Algorithm for Computing Approximate Nash Equilibrium in Continuous Games with Application to Continuous Blotto

Successful algorithms have been developed for computing Nash equilibrium in a variety of finite game classes. However, solving continuous games—in which the pure strategy space is (potentially uncountably) infinite—is far more challenging. Nonetheless, many real-world domains have continuous action spaces, e.g., where actions refer to an amount of time, money, or other resource that is naturally modeled as being real-valued as opposed to integral. We present a new algorithm for approximating Nash equilibrium strategies in continuous games. In addition to two-player zero-sum games, our algorithm also applies to multiplayer games and games with imperfect information. We experiment with our algorithm on a continuous imperfect-information Blotto game, in which two players distribute resources over multiple battlefields. Blotto games have frequently been used to model national security scenarios and have also been applied to electoral competition and auction theory. Experiments show that our algorithm is able to quickly compute close approximations of Nash equilibrium strategies for this game.

A Differential Game of Ecological Compensation Criterion for Transboundary Pollution Abatement under Learning by Doing

This paper studies a stochastic differential game of transboundary pollution abatement between two kinds of ecological compensation and the abatement policy, in which the learning by doing is taken into account. Emission and pollution abatement between upstream and downstream region in the same basin is a Stackelberg game, and the downstream regions provide economic compensation for pollution abatement in the upstream region. We discuss the feedback Nash equilibrium strategies of proportional compensation and investment compensation, and it is found that an appropriate ecological compensation ratio can improve the investment level of pollution abatement in the two regions by accumulating experience in the process of learning by doing. In the long term, the investment compensation mechanism is an effective transboundary pollution abatement measure that can continuously reduce the water pollution stock in the upstream and downstream.

NASH EQUILIBRIUM STRATEGIES REVISITED IN SOFTWARE RELEASE GAMES

A software release game was formulated by Zeephongsekul and Chiera [Zeephongsekul, P. & Chiera, C. (1995). Optimal software release policy based on a two-person game of timing. Journal of Applied Probability 32: 470–481] and was reconsidered by Dohi et al. [Dohi, T., Teraoka, Y., & Osaki, S. (2000). Software release games. Journal of Optimization Theory and Applications 105(2): 325–346] in a framework of two-person nonzero-sum games. In this paper, we further point out the faults in the above literature and revisit the Nash equilibrium strategies in the software release games from the viewpoints of both silent and noisy type of games. It is shown that the Nash equilibrium strategies in the silent and noisy of software release games exist under some parametric conditions.

Optimal Channel Design: A Game Theoretical Analysis

This paper studies the problem of optimal channel design. For a given input probability distribution and for hard and soft design constraints, the aim here is to design a (probabilistic) channel whose output leaks minimally from its input. To analyse this problem, general notions of entropy and information leakage are introduced. It can be shown that, for all notions of leakage here defined, the optimal channel design problem can be solved using convex programming with zero duality gap. Subsequently, the optimal channel design problem is studied in a game-theoretical framework: games allow for analysis of optimal strategies of both the defender and the adversary. It is shown that all channel design problems can be studied in this game-theoretical framework, and that the defender’s Bayes–Nash equilibrium strategies are equivalent to the solutions of the convex programming problem. Moreover, the adversary’s equilibrium strategies correspond to a robust inference problem.

Maximin and Minimax Strategies in Two-Players Game with Two Strategic Variables

We examine maximin and minimax strategies for players in a two-players game with two strategic variables, [Formula: see text] and [Formula: see text]. We consider two patterns of game; one is the [Formula: see text]-game in which the strategic variables of players are [Formula: see text]’s, and the other is the [Formula: see text]-game in which the strategic variables of players are [Formula: see text]’s. We call two players Players A and B, and will show that the maximin strategy and the minimax strategy in the [Formula: see text]-game, and the maximin strategy and the minimax strategy in the [Formula: see text]-game are all equivalent for each player. However, the maximin strategy for Player A and that for Player B are not necessarily equivalent, and they are not necessarily equivalent to their Nash equilibrium strategies in the [Formula: see text]-game nor the [Formula: see text]-game. But, in a special case, where the objective function of Player B is the opposite of the objective function of Player A, the maximin strategy for Player A and that for Player B are equivalent, and they constitute the Nash equilibrium both in the [Formula: see text]-game and the [Formula: see text]-game.

A Stochastic Game-Based Approach for Multiple Beyond-Visual-Range Air Combat

This paper addresses tactical decisions in beyond-visual-range (BVR) air combat between two adversarial teams of multiple (autonomous) aircraft. A BVR combat is formalized as a two-player stochastic game consisting of a sequence of normal-form games that determines on the number of missiles to be allocated to each adversary aircraft; within this normal-form game a continuous sub-game is embedded to determine the missile shooting times. The formulation reduces the size of decision space by taking advantage of the underlying symmetry of the combat scenario, and also facilitates incorporation of the effect of cooperative missile maneuvers and transition into within-visual-range (WVR) combat. The Nash equilibrium strategies and the associate value functions of the game are computed through linear-programming-based dynamic programming procedure. Numerical case studies on combat between airplanes with heterogeneous capabilities and cooperation effects demonstrate the validity of the proposed formulation and the effectiveness of the proposed solution scheme.

Super-Human AI for Strategic Reasoning: Beating Top Pros in Heads-Up No-Limit Texas Hold'em

Poker has been a challenge problem in AI and game theory for decades. As a game of imperfect information it involves obstacles not present in games like chess and Go, and requires totally different techniques. No program had been able to beat top players in large poker games. Until now! In January 2017, our AI, Libratus, beat a team of four top specialist professionals in heads-up no-limit Texas hold'em, which has 10^161 decision points. This game is the main benchmark challenge for imperfect-information game solving. Libratus is the only AI that has beat top humans at this game. Libratus is powered by new algorithms in each of its three main modules: 1. computing blueprint (approximate Nash equilibrium) strategies before the event, 2. novel nested endgame solving during play, and 3. fixing its own strategy to play even closer to equilibrium based on what holes the opponents have been able to identify and exploit. These domain-independent algorithms have potential applicability to a variety of real-world imperfect-information games such as negotiation, business strategy, cybersecurity, physical security, military applications, strategic pricing, product portfolio planning, certain areas of finance, auctions, political campaigns, and steering biological adaptation and evolution, for example, for medical treatment planning.