Sampling of alternatives is often required in discrete choice models to reduce the computational burden and to avoid describing a large number of attributes. This approach has been used in many areas, including modeling of route choice, vehicle ownership, trip destination, residential location, and activity scheduling. The need for sampling of alternatives is accentuated for Random Regret Minim...

We consider computationally tractable methods for the experimental design problem, where k out of n design points of dimension p are selected so that certain optimality criteria are approximately satisfied. Our algorithm finds a (1 + ε)approximate optimal design when k is a linear function of p; in contrast, existing results require k to be super-linear in p. Our algorithm also handles all popu...

This paper examines the problem of multi-agent learning in N -person non-cooperative games. For concreteness, we focus on the socalled “hedge” variant of the exponential weights (EW) algorithm, one of the most widely studied algorithmic schemes for regret minimization in online learning. In this multi-agent context, we show that a) dominated strategies become extinct (a.s.); and b) in generic g...

Approachability has become a standard tool in analyzing learning algorithms in the adversarial online learning setup. We develop a variant of approachability for games where there is ambiguity in the obtained reward that belongs to a set, rather than being a single vector. Using this variant we tackle the problem of approachability in games with partial monitoring and develop simple and efficie...

We study the generalization of counterfactual regret minimization (CFR) to partialinformation collaborative games with more than 2 players. For instance, many 4-player card games are structured as 2v2 games, with each player only knowing the contents of their own hand. To study this setting, we propose a multi-agent collaborative version of Kuhn Poker. We observe that a straightforward applicat...

Games are used to evaluate and advance Multiagent and Artificial Intelligence techniques. Most of these games are deterministic with perfect information (e.g. Chess and Checkers). A deterministic game has no chance element and in a perfect information game, all information is visible to all players. However, many real-world scenarios with competing agents are stochastic (non-deterministic) with...

j 6=i Aj . We let ai denote a pure action for player i, and let si ∈ ∆(Ai) denote a mixed action for player i. We will typically view si as a vector in R Ai , with si(ai) equal to the probability that player i places on ai. We let Πi(a) denote the payoff to player i when the composite pure action vector is a, and by an abuse of notation also let Πi(s) denote the expected payoff to player i when...

Counterfactual Regret Minimization and variants (e.g. Public Chance Sampling CFR and Pure CFR) have been known as the best approaches for creating approximate Nash equilibrium solutions for imperfect information games such as poker. This paper introduces CFR, a new algorithm that typically outperforms the previously known algorithms by an order of magnitude or more in terms of computation time ...