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You are offered a gamble (a “risky asset”) g in which it is equally likely that you gain $120 or lose $100. What is the risk in accepting g? Is there an objective way to measure the riskiness of g? “Objective” means that the measure should depend on the gamble itself and not on the decision maker; that is, only the outcomes and the probabilities (the(More)
We show that in an n-player m-action strategic form game, we can obtain an approximate equilibrium by sampling any mixed-action equilibrium a small number of times. We study three notions of equilibrium: Nash, correlated and coarse correlated. For each one of them we obtain upper and lower bounds on the asymptotic (where max(m,n) → ∞) worst-case number of(More)
We prove that in a normal form n-player game with m actions for each player, there exists an approximate Nash equilibrium where each player randomizes uniformly among a set of O(logm+log n) pure strategies. This result induces an N log logN algorithm for computing an approximate Nash equilibrium in games where the number of actions is polynomial in the(More)
Computable randomness is a central notion in the theory of algorithmic randomness. An infinite sequence of bits x is computably random if no computable betting strategy can win an infinite amount of money by betting on the values of the bits of x. In the classical model, the betting strategies considered take realvalued bets. We study two restricted models,(More)
A decision maker observes the evolving state of the world while constantly trying to predict the next state given the history of past states. The ability to benefit from such predictions depends not only on the ability to recognize patters in history, but also on the range of actions available to the decision maker. We assume there are two possible states(More)
A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle Zn. A hunter and a rabbit move on the nodes of Zn without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al (2003)(More)
We study the space-and-time automaton-complexity of two related problems concerning the cycle length of a periodic stream of input bits. One problem is to find the exact cycle length of a periodic stream of input bits provided that the cycle length is bounded by a known parameter n. The other problem is to find a large number k that divides the cycle(More)