Computational Complexity of Approximate Nash Equilibrium in Large Games

We prove that finding an epsilon-Nash equilibrium in a succinctly representable game with many players is PPAD-hard for constant epsilon. Our proof uses succinct games, i.e. games whose payoff function is represented by a circuit. Our techniques build on a recent query complexity lower bound by Babichenko.Comment: New version includes an addendum about subsequent work on the open problems propose

The complexity of the nucleolus in compact games

This is the author accepted manuscript. The final version is available from ACM via the DOI in this recordThe nucleolus is a well-known solution concept for coalitional games to fairly distribute the total available worth among the players. The nucleolus is known to be NP-hard to compute over compact coalitional games, that is, over games whose functions specifying the worth associated with each coalition are encoded in terms of polynomially computable functions over combinatorial structures. In particular, hardness results have been exhibited over minimum spanning tree games, threshold games, and flow games. However, due to its intricate definition involving reasoning over exponentially many coalitions, a nontrivial upper bound on its complexity was missing in the literature and looked for. This article faces this question and precisely characterizes the complexity of the nucleolus, by exhibiting an upper bound that holds on any class of compact games, and by showing that this bound is tight even on the (structurally simple) class of graph games. The upper bound is established by proposing a variant of the standard linear-programming based algorithm for nucleolus computation and by studying a framework for reasoning about succinctly specified linear programs, which are contributions of interest in their own. The hardness result is based on an elaborate combinatorial reduction, which is conceptually relevant for it provides a "measure" of the computational cost to be paid for guaranteeing voluntary participation to the distribution process. In fact, the pre-nucleolus is known to be efficiently computable over graph games, with this solution concept being defined as the nucleolus but without guaranteeing that each player is granted with it at least the worth she can get alone, that is, without collaborating with the other players. Finally, this article identifies relevant tractable classes of coalitional games, based on the notion of type of a player. Indeed, in most applications where many players are involved, it is often the case that such players do belong in fact to a limited number of classes, which is known in advance and may be exploited for computing the nucleolus in a fast way.Part of E. Malizia’s work was supported by the European Commission through the European Social Fund and by Calabria Regio

A study of one-turn quantum refereed games

This thesis studies one-turn quantum refereed games, which are abstract zero-sum games with two competing computationally unbounded quantum provers and a computationally bounded quantum referee. The provers send quantum states to the referee, who plugs the two states into his quantum circuit, measures the output of the circuit in the standard basis, and declares one of the two players as the winner depending on the outcome of the measurement. The complexity class QRG(1) comprises of those promise problems for which there exists a one-turn quantum refereed game such that one of the players wins with high probability for the yes-instances, and the other player wins with high probability for the no-instances, irrespective of the opponent’s strategy. QRG(1) is a generalization of QMA (or co-QMA), and can informally be viewed as QMA with a no-prover (or co-QMA with a yes-prover). We have given a full characterization of QRG(1), starting with appropriate definitions and known results, and building on to two new results about this class. Previously, the best known upper bound on QRG(1) was PSPACE. We have proved that if one of the provers is completely classical, sending a classical probability distribution instead of a quantum state, the new class, which we name CQRG(1), is contained in Ǝ · PP (non- deterministic polynomial-time operator applied to the class PP). We have also defined another restricted version of QRG(1) where both provers send quantum states, but the referee measures one of the quantum states first, and plugs the classical outcome into the measurement, along with the other prover’s quantum state, into a quantum circuit, before measuring the output of the quantum circuit in the standard basis. The new class, which we name MQRG(1), is contained in P · PP (the probabilistic polynomial time operator applied to PP). Ǝ · PP is contained in P · PP, which is, in turn, contained in PSPACE. Hence, our results give better containments than PSPACE for restricted versions of QRG(1)

The Complexity of angel-daemons and game isomorphism

The analysis of the computational aspects of strategic situations is a basic field in Computer Sciences. Two main topics related to strategic games have been developed. First, introduction and analysis of a class of games (so called angel/daemon games) designed to asses web applications, have been considered. Second, the problem of isomorphism between strategic games has been analysed. Both parts have been separately considered. Angel-Daemon Games A service is a computational method that is made available for general use through a wide area network. The performance of web-services may fluctuate; at times of stress the performance of some services may be degraded (in extreme cases, to the point of failure). In this thesis uncertainty profiles and Angel-Daemon games are used to analyse servicebased behaviours in situations where probabilistic reasoning may not be appropriate. In such a game, an angel player acts on a bounded number of ¿angelic¿ services in a beneficial way while a daemon player acts on a bounded number of ¿daemonic¿ services in a negative way. Examples are used to illustrate how game theory can be used to analyse service-based scenarios in a realistic way that lies between over-optimism and over-pessimism. The resilience of an orchestration to service failure has been analysed - here angels and daemons are used to model services which can fail when placed under stress. The Nash equilibria of a corresponding Angel-Daemon game may be used to assign a ¿robustness¿ value to an orchestration. Finally, the complexity of equilibria problems for Angel-Daemon games has been analysed. It turns out that Angel-Daemon games are, at the best of our knowledge, the first natural example of zero-sum succinct games. The fact that deciding the existence of a pure Nash equilibrium or a dominant strategy for a given player is Sp 2-complete has been proven. Furthermore, computing the value of an Angel-Daemon game is EXP-complete. Thus, matching the already known complexity results of the corresponding problems for the generic families of succinctly represented games with exponential number of actions. Game Isomorphism The question of whether two multi-player strategic games are equivalent and the computational complexity of deciding such a property has been addressed. Three notions of isomorphisms, strong, weak and local have been considered. Each one of these isomorphisms preserves a different structure of the game. Strong isomorphism is defined to preserve the utility functions and Nash equilibria. Weak isomorphism preserves only the player preference relations and thus pure Nash equilibria. Local isomorphism preserves preferences defined only on ¿close¿ neighbourhood of strategy profiles. The problem of the computational complexity of game isomorphism, which depends on the level of succinctness of the description of the input games but it is independent of the isomorphism to consider, has been shown. Utilities in games can be given succinctly by Turing machines, boolean circuits or boolean formulas, or explicitly by tables. Actions can be given also explicitly or succinctly. When the games are given in general form, an explicit description of actions and a succinct description of utilities have been assumed. It is has been established that the game isomorphism problem for general form games is equivalent to the circuit isomorphism when utilities are described by Turing Machines; and to the boolean formula isomorphism problem when utilities are described by formulas. When the game is given in explicit form, it is has been proven that the game isomorphism problem is equivalent to the graph isomorphism problem. Finally, an equivalence classes of small games and their graphical representation have been also examined.Postprint (published version

Generalized asset integrity games

Generalized assets represent a class of multi-scale adaptive state-transition systems with domain-oblivious performance criteria. The governance of such assets must proceed without exact specifications, objectives, or constraints. Decision making must rapidly scale in the presence of uncertainty, complexity, and intelligent adversaries. This thesis formulates an architecture for generalized asset planning. Assets are modelled as dynamical graph structures which admit topological performance indicators, such as dependability, resilience, and efficiency. These metrics are used to construct robust model configurations. A normalized compression distance (NCD) is computed between a given active/live asset model and a reference configuration to produce an integrity score. The utility derived from the asset is monotonically proportional to this integrity score, which represents the proximity to ideal conditions. The present work considers the situation between an asset manager and an intelligent adversary, who act within a stochastic environment to control the integrity state of the asset. A generalized asset integrity game engine (GAIGE) is developed, which implements anytime algorithms to solve a stochastically perturbed two-player zero-sum game. The resulting planning strategies seek to stabilize deviations from minimax trajectories of the integrity score. Results demonstrate the performance and scalability of the GAIGE. This approach represents a first-step towards domain-oblivious architectures for complex asset governance and anytime planning

On the Complexity of Succinct Zero-Sum Games

We study the complexity of solving succinct zero-sum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M(i,j) = C(i,j). We complement the known EXP-hardness of computing the exact value of a succinct zero-sum game by several results on approximating the value. (1) We prove that approximating the value of a succinct zero-sum game to within an additive error is complete for the class promise-$$S^{p}_{2}$$, the “promise” version of $$S^{p}_{2}$$. To the best of our knowledge, it is the first natural problem shown complete for this class. (2) We describe a ZPP NP algorithm for constructing approximately optimal strategies, and hence for approximating the value, of a given succinct zero-sum game. As a corollary, we obtain, in a uniform fashion, several complexity-theoretic results, e.g., a ZPP NP algorithm for learning circuits for SAT (Bshouty et al., JCSS, 1996) and a recent result by Cai (JCSS, 2007) that $$S^{p}_{2} ⊆ ZPP NP. (3) We observe that approximating the value of a succinct zero-sum game to within a multiplicative factor is in PSPACE, and that it cannot be in promise-$$S^{p}_{2}$$ unless the polynomial-time hierarchy collapses. Thus, under a reasonable complexity-theoretic assumption, multiplicative-factor approximation of succinct zero-sum games is strictly harder than additive-error approximation

On the complexity of succinct zero-sum games

We study the complexity of solving succinct zero-sum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M(i, j) = C(i, j). We complement the known EXP-hardness of computing the exact value of a succinct zero-sum game by several results on approximating the value. (1) We prove that approxi-mating the value of a succinct zero-sum game to within an additive factor is complete for the class promise-S p 2, the. To the best of our knowledge, it is “promise ” version of S p 2 the first natural problem shown complete for this class. (2) We describe a ZPP NP algorithm for constructing approximately optimal strategies, and hence for approximating the value, of a given succinct zero-sum game. As a corollary, we obtain, in a uniform fashion, several complexity-theoretic results, e.g., a ZPP NP algorithm for learning circuits for SAT [7] and a recent result by Cai [9] that S p 2 ⊆ ZPPNP. (3) We observe that approximating the value of a suc-cinct zero-sum game to within a multiplicative factor is in PSPACE, and that it cannot be in promise-S p 2 unless the polynomial-time hierarchy collapses. Thus, under a reasonable complexity-theoretic assumption, multiplicative-factor approximation of succinct zero-sum games is strictly harde