12 research outputs found
Computing large market equilibria using abstractions
Computing market equilibria is an important practical problem for market
design (e.g. fair division, item allocation). However, computing equilibria
requires large amounts of information (e.g. all valuations for all buyers for
all items) and compute power. We consider ameliorating these issues by applying
a method used for solving complex games: constructing a coarsened abstraction
of a given market, solving for the equilibrium in the abstraction, and lifting
the prices and allocations back to the original market. We show how to bound
important quantities such as regret, envy, Nash social welfare, Pareto
optimality, and maximin share when the abstracted prices and allocations are
used in place of the real equilibrium. We then study two abstraction methods of
interest for practitioners: 1) filling in unknown valuations using techniques
from matrix completion, 2) reducing the problem size by aggregating groups of
buyers/items into smaller numbers of representative buyers/items and solving
for equilibrium in this coarsened market. We find that in real data
allocations/prices that are relatively close to equilibria can be computed from
even very coarse abstractions
Robust Stackelberg Equilibria in Extensive-Form Games and Extension to Limited Lookahead
Stackelberg equilibria have become increasingly important as a solution
concept in computational game theory, largely inspired by practical problems
such as security settings. In practice, however, there is typically uncertainty
regarding the model about the opponent. This paper is, to our knowledge, the
first to investigate Stackelberg equilibria under uncertainty in extensive-form
games, one of the broadest classes of game. We introduce robust Stackelberg
equilibria, where the uncertainty is about the opponent's payoffs, as well as
ones where the opponent has limited lookahead and the uncertainty is about the
opponent's node evaluation function. We develop a new mixed-integer program for
the deterministic limited-lookahead setting. We then extend the program to the
robust setting for Stackelberg equilibrium under unlimited and under limited
lookahead by the opponent. We show that for the specific case of interval
uncertainty about the opponent's payoffs (or about the opponent's node
evaluations in the case of limited lookahead), robust Stackelberg equilibria
can be computed with a mixed-integer program that is of the same asymptotic
size as that for the deterministic setting.Comment: Published at AAAI1
Robust Stackelberg Equilibria in Extensive-Form Games and Extension to Limited Lookahead
Stackelberg equilibria have become increasingly important as a solution
concept in computational game theory, largely inspired by practical problems
such as security settings. In practice, however, there is typically uncertainty
regarding the model about the opponent. This paper is, to our knowledge, the
first to investigate Stackelberg equilibria under uncertainty in extensive-form
games, one of the broadest classes of game. We introduce robust Stackelberg
equilibria, where the uncertainty is about the opponent's payoffs, as well as
ones where the opponent has limited lookahead and the uncertainty is about the
opponent's node evaluation function. We develop a new mixed-integer program for
the deterministic limited-lookahead setting. We then extend the program to the
robust setting for Stackelberg equilibrium under unlimited and under limited
lookahead by the opponent. We show that for the specific case of interval
uncertainty about the opponent's payoffs (or about the opponent's node
evaluations in the case of limited lookahead), robust Stackelberg equilibria
can be computed with a mixed-integer program that is of the same asymptotic
size as that for the deterministic setting.Comment: Published at AAAI1
Imperfect-Recall Abstractions with Bounds in Games
Imperfect-recall abstraction has emerged as the leading paradigm for
practical large-scale equilibrium computation in incomplete-information games.
However, imperfect-recall abstractions are poorly understood, and only weak
algorithm-specific guarantees on solution quality are known. In this paper, we
show the first general, algorithm-agnostic, solution quality guarantees for
Nash equilibria and approximate self-trembling equilibria computed in
imperfect-recall abstractions, when implemented in the original
(perfect-recall) game. Our results are for a class of games that generalizes
the only previously known class of imperfect-recall abstractions where any
results had been obtained. Further, our analysis is tighter in two ways, each
of which can lead to an exponential reduction in the solution quality error
bound.
We then show that for extensive-form games that satisfy certain properties,
the problem of computing a bound-minimizing abstraction for a single level of
the game reduces to a clustering problem, where the increase in our bound is
the distance function. This reduction leads to the first imperfect-recall
abstraction algorithm with solution quality bounds. We proceed to show a divide
in the class of abstraction problems. If payoffs are at the same scale at all
information sets considered for abstraction, the input forms a metric space.
Conversely, if this condition is not satisfied, we show that the input does not
form a metric space. Finally, we use these results to experimentally
investigate the quality of our bound for single-level abstraction
Theoretical and Practical Advances on Smoothing for Extensive-Form Games
Sparse iterative methods, in particular first-order methods, are known to be
among the most effective in solving large-scale two-player zero-sum
extensive-form games. The convergence rates of these methods depend heavily on
the properties of the distance-generating function that they are based on. We
investigate the acceleration of first-order methods for solving extensive-form
games through better design of the dilated entropy function---a class of
distance-generating functions related to the domains associated with the
extensive-form games. By introducing a new weighting scheme for the dilated
entropy function, we develop the first distance-generating function for the
strategy spaces of sequential games that has no dependence on the branching
factor of the player. This result improves the convergence rate of several
first-order methods by a factor of , where is the branching
factor of the player, and is the depth of the game tree.
Thus far, counterfactual regret minimization methods have been faster in
practice, and more popular, than first-order methods despite their
theoretically inferior convergence rates. Using our new weighting scheme and
practical tuning we show that, for the first time, the excessive gap technique
can be made faster than the fastest counterfactual regret minimization
algorithm, CFR+, in practice