On "Indifference" and Backward Induction in Games with Perfect Information

Indifference of a player with respect to two distinct outcomes of a game cannot be handled by small perturbations, because the actual choice may have significant impact on other players, and cause them to act in a way that has significant impact of the indifferent player. It is argued that ties among rational choices can be resolved by refinements of the concept of rationality based on the utilities of other players. One such refinement is the concept of Tit-for-Tat

Remarks on Utility in Repeated Bets

The use of von Neumann -- Morgenstern utility is examined in the context of multiple choices between lotteries. Different conclusions are reached if the choices are simultaneous or sequential. It is demonstrated that utility cannot be additive

An Efficient Interior-Point Method for Online Convex Optimization

A new algorithm for regret minimization in online convex optimization is described. The regret of the algorithm after $T$ time periods is $O(\sqrt{T \log T})$ - which is the minimum possible up to a logarithmic term. In addition, the new algorithm is adaptive, in the sense that the regret bounds hold not only for the time periods $1,\ldots,T$ but also for every sub-interval $s,s+1,\ldots,t$. The running time of the algorithm matches that of newly introduced interior point algorithms for regret minimization: in $n$-dimensional space, during each iteration the new algorithm essentially solves a system of linear equations of order $n$, rather than solving some constrained convex optimization problem in $n$ dimensions and possibly many constraints