The Complexity of Simple Stochastic Games
Jonas Dieckelmann
在本文中，我们调查的计算时间复杂度的各种简单的随机博弈问题，我们给出了一个概述的最知名的算法与每个问题。
In this paper we survey the computational time complexity of assorted simple stochastic game problems, and we give an overview of the best known algorithms associated with each problem.
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On the Complexity of Deciding Degeneracy in Games
证明了判定双矩阵对策是否退化是NP完全的，判定双矩阵对策是否非退化是Co-NP完全的。
In this brief note, we prove that the existence of Nash equilibria on integer programming games is $Σ^p_2$-complete.
We prove communication complexity lower bounds for (possibly mixed) Nash equilibrium in potential games. In particular, we show that finding a Nash equilibrium requires $poly(N)$ communication in two-player $N \times N$ potential games, and $2^{poly(n)}$ communication in $n$-player two-action games. To the best of our knowledge, these are the first results to demonstrate hardness in any model of (possibly mixed) Nash equilibrium in potential games.
Communication Complexity of Correlated Equilibrium in Two-Player Games
我们发现了一个通信复杂性下界，找到一个相关的两个玩家的游戏均衡。更确切地说，我们定义了一个两个球员的$N \times N$游戏称为2-循环游戏，并表明，随机通信复杂性找到1/聚（$N$）近似相关平衡的2-循环游戏是$Ω(N)$。对于小的近似值，这回答了Babichenko和Rubinstein（STOC 2017）的一个悬而未决的问题。我们的下界是通过从唯一的集合不相交性问题的直接还原得到的。
Reducing Stochastic Games to Semidefinite Program Feasibility
通过多项式时间还原，将随机游戏问题转化为半定程序可行性问题，扩展了半定规划在博弈论中的应用范围。
This document collects the lecture notes from my mini-course "Complexity Theory, Game Theory, and Economics," taught at the Bellairs Research Institute of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th McGill Invitational Workshop on Computational Complexity. The goal of this mini-course is twofold: (i) to explain how complexity theory has helped illuminate several barriers in economics and game theory; and (ii) to illustrate how game-theoretic questions have led to new and interesting complexity theory, including recent several breakthroughs. It consists of two five…
Combinatorial Game Complexity: An Introduction with Poset Games
偏序集博弈作为数学研究的对象已经有一个多世纪了，但是很少有关于确定这些博弈的重要性质的计算复杂性的文章。在这篇介绍中，我们发展了组合博弈论的基本原理，并主要集中在偏序集博弈上，尼姆也许是其中最著名的例子。我们提出的复杂性结果已知的日期，最近发现的一些。
We study the complexity of computing stationary Nash equilibrium (NE) in n-player infinite-horizon general-sum stochastic games. We focus on the problem of computing NE in such stochastic games when each player is restricted to choosing a stationary policy and rewards are discounted. First, we prove that computing such NE is in PPAD (in addition to clearly being PPAD-hard). Second, we consider turn-based specializations of such games where at each state there is at most a single player that can take actions and show that these (seemingly-simpler) games remain PPAD-hard. Third, we show that und…
In this thesis, we survey techniques and results from the study of Complexity Theory and Games. We then apply these techniques to obtain new results for previously unstudied games. Our contributions in the games Hexiom, Cut the Rope, and Back to Bed may be helpful in further studies by exploiting structure common to several games. We also highlight some interesting paths for further study related to uncertainty that have yet to receive thorough study given their prevalence in today's games.
An $O(n^{0.4732})$ upper bound on the complexity of the GKS communication game
给出了G. Gilmer，M. Kouckelman和M. Saks \cite{saks}研究了灵敏度猜想\cite{linial}，改进了它们的$\sqrt{999\over 1000}\sqrt{n}$界。我们还确定了确切的复杂性的游戏高达$n\le 9$。
Computational Complexity of Game Boy Games
本文分析了任天堂Game Boy平台四款流行游戏的广义版本复杂度，通过Karp归约提供了其NP-hard的原创证明，并讨论了其他相关游戏的NP-hardness。
We study algorithms for solving Subtraction games, which sometimes are referred to as one-heap Nim games. We describe a quantum algorithm which is applicable to any game on DAG, and show that its query compexity for solving an arbitrary Subtraction game of $n$ stones is $O(n^{3/2}\log n)$. The best known deterministic algorithms for solving such games are based on the dynamic programming approach. We show that this approach is asymptotically optimal and that classical query complexity for solving a Subtraction game is generally $Θ(n^2)$. This paper perhaps is the first explicit "quantum" contr…
On the computational complexity of a game of cops and robbers
研究了Aigner和Fromme提出的完全信息二人对策的计算复杂性。游戏发生在一个无向图上，n个同时移动的警察试图抓住一个抢劫犯，所有人都以相同的速度移动。玩家可以在第一步选择他们的起始位置。决定这个博弈的计算复杂性的问题是由Goldstein和Reingold在90年代提出的。我们证明了这个博弈对于PSPACE是困难的。
The Computational Complexity of Portal and Other 3D Video Games
我们分类的计算复杂性流行的视频游戏门户和门户2。我们隔离个别机制的游戏，并证明NP-硬度，PSPACE-完整性，或（伪）多项式取决于特定的游戏机制允许。我们的一个证明推广到证明许多其他视频游戏的NP-硬度，如半条命2，光环，末日，上古卷轴，辐射，侠盗猎车手，左4死，质量效应，Deus Ex，合金装备固体和生化危机。这些结果建立在视频游戏复杂性的现有文献基础上。
Complexity of the Game Domination Problem
博弈控制数是由博弈产生的图不变量，它与图控制的关系类似于博弈色数与图着色的关系。在本文中，我们证明了验证一个图的博弈控制数是否有界于一个给定的整数是PSPACE-完全的。这与复杂性仍然未知的博弈着色问题的情况形成对比。
This paper is concerned with complexity theoretic aspects of a general formulation of quantum game theory that models strategic interactions among rational agents that process and exchange quantum information. In particular, we prove that the computational problem of finding an approximate Nash equilibrium in a broad class of quantum games is, like the analogous problem for classical games, included in (and therefore complete for) the complexity class PPAD. Our main technical contribution, which facilitates this inclusion, is an extension of prior methods in computational game theory to strate…
Algorithmic Complexity in Minority Game
在本文中，我们介绍了一种新的方法来研究的复杂性行为的少数者游戏使用的工具，算法的复杂性，物理熵和信息论。我们表明，物理复杂性和互信息函数强烈依赖于内存大小的代理，并产生更多的信息比波动性本身的复杂功能的流的二进制结果的游戏。
Recently, a standardized framework was proposed for introducing quantum-inspired moves in mathematical games with perfect information and no chance. The beauty of quantum games-succinct in representation, rich in structures, explosive in complexity, dazzling for visualization, and sophisticated for strategic reasoning-has drawn us to play concrete games full of subtleties and to characterize abstract properties pertinent to complexity consequence. Going beyond individual games, we explore the tractability of quantum combinatorial games as whole, and address fundamental questions including: Qua…
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