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Completeness in the polynomialtime hierarchy: A compendium
 SIGACT News
"... We present a Garey/Johnsonstyle list of problems known to be complete for the second and higher levels of the polynomialtime Hierarchy (polynomial hierarchy, or PH for short). We also include the bestknown hardness of approximation results. The list will be updated as necessary. Updates The compe ..."
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We present a Garey/Johnsonstyle list of problems known to be complete for the second and higher levels of the polynomialtime Hierarchy (polynomial hierarchy, or PH for short). We also include the bestknown hardness of approximation results. The list will be updated as necessary. Updates The compendium currently lists more than 80 problems. Latest changes include: • added [GT26] SUCCINCT kKING, • added [GT25] SUCCINCT kDIAMETER, • added [GT4] SUCCINCT kRADIUS at third level, • added [GT24] MINIMUM VERTEX COLORING DEFINING SET, • added [GT23] GRAPH SANDWICH PROBLEM FOR Π, • added [L24] MINIMUM 3SAT DEFINING SET,
Symmetries and the Complexity of Pure Nash Equilibrium
, 2006
"... Strategic games may exhibit symmetries in a variety of ways. A common aspect of symmetry, 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 define four classes of symmetric games b ..."
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Cited by 22 (3 self)
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Strategic games may exhibit symmetries in a variety of ways. A common aspect of symmetry, 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 define four classes of symmetric games by considering two additional properties: identical payoff functions for all players and the ability to distinguish oneself from the other players. Based on these varying notions of symmetry, we investigate the computational complexity of pure Nash equilibria. It turns out that in all four classes of games equilibria can be found efficiently when only a constant number of actions is available to each player, a problem that has been shown intractable for other succinct representations of multiplayer games. We further show that identical payoff functions simplify the search for equilibria, while a growing number of actions renders it intractable. Finally, we show that our results extend to wider classes of threshold symmetric games where players are unable to determine the exact number of players playing a certain action.
The Computational Complexity of Nash Equilibria in Concisely Represented Games
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 52
, 2005
"... Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blo ..."
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Cited by 21 (1 self)
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Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payoffs are computed by a given boolean circuit, and graph games, where each agent’s payoff is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes.
On the Complexity of PureStrategy Nash Equilibria in Congestion and LocalEffect Games
 In Proc. of the 2nd Int. Workshop on Internet and Network Economics (WINE
, 2006
"... doi 10.1287/moor.1080.0322 ..."
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Dependencies between players in Boolean games
 In Proc. ECSQARU ’07, volume 4724 of LNCS
, 2007
"... Boolean games are a logical setting for representing static games in a succinct way, taking advantage of the expressive power and succinctness of propositional logic. A Boolean game consists of a set of players, each of them controlling a set of propositional variables and having a specific goal ex ..."
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Boolean games are a logical setting for representing static games in a succinct way, taking advantage of the expressive power and succinctness of propositional logic. A Boolean game consists of a set of players, each of them controlling a set of propositional variables and having a specific goal expressed by a propositional formula, or more generally a specification of the player’s preference relation in some logical language for compact preference representation, such as prioritized goals. There is a lot of graphical structure hidden in a Boolean game: the satisfaction of each player’s goal depends on players whose actions have an influence on her goals. Exploiting this dependency structure facilitates the computation of pure Nash equilibria, by partly decomposing a game into several subgames that are only loosely related. Key words: Game theory, compact preference representation, problem decomposition 1
Hypertree Decompositions: Structure, Algorithms, and Applications
 In Proc. of WG’05
, 2005
"... Abstract We review the concepts of hypertree decomposition and hypertree width from a graph theoretical perspective and report on a number of recent results related to these concepts. We also show – as a new result – that computing hypertree decompositions is fixedparameter intractable. 1 Hypertre ..."
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Abstract We review the concepts of hypertree decomposition and hypertree width from a graph theoretical perspective and report on a number of recent results related to these concepts. We also show – as a new result – that computing hypertree decompositions is fixedparameter intractable. 1 Hypertree Decompositions: Definition and Basics This paper reports about the recently introduced concept of hypertree decomposition and the associated notion of hypertree width. The latter is a cyclicity measure for hypergraphs, and constitutes a hypergraph invariant as it is preserved under hypergraph isomorphisms. Many interesting NPhard problems are polynomially solvable for classes of instances are associated with hypergraphs of bounded width. This is also true for other hypergraph invariants such as treewidth, cutsetwidth, and so on. However, the advantage of hypertree width with respect to other known hypergraph invariants is that it is more general and covers larger classes of instances of bounded width. The main concepts of hypertree decomposition and hypertree width are introduced in the present section. A normal form for hypertree decompositions is described in Section 2. Section 3 describes the Robbers and Marshals game which caracterizes hypertreewidth. In Section 4 we use this game to explain why the problem of checking whether the hypertree width of a hypergraph is k is feasible in polynopmial time for each constant k. However, in Section 5 we show that this
A generalized strategy eliminability criterion and computational methods for applying it
 In Proceedings of the National Conference on Artificial Intelligence (AAAI
, 2005
"... We define a generalized strategy eliminability criterion for bimatrix games that considers whether a given strategy is eliminable relative to given dominator & eliminee subsets of the players ’ strategies. We show that this definition spans a spectrum of eliminability criteria from strict domina ..."
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We define a generalized strategy eliminability criterion for bimatrix games that considers whether a given strategy is eliminable relative to given dominator & eliminee subsets of the players ’ strategies. We show that this definition spans a spectrum of eliminability criteria from strict dominance (when the sets are as small as possible) to Nash equilibrium (when the sets are as large as possible). We show that checking whether a strategy is eliminable according to this criterion is coNPcomplete (both when all the sets are as large as possible and when the dominator sets each have size 1). We then give an alternative definition of the eliminability criterion and show that it is equivalent using the Minimax Theorem. We show how this alternative definition can be translated into a mixed integer program of polynomial size with a number of (binary) integer variables equal to the sum of the sizes of the eliminee sets, implying that checking whether a strategy is eliminable according to the criterion can be done in polynomial time, given that the eliminee sets are small. Finally, we study using the criterion for iterated elimination of strategies. Categories and Subject Descriptors
Zip60: Further explorations in the evolutionary design of online auction market mechanisms
, 2005
"... The “ZIP ” adaptive automated trading algorithm has been demonstrated to outperform human traders in experimental studies of continuous double auction (CDA) markets populated by mixtures of human and “software robot ” traders. Previous papers have shown that values of the eight parameters governing ..."
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The “ZIP ” adaptive automated trading algorithm has been demonstrated to outperform human traders in experimental studies of continuous double auction (CDA) markets populated by mixtures of human and “software robot ” traders. Previous papers have shown that values of the eight parameters governing behavior of ZIP traders can be automatically optimized using a genetic algorithm (GA), and that markets populated by GAoptimized traders perform better than those populated by ZIP traders with manuallyset parameter values. This paper introduces a more sophisticated version of the ZIP algorithm, called “ZIP60”, which requires the values of 60 parameters to be set correctly. ZIP60 is shown here to produce significantly better results in comparison to the original ZIP algorithm (called “ZIP8 ” hereafter) when a GA is used to search the 60dimensional parameter space. It is also demonstrated here that this works best when the GA itself has control over the dimensionality of the searchspace, allowing evolution to guide the expansion of the searchspace up from 8 parameters to 60 via intermediate steps. Principal component analysis of the best evolved ZIP60 parametersets establishes that no ZIP8 solutions are embedded in the 60dimensional space. Moreover, some of the results and analysis presented here
Compact preference representation for boolean games
 In Proceedings of the 9th Pacific Rim International Conference on Artificial Intelligence (PRICAI
, 2006
"... Abstract. Boolean games, introduced by [15, 14], allow for expressing compactly twoplayers zerosum static games with binary preferences: an agent’s strategy consists of a truth assignment of the propositional variables she controls, and a player’s preferences is expressed by a plain propositional ..."
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Cited by 13 (4 self)
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Abstract. Boolean games, introduced by [15, 14], allow for expressing compactly twoplayers zerosum static games with binary preferences: an agent’s strategy consists of a truth assignment of the propositional variables she controls, and a player’s preferences is expressed by a plain propositional formula. These restrictions (twoplayers, zerosum, binary preferences) strongly limit the expressivity of the framework. While the first two can be easily encompassed by defining the agents ’ preferences as an arbitrary nuple of propositional formulas, relaxing the last one needs Boolean games to be coupled with a propositional language for compact preference representation. In this paper, we consider generalized Boolean games where players ’ preferences are expressed within two of these languages: prioritized goals and propositionalized CPnets. 1