Long games and sigma-projective sets

We prove a number of results on the determinacy of $\sigma$-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between $\sigma$-projective determinacy and the determinacy of certain classes of games of variable length ${<}\omega^2$ (Theorem 2.4). We then give an elementary proof of the determinacy of $\sigma$-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the proof to obtain proofs of the determinacy of $\sigma$-projective games of a given countable length and of games with payoff in the smallest $\sigma$-algebra containing the projective sets, from corresponding assumptions (Theorems 5.1 and 5.4)

Blackwell Games

Blackwell games are infinite games of imperfect information. The two players simultaneously make their moves, and are then informed of each other's moves. Payoff is determined by a Borel measurable function $f$ on the set of possible resulting sequences of moves. A standard result in Game Theory is that finite games of this type are determined. Blackwell proved that infinite games are determined, but only for the cases where the payoff function is the indicator function of an open or $G_\delta$ set. For games of perfect information, determinacy has been proven for games of arbitrary Borel complexity. In this paper I prove the determinacy of Blackwell games over a $G_{\delta\sigma}$ set, in a manner similar to Davis' proof of determinacy of games of $G_{\delta\sigma}$ complexity of perfect information. There is also extensive literature about the consequences of assuming AD, the axiom that _all_ such games of perfect information are determined. In the final section of this paper I formulate an analogous axiom for games of imperfect information, and explore some of the consequences of this axiom

The Natural Rate Hypothesis and Real Determinacy

The uniqueness of bounded local equilibria under interest rate rules is analyzed in a model with sticky information `a la Mankiw and Reis (2002). The main results are tighter bounds on monetary policy than in sticky-price models, irrelevance of the degree of output-gap targeting for determinacy, independence of determinacy regions from parameters outside the interest-rate rule, and equivalence between real determinacy in models satisfying the natural rate hypothesis and nominal determinacy in the associated full-information, flex-price equivalent. The analysis follows from boundedness considerations on the nonautonomous recursion that describe the MA(¥) representation of variables’ reaction to endogenous fluctuations.Nonautonomous difference equations; Indeterminacy; Taylor rule; Sticky information; Sticky prices

Average-energy games

Two-player quantitative zero-sum games provide a natural framework to synthesize controllers with performance guarantees for reactive systems within an uncontrollable environment. Classical settings include mean-payoff games, where the objective is to optimize the long-run average gain per action, and energy games, where the system has to avoid running out of energy. We study average-energy games, where the goal is to optimize the long-run average of the accumulated energy. We show that this objective arises naturally in several applications, and that it yields interesting connections with previous concepts in the literature. We prove that deciding the winner in such games is in NP inter coNP and at least as hard as solving mean-payoff games, and we establish that memoryless strategies suffice to win. We also consider the case where the system has to minimize the average-energy while maintaining the accumulated energy within predefined bounds at all times: this corresponds to operating with a finite-capacity storage for energy. We give results for one-player and two-player games, and establish complexity bounds and memory requirements.Comment: In Proceedings GandALF 2015, arXiv:1509.0685

Recursive Concurrent Stochastic Games

We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent analysis of recursive simple stochastic games to a concurrent setting where the two players choose moves simultaneously and independently at each state. For multi-exit games, our earlier work already showed undecidability for basic questions like termination, thus we focus on the important case of single-exit RCSGs (1-RCSGs). We first characterize the value of a 1-RCSG termination game as the least fixed point solution of a system of nonlinear minimax functional equations, and use it to show PSPACE decidability for the quantitative termination problem. We then give a strategy improvement technique, which we use to show that player 1 (maximizer) has \epsilon-optimal randomized Stackless & Memoryless (r-SM) strategies for all \epsilon > 0, while player 2 (minimizer) has optimal r-SM strategies. Thus, such games are r-SM-determined. These results mirror and generalize in a strong sense the randomized memoryless determinacy results for finite stochastic games, and extend the classic Hoffman-Karp strategy improvement approach from the finite to an infinite state setting. The proofs in our infinite-state setting are very different however, relying on subtle analytic properties of certain power series that arise from studying 1-RCSGs. We show that our upper bounds, even for qualitative (probability 1) termination, can not be improved, even to NP, without a major breakthrough, by giving two reductions: first a P-time reduction from the long-standing square-root sum problem to the quantitative termination decision problem for finite concurrent stochastic games, and then a P-time reduction from the latter problem to the qualitative termination problem for 1-RCSGs.Comment: 21 pages, 2 figure