13,337 research outputs found
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
Quasi-Perfect Stackelberg Equilibrium
Equilibrium refinements are important in extensive-form (i.e., tree-form)
games, where they amend weaknesses of the Nash equilibrium concept by requiring
sequential rationality and other beneficial properties. One of the most
attractive refinement concepts is quasi-perfect equilibrium. While
quasi-perfection has been studied in extensive-form games, it is poorly
understood in Stackelberg settings---that is, settings where a leader can
commit to a strategy---which are important for modeling, for example, security
games. In this paper, we introduce the axiomatic definition of quasi-perfect
Stackelberg equilibrium. We develop a broad class of game perturbation schemes
that lead to them in the limit. Our class of perturbation schemes strictly
generalizes prior perturbation schemes introduced for the computation of
(non-Stackelberg) quasi-perfect equilibria. Based on our perturbation schemes,
we develop a branch-and-bound algorithm for computing a quasi-perfect
Stackelberg equilibrium. It leverages a perturbed variant of the linear program
for computing a Stackelberg extensive-form correlated equilibrium. Experiments
show that our algorithm can be used to find an approximate quasi-perfect
Stackelberg equilibrium in games with thousands of nodes
Leadership in Singleton Congestion Games: What is Hard and What is Easy
We study the problem of computing Stackelberg equilibria Stackelberg games
whose underlying structure is in congestion games, focusing on the case where
each player can choose a single resource (a.k.a. singleton congestion games)
and one of them acts as leader. In particular, we address the cases where the
players either have the same action spaces (i.e., the set of resources they can
choose is the same for all of them) or different ones, and where their costs
are either monotonic functions of the resource congestion or not. We show that,
in the case where the players have different action spaces, the cost the leader
incurs in a Stackelberg equilibrium cannot be approximated in polynomial time
up to within any polynomial factor in the size of the game unless P = NP,
independently of the cost functions being monotonic or not. We show that a
similar result also holds when the players have nonmonotonic cost functions,
even if their action spaces are the same. Differently, we prove that the case
with identical action spaces and monotonic cost functions is easy, and propose
polynomial-time algorithm for it. We also improve an algorithm for the
computation of a socially optimal equilibrium in singleton congestion games
with the same action spaces without leadership, and extend it to the
computation of a Stackelberg equilibrium for the case where the leader is
restricted to pure strategies. For the cases in which the problem of finding an
equilibrium is hard, we show how, in the optimistic setting where the followers
break ties in favor of the leader, the problem can be formulated via
mixed-integer linear programming techniques, which computational experiments
show to scale quite well
Equilibria in Sequential Allocation
Sequential allocation is a simple mechanism for sharing multiple indivisible
items. We study strategic behavior in sequential allocation. In particular, we
consider Nash dynamics, as well as the computation and Pareto optimality of
pure equilibria, and Stackelberg strategies. We first demonstrate that, even
for two agents, better responses can cycle. We then present a linear-time
algorithm that returns a profile (which we call the "bluff profile") that is in
pure Nash equilibrium. Interestingly, the outcome of the bluff profile is the
same as that of the truthful profile and the profile is in pure Nash
equilibrium for \emph{all} cardinal utilities consistent with the ordinal
preferences. We show that the outcome of the bluff profile is Pareto optimal
with respect to pairwise comparisons. In contrast, we show that an assignment
may not be Pareto optimal with respect to pairwise comparisons even if it is a
result of a preference profile that is in pure Nash equilibrium for all
utilities consistent with ordinal preferences. Finally, we present a dynamic
program to compute an optimal Stackelberg strategy for two agents, where the
second agent has a constant number of distinct values for the items
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