We propose a measure of riskiness of " gambles " (risky assets) that is objective: it depends only on the gamble and not on the decision maker. The measure is based on identifying for every gamble the critical wealth level below which it becomes " risky " to accept the gamble.
We prove that in every normal form <i>n</i>-player game with <i>m</i> actions for each player, there exists an approximate Nash equilibrium in which each player randomizes uniformly among a set of <i>O</i>(log <i>m</i> + log <i>n</i>) pure actions. This result induces an <i>O</i>(<i>N</i> <sup>log log <i>N</i></sup>)-time algorithm for computing an… (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(log m+log n) pure strategies. This result induces an N log log N algorithm for computing an approximate Nash equilibrium in games where the number of actions is polynomial in the… (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 Z n. A hunter and a rabbit move on the nodes of Z n 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 compare the prediction power of betting strategies (aka martingales) whose wagers take values in different sets of reals. A mar-tingale whose wagers take values in a set A is called an A-martingale. A set of reals B anticipates a set A, if for every A-martingale there is a countable set of B-martingales, such that on every binary sequence on which the… (More)
Computable randomness is a central notion in the theory of algorithmic ran-domness. 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 real-valued bets. We study two restricted… (More)
We prove the existence of approximate correlated equilibrium of support size polylogarithmic in the number of players and the number of actions per player. In particular, using the probabilistic method, we show that there exists a multiset of polylogarithmic size such that the uniform distribution over this multiset forms an approximate correlated… (More)