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Knapsack Auctions
 Proceedings of the Seventeenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2006
"... We consider a game theoretic knapsack problem that has application to auctions for selling advertisements on Internet search engines. Consider n agents each wishing to place an object in the knapsack. Each agent has a private valuation for having their object in the knapsack and each object has a pu ..."
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Cited by 61 (12 self)
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We consider a game theoretic knapsack problem that has application to auctions for selling advertisements on Internet search engines. Consider n agents each wishing to place an object in the knapsack. Each agent has a private valuation for having their object in the knapsack and each object has a publicly known size. For this setting, we consider the design of auctions in which agents have an incentive to truthfully reveal their private valuations. Following the framework of Goldberg et al. [10], we look to design an auction that obtains a constant fraction of the profit obtainable by a natural optimal pricing algorithm that knows the agents ’ valuations and object sizes. We give an auction that obtains a constant factor approximation in the nontrivial special case where the knapsack has unlimited capacity. We then reduce the limited capacity version of the problem to the unlimited capacity version via an approximately efficient auction (i.e., one that maximizes the social welfare). This reduction follows from generalizable principles. 1
InfinitelyOften Autoreducible Sets
 IN PROCEEDINGS OF THE 14TH ANNUAL INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION
, 2003
"... A set A is autoreducible if one can compute, for all x, the value A(x) by querying A only at places y x. Furthermore, A is infinitelyoften autoreducible if, for infinitely many x, the value A(x) can be computed by querying A only at places y x. For all other x, the computation outputs a sp ..."
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Cited by 6 (0 self)
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A set A is autoreducible if one can compute, for all x, the value A(x) by querying A only at places y x. Furthermore, A is infinitelyoften autoreducible if, for infinitely many x, the value A(x) can be computed by querying A only at places y x. For all other x, the computation outputs a special symbol to signal that the reduction is undefined.
A Puzzle to Challenge Genetic Programming
, 2002
"... This report represents an initial investigation into the use of genetic programming to solve the Nprisoners puzzle. The puzzle has generated a certain level of interest among the mathematical community. We believe ..."
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Cited by 6 (0 self)
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This report represents an initial investigation into the use of genetic programming to solve the Nprisoners puzzle. The puzzle has generated a certain level of interest among the mathematical community. We believe
On the construction of effective random sets
 MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE 2002, LECTURE NOTES IN COMPUTER SCIENCE
, 2002
"... We give a direct and rather simple construction of MartinLöf random and recrandom sets with certain additional properties. First, reviewing the result of Gacs and Kucera, given any set X we construct a MartinLöf random set R from which X can be decoded effectively. Second, by essentially the same ..."
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We give a direct and rather simple construction of MartinLöf random and recrandom sets with certain additional properties. First, reviewing the result of Gacs and Kucera, given any set X we construct a MartinLöf random set R from which X can be decoded effectively. Second, by essentially the same construction we obtain a MartinLöf random set R that is computably enumerable selfreducible. Alternatively, using the observation that a set is computably enumerable selfreducible if and only if its associated real is computably enumerable, the existence of such a set R follows from the known fact that every Chaitin real is MartinLöf random and computably enumerable. Third, by a variant of the basic construction we obtain a recrandom set that is weak truthtable autoreducible. The mentioned results on self and autoreducibility complement work of Ebert, Merkle, and Vollmer [79], from which it follows that no MartinLöf random set is Turingautoreducible and that no recrandom set is truthtable autoreducible.
On Hats and other Covers
, 2005
"... We study a game puzzle that has enjoyed recent popularity among mathematicians, computer scientist, coding theorists and even the mass press. In the game, n players are fitted with randomly assigned colored hats. Individual players can see their teammates ’ hat colors, but not their own. Based on t ..."
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We study a game puzzle that has enjoyed recent popularity among mathematicians, computer scientist, coding theorists and even the mass press. In the game, n players are fitted with randomly assigned colored hats. Individual players can see their teammates ’ hat colors, but not their own. Based on this information, and without any further communication, each player must attempt to guess his hat color, or pass. The team wins if there is at least one correct guess, and no incorrect ones. The goal is to devise guessing strategies that maximize the team winning probability. We show that for the case of two hat colors, and for any value of n, playing strategies are equivalent to binary covering codes of radius one. This link, in particular with Hamming codes, had been observed for values of n of the form 2 m − 1. We extend the analysis to games with hats of q colors, q ≥ 2, where 1coverings are not sufficient to characterize the best strategies. Instead, we introduce the more appropriate notion of a strong covering, and show efficient constructions of these coverings, which achieve winning probabilities approaching unity. Finally, we briefly discuss results on variants of the problem, including arbitrary input distributions, randomized playing strategies, and symmetric strategies.
Computing With Strategic Agents
, 2005
"... This dissertation studies mechanism design for various combinatorial problems in the presence of strategic agents. A mechanism is an algorithm for allocating a resource among a group of participants, each of which has a privatelyknown value for any particular allocation. A mechanism is truthful if ..."
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This dissertation studies mechanism design for various combinatorial problems in the presence of strategic agents. A mechanism is an algorithm for allocating a resource among a group of participants, each of which has a privatelyknown value for any particular allocation. A mechanism is truthful if it is in each participant’s best interest to reveal his private information truthfully regardless of the strategies of the other participants. First, we explore a competitive auction framework for truthful mechanism design in the setting of multiunit auctions, or auctions which sell multiple identical copies of a good. In this framework, the goal is to design a truthful auction whose revenue approximates that of an omniscient auction for any set of bids. We focus on two natural settings — the limited demand setting where bidders desire at most a fixed number of copies and the limited budget setting where bidders can spend at most a fixed amount of money. In the limit demand setting, all prior auctions employed the use of randomization in the computation of the allocation and prices. Randomization
Yet Another Hat Game
, 2010
"... Several different “hat games ” have recentlyreceived afair amount of attention. Typically, in a hat game, one or more players are required to correctly guess their hat colour when given some information about other players ’ hat colours. Some versions of these games have been motivated by research i ..."
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Several different “hat games ” have recentlyreceived afair amount of attention. Typically, in a hat game, one or more players are required to correctly guess their hat colour when given some information about other players ’ hat colours. Some versions of these games have been motivated by research in complexity theory and have ties to wellknown research problems in coding theory, and some variations have led to interesting new research. In this paper, we review Ebert’s Hat Game [5, 6] which garnered a considerable amount of publicity in the late 90’s and early 00’s [9], and the Hatsonaline Game [2, 3]. Then we introduce a new hat game which is a “hybrid ” of these two games and provide an optimal strategy for playing the new game. The optimal strategy is quite simple, but the proof involves an interesting combinatorial argument. 1
New Surprises from SelfReducibility
"... Abstract. Selfreducibility continues to give us new angles on attacking some of the fundamental questions about computation and complexity. 1 ..."
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Abstract. Selfreducibility continues to give us new angles on attacking some of the fundamental questions about computation and complexity. 1
A construction for the hat problem on a directed graph
"... A team of n players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. Visibility is defined by ..."
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A team of n players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. Visibility is defined by a directed graph; that is, vertices correspond to players, and a player can see each player to whom he is connected by an arc. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The team aims to maximize the probability of a win, and this maximum is called the hat number of the graph. Previous works focused on the hat problem on complete graphs and on undirected graphs. Some cases were solved, e.g., complete graphs of certain orders, trees, cycles, and bipartite graphs. These led Uriel Feige to conjecture that the hat number of any graph is equal to the hat number of its maximum clique. We show that the conjecture does not hold for directed graphs. Moreover, for every value of the maximum clique size, we provide a tight characterization of the range of possible values of the hat number. We construct families of directed graphs with a fixed clique number the hat number of which is asymptotically optimal. We also determine the hat number of tournaments to be one half.