8 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
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
Robust Stackelberg Equilibria
This paper provides a systematic study of the robust Stackelberg equilibrium
(RSE), which naturally generalizes the widely adopted solution concept of the
strong Stackelberg equilibrium (SSE). The RSE accounts for any possible
up-to- suboptimal follower responses in Stackelberg games and is
adopted to improve the robustness of the leader's strategy. While a few
variants of robust Stackelberg equilibrium have been considered in previous
literature, the RSE solution concept we consider is importantly different -- in
some sense, it relaxes previously studied robust Stackelberg strategies and is
applicable to much broader sources of uncertainties.
We provide a thorough investigation of several fundamental properties of RSE,
including its utility guarantees, algorithmics, and learnability. We first show
that the RSE we defined always exists and thus is well-defined. Then we
characterize how the leader's utility in RSE changes with the robustness level
considered. On the algorithmic side, we show that, in sharp contrast to the
tractability of computing an SSE, it is NP-hard to obtain a fully polynomial
approximation scheme (FPTAS) for any constant robustness level. Nevertheless,
we develop a quasi-polynomial approximation scheme (QPTAS) for RSE. Finally, we
examine the learnability of the RSE in a natural learning scenario, where both
players' utilities are not known in advance, and provide almost tight sample
complexity results on learning the RSE. As a corollary of this result, we also
obtain an algorithm for learning SSE, which strictly improves a key result of
Bai et al. in terms of both utility guarantee and computational efficiency
Faster Game Solving via Predictive Blackwell Approachability: Connecting Regret Matching and Mirror Descent
Blackwell approachability is a framework for reasoning about repeated games
with vector-valued payoffs. We introduce predictive Blackwell approachability,
where an estimate of the next payoff vector is given, and the decision maker
tries to achieve better performance based on the accuracy of that estimator. In
order to derive algorithms that achieve predictive Blackwell approachability,
we start by showing a powerful connection between four well-known algorithms.
Follow-the-regularized-leader (FTRL) and online mirror descent (OMD) are the
most prevalent regret minimizers in online convex optimization. In spite of
this prevalence, the regret matching (RM) and regret matching+ (RM+) algorithms
have been preferred in the practice of solving large-scale games (as the local
regret minimizers within the counterfactual regret minimization framework). We
show that RM and RM+ are the algorithms that result from running FTRL and OMD,
respectively, to select the halfspace to force at all times in the underlying
Blackwell approachability game. By applying the predictive variants of FTRL or
OMD to this connection, we obtain predictive Blackwell approachability
algorithms, as well as predictive variants of RM and RM+. In experiments across
18 common zero-sum extensive-form benchmark games, we show that predictive RM+
coupled with counterfactual regret minimization converges vastly faster than
the fastest prior algorithms (CFR+, DCFR, LCFR) across all games but two of the
poker games and Liar's Dice, sometimes by two or more orders of magnitude
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