14,951 research outputs found
Spanning connectivity games
The Banzhaf index, Shapley-Shubik index and other voting power indices measure the importance of a player in a coalitional game. We consider a simple coalitional game called the spanning connectivity game (SCG) based on an undirected, unweighted multigraph, where edges are players. We examine the computational complexity of computing the voting power indices of edges in the SCG. It is shown that computing Banzhaf values and Shapley-Shubik indices is #P-complete for SCGs. Interestingly, Holler indices and Deegan-Packel indices can be computed in polynomial time. Among other results, it is proved that Banzhaf indices can be computed in polynomial time for graphs with bounded treewidth. It is also shown that for any reasonable representation of a simple game, a polynomial time algorithm to compute the Shapley-Shubik indices implies a polynomial time algorithm to compute the Banzhaf indices. As a corollary, computing the Shapley value is #P-complete for simple games represented by the set of minimal winning coalitions, Threshold Network Flow Games, Vertex Connectivity Games and Coalitional Skill Games
Positional games on random graphs
We introduce and study Maker/Breaker-type positional games on random graphs.
Our main concern is to determine the threshold probability for the
existence of Maker's strategy to claim a member of in the unbiased game
played on the edges of random graph , for various target families
of winning sets. More generally, for each probability above this threshold we
study the smallest bias such that Maker wins the biased game. We
investigate these functions for a number of basic games, like the connectivity
game, the perfect matching game, the clique game and the Hamiltonian cycle
game
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
Wiretapping a hidden network
We consider the problem of maximizing the probability of hitting a
strategically chosen hidden virtual network by placing a wiretap on a single
link of a communication network. This can be seen as a two-player win-lose
(zero-sum) game that we call the wiretap game. The value of this game is the
greatest probability that the wiretapper can secure for hitting the virtual
network. The value is shown to equal the reciprocal of the strength of the
underlying graph.
We efficiently compute a unique partition of the edges of the graph, called
the prime-partition, and find the set of pure strategies of the hider that are
best responses against every maxmin strategy of the wiretapper. Using these
special pure strategies of the hider, which we call
omni-connected-spanning-subgraphs, we define a partial order on the elements of
the prime-partition. From the partial order, we obtain a linear number of
simple two-variable inequalities that define the maxmin-polytope, and a
characterization of its extreme points.
Our definition of the partial order allows us to find all equilibrium
strategies of the wiretapper that minimize the number of pure best responses of
the hider. Among these strategies, we efficiently compute the unique strategy
that maximizes the least punishment that the hider incurs for playing a pure
strategy that is not a best response. Finally, we show that this unique
strategy is the nucleolus of the recently studied simple cooperative spanning
connectivity game
Enforcing efficient equilibria in network design games via subsidies
The efficient design of networks has been an important engineering task that
involves challenging combinatorial optimization problems. Typically, a network
designer has to select among several alternatives which links to establish so
that the resulting network satisfies a given set of connectivity requirements
and the cost of establishing the network links is as low as possible. The
Minimum Spanning Tree problem, which is well-understood, is a nice example.
In this paper, we consider the natural scenario in which the connectivity
requirements are posed by selfish users who have agreed to share the cost of
the network to be established according to a well-defined rule. The design
proposed by the network designer should now be consistent not only with the
connectivity requirements but also with the selfishness of the users.
Essentially, the users are players in a so-called network design game and the
network designer has to propose a design that is an equilibrium for this game.
As it is usually the case when selfishness comes into play, such equilibria may
be suboptimal. In this paper, we consider the following question: can the
network designer enforce particular designs as equilibria or guarantee that
efficient designs are consistent with users' selfishness by appropriately
subsidizing some of the network links? In an attempt to understand this
question, we formulate corresponding optimization problems and present positive
and negative results.Comment: 30 pages, 7 figure
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
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