53,772 research outputs found
A Law of Large Numbers for Weighted Majority
Consider an election between two candidates in which the voters' choices are
random and independent and the probability of a voter choosing the first
candidate is . Condorcet's Jury Theorem which he derived from the weak
law of large numbers asserts that if the number of voters tends to infinity
then the probability that the first candidate will be elected tends to one. The
notion of influence of a voter or its voting power is relevant for extensions
of the weak law of large numbers for voting rules which are more general than
simple majority. In this paper we point out two different ways to extend the
classical notions of voting power and influences to arbitrary probability
distributions. The extension relevant to us is the ``effect'' of a voter, which
is a weighted version of the correlation between the voter's vote and the
election's outcomes. We prove an extension of the weak law of large numbers to
weighted majority games when all individual effects are small and show that
this result does not apply to any voting rule which is not based on weighted
majority
Single-Elimination Brackets Fail to Approximate Copeland Winner
Single-elimination (SE) brackets appear commonly in both sports tournaments and the voting theory literature. In certain tournament models, they have been shown to select the unambiguously-strongest competitor with optimum probability. By contrast, we reevaluate SE brackets through the lens of approximation, where the goal is to select a winner who would beat the most other competitors in a round robin (i.e., maximize the Copeland score), and find them lacking. Our primary result establishes the approximation ratio of a randomly-seeded SE bracket is 2^{- Theta(sqrt{log n})}; this is underwhelming considering a 1/2 ratio is achieved by choosing a winner uniformly at random. We also establish that a generalized version of the SE bracket performs nearly as poorly, with an approximation ratio of 2^{- Omega(sqrt[4]{log n})}, addressing a decade-old open question in the voting tree literature
Human gesture classification by brute-force machine learning for exergaming in physiotherapy
In this paper, a novel approach for human gesture classification on skeletal data is proposed for the application of exergaming in physiotherapy. Unlike existing methods, we propose to use a general classifier like Random Forests to recognize dynamic gestures. The temporal dimension is handled afterwards by majority voting in a sliding window over the consecutive predictions of the classifier. The gestures can have partially similar postures, such that the classifier will decide on the dissimilar postures. This brute-force classification strategy is permitted, because dynamic human gestures show sufficient dissimilar postures. Online continuous human gesture recognition can classify dynamic gestures in an early stage, which is a crucial advantage when controlling a game by automatic gesture recognition. Also, ground truth can be easily obtained, since all postures in a gesture get the same label, without any discretization into consecutive postures. This way, new gestures can be easily added, which is advantageous in adaptive game development. We evaluate our strategy by a leave-one-subject-out cross-validation on a self-captured stealth game gesture dataset and the publicly available Microsoft Research Cambridge-12 Kinect (MSRC-12) dataset. On the first dataset we achieve an excellent accuracy rate of 96.72%. Furthermore, we show that Random Forests perform better than Support Vector Machines. On the second dataset we achieve an accuracy rate of 98.37%, which is on average 3.57% better then existing methods
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 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
A note on Condorcet consistency and the median voter
We discuss to which extent the median voter theorem extends to the domain of single-peaked preferences on median spaces. After observing that on this domain a Condorcet winner need not exist, we show that if a Condorcet winner does exist, then it coincides with the median alternative ('the median voter'). Based on this result, we propose two non-cooperative games that implement the unique strategy-proof social choice rule on this domain. --
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