541 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
Solving ill-posed bilevel programs
This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem an a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem
Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems
We present optimality conditions for bilevel optimal control problems where the upper level, to be solved by a leader, is a scalar optimal control problem and the lower level, to be solved by several followers, is a multiobjective convex optimal control problem. Multiobjective optimal control problems arise in many application areas where several conflicting objectives need to be considered. Minimize several objective functionals leads to solutions such that none of the objective functional values can be improved further without deteriorating another. The set of all such solutions is referred to as efficient (also called Pareto optimal, noninferior, or nondominated) set of solutions. The lower level of the semivectorial bilevel optimal control problems can be considered to be associated to a ”grande coalition” of a p-player cooperative differential game, every player having its own objective and control function. We consider situations in which these p-?players react as ”followers” to every decision imposed by a ”leader” (who acts at the so-called upper level). The best reply correspondence of the followers being in general non uniquely determined, the leader cannot predict the followers choice simply on the basis of his rational behavior. So, the choice of the best strategy from the leader point of view depends of how the followers choose a strategy among his best responses. In this paper, we will consider two (extreme) possibilities: (i) the optimistic situation, when for every decison of the leader, the followers will choose a strategy amongst the efficient controls which minimizes the (scalar) objective of the leader; in this case the leader will choose a strategy which minimizes the best he can obtain amongst all the best responses of the followers: (ii) the pessimistic situation, when the followers can choose amongst the efficient controls one which maximizes the (scalar) objective of the leader; in this case the leader will choose a strategy which minimizes the worst he could obtain amongst all the best responses of the followers. This paper continues the research initiated in [17] where existence results for these problems have been obtained.
Imitative Follower Deception in Stackelberg Games
Information uncertainty is one of the major challenges facing applications of
game theory. In the context of Stackelberg games, various approaches have been
proposed to deal with the leader's incomplete knowledge about the follower's
payoffs, typically by gathering information from the leader's interaction with
the follower. Unfortunately, these approaches rely crucially on the assumption
that the follower will not strategically exploit this information asymmetry,
i.e., the follower behaves truthfully during the interaction according to their
actual payoffs. As we show in this paper, the follower may have strong
incentives to deceitfully imitate the behavior of a different follower type
and, in doing this, benefit significantly from inducing the leader into
choosing a highly suboptimal strategy. This raises a fundamental question: how
to design a leader strategy in the presence of a deceitful follower? To answer
this question, we put forward a basic model of Stackelberg games with
(imitative) follower deception and show that the leader is indeed able to
reduce the loss due to follower deception with carefully designed policies. We
then provide a systematic study of the problem of computing the optimal leader
policy and draw a relatively complete picture of the complexity landscape;
essentially matching positive and negative complexity results are provided for
natural variants of the model. Our intractability results are in sharp contrast
to the situation with no deception, where the leader's optimal strategy can be
computed in polynomial time, and thus illustrate the intrinsic difficulty of
handling follower deception. Through simulations we also examine the benefit of
considering follower deception in randomly generated games
On the Inducibility of Stackelberg Equilibrium for Security Games
Strong Stackelberg equilibrium (SSE) is the standard solution concept of
Stackelberg security games. As opposed to the weak Stackelberg equilibrium
(WSE), the SSE assumes that the follower breaks ties in favor of the leader and
this is widely acknowledged and justified by the assertion that the defender
can often induce the attacker to choose a preferred action by making an
infinitesimal adjustment to her strategy. Unfortunately, in security games with
resource assignment constraints, the assertion might not be valid; it is
possible that the defender cannot induce the desired outcome. As a result, many
results claimed in the literature may be overly optimistic. To remedy, we first
formally define the utility guarantee of a defender strategy and provide
examples to show that the utility of SSE can be higher than its utility
guarantee. Second, inspired by the analysis of leader's payoff by Von Stengel
and Zamir (2004), we provide the solution concept called the inducible
Stackelberg equilibrium (ISE), which owns the highest utility guarantee and
always exists. Third, we show the conditions when ISE coincides with SSE and
the fact that in general case, SSE can be extremely worse with respect to
utility guarantee. Moreover, introducing the ISE does not invalidate existing
algorithmic results as the problem of computing an ISE polynomially reduces to
that of computing an SSE. We also provide an algorithmic implementation for
computing ISE, with which our experiments unveil the empirical advantage of the
ISE over the SSE.Comment: The Thirty-Third AAAI Conference on Artificial Intelligenc
Optimistic minimax search for noncooperative switched control with or without dwell time
International audienceWe consider adversarial problems in which two agents control two switching signals, the first agent aiming to maximize a discounted sum of rewards, and the second aiming to minimize it. Both signals may be subject to constraints on the dwell time after a switch. We search the tree of possible mode sequences with an algorithm called optimistic minimax search with dwell time (OMSd), showing that it obtains a solution close to the minimax-optimal one, and we characterize the rate at which the suboptimality goes to zero. The analysis is driven by a novel measure of problem complexity, and it is first given in the general dwell-time case, after which it is specialized to the unconstrained case. We exemplify the framework for networked control systems where the minimizer signal is a discrete time delay on the control channel, and we provide extensive simulations and a real-time experiment for nonlinear systems of this type
Fuzzy Bi-level Decision-Making Techniques: A Survey
© 2016 the authors. Bi-level decision-making techniques aim to deal with decentralized management problems that feature interactive decision entities distributed throughout a bi-level hierarchy. A challenge in handling bi-level decision problems is that various uncertainties naturally appear in decision-making process. Significant efforts have been devoted that fuzzy set techniques can be used to effectively deal with uncertain issues in bi-level decision-making, known as fuzzy bi-level decision-making techniques, and researchers have successfully gained experience in this area. It is thus vital that an instructive review of current trends in this area should be conducted, not only of the theoretical research but also the practical developments. This paper systematically reviews up-to-date fuzzy bi-level decisionmaking techniques, including models, approaches, algorithms and systems. It also clusters related technique developments into four main categories: basic fuzzy bi-level decision-making, fuzzy bi-level decision-making with multiple optima, fuzzy random bi-level decision-making, and the applications of bi-level decision-making techniques in different domains. By providing state-of-the-art knowledge, this survey paper will directly support researchers and practitioners in their understanding of developments in theoretical research results and applications in relation to fuzzy bi-level decision-making techniques
Stochastic Multilevel Programming with a Hybrid Intelligent Algorithm
A framework of stochastic multilevel programming is proposed for modelling decentralized decision-making problem in stochastic environment. According to different decision criteria, the stochastic decentralized decision-making problem is formulated as expected value multilevel programming, and chanceconstrained multilevel programming. In order to solve the proposed stochastic multilevel programming models for the Stackelberg-Nash equilibriums, genetic algorithms, neural networks and stochastic simulation are integrated to produce a hybrid intelligent algorithm. Finally, two numerical examples are provided to illustrate the effectiveness of the hybrid intelligent algorithm
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