A Network Game with Attackers and a Defender

Proceedings of the 9th conference on Theoretical aspects of rationality and knowledge - TARK '03, 2003

In this paper 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 St-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 player's move 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 actions of) a logarithmically small number of other players, The dependency structure of a game G can he expressed by a graph G(G) or by a hypergraph I-I(G). Among other results, we show that if jC has small neighborhood and if I-I(G) has botmdecl hypertree width (or if G(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class _NC~ of highly parallelizable problems.

arXiv: Combinatorics, 2020

We explore the uniqueness of pure strategy Nash equilibria in the Netflix Games of Gerke et al. (arXiv:1905.01693, 2019). Let $G=(V,E)$ be a graph and $\kappa:\ V\to \mathbb{Z}_{\ge 0}$ a function, and call the pair $(G, \kappa)$ a capacitated graph. A spanning subgraph $H$ of $(G, \kappa)$ is called a $DP$-Nash subgraph if $H$ is bipartite with partite sets $X,Y$ called the $D$-set and $P$-set of $H$, respectively, such that no vertex of $P$ is isolated and for every $x\in X,$ $d_H(x)=\min\{d_G(x),\kappa(x)\}.$ We prove that whether $(G,\kappa)$ has a unique $DP$-Nash subgraph can be decided in polynomial time. We also show that when $\kappa(v)=k$ for every $v\in V$, the problem of deciding whether $(G,\kappa)$ has a unique $D$-set is polynomial time solvable for $k=0$ and 1, and co-NP-complete for $k\ge2.$

We prove existence of equilibria in bipartite social games, where players choose both a strategy in a game and a partner with whom to play the game. Such social games generalize the well-known marriage problem where players choose partners but do not take actions subsequent to matching.

2015

Consider a distributed information network with harmful procedures called attackers (e.g., viruses); each attacker uses a probability distribution to choose a node of the network to damage. Opponent to the attackers is the system protector scanning and cleaning from attackers some part of the network (e.g., an edge or a simple path), which it chooses independently using an-other probability distribution. Each attacker wishes to maximize the proba-bility of escaping its cleaning by the system protector; towards a conflicting objective, the system protector aims at maximizing the expected number of cleaned attackers. In [8, 9], we model this network scenario as a non-cooperative strategic game on graphs. We focus on two basic cases for the protector; where it may choose a single edge or a simple path of the network. The two games obtained are called as the Path and the Edge model, respec-tively. For these games, we are interested in the associated Nash equilibria, where no network ent...

Journal of The ACM, 2009

We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991. Our result, building upon the work of Daskalakis et al. [2006a] on the complexity of four-player Nash equilibria, settles a long standing open problem in algorithmic game theory.

2007

We analyze the problem of computing pure Nash equilibria in action graph games (AGGs), which are a compact gametheoretic representation. While the problem is NP-complete in general, for certain classes of AGGs there exist polynomial time algorithms. We propose a dynamic-programming approach that constructs equilibria of the game from equilibria of restricted games played on subgraphs of the action graph. In particular, if the game is symmetric and the action graph has bounded treewidth, our algorithm determines the existence of pure Nash equilibrium in polynomial time.

2006

We prove that the problem of finding a Nash equilibrium in a two-player game is PPAD-complete.

2009

In this paper we show that some decision problems regarding the computation of Nash equilibria are to be considered particularly hard. Most decision problems regarding Nash equilibria have been shown to be NP-complete. While some NP-complete problems can find an alternative to tractability with the tools of Parameterized Complexity Theory, it is also the case that some classes of problems do not seem to have fixed-parameter tractable algorithms. We show that k-Uniform Nash and k-Minimal Nash support are W [2]-hard. Given a game G=(A,B) and a nonnegative integer k, the k-Uniform Nash problem asks whether G has a uniform Nash equilibrium of size k. The k-Minimal Nash support asks whether G has Nash equilibrium such that the support of each player's Nash strategy has size equal to or less than k. First, we show that k-Uniform Nash (with k as the parameter) is W [2]-hard even when we have 2 players, or fewer than 4 different integer values in the matrices. Second, we illustrate that even in zerosum games k-Minimal Nash support is W [2]-hard (a sample Nash equilibrium in a zero-sum 2-player game can be found in polynomial time (von Stengel 2002)). Thus, it must be the case that other more general decision problems are also W [2]-hard. Therefore, the possible parameters for fixed parameter tractability in those decision problems regarding Nash equilibria seem elusive.

2010

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 (ie, 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 ...

Information Processing Letters, 2006