The list update problem is a classical online problem, with an optimal competitive ratio that is still open, known to be somewhere between 1.5 and 1.6. An algorithm with competitive ratio 1.6, the smallest known to date, is COMB, a randomized combination of BIT and the TIMESTAMP algorithm TS. This and almost all other list update algorithms, like MTF, are projective in the sense that they can be defined by looking only at any pair of list items at a time. Projectivity (also known as “list factoring”) simplifies both the description of the algorithm and its analysis, and so far seems to be the only way to define a good online algorithm for lists of arbitrary length. In this article, we characterize all projective list update algorithms and show that their competitive ratio is never smaller than 1.6 in the partial cost model. Therefore, COMB is a best possible projective algorithm in this model.

Game-theoretic problems have found massive recent interest in computer science. Obvious applications arise from the internet and electronic commerce, for example the study of online auctions.

The Lemke-Howson algorithm is the classical algorithm for the problem NASH of finding one Nash equilibrium of a bimatrix game. It provides a constructive and elementary proof of existence of an equilibrium, by a typical "directed parity argument", which puts NASH into the complexity class PPAD. This paper presents a class of bimatrix games for which the Lemke-Howson algorithm takes, even in the best case, exponential time in the dimension d of the game, requiring /spl Omega/((/spl theta//sup 3/4/)/sup d/) many steps, where /spl theta/ is the golden ratio. The "parity argument" for NASH is thus explicitly shown to be inefficient. The games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in d-space.

Interactions among agents can be conveniently described by game trees. In order to analyze a game, it is important to derive optimal (or equilibrium) strategies for the different players. The standard approach to finding such strategies in games with imperfect information is, in general, computationally intractable. The approach is to generate the normal form of the game (the matrix containing the payoff for each strategy combination), and then solve a linear program (LP) or a linear complementarity problem (LCP). The size of the normal form, however, is typically exponential in the size of the game tree, thus making this method impractical in all but the simplest cases. This paper describes a new representation of strategies which results in a practical linear formulation of the problem of two-player games with perfect recall (i.e., games where players never forget anything, which is a standard assumption). Standard LP or LCP solvers can then be applied to find optimal randomized strategies. The resulting algorithms are, in general, exponentially better than the standard ones, both in terms of time and in terms of space.