Strategic Reasoning in Game Theory

Game theory in AI is a powerful mathematical framework largely applied in the last three decades for the strategic reasoning in multi-agent systems. Seminal works along this line started with turn-based two-player games (under perfect and imperfect information) to check the correctness of a system against an unpredictable environment. Then, large effort has been devoted to extend those works to the multi-agent setting and, specifically, to efficiently reasoning about important solution concepts such as Nash Equilibria and the like. Breakthrough contributions along this direction concern the introduction of logics for the strategic reasoning such as Alternating-time Temporal Logic (ATL), Strategy Logic (SL), and their extensions. Two-player games and logics for the strategic reasoning are nowadays very active areas of research. In this thesis we significantly advance the work along both these two lines of research by providing fresh studies and results of practical application. We start with two-player reachability games and first investigate the problem of checking whether a designed player has more than a winning strategy to reach a target. We investigate this question under both perfect and imperfect information. We provide an automata-based solution that requires linear-time, in the perfect information setting, and exponential-time, in the imperfect one. In both cases, the results are tight. Then, we move to multi-player concurrent games and study the following specific setting: (i) Player_0's objective is to reach a target W, and (ii) the opponents are trying to stop this but have partial observation about Player_0's actions. We study the problem of deciding whether the opponents can prevent Player_0 to reach W. We show, using an automata-theoretic approach that, assuming the opponents have the same partial observation and play under uniformity, the problem is in ExpTime. We recall that, in general, multi-player reachability games with imperfect information are undecidable. Then, we move to the more expressive framework of logics for the strategic reasoning. We first introduce and study two graded extensions of SL, namely GSL and GradedSL. By the former, we define a graded version over single strategy variables, i.e. "there exist at least g different strategies", where the strategies are counted semantically. We study the model checking-problem for GSL and show that for its fragment called vanilla GSL[1G] the problem is PTime-complete. By GradedSL, we consider a graded version over tuple of strategy variables and use a syntactic counting over strategies. By means of GradedSL we show how to count the number of different strategy profiles that are Nash equilibria (NE). By analyzing the structure of the specific formulas involved, we conclude that the important problem of checking for the existence of a unique NE can be solved in 2ExpTime, which is not harder than merely checking for the existence of such an equilibrium. Finally, we adopt the view of bounded rationality, and look only at "simple" strategies in specifications of agents’ abilities. We formally define what "simple" means, and propose a variant of plain ATL, namely NatATL, that takes only such strategies into account. We study the model checking problem for the resulting semantics of ability and obtain tight results. The positive results achieved with NatATL encourage for the investigation of simple strategies over more powerful logics, including SL

Searching for a Solution to Program Verification=Equation Solving in CCS

International audienceUnder non-exponential discounting, we develop a dynamic theory for stopping problems in continuous time. Our framework covers discount functions that induce decreasing impatience. Due to the inherent time inconsistency, we look for equilibrium stopping policies, formulated as fixed points of an operator. Under appropriate conditions, fixed-point iterations converge to equilibrium stopping policies. This iterative approach corresponds to the hierarchy of strategic reasoning in game theory and provides “agent-specific” results: it assigns one specific equilibrium stopping policy to each agent according to her initial behavior. In particular, it leads to a precise mathematical connection between the naive behavior and the sophisticated one. Our theory is illustrated in a real options model

Dumbing down rational players: Learning and teaching in an experimental game

This paper uses experimental data to examine the existence of a teaching strategy among boundedly rational players. If players realize that their own actions modify their opponents' beliefs and actions, they might play certain actions to this specific end and forego immedi- ate payoffs if the expected payoff gain from a teaching strategy is high enough. Our results support the existence of a teaching strategy in several ways. After exhibiting some regular- ities consistent with teaching, we examine more precisely the existence of such a strategy. First we show that players update their beliefs in order to take account of the reaction of their opponents to their own action. Second, we examine whether players actually use a teaching strategy by playing an action that induces a poor immediate payoff but is likely to modify the opponent's behavior so that a preferable outcome might emerge in the future. We find strong evidence of such a strategy in the data and confirm this finding within a logistic model that suggests that the future expected payoff that could arise from a teach- ing strategy has indeed a significant impact on choice probabilities. Finally, we investigate the effective impact of a teaching strategy on achieved outcomes and find that more tena- cious teachers can successfully use such a strategy in order to reach their favorite outcome at the expense of their opponents.Game theory; Teaching; Beliefs; Experiment

Discounting in Strategy Logic

Discounting is an important dimension in multi-agent systems as long as we want to reason about strategies and time. It is a key aspect in economics as it captures the intuition that the far-away future is not as important as the near future. Traditional verification techniques allow to check whether there is a winning strategy for a group of agents but they do not take into account the fact that satisfying a goal sooner is different from satisfying it after a long wait. In this paper, we augment Strategy Logic with future discounting over a set of discounted functions D, denoted SLdisc[D]. We consider "until" operators with discounting functions: the satisfaction value of a specification in SLdisc[D] is a value in [0, 1], where the longer it takes to fulfill requirements, the smaller the satisfaction value is. We motivate our approach with classical examples from Game Theory and study the complexity of model-checking SLdisc[D]-formulas.Comment: Extended version of the paper accepted at IJCAI 202

Dumbing down rational players: learning and teaching in an experimental game

"This paper uses experimental data to examine the existence of a teaching strategy among boundedly rational players. If players realize that their own actions modify their opponents’ beliefs and actions, they might play certain actions to this specific end and forego immediate payoffs if the expected payoff gain from a teaching strategy is high enough. Our results support the existence of a teaching strategy in several ways. After exhibiting some regularities consistent with teaching, we examine more precisely the existence of such a strategy. First we show that players update their beliefs in order to take account of the reaction of their opponents to their own action. Second, we examine whether players actually use a teaching strategy by playing an action that induces a poor immediate payoff but is likely to modify the opponent's behavior so that a preferable outcome might emerge in the future. We find strong evidence of such a strategy in the data and confirm this finding within a logistic model that suggests that the future expected payoff that could arise from a teaching strategy has indeed a significant impact on choice probabilities. Finally, we investigate the effective impact of a teaching strategy on achieved outcomes and find that more tenacious teachers can successfully use such a strategy in order to reach their favorite outcome at the expense of their opponents." [author's abstract

The Optimal Equilibrium for Time-Inconsistent Stopping Problems - the Discrete-Time Case

We study an infinite-horizon discrete-time optimal stopping problem under non-exponential discounting. A new method, which we call the iterative approach, is developed to find subgame perfect Nash equilibria. When the discount function induces decreasing impatience, we establish the existence of an equilibrium through fixed-point iterations. Moreover, we show that there exists a unique optimal equilibrium, which generates larger value than any other equilibrium does at all times. To the best of our knowledge, this is the first time a dominating subgame perfect Nash equilibrium is shown to exist in the literature of time-inconsistency.</p