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Register a new userOn the Complexity of Nucleolus Computation for Bipartite b-Matching Games
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Added by
Felix Zhou
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We explore the complexity of nucleolus computation in b-matching games on bipartite graphs. We show that computing the nucleolus of a simple b-matching game is NP-hard even on bipartite graphs of maximum degree 7. We complement this with partial positive results in the special case where b values are bounded by 2. In particular, we describe an efficient algorithm when a constant number of vertices satisfy b(v) = 2 as well as an efficient algorithm for computing the non-simple b-matching nucleolus when b = 2.
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Weighted voting games (WVG) are coalitional games in which an agents contribution to a coalition is given by his it weight, and a coalition wins if its total weight meets or exceeds a given quota. These games model decision-making in political bodies as well as collaboration and surplus division in multiagent domains. The computational complexity of various solution concepts for weighted voting games received a lot of attention in recent years. In particular, Elkind et al.(2007) studied the complexity of stability-related solution concepts in WVGs, namely, of the core, the least core, and the nucleolus. While they have completely characterized the algorithmic complexity of the core and the least core, for the nucleolus they have only provided an NP-hardness result. In this paper, we solve an open problem posed by Elkind et al. by showing that the nucleolus of WVGs, and, more generally, k-vector weighted voting games with fixed k, can be computed in pseudopolynomial time, i.e., there exists an algorithm that correctly computes the nucleolus and runs in time polynomial in the number of players and the maximum weight. In doing so, we propose a general framework for computing the nucleolus, which may be applicable to a wider of class of games.
We investigate the problem of equilibrium computation for large $n$-player games. Large games have a Lipschitz-type property that no single players utility is greatly affected by any other individual players actions. In this paper, we mostly focus on the case where any change of strategy by a player causes other players payoffs to change by at most $frac{1}{n}$. We study algorithms having query access to the games payoff function, aiming to find $epsilon$-Nash equilibria. We seek algorithms that obtain $epsilon$ as small as possible, in time polynomial in $n$. Our main result is a randomised algorithm that achieves $epsilon$ approaching $frac{1}{8}$ for 2-strategy games in a {em completely uncoupled} setting, where each player observes her own payoff to a query, and adjusts her behaviour independently of other players payoffs/actions. $O(log n)$ rounds/queries are required. We also show how to obtain a slight improvement over $frac{1}{8}$, by introducing a small amount of communication between the players. Finally, we give extension of our results to large games with more than two strategies per player, and alternative largeness parameters.
Let $(f,P)$ be an incentive compatible mechanism where $f$ is the social choice function and $P$ is the payment function. In many important settings, $f$ uniquely determines $P$ (up to a constant) and therefore a common approach is to focus on the design of $f$ and neglect the role of the payment function. Fadel and Segal [JET, 2009] question this approach by taking the lenses of communication complexity: can it be that the communication complexity of an incentive compatible mechanism that implements $f$ (that is, computes both the output and the payments) is much larger than the communication complexity of computing the output? I.e., can it be that $cc_{IC}(f)>>cc(f)$? Fadel and Segal show that for every $f$, $cc_{IC}(f)leq exp(cc(f))$. They also show that fully computing the incentive compatible mechanism is strictly harder than computing only the output: there exists a social choice function $f$ such that $cc_{IC}(f)=cc(f)+1$. In a follow-up work, Babaioff, Blumrosen, Naor, and Schapira [EC08] provide a social choice function $f$ such that $cc_{IC}(f)=Theta(ncdot cc(f))$, where $n$ is the number of players. The question of whether the exponential upper bound of Fadel and Segal is tight remained wide open. In this paper we solve this question by explicitly providing an $f$ such that $cc_{IC}(f)= exp(cc(f))$. In fact, we establish this via two very different proofs. In contrast, we show that if the players are risk-neutral and we can compromise on a randomized truthful-in-expectation implementation (and not on deterministic ex-post implementation) gives that $cc_{TIE}(f)=poly(n,cc(f))$ for every function $f$, as long as the domain of $f$ is single parameter or a convex multi-parameter domain. We also provide efficient algorithms for deterministic computation of payments in several important domains.
Nearly a decade ago, Azrieli and Shmaya introduced the class of $lambda$-Lipschitz games in which every players payoff function is $lambda$-Lipschitz with respect to the actions of the other players. They showed that such games admit $epsilon$-approximate pure Nash equilibria for certain settings of $epsilon$ and $lambda$. 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 $epsilon$-approximate pure Nash equilibria of $n$-player, binary-action, $lambda$-Lipschitz games that is exponential in $frac{nlambda}{epsilon}$. In addition, we introduce ``Multi-Lipschitz games, a generalization involving player-specific Lipschitz values, and provide a reduction from finding equilibria of these games to finding equilibria of Lipschitz games, showing that the value of interest is the sum of the individual Lipschitz parameters. Finally, we provide an exponential lower bound on the deterministic query complexity of finding $epsilon$-approximate correlated equilibria of $n$-player, $m$-action, $lambda$-Lipschitz games for strong values of $epsilon$, motivating the consideration of explicitly randomized algorithms in the above results. Our proof is arguably simpler than those previously used to show similar results.
Similar to the role of Markov decision processes in reinforcement learning, Stochastic Games (SGs) lay the foundation for the study of multi-agent reinforcement learning (MARL) and sequential agent interactions. In this paper, we derive that computing an approximate Markov Perfect Equilibrium (MPE) in a finite-state discounted Stochastic Game within the exponential precision is textbf{PPAD}-complete. We adopt a function with a polynomially bounded description in the strategy space to convert the MPE computation to a fixed-point problem, even though the stochastic game may demand an exponential number of pure strategies, in the number of states, for each agent. The completeness result follows the reduction of the fixed-point problem to {sc End of the Line}. Our results indicate that finding an MPE in SGs is highly unlikely to be textbf{NP}-hard unless textbf{NP}=textbf{co-NP}. Our work offers confidence for MARL research to study MPE computation on general-sum SGs and to develop fruitful algorithms as currently on zero-sum SGs.
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