57 research outputs found
Computing the Least-core and Nucleolus for Threshold Cardinality Matching Games
Cooperative games provide a framework for fair and stable profit allocation
in multi-agent systems. \emph{Core}, \emph{least-core} and \emph{nucleolus} are
such solution concepts that characterize stability of cooperation. In this
paper, we study the algorithmic issues on the least-core and nucleolus of
threshold cardinality matching games (TCMG). A TCMG is defined on a graph
and a threshold , in which the player set is and the profit of
a coalition is 1 if the size of a maximum matching in
meets or exceeds , and 0 otherwise. We first show that for a TCMG, the
problems of computing least-core value, finding and verifying least-core payoff
are all polynomial time solvable. We also provide a general characterization of
the least core for a large class of TCMG. Next, based on Gallai-Edmonds
Decomposition in matching theory, we give a concise formulation of the
nucleolus for a typical case of TCMG which the threshold equals . When
the threshold is relevant to the input size, we prove that the nucleolus
can be obtained in polynomial time in bipartite graphs and graphs with a
perfect matching
The Least-core and Nucleolus of Path Cooperative Games
Cooperative games provide an appropriate framework for fair and stable profit
distribution in multiagent systems. In this paper, we study the algorithmic
issues on path cooperative games that arise from the situations where some
commodity flows through a network. In these games, a coalition of edges or
vertices is successful if it enables a path from the source to the sink in the
network, and lose otherwise. Based on dual theory of linear programming and the
relationship with flow games, we provide the characterizations on the CS-core,
least-core and nucleolus of path cooperative games. Furthermore, we show that
the least-core and nucleolus are polynomially solvable for path cooperative
games defined on both directed and undirected network
On the Complexity of Nucleolus Computation for Bipartite b-Matching Games
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
Complexity of coalition structure generation
We revisit the coalition structure generation problem in which the goal is to
partition the players into exhaustive and disjoint coalitions so as to maximize
the social welfare. One of our key results is a general polynomial-time
algorithm to solve the problem for all coalitional games provided that player
types are known and the number of player types is bounded by a constant. As a
corollary, we obtain a polynomial-time algorithm to compute an optimal
partition for weighted voting games with a constant number of weight values and
for coalitional skill games with a constant number of skills. We also consider
well-studied and well-motivated coalitional games defined compactly on
combinatorial domains. For these games, we characterize the complexity of
computing an optimal coalition structure by presenting polynomial-time
algorithms, approximation algorithms, or NP-hardness and inapproximability
lower bounds.Comment: 17 page
Stability and Fairness in Models with a Multiple Membership
This article studies a model of coalition formation for the joint production (and finance) of public projects, in which agents may belong to multiple coalitions. We show that, if projects are divisible, there always exists a stable (secession-proof) structure, i.e., a structure in which no coalition would reject a proposed arrangement. When projects are indivisible, stable allocations may fail to exist and, for those cases, we resort to the least core in order to estimate the degree of instability. We also examine the compatibility of stability and fairness in metric environments with indivisible projects, where we also explore the performance of well-known solutions, such as the Shapley value and the nucleolus.Stability, Fairness, Membership, Coalition Formation
Stability and fairness in models with a multiple membership
This article studies a model of coalition formation for the joint production (and finance) of public projects, in which agents may belong to multiple coalitions. We show that, if projects are divisible, there always exists a stable (secession-proof) structure, i.e., a structure in which no coalition would reject a proposed arrangement. When projects are in- divisible, stable allocations may fail to exist and, for those cases, we resort to the least core in order to estimate the degree of instability. We also examine the compatibility of stability and fairness on metric environments with indivisible projects. To do so, we explore, among other things, the performance of several well-known solutions (such as the Shapley value, the nucleolus, or the Dutta-Ray value) in these environments.stability, fairness, membership, coalition formation
Complementary cooperation, minimal winning coalitions, and power indices
We introduce a new simple game, which is referred to as the complementary
weighted multiple majority game (C-WMMG for short). C-WMMG models a basic
cooperation rule, the complementary cooperation rule, and can be taken as a
sister model of the famous weighted majority game (WMG for short). In this
paper, we concentrate on the two dimensional C-WMMG. An interesting property of
this case is that there are at most minimal winning coalitions (MWC for
short), and they can be enumerated in time , where is the
number of players. This property guarantees that the two dimensional C-WMMG is
more handleable than WMG. In particular, we prove that the main power indices,
i.e. the Shapley-Shubik index, the Penrose-Banzhaf index, the Holler-Packel
index, and the Deegan-Packel index, are all polynomially computable. To make a
comparison with WMG, we know that it may have exponentially many MWCs, and none
of the four power indices is polynomially computable (unless P=NP). Still for
the two dimensional case, we show that local monotonicity holds for all of the
four power indices. In WMG, this property is possessed by the Shapley-Shubik
index and the Penrose-Banzhaf index, but not by the Holler-Packel index or the
Deegan-Packel index. Since our model fits very well the cooperation and
competition in team sports, we hope that it can be potentially applied in
measuring the values of players in team sports, say help people give more
objective ranking of NBA players and select MVPs, and consequently bring new
insights into contest theory and the more general field of sports economics. It
may also provide some interesting enlightenments into the design of
non-additive voting mechanisms. Last but not least, the threshold version of
C-WMMG is a generalization of WMG, and natural variants of it are closely
related with the famous airport game and the stable marriage/roommates problem.Comment: 60 page
Stability and Fairness in Models with a Multiple Membership
This article studies a model of coalition formation for the joint production (and finance) of public projects, in which agents may belong to multiple coalitions. We show that, if projects are divisible, there always exists a stable (secession-proof) structure, i.e., a structure in which no coalition would reject a proposed arrangement. When projects are indivisible, stable allocations may fail to exist and, for those cases, we resort to the least core in order to estimate the degree of instability. We also examine the compatibility of stability and fairness in metric environments with indivisible projects, where we also explore the performance of well-known solutions, such as the Shapley value and the nucleolus.Stability, Fairness, Membership, Coalition Formation
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