Dynamic Zero-sum Research Articles

An empirical Bayes approach for estimating skill models for professional darts players

Abstract We perform an exploratory data analysis on a data-set for the top 16 professional darts players from the 2019 season. We use this data-set to fit player skill models which can then be used in dynamic zero-sum games (ZSGs) that model real-world matches between players. We propose an empirical Bayesian approach based on the Dirichlet-Multinomial (DM) model that overcomes limitations in the data. Specifically we introduce two DM-based skill models where the first model borrows strength from other darts players and the second model borrows strength from other regions of the dartboard. We find these DM-based models outperform simpler benchmark models with respect to Brier and Spherical scores, both of which are proper scoring rules. We also show in ZSGs settings that the difference between DM-based skill models and the simpler benchmark models is practically significant. Finally, we use our DM-based model to analyze specific situations that arose in real-world darts matches during the 2019 season.

Research on dynamic robust planning method for active distribution network considering correlation

The universality of load subjects in distribution network brings challenges to the reliability of distribution network planning results. In this paper, a two-stage dynamic robust distribution network planning method considering correlation is proposed. The method evaluates the correlation between random variables using the Spearman rank correlation coefficient, and converts the correlated random variables into mutually independent random variables by Cholesky decomposition and independent transformation; expresses the source-load uncertainty by a bounded interval without distribution, and describes the active distribution network planning as a dynamic zero-sum game problem by combining with the two-phase dynamic robust planning; use the Benders decomposition approach to tackle the issue; mathematical simulation is used to confirm the accuracy and efficacy of the method. The results show that the dynamic robustness planning method of active distribution network taking into account the correlation can accurately simulate the operation of active distribution network with uncertain boundaries, which enhances the reliability and economy of the active distribution network planning results.

Optimal defense strategy for AC/DC hybrid power grid cascading failures based on game theory and deep reinforcement learning

This paper proposes a two-person multi-stage zero-sum game model considering the confrontation between cascading failures and control strategies in an AC/DC hybrid system to solve the blocking problem of DC systems caused by successive failures at the receiving end of an AC/DC system. A game model is established between an attacker (power grid failure) and a defender (dispatch side). From the attacker’s perspective, this study mainly investigates the problem of system line failures caused by AC or DC blockages. From the perspective of dispatch-side defense, the multiple-feed short-circuit ratio constraint method, output adjustment measures of the energy storage system, sensitivity control, and distance third-segment protection adjustment are used as strategies to reduce system losses. Using as many line return data as possible as samples, the deep Q-network (DQN), a deep reinforcement learning algorithm, is used to obtain the Nash equilibrium of the game model. The corresponding optimal dispatch and defense strategies are also obtained while obtaining the optimal sequence of tripping failures for AC/DC hybrid system cascading failures. Using the improved IEEE 39-node system as an example, the simulation results verify the appropriateness of the two-stage dynamic zero-sum game model to schedule online defense strategies and the effectiveness and superiority of the energy storage system participating in defense adjustment.

Play Like the Pros? Solving the Game of Darts as a Dynamic Zero-Sum Game

The game of darts has enjoyed great growth over the past decade with the perception of darts moving from that of a pub game to a game that is regularly scheduled on prime-time television in many countries such as the United Kingdom, Germany, the Netherlands, and Australia, among others. It involves strategic interactions between two players, but to date, the literature has ignored these interactions. In this paper, we formulate and solve the game of darts as a dynamic zero-sum game (ZSG), and to the best of our knowledge, we are the first to do so. We also estimate individual skill models using a novel data set based on darts matches that were played by the top 16 professional players in the world during the 2019 season. Using the fitted skill models and our ZSG problem formulation, we quantify the importance of playing strategically—that is, taking into account the score and strategy of one’s opponent—when computing an optimal strategy. For top professionals, we find that playing strategically results in an increase in win probability of just 0.2%–0.6% over a single leg but as much as 2.2% over a best-of-31-legs match. Summary of Contribution: Dynamic zero-sum games (ZSGs) are of considerable interest, as they arise in many applications including sports, the management of communication networks, interdiction games, and heads-up poker—an important topic in modern artificial intelligence. In this study we consider the game of darts, which is growing increasingly popular around the world today. We formulate the game of darts as a ZSG and solve it iteratively by formulating each player’s best-response problem as a stochastic shortest-path (SSP) problem. We then solve these SSPs using standard dynamic programming methods. In solving the ZSG, we are able to accurately quantify the importance of top professionals playing strategically.

Tripartite Dynamic Zero-Sum Quantum Games.

The Nash equilibrium plays a crucial role in game theory. Most of results are based on classical resources. Our goal in this paper is to explore multipartite zero-sum game with quantum settings. We find that in two different settings there is no strategy for a tripartite classical game being fair. Interestingly, this is resolved by providing dynamic zero-sum quantum games using single quantum state. Moreover, the gains of some players may be changed dynamically in terms of the committed state. Both quantum games are robust against the preparation noise and measurement errors.

Tauberian Theorem for Value Functions

For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long run average cost as the time horizon tends to infinity and the limit of value functions of $\lambda$-discounted games as the discount tends to zero. We prove that the Dynamic Programming Principle for value functions directly leads to the Tauberian Theorem---that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub- and super- optimality principles (the Dynamic Programming Principle). With their aid, we obtain certain inequalities on asymptotics of sub- and super- solutions, which lead to the Tauberian Theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point; a very simple stochastic game model is also considered.

Limit Value of Dynamic Zero-Sum Games with Vanishing Stage Duration

We consider two-person zero-sum games where the players control, at discrete times {tn} induced by a partition Π of ℝ+, a continuous time Markov process. We prove that the limit of the values υΠ exist as the mesh of Π goes to 0. The analysis covers the cases of (1) stochastic games (where both players know the state), and (2) games with unknown state and symmetric signals. The proof is by reduction to deterministic differential games.

Approximations in Dynamic Zero-Sum Games II

We pursue in this paper our study of approximations of values and $\epsilon$-saddle-point policies in dynamic zero-sum games. After extending the general theorem for approximation, we study zero-sum stochastic games with countable state space and unbounded immediate reward. We focus on the expected average payoff criterion. We use some tools developed in [M. M. Tidball and E. Altman, SIAM J. Control Optim., 34 (1996), pp. 311--328] to obtain the convergence of the values as well as the convergence of the $\epsilon$ saddle-point policies in various approximation problems. We consider several schemes of truncation of the state space (e.g., finite state approximation) and approximations of games with discount factor close to one for the game with expected average cost. We use the extension of the general theorem for approximation to study approximations in stochastic games with complete information. Finally, we consider the problem of approximating the sets of policies. We obtain some general results that we apply to a pursuit evasion differential game.

Approximations in Dynamic Zero-Sum Games I

We develop a unifying approach for approximating a “limit” Zero-sum game by a sequence of approximating games. We discuss both the convergence of the values and the convergence of optimal (or “almost” optimal) strategies. Moreover, based on optimal policies for the limit game, we construct policies which are almost optimal for the approximating games. We then apply the general framework to state approximations of stochastic games, to convergence of finite horizon problems to infinite horizon problems, and to convergence in the discount factor and in the immediate reward.

Über die optimalität von strategien in stochastischen dynamischen minimax-entscheidungsmodellen I

Stochastic dynamic zero-sum two-person games are considered. The maximal total expected costs of the first player are the criterion where the maximum is taken on all of the strategies of the second player. Sufficient conditions of the existence of optimal resp. (μ, ∊)-optimal strategies are given by methods of dynamic programming.