Computing Majority with Triple Queries

Consider a bin containing $n$ balls colored with two colors. In a $k$-query, $k$ balls are selected by a questioner and the oracle's reply is related (depending on the computation model being considered) to the distribution of colors of the balls in this $k$-tuple; however, the oracle never reveals the colors of the individual balls. Following a number of queries the questioner is said to determine the majority color if it can output a ball of the majority color if it exists, and can prove that there is no majority if it does not exist. We investigate two computation models (depending on the type of replies being allowed). We give algorithms to compute the minimum number of 3-queries which are needed so that the questioner can determine the majority color and provide tight and almost tight upper and lower bounds on the number of queries needed in each case.Comment: 22 pages, 1 figure, conference version to appear in proceedings of the 17th Annual International Computing and Combinatorics Conference (COCOON 2011

Playing off-line games with bounded rationality

We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payo is the average of a one-shot payo over the joint sequence. We consider the maxmin value of the game played in pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First we dene the complexity of a sequence by its smallest period (a non-periodic sequence being of innite complexity) and study the maxmin of the game where player 1 is restricted to strategies with complexity at most n and player 2 is restricted to strategies with complexity at most m. We study the asymptotics of this value and a complete characterization in the matching pennies case. We extend the analysis of matching pennies to strategies with bounded recall.We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payo is the average of a one-shot payo over the joint sequence. We consider the maxmin value of the game played in pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First we dene the complexity of a sequence by its smallest period (a non-periodic sequence being of innite complexity) and study the maxmin of the game where player 1 is restricted to strategies with complexity at most n and player 2 is restricted to strategies with complexity at most m. We study the asymptotics of this value and a complete characterization in the matching pennies case. We extend the analysis of matching pennies to strategies with bounded recall.Refereed Working Papers / of international relevanc

ADAPTIVE MAJORITY PROBLEMS FOR RESTRICTED QUERY GRAPHS AND FOR WEIGHTED SETS

Suppose that the vertices of a graph G are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists), if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by m(G). It was shown by Saks and Werman that m(K-n) = n - b(n) where b(n) is the number of 1's in the binary representation of n. In this paper we initiate the study of the problem for general graphs. The obvious bounds for a connected graph G on n vertices are n - b(n) <= m(G) <= n - 1. We show that for any tree T on an even number of vertices we have m(T) = n - 1, and that for any tree T on an odd number of vertices, we have n - 65 <= m (T) <= n - 2. Our proof uses results about the weighted version of the problem for K-n, which may be of independent interest. We also exhibit a sequence G(n) of graphs with m(G(n)) = n - b(n) such that the number of edges in G(n) is O(nb(n))

Adaptive Majority Problemsfor Restricted Query Graphsand for Weighted Sets

Suppose that the vertices of a graph G are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists), if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by m(G). It was shown by Saks and Werman that m(K-n) = n - b(n) where b(n) is the number of 1's in the binary representation of n. In this paper we initiate the study of the problem for general graphs. The obvious bounds for a connected graph G on n vertices are n - b(n) <= m(G) <= n - 1. We show that for any tree T on an even number of vertices we have m(T) = n - 1, and that for any tree T on an odd number of vertices, we have n - 65 <= m (T) <= n - 2. Our proof uses results about the weighted version of the problem for K-n, which may be of independent interest. We also exhibit a sequence G(n) of graphs with m(G(n)) = n - b(n) such that the number of edges in G(n) is O(nb(n))

Oblivious and adaptive strategies for the majority and plurality problems

In the well-studied Majority problem, we are given a set of n balls colored with two or more colors, and the goal is to use the minimum number of color comparisons to find a ball of the majority color (i.e., a color that occurs for more than ⌈n/2⌉ times). The Plurality problem has exactly the same setting while the goal is to find a ball of the dominant color (i.e., a color that occurs most often). Previous literature regarding this topic dealt mainly with adaptive strategies, whereas in this paper we focus more on the oblivious (i.e., non-adaptive) strategies. Given that our strategies are oblivious, we establish a linear upper bound for the Majority problem with arbitrarily many different colors. We then show that the Plurality problem is significantly more difficult by establishing quadratic lower and upper bounds. In the end, we also discuss some generalized upper bounds for adaptive strategies in the k-color Plurality problem