Two-terminal routing games with unknown active players

We analyze 2-terminal routing games with linear cost functions and with unknown number of active players. We deal with both splittable and unsplittable models. We prove the existence and uniqueness of a symmetric safety-level equilibrium in such games and show that in many cases every player benefits from the common ignorance about the number of players. Furthermore, we prove new theorems on existence and uniqueness of equilibrium in 2-terminal convex routing games with complete information.     

Online calibrated forecasts: Memory efficiency versus universality for learning in games

We provide a simple learning process that enables an agent to forecast a sequence of outcomes. Our forecasting scheme, termed tracking forecast, is based on tracking the past observations while emphasizing recent outcomes. As opposed to other forecasting schemes, we sacrifice universality in favor of a significantly reduced memory requirements. We show that if the sequence of outcomes has certain properties—it has some internal (hidden) state that does not change too rapidly—then the tracking forecast is weakly calibrated so that the forecast appears to be correct most of the time. For binary outcomes, this result holds without any internal state assumptions. We consider learning in a repeated strategic game where each player attempts to compute some forecast of the opponent actions and play a best response to it. We show that if one of the players uses a tracking forecast, while the other player uses a standard learning algorithm (such as exponential regret matching or smooth fictitious play), then the player using the tracking forecast obtains the best response to the actual play of the other players. We further show that if both players use tracking forecast, then under certain conditions on the game matrix, convergence to a Nash equilibrium is possible with positive probability for a larger class of games than the class of games for which smooth fictitious play converges to a Nash equilibrium.     

Partition Equilibrium Always Exists in Resource Selection Games

We consider the existence of Partition Equilibrium in Resource Selection Games. Super-strong equilibrium, where no subset of players has an incentive to change their strategies collectively, does not always exist in such games. We show, however, that partition equilibrium (introduced in (Feldman and Tennenholtz in SAGT, pp. 48–59, 2009) to model coalitions arising in a social context) always exists in general resource selection games, as well as how to compute it efficiently. In a partition equilibrium, the set of players has a fixed partition into coalitions, and the only deviations considered are by coalitions that are sets in this partition. Our algorithm to compute a partition equilibrium in any resource selection game (i.e., load balancing game) settles the open question from (Feldman and Tennenholtz in SAGT, pp. 48–59, 2009) about existence of partition equilibrium in general resource selection games. Moreover, we show how to always find a partition equilibrium which is also a Nash equilibrium. This implies that in resource selection games, we do not need to sacrifice the stability of individual players when forming solutions stable against coalitional deviations. In addition, while super-strong equilibrium may not exist in resource selection games, we show that its existence can be decided efficiently, and how to find one if it exists.     

When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the n-stage repeated version of the game exist if and only if both players have Ω(n) random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the n-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on “random-like” sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators do not exist, then Ω(n) random bits remain necessary for equilibria to exist.     

Symmetries and the complexity of pure Nash equilibrium

Strategic games may exhibit symmetries in a variety of ways. A characteristic feature, enabling the compact representation of games even when the number of players is unbounded, is that players cannot, or need not, distinguish between the other players. We investigate the computational complexity of pure Nash equilibria in four classes of symmetric games obtained by considering two additional properties: identical payoff functions for all players and the ability to distinguish oneself from the other players. In contrast to other types of succinctly representable multi-player games, the pure equilibrium problem is tractable in all four classes when only a constant number of actions is available to each player. Identical payoff functions make the difference between TC⁰-completeness and membership in AC⁰, while a growing number of actions renders the equilibrium problem NP-hard for three of the classes and PLS-hard for the most restricted class for which the existence of a pure equilibrium is guaranteed. Our results also extend to larger classes of threshold symmetric games where players are unable to determine the exact number of players playing a certain action.     

On best response dynamics in weighted congestion games with polynomial delays

We investigate the speed of convergence of best response dynamics to approximately optimal solutions in weighted congestion games with polynomial delay functions. Awerbuch et?al. (Fast convergence to nearly optimal solutions in potential games. ACM Conference on Electronic Commerce, 2008) showed that the convergence time of such dynamics to Nash equilibrium may be exponential in the number of players n even for unweighted players and linear delay functions. Nevertheless, we show that ??(n log log W) (where W is the sum of all the players?? weights) best responses are necessary and sufficient to achieve states that approximate the optimal solution by a constant factor, under the assumption that every O(n) steps each player performs a constant (and non-null) number of best responses. For congestion games in which computing a best response is a polynomial time solvable problem, such a dynamics naturally implies a polynomial time distributed algorithm for the problem of computing the social optimum in congestion games, approximating the optimal solution by a constant factor.     

On the Rate of Convergence of Fictitious Play

Fictitious play is a simple learning algorithm for strategic games that proceeds in rounds. In each round, the players play a best response to a mixed strategy that is given by the empirical frequencies of actions played in previous rounds. There is a close relationship between fictitious play and the Nash equilibria of a game: if the empirical frequencies of fictitious play converge to a strategy profile, this strategy profile is a Nash equilibrium. While fictitious play does not converge in general, it is known to do so for certain restricted classes of games, such as constant-sum games, non-degenerate 2×n games, and potential games. We study the rate of convergence of fictitious play and show that, in all the classes of games mentioned above, fictitious play may require an exponential number of rounds (in the size of the representation of the game) before some equilibrium action is eventually played. In particular, we show the above statement for symmetric constant-sum win-lose-tie games.     

Competitive routing over time

Congestion games are a fundamental and widely studied model for selfish allocation problems like routing and load balancing. An intrinsic property of these games is that players allocate resources simultaneously and instantly. This is particularly unrealistic for many network routing scenarios, which are one of the prominent application scenarios of congestion games. In many networks, load travels along routes over time and allocation of edges happens sequentially. In this paper, we consider two frameworks that enhance network congestion games with a notion of time. We introduce temporal network congestion games that are based on coordination mechanisms — local policies that allow to sequentialize traffic on the edges. In addition, we consider congestion games with time-dependent costs, in which travel times are fixed but quality of service of transmission varies with load over time. We study existence and complexity properties of pure Nash equilibria and best-response strategies in both frameworks for the special case of linear latency functions. In some cases our results can be used to characterize convergence properties of various improvement dynamics, by which the population of players can reach equilibrium in a distributed fashion.     

Graphical Congestion Games

We consider congestion games with linear latency functions in which each player is aware only of a subset of all the other players. This is modeled by means of a social knowledge graph G in which nodes represent players and there is an edge from i to j if i knows j. Under the assumption that the payoff of each player is affected only by the strategies of the adjacent ones, we first give a complete characterization of the games possessing pure Nash equilibria. Namely, if the social graph G is undirected, the game is an exact potential game and thus isomorphic to a classical congestion game. As a consequence, it always converges and possesses Nash equilibria. On the other hand, if G is directed an equilibrium is not guaranteed to exist, but the game is always convergent and an equilibrium can be found in polynomial time if G is acyclic, even if finding the best equilibrium remains an intractable problem.     

On the variational equilibrium as a refinement of the generalized Nash equilibrium

We are concerned with a class of games in which the players’ strategy sets are coupled by a shared constraint. A widely employed solution concept for such generalized Nash games is the generalized Nash equilibrium (GNE). The variational equilibrium (VE) (Facchinei & Kanzow, 2007) is a specific kind of GNE characterized by the solution of the variational inequality formed from the common constraint and the mapping of the gradients of player objectives. Our contribution is a theory that provides sufficient conditions for ensuring that the existence of a GNE implies the existence of a VE; in such an instance, the VE is said to be a refinement of the GNE. For certain games, these conditions are shown to be necessary. This theory rests on a result showing the equality of the Brouwer degree of two suitably defined functions, whose zeros are the GNE and VE, respectively. This theory has a natural extension to the primal–dual space of strategies and Lagrange multipliers corresponding to the shared constraint. Our results unify some known results pertaining to such equilibria and provide mathematical substantiation for ideas that were known to be appealing to economic intuition.     

Ranking games

The outcomes of many strategic situations such as parlor games or competitive economic scenarios are rankings of the participants, with higher ranks generally at least as desirable as lower ranks. Here we define ranking games as a class of n-player normal-form games with a payoff structure reflecting the players' von Neumann-Morgenstern preferences over their individual ranks. We investigate the computational complexity of a variety of common game-theoretic solution concepts in ranking games and deliver hardness results for iterated weak dominance and mixed Nash equilibrium when there are more than two players, and for pure Nash equilibrium when the number of players is unbounded but the game is described succinctly. This dashes hope that multi-player ranking games can be solved efficiently, despite their profound structural restrictions. Based on these findings, we provide matching upper and lower bounds for three comparative ratios, each of which relates two different solution concepts: the price of cautiousness, the mediation value, and the enforcement value.     

Convergence analysis for pure stationary strategies in repeated potential games: Nash,Lyapunov and correlated equilibria

In game theory the interaction among players obligates each player to develop a belief about the possible strategies of the other players, to choose a best-reply given those beliefs, and to look for an adjustment of the best-reply and the beliefs using a learning mechanism until they reach an equilibrium point. Usually, the behavior of an individual cost-function, when such best-reply strategies are applied, turns out to be non-monotonic and concluding that such strategies lead to some equilibrium point is a non-trivial task. Even in repeated games the convergence to a stationary equilibrium is not always guaranteed. The best-reply strategies analyzed in this paper represent the most frequent type of behavior applied in practice in problems of bounded rationality of agents considered within the Artificial Intelligence research area. They are naturally related with the, so-called, fixed-local-optimal actions or, in other words, with one step-ahead optimization algorithms widely used in the modern Intelligent Systems theory.This paper shows that for an ergodic class of finite controllable Markov games the best-reply strategies lead necessarily to a Lyapunov/Nash equilibrium point. One of the most interesting properties of this approach is that an expedient (or absolutely expedient) behavior of an ergodic system (repeated game) can be represented by a Lyapunov-like function non-decreasing in time. We present a method for constructing a Lyapunov-like function: the Lyapunov-like function replaces the recursive mechanism with the elements of the ergodic system that model how players are likely to behave in one-shot games. To show our statement, we first propose a non-converging state-value function that fluctuates (increases and decreases) between states of the Markov game. Then, we prove that it is possible to represent that function in a recursive format using a one-step-ahead fixed-local-optimal strategy. As a result, we prove that a Lyapunov-like function can be built using the previous recursive expression for the Markov game, i.e., the resulting Lyapunov-like function is a monotonic function which can only decrease (or remain the same) over time, whatever the initial distribution of probabilities. As a result, a new concept called Lyapunov games is suggested for a class of repeated games. Lyapunov games allow to conclude during the game whether the applied strategy provides the convergence to an equilibrium point (or not). The time for constructing a Potential (Lyapunov-like) function is exponential. Our algorithm tractably computes the Nash, Lyapunov and the correlated equilibria: a Lyapunov equilibrium is a Nash equilibrium, as well it is also a correlated equilibrium. Validity of the proposed method is successfully demonstrated both theoretically and practically by a simulated experiment related to the Duel game.     

When ignorance helps: Graphical multicast cost sharing games

In non-cooperative games played on highly decentralized networks the assumption that each player knows the strategy adopted by any other player may be too optimistic or even infeasible. In such situations, the set of players of which each player knows the chosen strategy can be modeled by means of a social knowledge graph in which nodes represent players and there is an edge from i to j if i knows j. Following the framework introduced in [7], we study the impact of social knowledge graphs on the fundamental multicast cost sharing game in which all the players want to receive the same communication from a given source in an undirected network. In the classical complete information case, such a game is known to be highly inefficient, since its price of anarchy can be as high as the total number of players ρ. We first show that, under our incomplete information setting, pure Nash equilibria always exist only if the social knowledge graph is directed acyclic (DAG). We then prove that the price of stability of any DAG is at least and provide a DAG lowering the classical price of anarchy to a value between and log²ρ. If specific instances of the game are concerned, that is if the social knowledge graph can be selected as a function of the instance, we show that the price of stability is at least , and that the same bound holds also for the price of anarchy of any social knowledge graph (not only DAGs). Moreover, we provide a nearly matching upper bound by proving that, for any fixed instance, there always exists a DAG yielding a price of anarchy less than 4. Our results open a new window on how the performances of non-cooperative systems may benefit from the lack of total knowledge among players.     

On the Structure of Weakly Acyclic Games

The class of weakly acyclic games, which includes potential games and dominance-solvable games, captures many practical application domains. In a weakly acyclic game, from any starting state, there is a sequence of better-response moves that leads to a pure Nash equilibrium; informally, these are games in which natural distributed dynamics, such as better-response dynamics, cannot enter inescapable oscillations. We establish a novel link between such games and the existence of pure Nash equilibria in subgames. Specifically, we show that the existence of a unique pure Nash equilibrium in every subgame implies the weak acyclicity of a game. In contrast, the possible existence of multiple pure Nash equilibria in every subgame is insufficient for weak acyclicity in general; here, we also systematically identify the special cases (in terms of the number of players and strategies) for which this is sufficient to guarantee weak acyclicity.     

The Impact of Social Ignorance on Weighted Congestion Games

We consider weighted linear congestion games, and investigate how social ignorance, namely lack of information about the presence of some players, affects the inefficiency of pure Nash equilibria (PNE) and the convergence rate of the ε-Nash dynamics. To this end, we adopt the model of graphical linear congestion games with weighted players, where the individual cost and the strategy selection of each player only depends on his neighboring players in the social graph. We show that such games admit a potential function, and thus a PNE. Next, we investigate the Price of Anarchy (PoA) and the Price of Stability (PoS) of graphical linear congestion games with respect to the players’ total actual cost. Our main result is that the impact of social ignorance on the PoA and on the PoS is naturally quantified by the independence number α(G) of the social graph G. In particular, we show that the PoA grows roughly as α(G)(α(G)+2), which is essentially tight as long as α(G) does not exceed half the number of players, and that the PoS lies between α(G) and 2α(G). Moreover, we show that the ε-Nash dynamics reaches an α(G)(α(G)+2)-approximate configuration in polynomial time that does not directly depend on the social graph. For unweighted graphical linear games with symmetric strategies, we show that the ε-Nash dynamics reaches an ε-approximate PNE in polynomial time that exceeds the corresponding time for symmetric linear games by a factor at most as large as the number of players.     

Strategic stability in linear-quadratic differential games with nontransferable payoffs

We address the problem of strategically supported cooperation for linear-quadratic differential games with nontransferable payoffs. As an optimality principle, we study Pareto-optimal solutions. It is assumed that players use a payoff distribution procedure guaranteeing individual rationality of a cooperative solution over the entire game horizon. We prove that under these conditions a Pareto-optimal solution can be strategically supported by an ε-Nash equilibrium. An example is considered.     

Bounded Budget Connection (BBC) games or how to make friends and influence people,on a budget

Motivated by applications in social and peer-to-peer networks, we introduce the Bounded Budget Connection (BBC) game and study its pure Nash equilibria. We have a collection of n   players, each with a budget for purchasing links. Each link has a cost and a length. Each node has a preference weight for each node, and its objective is to purchase outgoing links within its budget to minimize its sum of preference-weighted distances to the nodes. We show that determining if a BBC game has pure Nash equilibria is NP-hard. We study (n,k)$(n,k)$-uniform BBC games, where all link costs, lengths, and preferences are equal and every budget equals k  . We show that pure Nash equilibria exist for all (n,k)$(n,k)$-uniform BBC games and all equilibria are essentially fair. We construct a family of equilibria spanning the spectrum from minimum to maximum social cost. We also analyze best-response walks and alternative node objectives.     

Equilibrium strategies in dynamic games with multi-levels of hierarchy

This paper considers noncooperative equilibria of three-player dynamic games with three levels of hierarchy in decision making. In this context, first a general definition of a hierarchical equilibrium solution is given, which also accounts for nonunique responses of the players who are not at the top of the hierarchy. Then, a general theorem is proven which provides a set of sufficient conditions for a triple of strategies to be in hierarchical equilibrium. When applied to linear-quadratic games, this theorem provides conditions under which there exists a linear one-step memory strategy for the player (say, $\text{J}$1) at the top of the hierarchy, which forces the other two players to act in such a way so as to jointly minimize the cost function of $\text{J}$1. Furthermore, there exists a linear one-step memory strategy for the second-level player (say, $\text{J}$2), which forces the remaining player to jointly minimize the cost function of $\text{J}$2 under the declared equilibrium strategy of $\text{J}$1. A numerical example included in the paper illustrates the results and the convergence property of the equilibrium strategies, as the number of stages in the game becomes arbitrarily large.     

Improved Lower Bounds on the Price of Stability of Undirected Network Design Games

Bounding the price of stability of undirected network design games with fair cost allocation is a challenging open problem in the Algorithmic Game Theory research agenda. Even though the generalization of such games in directed networks is well understood in terms of the price of stability (it is exactly H _n , the n-th harmonic number, for games with n players), far less is known for network design games in undirected networks. The upper bound carries over to this case as well while the best known lower bound is 42/23≈1.826. For more restricted but interesting variants of such games such as broadcast and multicast games, sublogarithmic upper bounds are known while the best known lower bound is 12/7≈1.714. In the current paper, we improve the lower bounds as follows. We break the psychological barrier of 2 by showing that the price of stability of undirected network design games is at least 348/155≈2.245. Our proof uses a recursive construction of a network design game with a simple gadget as the main building block. For broadcast and multicast games, we present new lower bounds of 20/11≈1.818 and 1.862, respectively.