games, which has been shown to be PLS-complete. With the help of $\hatΨ$-game and approximate potential functions, we propose two algorithms based on best response dynamics, and prove that they efficiently compute $\fracρ{1-ε}$-approximate pure Nash equilibria for $ρ= d!$ and $ρ=\frac{2\cdot W\cdot(d+1)}{2\cdot W+d+1}\le {d + 1}$, respectively, when the weighted congestion game has polynomial latency functions of degree at most $d \ge 1$ and players' weights are bounded from above by a constant $W \ge 1$. Thi…
Sequential allocation is a simple mechanism for sharing multiple indivisible items. We study strategic behavior in sequential allocation. In particular, we consider Nash dynamics, as well as the computation and Pareto optimality of pure equilibria, and Stackelberg strategies. We first demonstrate that, even for two agents, better responses can cycle. We then present a linear-time algorithm that returns a profile (which we call the "bluff profile") that is in pure Nash equilibrium. Interestingly, the outcome of the bluff profile is the same as that of the truthful profile and the profile is in …
Designing efficient algorithms to compute Nash equilibria poses considerable challenges in Algorithmic Game Theory and Optimization. In this work, we employ integer programming techniques to compute Nash equilibria in Integer Programming Games, a class of simultaneous and non-cooperative games where each player solves a parametrized integer program. We introduce ZERO Regrets, a general and efficient cutting plane algorithm to compute, enumerate, and select Nash equilibria. Our framework leverages the concept of equilibrium inequality, an inequality valid for any Nash equilibrium, and the assoc…
We show that the problem of finding an ε-approximate Nash equilibrium in an anonymous game with seven pure strategies is complete in PPAD, when the approximation parameter ε is exponentially small in the number of players.
Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general {\sf PLS}-complete. We present a surprisingly simple polynomial-time algorithm that computes O(1)-approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes $(2+ε)$-approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and $1/ε$. It also applies to games with polynomial latency functions with constant maximum degree $d$; there, the app…
The problem of finding pure strategy Nash equilibria in multiagent concurrent games with finite-horizon temporal goals has received some recent attention. Earlier work solved this problem through the use of Rabin automata. In this work, we take advantage of the finite-horizon nature of the agents' goals and show that checking for and finding pure strategy Nash equilibria can be done using a combination of safety games and lasso testing in Büchi automata. To separate strategic reasoning from temporal reasoning, we model agents' goals by deterministic finite-word automata (DFAs), since finite-ho…
We study Fisher markets that admit equilibria wherein each good is integrally assigned to some agent. While strong existence and computational guarantees are known for equilibria of Fisher markets with additive valuations, such equilibria, in general, assign goods fractionally to agents. Hence, Fisher markets are not directly applicable in the context of indivisible goods. In this work we show that one can always bypass this hurdle and, up to a bounded change in agents' budgets, obtain markets that admit an integral equilibrium. We refer to such markets as pure markets and show that, for any g…
We study pure-strategy Nash equilibria in multi-player concurrent deterministic games, for a variety of preference relations. We provide a novel construction, called the suspect game, which transforms a multi-player concurrent game into a two-player turn-based game which turns Nash equilibria into winning strategies (for some objective that depends on the preference relations of the players in the original game). We use that transformation to design algorithms for computing Nash equilibria in finite games, which in most cases have optimal worst-case complexity, for large classes of preference …
We investigate the degree of discontinuity of several solution concepts from non-cooperative game theory. While the consideration of Nash equilibria forms the core of our work, also pure and correlated equilibria are dealt with. Formally, we restrict the treatment to two player games, but results and proofs extend to the n-player case. As a side result, the degree of discontinuity of solving systems of linear inequalities is settled.
Competing firms tend to select similar locations for their stores. This phenomenon, called the principle of minimum differentiation, was captured by Hotelling with a landmark model of spatial competition but is still the object of an ongoing scientific debate. Although consistently observed in practice, many more realistic variants of Hotelling's model fail to support minimum differentiation or do not have pure equilibria at all. In particular, it was recently proven for a generalized model which incorporates negative network externalities and which contains Hotelling's model and classical sel…
We consider the computational complexity of pure Nash equilibria in graphical games. It is known that the problem is NP-complete in general, but tractable (i.e., in P) for special classes of graphs such as those with bounded treewidth. It is then natural to ask: is it possible to characterize all tractable classes of graphs for this problem? In this work, we provide such a characterization for the case of bounded in-degree graphs, thereby resolving the gap between existing hardness and tractability results. In particular, we analyze the complexity of PUREGG(C, -), the problem of deciding the e…
We study the computational complexity of finding Stackelberg Equilibria in general-sum games, where the set of pure strategies of the leader and the followers are exponentially large in a natrual representation of the problem. In \emph{zero-sum} games, the notion of a Stackelberg equilibrium coincides with the notion of a \emph{Nash Equilibrium}~\cite{korzhyk2011stackelberg}. Finding these equilibrium concepts in zero-sum games can be efficiently done when the players have polynomially many pure strategies or when (in additional to some structural properties) a best-response oracle is availabl…
We study the efficiency of the proportional allocation mechanism, that is widely used to allocate divisible resources. Each agent submits a bid for each divisible resource and receives a fraction proportional to her bids. We quantify the inefficiency of Nash equilibria by studying the Price of Anarchy (PoA) of the induced game under complete and incomplete information. When agents' valuations are concave, we show that the Bayesian Nash equilibria can be arbitrarily inefficient, in contrast to the well-known 4/3 bound for pure equilibria. Next, we upper bound the PoA over Bayesian equilibria by…
We consider solutions of normal form games that are invariant under strategic equivalence. We consider additional properties that can be expected (or be desired) from a solution of a game, and we observe the following: - Even the weakest notion of individual rationality restricts the set of solutions to be equilibria. This observation holds for all types of solutions: in pure-strategies, in mixed strategies, and in correlated strategies where the corresponding notions of equilibria are pure-Nash, Nash and coarse-correlated. An action profile is (strict) simultaneous maximizer if it simultaneou…
Nearly a decade ago, Azrieli and Shmaya introduced the class of $λ$-Lipschitz games in which every player's payoff function is $λ$-Lipschitz with respect to the actions of the other players. They showed that such games admit $ε$-approximate pure Nash equilibria for certain settings of $ε$ and $λ$. They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop a query-efficient reduction from more general games to Lipschitz games. We use this reduction to show a query lower bound for any randomized algorithm finding $ε$-approximate pure Nash equili…
We study pure Nash equilibria in infinite-duration games on graphs, with partial visibility of actions but communication (based on a graph) among the players. We show that a simple communication mechanism consisting in reporting the deviator when seeing it and propagating this information is sufficient for characterizing Nash equilibria. We propose an epistemic game construction, which conveniently records important information about the knowledge of the players. With this abstraction, we are able to characterize Nash equilibria which follow the simple communication pattern via winning strateg…
In finite games mixed Nash equilibria always exist, but pure equilibria may fail to exist. To assess the relevance of this nonexistence, we consider games where the payoffs are drawn at random. In particular, we focus on games where a large number of players can each choose one of two possible strategies, and the payoffs are i.i.d. with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem, with bounds for the approximation error. Moreover, by using a new link between percolation models and game t…
In this paper we extend a popular non-cooperative network creation game (NCG) to allow for disconnected equilibrium networks. There are n players, each is a vertex in a graph, and a strategy is a subset of players to build edges to. For each edge a player must pay a cost α, and the individual cost for a player represents a trade-off between edge costs and shortest path lengths to all other players. We extend the model to a penalized game (PCG), for which we reduce the penalty counted towards the individual cost for a pair of disconnected players to a finite value β. Our analysis concentrates o…
In this paper we consider strategic cost sharing games with so-called arbitrary sharing based on various combinatorial optimization problems, such as vertex and set cover, facility location, and network design problems. We concentrate on the existence and computational complexity of strong equilibria, in which no coalition can improve the cost of each of its members. Our main result reveals a connection between strong equilibrium in strategic games and the core in traditional coalitional cost sharing games studied in economics. For set cover and facility location games this results in a tight …
A recent body of experimental literature has studied empirical game-theoretical analysis, in which we have partial knowledge of a game, consisting of observations of a subset of the pure-strategy profiles and their associated payoffs to players. The aim is to find an exact or approximate Nash equilibrium of the game, based on these observations. It is usually assumed that the strategy profiles may be chosen in an on-line manner by the algorithm. We study a corresponding computational learning model, and the query complexity of learning equilibria for various classes of games. We give basic res…
We introduce natural strategic games on graphs, which capture the idea of coordination in a local setting. We study the existence of equilibria that are resilient to coalitional deviations of unbounded and bounded size (i.e., strong equilibria and k-equilibria respectively). We show that pure Nash equilibria and 2-equilibria exist, and give an example in which no 3-equilibrium exists. Moreover, we prove that strong equilibria exist for various special cases. We also study the price of anarchy (PoA) and price of stability (PoS) for these solution concepts. We show that the PoS for strong equili…
In this paper we study a game where every player is to choose a vertex (facility) in a given undirected graph. All vertices (customers) are then assigned to closest facilities and a player's payoff is the number of customers assigned to it. We show that deciding the existence of a Nash equilibrium for a given graph is NP-hard which to our knowledge is the first result of this kind for a zero-sum game. We also introduce a new measure, the social cost discrepancy, defined as the ratio of the costs between the worst and the best Nash equilibria. We show that the social cost discrepancy in our gam…
Boolean games are an expressive and natural formalism through which to investigate problems of strategic interaction in multiagent systems. Although they have been widely studied, almost all previous work on Nash equilibria in Boolean games has focused on the restricted setting of pure strategies. This is a shortcoming as finite games are guaranteed to have at least one equilibrium in mixed strategies, but many simple games fail to have pure strategy equilibria at all. We address this by showing that a natural decision problem about mixed equilibria: determining whether a Boolean game has a mi…
Public goods games in undirected networks are generally known to have pure Nash equilibria, which are easy to find. In contrast, we prove that, in directed networks, a broad range of public goods games have intractable equilibrium problems: The existence of pure Nash equilibria is NP-hard to decide, and mixed Nash equilibria are PPAD-hard to find. We define general utility public goods games, and prove a complexity dichotomy result for finding pure equilibria, and a PPAD-completeness proof for mixed Nash equilibria. Even in the divisible goods variant of the problem, where existence is easy to…
In this short note we give an example of a four-person finite positional game with perfect information that has no positions of chance and no Nash equilibria in pure stationary strategies. The corresponding directed graph has only one directed cycle and only five terminal positions. It remains open: (i) if the number $n$ of the players can be reduced from $4$ to $3$, (ii) if the number $p$ of the terminals can be reduced from $5$ to $4$, and most important, (iii) whether it is possible to get a similar example in which the outcome $c$ corresponding to all (possibly, more than one) directed cyc…
Observable games are game situations that reach one of possibly many Nash equilibria. Before an instance of the game starts, an external observer does not know, a priori, what is the exact profile of actions that will occur; thus, he assigns subjective probabilities to players' actions. However, not all probabilistic assignments are coherent with a given game. We study the decision problem of determining if a given set of probabilistic constraints assigned a priori by the observer to a given game is coherent, which we call the Coherence of Probabilistic Constraints on Equilibria, or PCE-Cohere…
We consider normal-form games with $n$ players and two strategies for each player, where the payoffs are i.i.d. random variables with some distribution $F$ and we consider issues related to the pure equilibria in the game as the number of players diverges. It is well-known that, if the distribution $F$ has no atoms, the random number of pure equilibria is asymptotically Poisson$(1)$. In the presence of atoms, it diverges. For each strategy profile, we consider the (random) average payoff of the players, called Average Social Utility (ASU). In particular, we examine the asymptotic behavior of t…
