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Register a new userA General Framework for Computing the Nucleolus Via Dynamic Programming
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William Justin Toth
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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This paper defines a general class of cooperative games for which the nucleolus is efficiently computable. This class includes new members for which the complexity of computing their nucleolus was not previously known. We show that when the minimum excess coalition problem of a cooperative game can be formulated as a hypergraph dynamic program its nucleolus is efficiently computable. This gives a general technique for designing efficient algorithms for computing the nucleolus of a cooperative game. This technique is inspired by a recent result of Pashkovich (2018) on weighted voting games. However our technique significantly extends beyond the capabilities of previous work. We demonstrate this by applying it to give an algorithm for computing the nucleolus of b-matching games in polynomial time on graphs of bounded treewidth.
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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.
For years, there has been interest in approximation methods for solving dynamic programming problems, because of the inherent complexity in computing optimal solutions characterized by Bellmans principle of optimality. A wide range of approximate dynamic programming (ADP) methods now exists. It is of great interest to guarantee that the performance of an ADP scheme be at least some known fraction, say $beta$, of optimal. This paper introduces a general approach to bounding the performance of ADP methods, in this sense, in the stochastic setting. The approach is based on new results for bounding greedy solutions in string optimization problems, where one has to choose a string (ordered set) of actions to maximize an objective function. This bounding technique is inspired by submodularity theory, but submodularity is not required for establishing bounds. Instead, the bounding is based on quantifying certain notions of curvature of string functions; the smaller the curvatures the better the bound. The key insight is that any ADP scheme is a greedy scheme for some surrogate string objective function that coincides in its optimal solution and value with those of the original optimal control problem. The ADP scheme then yields to the bounding technique mentioned above, and the curvatures of the surrogate objective determine the value $beta$ of the bound. The surrogate objective and its curvatures depend on the specific ADP.
The endowment effect, coined by Nobel Laureate Richard Thaler, posits that people tend to inflate the value of items they own. This bias was studied, both theoretically and empirically, with respect to a single item. Babaioff et al. [EC18] took a first step at extending this study beyond a single item. They proposed a specific formulation of the endowment effect in combinatorial settings, and showed that equilibrium existence with respect to the endowed valuations extends from gross substitutes to submodular valuations, but provably fails to extend to XOS valuations. Extending the endowment effect to combinatorial settings can take different forms. In this work, we devise a framework that captures a space of endowment effects, upon which we impose a partial order, which preserves endowment equilibrium existence. Within this framework, we provide existence and welfare guarantees for endowment equilibria corresponding to various endowment effects. Our main results are the following: (1) For markets with XOS valuations, we introduce an endowment effect that is stronger than that of Babaioff et al., for which an endowment equilibrium is guaranteed to exist and gives at least half of the optimal welfare. Moreover, this equilibrium can be reached via a variant of the flexible ascent auction. (2) For markets with arbitrary valuations, we show that bundling leads to a sweeping positive result. In particular, if items can be prepacked into indivisible bundles, there always exists an endowment equilibrium with optimal welfare. Moreover, we provide a polynomial algorithm that given an arbitrary allocation $S$, computes an endowment equilibrium with the same welfare guarantee as in $S$.
In this paper, we will develop a systematic approach to deriving guaranteed bounds for approximate dynamic programming (ADP) schemes in optimal control problems. Our approach is inspired by our recent results on bounding the performance of greedy strategies in optimization of string-submodular functions over a finite horizon. The approach is to derive a string-submodular optimization problem, for which the optimal strategy is the optimal control solution and the greedy strategy is the ADP solution. Using this approach, we show that any ADP solution achieves a performance that is at least a factor of $beta$ of the performance of the optimal control solution, which satisfies Bellmans optimality principle. The factor $beta$ depends on the specific ADP scheme, as we will explicitly characterize. To illustrate the applicability of our bounding technique, we present examples of ADP schemes, including the popular rollout method.
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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