353 research outputs found
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
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
Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case
The search problem of computing a Stackelberg (or leader-follower)equilibrium (also referred to as an optimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the optimistic and pessimistic versions of the problem, i.e., those where the single follower breaks any ties among multiple equilibria either in favour or against the leader, are solved with different methodologies, both cases allow for efficient, polynomial-time algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers' game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader's commitment, play a Nash equilibrium. When both leader and followers are allowed to play mixed strategies, the corresponding search problem, both in the optimistic and pessimistic versions, is known to be inapproximable in polynomial time to within any multiplicative polynomial factor unless . Exact algorithms are known only for the optimistic case. We focus on the case where the followers play pure strategies—a restriction that applies to a number of real-world scenarios and which, in principle, makes the problem easier—under the assumption of pessimism (the optimistic version of the problem can be straightforwardly solved in polynomial time). After casting this search problem (with followers playing pure strategies) as a pessimistic bilevel programming problem, we show that, with two followers, the problem is NP-hard and, with three or more followers, it cannot be approximated in polynomial time to within any multiplicative factor which is polynomial in the size of the normal-form game, nor, assuming utilities in [0, 1], to within any constant additive loss stricly smaller than 1 unless . This shows that, differently from what happens in the optimistic version, hardness and inapproximability in the pessimistic problem are not due to the adoption of mixed strategies. We then show that the problem admits, in the general case, a supremum but not a maximum, and we propose a single-level mathematical programming reformulation which asks for the maximization of a nonconcave quadratic function over an unbounded nonconvex feasible region defined by linear and quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-the-art methods, we propose an exact ad hoc algorithm (which we also embed within a branch-and-bound scheme) capable of computing the supremum of the problem and, for cases where there is no leader's strategy where such value is attained, also an -approximate strategy where is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computational experiments on a well-established testbed of game instances
Leadership Games: Multiple Followers, Multiple Leaders, and Perfection
AbstractOver the last years,algorithmic game theoryhas received growing interest in AI, as it allows to tackle complex real-world scenarios involving multiple artificial agents engaged in a competitive interaction. These settings call for rational agents endowed with the capability of reasoning strategically, which is achieved by exploitingequilibriumconcepts from game theory
Signaling in Bayesian Network Congestion Games: the Subtle Power of Symmetry
Network congestion games are a well-understood model of multi-agent strategic
interactions. Despite their ubiquitous applications, it is not clear whether it
is possible to design information structures to ameliorate the overall
experience of the network users. We focus on Bayesian games with atomic
players, where network vagaries are modeled via a (random) state of nature
which determines the costs incurred by the players. A third-party entity---the
sender---can observe the realized state of the network and exploit this
additional information to send a signal to each player. A natural question is
the following: is it possible for an informed sender to reduce the overall
social cost via the strategic provision of information to players who update
their beliefs rationally? The paper focuses on the problem of computing optimal
ex ante persuasive signaling schemes, showing that symmetry is a crucial
property for its solution. Indeed, we show that an optimal ex ante persuasive
signaling scheme can be computed in polynomial time when players are symmetric
and have affine cost functions. Moreover, the problem becomes NP-hard when
players are asymmetric, even in non-Bayesian settings
Signaling in Posted Price Auctions
We study single-item single-unit Bayesian posted price auctions, where buyers
arrive sequentially and their valuations for the item being sold depend on a
random, unknown state of nature. The seller has complete knowledge of the
actual state and can send signals to the buyers so as to disclose information
about it. For instance, the state of nature may reflect the condition and/or
some particular features of the item, which are known to the seller only. The
problem faced by the seller is about how to partially disclose information
about the state so as to maximize revenue. Unlike classical signaling problems,
in this setting, the seller must also correlate the signals being sent to the
buyers with some price proposals for them. This introduces additional
challenges compared to standard settings. We consider two cases: the one where
the seller can only send signals publicly visible to all buyers, and the case
in which the seller can privately send a different signal to each buyer. As a
first step, we prove that, in both settings, the problem of maximizing the
seller's revenue does not admit an FPTAS unless P=NP, even for basic instances
with a single buyer. As a result, in the rest of the paper, we focus on
designing PTASs. In order to do so, we first introduce a unifying framework
encompassing both public and private signaling, whose core result is a
decomposition lemma that allows focusing on a finite set of possible buyers'
posteriors. This forms the basis on which our PTASs are developed. In
particular, in the public signaling setting, our PTAS employs some ad hoc
techniques based on linear programming, while our PTAS for the private setting
relies on the ellipsoid method to solve an exponentially-sized LP in polynomial
time. In the latter case, we need a custom approximate separation oracle, which
we implement with a dynamic programming approach
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