Incentive Stackelberg Mean-payoff Games

We introduce and study incentive equilibria for multi-player meanpayoff games. Incentive equilibria generalise well-studied solution concepts such as Nash equilibria and leader equilibria (also known as Stackelberg equilibria). Recall that a strategy profile is a Nash equilibrium if no player can improve his payoff by changing his strategy unilaterally. In the setting of incentive and leader equilibria, there is a distinguished player called the leader who can assign strategies to all other players, referred to as her followers. A strategy profile is a leader strategy profile if no player, except for the leader, can improve his payoff by changing his strategy unilaterally, and a leader equilibrium is a leader strategy profile with a maximal return for the leader. In the proposed case of incentive equilibria, the leader can additionally influence the behaviour of her followers by transferring parts of her payoff to her followers. The ability to incentivise her followers provides the leader with more freedom in selecting strategy profiles, and we show that this can indeed improve the payoff for the leader in such games. The key fundamental result of the paper is the existence of incentive equilibria in mean-payoff games. We further show that the decision problem related to constructing incentive equilibria is NP-complete. On a positive note, we show that, when the number of players is fixed, the complexity of the problem falls in the same class as two-player mean-payoff games. We also present an implementation of the proposed algorithms, and discuss experimental results that demonstrate the feasibility of the analysis of medium sized games.Comment: 15 pages, references, appendix, 5 figure

Rage Against the Machines: How Subjects Learn to Play Against Computers

We use an experiment to explore how subjects learn to play against computers which are programmed to follow one of a number of standard learning algorithms. The learning theories are (unbeknown to subjects) a best response process, fictitious play, imitation, reinforcement learning, and a trial & error process. We test whether subjects try to influence those algorithms to their advantage in a forward-looking way (strategic teaching). We find that strategic teaching occurs frequently and that all learning algorithms are subject to exploitation with the notable exception of imitation. The experiment was conducted, both, on the internet and in the usual laboratory setting. We find some systematic differences, which however can be traced to the different incentives structures rather than the experimental environment

Privatization in oligopoly : the impact of the shadow cost of public funds

The aim of this paper is to investigate the welfare effect of privatization in oligopoly when the government takes into account the distortionary effect of raising funds by taxation (shadow cost of public funds). We analyze the impact of the change in ownership not only on the objective function of the firms, but also on the timing of competition by endogenizing the determination of simultaneous (Nash-Cournot) versus sequential (Stackelberg) games. We show that, absent efficiency gains, privatization never increases welfare. Moreover, even when large efficiency gains are realized, an inefficient public firm may be preferred

Designing the Game to Play: Optimizing Payoff Structure in Security Games

Effective game-theoretic modeling of defender-attacker behavior is becoming increasingly important. In many domains, the defender functions not only as a player but also the designer of the game's payoff structure. We study Stackelberg Security Games where the defender, in addition to allocating defensive resources to protect targets from the attacker, can strategically manipulate the attacker's payoff under budget constraints in weighted L^p-norm form regarding the amount of change. Focusing on problems with weighted L^1-norm form constraint, we present (i) a mixed integer linear program-based algorithm with approximation guarantee; (ii) a branch-and-bound based algorithm with improved efficiency achieved by effective pruning; (iii) a polynomial time approximation scheme for a special but practical class of problems. In addition, we show that problems under budget constraints in L^0-norm form and weighted L^\infty-norm form can be solved in polynomial time. We provide an extensive experimental evaluation of our proposed algorithms

Endogenous Timing in Pollution Control: Stackelberg versus Cournot-Nash Equilibria

In the framework of international cooperation on climate change to control greenhouse gas emissions (GHG), this paper aims to shed new light on the eventuality of the emergence of a country (or a group of countries) behaving as a leader in the implementation of its environmental policy. The sequence of moves in the existing literature is usually an exogenous assumption, – known as the Cournot assumption (if countries take action simultaneously) and the Stackelberg assumption (if they act sequentially, the latter observing the strategy of the former). The main purpose here is to make the timing endogenous. To do so, we introduce a pre-play stage in the basic two-country game. Then we provide different sets of minimal conditions – on the benefit and damage functions linked to GHG emissions into the atmosphere, yielding respectively the simultaneous and the two sequential modes of play. While the results essentially confirm the prevalence of the former, they also indicate that the latter are natural under some robust conditions: a leader can emerge endogenously when implementing its environmental policy. Finally we provide sufficient conditions for a specific leader to appear. All the results come with an analysis in terms of global emissions and global welfare. No extraneous assumptions such as concavity, existence, or uniqueness of equilibria are needed, and the analysis makes crucial use of the basic results from the theory of supermodular games.Climate change; non cooperative game; global pollution; strategic interactions; endogenous timing; supermodular game theory

When is Reputation Bad?

In traditional reputation theory, reputation is good for the long-run player. In “Bad Reputation,” Ely and Valimaki give an example in which reputation is unambiguously bad. This paper characterizes a more general class of games in which that insight holds, and presents some examples to illustrate when the bad reputation effect does and does not play a role. The key properties are that participation is optional for the short-run players, and that every action of the long-run player that makes the short-run players want to participate has a chance of being interpreted as a signal that the long-run player is “bad. ” We also broaden the set of commitment types, allowing many types, including the “Stackelberg type” used to prove positive results on reputation. Although reputation need not be bad if the probability of the Stackelberg type is too high, the relative probability of the Stackelberg type can be high when all commitment types are unlikely.