??-Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games

We show that the problem of deciding whether in a multi-player perfect information recursive game (i.e. a stochastic game with terminal rewards) there exists a stationary Nash equilibrium ensuring each player a certain payoff is ??-complete. Our result holds for acyclic games, where a Nash equilibrium may be computed efficiently by backward induction, and even for deterministic acyclic games with non-negative terminal rewards. We further extend our results to the existence of Nash equilibria where a single player is surely winning. Combining our result with known gadget games without any stationary Nash equilibrium, we obtain that for cyclic games, just deciding existence of any stationary Nash equilibrium is ??-complete. This holds for reach-a-set games, stay-in-a-set games, and for deterministic recursive games

Depth Reduction for Circuits with a Single Layer of Modular Counting Gates

We consider the class of constant depth AND/OR circuits augmented with a layer of modular counting gates at the bottom layer, i.e ${AC}^0 circ {MOD}_m$ circuits. We show that the following holds for several types of gates $G$: by adding a gate of type $G$ at the output, it is possible to obtain an equivalent randomized depth 2 circuit of quasipolynomial size consisting of a gate of type $G$ at the output and a layer of modular counting gates, i.e $G circ {MOD}_m$ circuits. The types of gates $G$ we consider are modular counting gates and threshold-style gates. For all of these, strong lower bounds are known for (deterministic) $G circ {MOD}_m$ circuits

The Big Match in Small Space

In this paper we study how to play (stochastic) games optimally using little space. We focus on repeated games with absorbing states, a type of two-player, zero-sum concurrent mean-payoff games. The prototypical example of these games is the well known Big Match of Gillete (1957). These games may not allow optimal strategies but they always have {\epsilon}-optimal strategies. In this paper we design {\epsilon}-optimal strategies for Player 1 in these games that use only O(log log T ) space. Furthermore, we construct strategies for Player 1 that use space s(T), for an arbitrary small unbounded non-decreasing function s, and which guarantee an {\epsilon}-optimal value for Player 1 in the limit superior sense. The previously known strategies use space {\Omega}(logT) and it was known that no strategy can use constant space if it is {\epsilon}-optimal even in the limit superior sense. We also give a complementary lower bound. Furthermore, we also show that no Markov strategy, even extended with finite memory, can ensure value greater than 0 in the Big Match, answering a question posed by Abraham Neyman

Computational Complexity of Computing a Quasi-Proper Equilibrium

We study the computational complexity of computing or approximating a quasi-proper equilibrium for a given finite extensive form game of perfect recall. We show that the task of computing a symbolic quasi-proper equilibrium is $\mathrm{PPAD}$-complete for two-player games. For the case of zero-sum games we obtain a polynomial time algorithm based on Linear Programming. For general $n$-player games we show that computing an approximation of a quasi-proper equilibrium is $\mathrm{FIXP}_a$-complete.Comment: Full version of paper to appear at the 23rd International Symposium on Fundamentals of Computation Theory (FCT 2021

On the complexity of Pareto-optimal and envy-free lotteries

We study the classic problem of dividing a collection of indivisible resources in a fair and efficient manner among a set of agents having varied preferences. Pareto optimality is a standard notion of economic efficiency, which states that it should be impossible to find an allocation that improves some agent's utility without reducing any other's. On the other hand, a fundamental notion of fairness in resource allocation settings is that of envy-freeness, which renders an allocation to be fair if every agent (weakly) prefers her own bundle over that of any other agent's bundle. Unfortunately, an envy-free allocation may not exist if we wish to divide a collection of indivisible items. Introducing randomness is a typical way of circumventing the non-existence of solutions, and therefore, allocation lotteries, i.e., distributions over allocations have been explored while relaxing the notion of fairness to ex-ante envy freeness. We consider a general fair division setting with $n$ agents and a family of admissible $n$-partitions of an underlying set of items. Every agent is endowed with partition-based utilities, which specify her cardinal utility for each bundle of items in every admissible partition. In such fair division instances, Cole and Tao (2021) have proved that an ex-ante envy-free and Pareto-optimal allocation lottery is always guaranteed to exist. We strengthen their result while examining the computational complexity of the above total problem and establish its membership in the complexity class PPAD. Furthermore, for instances with a constant number of agents, we develop a polynomial-time algorithm to find an ex-ante envy-free and Pareto-optimal allocation lottery. On the negative side, we prove that maximizing social welfare over ex-ante envy-free and Pareto-optimal allocation lotteries is NP-hard.Comment: 22 page