15,175 research outputs found
Computing power indices for weighted voting games via dynamic programming
We study the efficient computation of power indices for weighted voting games using the paradigm of dynamic programming. We survey the state-of-the-art algorithms for computing the Banzhaf and Shapley-Shubik indices and point out how these approaches carry over to related power indices. Within a unified framework, we present new efficient algorithms for the Public Good index and a recently proposed power index based on minimal winning coalitions of the smallest size, as well as a very first method for computing the Johnston indices for weighted voting games efficiently. We introduce a software package providing fast C++ implementations of all the power indices mentioned in this article, discuss computing times, as well as storage requirements
Coalitional power indices applied to voting systems
We describe voting mechanisms to study voting systems. The classical power indices applied to simple games
just consider parties, players or voters. Here, we also consider games with a priori unions, i.e., coalitions
among parties, players or voters. We measure the power of each party, player or voter when there are coalitions
among them. In particular, we study real situations of voting systems using extended Shapley–Shubik
and Banzhaf indices, the so-called coalitional power indices. We also introduce a dynamic programming to
compute them.Peer ReviewedPostprint (published version
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
Cooperative Games with Bounded Dependency Degree
Cooperative games provide a framework to study cooperation among
self-interested agents. They offer a number of solution concepts describing how
the outcome of the cooperation should be shared among the players.
Unfortunately, computational problems associated with many of these solution
concepts tend to be intractable---NP-hard or worse. In this paper, we
incorporate complexity measures recently proposed by Feige and Izsak (2013),
called dependency degree and supermodular degree, into the complexity analysis
of cooperative games. We show that many computational problems for cooperative
games become tractable for games whose dependency degree or supermodular degree
are bounded. In particular, we prove that simple games admit efficient
algorithms for various solution concepts when the supermodular degree is small;
further, we show that computing the Shapley value is always in FPT with respect
to the dependency degree. Finally, we note that, while determining the
dependency among players is computationally hard, there are efficient
algorithms for special classes of games.Comment: 10 pages, full version of accepted AAAI-18 pape
Computing the Power Distribution in the IMF
The International Monetary Fund is one of the largest international
organizations using a weighted voting system. The weights of its 188 members
are determined by a fixed amount of basic votes plus some extra votes for
so-called Special Drawing Rights (SDR). On January 26, 2016, the conditions for
the SDRs were increased at the 14th General Quota Review, which drastically
changed the corresponding voting weights. However, since the share of voting
weights in general is not equal to the influence, of a committee member on the
committees overall decision, so-called power indices were introduced. So far
the power distribution of the IMF was only computed by either approximation
procedures or smaller games than then entire Board of Governors consisting of
188 members. We improve existing algorithms, based on dynamic programming, for
the computation of power indices and provide the exact results for the IMF
Board of Governors before and after the increase of voting weights. Tuned
low-level details of the algorithms allow the repeated routine with sparse
computational resources and can of course be applied to other large voting
bodies. It turned out that the Banzhaf power shares are rather sensitive to
changes of the quota.Comment: 19 pages, 2 figures, 13 table
Measuring voting power in convex policy spaces
Classical power index analysis considers the individual's ability to
influence the aggregated group decision by changing its own vote, where all
decisions and votes are assumed to be binary. In many practical applications we
have more options than either "yes" or "no". Here we generalize three important
power indices to continuous convex policy spaces. This allows the analysis of a
collection of economic problems like e.g. tax rates or spending that otherwise
would not be covered in binary models.Comment: 31 pages, 9 table
Average Weights and Power in Weighted Voting Games
We investigate a class of weighted voting games for which weights are
randomly distributed over the standard probability simplex. We provide
close-formed formulae for the expectation and density of the distribution of
weight of the -th largest player under the uniform distribution. We analyze
the average voting power of the -th largest player and its dependence on the
quota, obtaining analytical and numerical results for small values of and a
general theorem about the functional form of the relation between the average
Penrose--Banzhaf power index and the quota for the uniform measure on the
simplex. We also analyze the power of a collectivity to act (Coleman efficiency
index) of random weighted voting games, obtaining analytical upper bounds
therefor.Comment: 12 pages, 7 figure
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