Quantitative games with interval objectives

Traditionally quantitative games such as mean-payoff games and discount sum games have two players -- one trying to maximize the payoff, the other trying to minimize it. The associated decision problem, "Can Eve (the maximizer) achieve, for example, a positive payoff?" can be thought of as one player trying to attain a payoff in the interval $(0,\infty)$. In this paper we consider the more general problem of determining if a player can attain a payoff in a finite union of arbitrary intervals for various payoff functions (liminf, mean-payoff, discount sum, total sum). In particular this includes the interesting exact-value problem, "Can Eve achieve a payoff of exactly (e.g.) 0?"Comment: Full version of CONCUR submissio

Role of the effective payoff function in evolutionary game dynamics

In most studies regarding evolutionary game dynamics, the effective payoff, a quantity that translates the payoff derived from game interactions into reproductive success, is usually assumed to be a specific function of the payoff. Meanwhile, the effect of different function forms of effective payoff on evolutionary dynamics is always left in the basket. With introducing a generalized mapping that the effective payoff of individuals is a non-negative function of two variables on selection intensity and payoff, we study how different effective payoff functions affect evolutionary dynamics in a symmetrical mutation-selection process. For standard two-strategy two-player games, we find that under weak selection the condition for one strategy to dominate the other depends not only on the classical {\sigma}-rule, but also on an extra constant that is determined by the form of the effective payoff function. By changing the sign of the constant, we can alter the direction of strategy selection. Taking the Moran process and pairwise comparison process as specific models in well-mixed populations, we find that different fitness or imitation mappings are equivalent under weak selection. Moreover, the sign of the extra constant determines the direction of one-third law and risk-dominance for sufficiently large populations. This work thus helps to elucidate how the effective payoff function as another fundamental ingredient of evolution affect evolutionary dynamics.Comment: This paper has been accepted to publish on EP

Looking at Mean-Payoff through Foggy Windows

Mean-payoff games (MPGs) are infinite duration two-player zero-sum games played on weighted graphs. Under the hypothesis of perfect information, they admit memoryless optimal strategies for both players and can be solved in NP-intersect-coNP. MPGs are suitable quantitative models for open reactive systems. However, in this context the assumption of perfect information is not always realistic. For the partial-observation case, the problem that asks if the first player has an observation-based winning strategy that enforces a given threshold on the mean-payoff, is undecidable. In this paper, we study the window mean-payoff objectives that were introduced recently as an alternative to the classical mean-payoff objectives. We show that, in sharp contrast to the classical mean-payoff objectives, some of the window mean-payoff objectives are decidable in games with partial-observation

Mean-payoff Automaton Expressions

Quantitative languages are an extension of boolean languages that assign to each word a real number. Mean-payoff automata are finite automata with numerical weights on transitions that assign to each infinite path the long-run average of the transition weights. When the mode of branching of the automaton is deterministic, nondeterministic, or alternating, the corresponding class of quantitative languages is not robust as it is not closed under the pointwise operations of max, min, sum, and numerical complement. Nondeterministic and alternating mean-payoff automata are not decidable either, as the quantitative generalization of the problems of universality and language inclusion is undecidable. We introduce a new class of quantitative languages, defined by mean-payoff automaton expressions, which is robust and decidable: it is closed under the four pointwise operations, and we show that all decision problems are decidable for this class. Mean-payoff automaton expressions subsume deterministic mean-payoff automata, and we show that they have expressive power incomparable to nondeterministic and alternating mean-payoff automata. We also present for the first time an algorithm to compute distance between two quantitative languages, and in our case the quantitative languages are given as mean-payoff automaton expressions