COMBINATORIAL GAME THEORY in CLOBBER Jeroen Claessen
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Gradient Play in N-Cluster Games with Zero-Order Information
Gradient Play in n-Cluster Games with Zero-Order Information Tatiana Tatarenko, Jan Zimmermann, Jurgen¨ Adamy Abstract— We study a distributed approach for seeking a models, each agent intends to find a strategy to achieve a Nash equilibrium in n-cluster games with strictly monotone Nash equilibrium in the resulting n-cluster game, which is mappings. Each player within each cluster has access to the a stable state that minimizes the cluster’s cost functions in current value of her own smooth local cost function estimated by a zero-order oracle at some query point. We assume the response to the actions of the agents from other clusters. agents to be able to communicate with their neighbors in Continuous time algorithms for the distributed Nash equi- the same cluster over some undirected graph. The goal of libria seeking problem in multi-cluster games were proposed the agents in the cluster is to minimize their collective cost. in [17]–[19]. The paper [17] solves an unconstrained multi- This cost depends, however, on actions of agents from other cluster game by using gradient-based algorithms, whereas the clusters. Thus, a game between the clusters is to be solved. We present a distributed gradient play algorithm for determining works [18] and [19] propose a gradient-free algorithm, based a Nash equilibrium in this game. The algorithm takes into on zero-order information, for seeking Nash and generalized account the communication settings and zero-order information Nash equilibria respectively. In discrete time domain, the under consideration. We prove almost sure convergence of this work [5] presents a leader-follower based algorithm, which algorithm to a Nash equilibrium given appropriate estimations can solve unconstrained multi-cluster games in linear time. -
Alpha-Beta Pruning
Alpha-Beta Pruning Carl Felstiner May 9, 2019 Abstract This paper serves as an introduction to the ways computers are built to play games. We implement the basic minimax algorithm and expand on it by finding ways to reduce the portion of the game tree that must be generated to find the best move. We tested our algorithms on ordinary Tic-Tac-Toe, Hex, and 3-D Tic-Tac-Toe. With our algorithms, we were able to find the best opening move in Tic-Tac-Toe by only generating 0.34% of the nodes in the game tree. We also explored some mathematical features of Hex and provided proofs of them. 1 Introduction Building computers to play board games has been a focus for mathematicians ever since computers were invented. The first computer to beat a human opponent in chess was built in 1956, and towards the late 1960s, computers were already beating chess players of low-medium skill.[1] Now, it is generally recognized that computers can beat even the most accomplished grandmaster. Computers build a tree of different possibilities for the game and then work backwards to find the move that will give the computer the best outcome. Al- though computers can evaluate board positions very quickly, in a game like chess where there are over 10120 possible board configurations it is impossible for a computer to search through the entire tree. The challenge for the computer is then to find ways to avoid searching in certain parts of the tree. Humans also look ahead a certain number of moves when playing a game, but experienced players already know of certain theories and strategies that tell them which parts of the tree to look. -
Game Theory with Translucent Players
Game Theory with Translucent Players Joseph Y. Halpern and Rafael Pass∗ Cornell University Department of Computer Science Ithaca, NY, 14853, U.S.A. e-mail: fhalpern,[email protected] November 27, 2012 Abstract A traditional assumption in game theory is that players are opaque to one another—if a player changes strategies, then this change in strategies does not affect the choice of other players’ strategies. In many situations this is an unrealistic assumption. We develop a framework for reasoning about games where the players may be translucent to one another; in particular, a player may believe that if she were to change strategies, then the other player would also change strategies. Translucent players may achieve significantly more efficient outcomes than opaque ones. Our main result is a characterization of strategies consistent with appro- priate analogues of common belief of rationality. Common Counterfactual Belief of Rationality (CCBR) holds if (1) everyone is rational, (2) everyone counterfactually believes that everyone else is rational (i.e., all players i be- lieve that everyone else would still be rational even if i were to switch strate- gies), (3) everyone counterfactually believes that everyone else is rational, and counterfactually believes that everyone else is rational, and so on. CCBR characterizes the set of strategies surviving iterated removal of minimax dom- inated strategies: a strategy σi is minimax dominated for i if there exists a 0 0 0 strategy σ for i such that min 0 u (σ ; µ ) > max u (σ ; µ ). i µ−i i i −i µ−i i i −i ∗Halpern is supported in part by NSF grants IIS-0812045, IIS-0911036, and CCF-1214844, by AFOSR grant FA9550-08-1-0266, and by ARO grant W911NF-09-1-0281. -
Game Playing
Game Playing Philipp Koehn 29 September 2015 Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Outline 1 ● Games ● Perfect play – minimax decisions – α–β pruning ● Resource limits and approximate evaluation ● Games of chance ● Games of imperfect information Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 2 games Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Games vs. Search Problems 3 ● “Unpredictable” opponent ⇒ solution is a strategy specifying a move for every possible opponent reply ● Time limits ⇒ unlikely to find goal, must approximate ● Plan of attack: – computer considers possible lines of play (Babbage, 1846) – algorithm for perfect play (Zermelo, 1912; Von Neumann, 1944) – finite horizon, approximate evaluation (Zuse, 1945; Wiener, 1948; Shannon, 1950) – first Chess program (Turing, 1951) – machine learning to improve evaluation accuracy (Samuel, 1952–57) – pruning to allow deeper search (McCarthy, 1956) Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Types of Games 4 deterministic chance perfect Chess Backgammon information Checkers Monopoly Go Othello imperfect battleships Bridge information Blind Tic Tac Toe Poker Scrabble Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Game Tree (2-player, Deterministic, Turns) 5 Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Simple Game Tree 6 ● 2 player game ● Each player has one move ● You move first ● Goal: optimize your payoff (utility) Start Your move Opponent -
The Effective Minimax Value of Asynchronously Repeated Games*
Int J Game Theory (2003) 32: 431–442 DOI 10.1007/s001820300161 The effective minimax value of asynchronously repeated games* Kiho Yoon Department of Economics, Korea University, Anam-dong, Sungbuk-gu, Seoul, Korea 136-701 (E-mail: [email protected]) Received: October 2001 Abstract. We study the effect of asynchronous choice structure on the possi- bility of cooperation in repeated strategic situations. We model the strategic situations as asynchronously repeated games, and define two notions of effec- tive minimax value. We show that the order of players’ moves generally affects the effective minimax value of the asynchronously repeated game in significant ways, but the order of moves becomes irrelevant when the stage game satisfies the non-equivalent utilities (NEU) condition. We then prove the Folk Theorem that a payoff vector can be supported as a subgame perfect equilibrium out- come with correlation device if and only if it dominates the effective minimax value. These results, in particular, imply both Lagunoff and Matsui’s (1997) result and Yoon (2001)’s result on asynchronously repeated games. Key words: Effective minimax value, folk theorem, asynchronously repeated games 1. Introduction Asynchronous choice structure in repeated strategic situations may affect the possibility of cooperation in significant ways. When players in a repeated game make asynchronous choices, that is, when players cannot always change their actions simultaneously in each period, it is quite plausible that they can coordinate on some particular actions via short-run commitment or inertia to render unfavorable outcomes infeasible. Indeed, Lagunoff and Matsui (1997) showed that, when a pure coordination game is repeated in a way that only *I thank three anonymous referees as well as an associate editor for many helpful comments and suggestions. -
570: Minimax Sample Complexity for Turn-Based Stochastic Game
Minimax Sample Complexity for Turn-based Stochastic Game Qiwen Cui1 Lin F. Yang2 1School of Mathematical Sciences, Peking University 2Electrical and Computer Engineering Department, University of California, Los Angeles Abstract guarantees are rather rare due to complex interaction be- tween agents that makes the problem considerably harder than single agent reinforcement learning. This is also known The empirical success of multi-agent reinforce- as non-stationarity in MARL, which means when multi- ment learning is encouraging, while few theoret- ple agents alter their strategies based on samples collected ical guarantees have been revealed. In this work, from previous strategy, the system becomes non-stationary we prove that the plug-in solver approach, proba- for each agent and the improvement can not be guaranteed. bly the most natural reinforcement learning algo- One fundamental question in MBRL is that how to design rithm, achieves minimax sample complexity for efficient algorithms to overcome non-stationarity. turn-based stochastic game (TBSG). Specifically, we perform planning in an empirical TBSG by Two-players turn-based stochastic game (TBSG) is a two- utilizing a ‘simulator’ that allows sampling from agents generalization of Markov decision process (MDP), arbitrary state-action pair. We show that the em- where two agents choose actions in turn and one agent wants pirical Nash equilibrium strategy is an approxi- to maximize the total reward while the other wants to min- mate Nash equilibrium strategy in the true TBSG imize it. As a zero-sum game, TBSG is known to have and give both problem-dependent and problem- Nash equilibrium strategy [Shapley, 1953], which means independent bound. -
Tic-Tac-Toe and Machine Learning David Holmstedt Davho304 729G43
Tic-Tac-Toe and machine learning David Holmstedt Davho304 729G43 Table of Contents Introduction ............................................................................................................................... 1 What is tic-tac-toe.................................................................................................................. 1 Tic-tac-toe Strategies ............................................................................................................. 1 Search-Algorithms .................................................................................................................. 1 Machine learning ................................................................................................................... 2 Weights .............................................................................................................................. 2 Training data ...................................................................................................................... 3 Bringing them together ...................................................................................................... 3 Implementation ......................................................................................................................... 4 Overview ................................................................................................................................ 4 UI ....................................................................................................................................... -
Games with Hidden Information
Games with Hidden Information R&N Chapter 6 R&N Section 17.6 • Assumptions so far: – Two-player game : Player A and B. – Perfect information : Both players see all the states and decisions. Each decision is made sequentially . – Zero-sum : Player’s A gain is exactly equal to player B’s loss. • We are going to eliminate these constraints. We will eliminate first the assumption of “perfect information” leading to far more realistic models. – Some more game-theoretic definitions Matrix games – Minimax results for perfect information games – Minimax results for hidden information games 1 Player A 1 L R Player B 2 3 L R L R Player A 4 +2 +5 +2 L Extensive form of game: Represent the -1 +4 game by a tree A pure strategy for a player 1 defines the move that the L R player would make for every possible state that the player 2 3 would see. L R L R 4 +2 +5 +2 L -1 +4 2 Pure strategies for A: 1 Strategy I: (1 L,4 L) Strategy II: (1 L,4 R) L R Strategy III: (1 R,4 L) Strategy IV: (1 R,4 R) 2 3 Pure strategies for B: L R L R Strategy I: (2 L,3 L) Strategy II: (2 L,3 R) 4 +2 +5 +2 Strategy III: (2 R,3 L) L R Strategy IV: (2 R,3 R) -1 +4 In general: If N states and B moves, how many pure strategies exist? Matrix form of games Pure strategies for A: Pure strategies for B: Strategy I: (1 L,4 L) Strategy I: (2 L,3 L) Strategy II: (1 L,4 R) Strategy II: (2 L,3 R) Strategy III: (1 R,4 L) Strategy III: (2 R,3 L) 1 Strategy IV: (1 R,4 R) Strategy IV: (2 R,3 R) R L I II III IV 2 3 L R I -1 -1 +2 +2 L R 4 II +4 +4 +2 +2 +2 +5 +1 L R III +5 +1 +5 +1 IV +5 +1 +5 +1 -1 +4 3 Pure strategies for Player B Player A’s payoff I II III IV if game is played I -1 -1 +2 +2 with strategy I by II +4 +4 +2 +2 Player A and strategy III by III +5 +1 +5 +1 Player B for Player A for Pure strategies IV +5 +1 +5 +1 • Matrix normal form of games: The table contains the payoffs for all the possible combinations of pure strategies for Player A and Player B • The table characterizes the game completely, there is no need for any additional information about rules, etc. -
Combinatorial Games
Midterm 1 Stat155 Game Theory Lecture 11: Midterm 1 Review “This is an open book exam: you can use any printed or written material, but you cannot use a laptop, tablet, or phone (or any device that can communicate). There are three questions, each consisting of three parts. Peter Bartlett Each part of each question carries equal weight. Answer each question in the space provided.” September 29, 2016 1 / 35 2 / 35 Topics Definitions: Combinatorial games Combinatorial games Positions, moves, terminal positions, impartial/partisan, progressively A combinatorial game has: bounded Two players, Player I and Player II. Progressively bounded impartial and partisan games The sets N and P A set of positions X . Theorem: Someone can win For each player, a set of legal moves between positions, that is, a set Examples: Subtraction, Chomp, Nim, Rims, Staircase Nim, Hex of ordered pairs, (current position, next position): Zero sum games Payoff matrices, pure and mixed strategies, safety strategies MI , MII X X . Von Neumann’s minimax theorem ⊂ × Solving two player zero-sum games Saddle points Equalizing strategies Players alternately choose moves. Solving2 2 games × Play continues until some player cannot move. Dominated strategies 2 n and m 2 games Normal play: the player that cannot move loses the game. × × Principle of indifference Symmetry: Invariance under permutations 3 / 35 4 / 35 Definitions: Combinatorial games Impartial combinatorial games and winning strategies Terminology: An impartial game has the same set of legal moves for both players: MI = MII . Theorem A partisan game has different sets of legal moves for the players. In a progressively bounded impartial A terminal position for a player has no legal move to another position. -
Recursion & the Minimax Algorithm Winning Tic-Tac-Toe How to Pass The
Recursion & the Minimax Algorithm Winning Tic-tac-toe Key to Acing Computer Science Anticipate the implications of your move. If you understand everything, ace your computer science course. Avoid moves which could enable your opponent to win. Otherwise, study & program something you don't understand, Attempt moves which would then see force your opponent to lose, so you win. Key to Acing Computer Science CompSci 4 Recursion & Minimax 27.1 CompSci 4 Recursion & Minimax 27.2 How to pass the buck: recursion! How could this approach work? Imagine this: What actually happens is that your friends doing most of You have all sorts of friends who are willing to help you do your work have friends of their own. Everyone your work with two caveats: eventually does a small part of the work and piece it o They won't do all of your work back together to do the work in its entirity. o They will only do the type of work you have to do In the last example, 100 total people would be working Example: on the homework, each person doing only one problem How would you do your homework consisting of 100 calculus each and passing on the rest. problems? Important: the person to do the last homework problem Easy – do the first problem yourself and ask your friend to do does it entirely without help. the rest CompSci 4 Recursion & Minimax 27.3 CompSci 4 Recursion & Minimax 27.4 The Parts of Recursion Example: Finding the Maximum ÿ Base Case – this is the simplest form of the problem ÿ Imagine you are given a stack of 1000 unsorted papers which can be solved directly. -
Graph Ramsey Games
TECHNICAL R EPORT INSTITUT FUR¨ INFORMATIONSSYSTEME ABTEILUNG DATENBANKEN UND ARTIFICIAL INTELLIGENCE Graph Ramsey games DBAI-TR-99-34 Wolfgang Slany arXiv:cs/9911004v1 [cs.CC] 10 Nov 1999 DBAI TECHNICAL REPORT Institut f¨ur Informationssysteme NOVEMBER 5, 1999 Abteilung Datenbanken und Artificial Intelligence Technische Universit¨at Wien Favoritenstr. 9 A-1040 Vienna, Austria Tel: +43-1-58801-18403 Fax: +43-1-58801-18492 [email protected] http://www.dbai.tuwien.ac.at/ DBAI TECHNICAL REPORT DBAI-TR-99-34, NOVEMBER 5, 1999 Graph Ramsey games Wolfgang Slany1 Abstract. We consider combinatorial avoidance and achievement games based on graph Ramsey theory: The players take turns in coloring still uncolored edges of a graph G, each player being assigned a distinct color, choosing one edge per move. In avoidance games, completing a monochromatic subgraph isomorphic to another graph A leads to immedi- ate defeat or is forbidden and the first player that cannot move loses. In the avoidance+ variants, both players are free to choose more than one edge per move. In achievement games, the first player that completes a monochromatic subgraph isomorphic to A wins. Erd˝os & Selfridge [16] were the first to identify some tractable subcases of these games, followed by a large number of further studies. We complete these investigations by settling the complexity of all unrestricted cases: We prove that general graph Ramsey avoidance, avoidance+, and achievement games and several variants thereof are PSPACE-complete. We ultra-strongly solve some nontrivial instances of graph Ramsey avoidance games that are based on symmetric binary Ramsey numbers and provide strong evidence that all other cases based on symmetric binary Ramsey numbers are effectively intractable. -
Maximin Equilibrium∗
Maximin equilibrium∗ Mehmet ISMAILy March, 2014. This version: June, 2014 Abstract We introduce a new theory of games which extends von Neumann's theory of zero-sum games to nonzero-sum games by incorporating common knowledge of individual and collective rationality of the play- ers. Maximin equilibrium, extending Nash's value approach, is based on the evaluation of the strategic uncertainty of the whole game. We show that maximin equilibrium is invariant under strictly increasing transformations of the payoffs. Notably, every finite game possesses a maximin equilibrium in pure strategies. Considering the games in von Neumann-Morgenstern mixed extension, we demonstrate that the maximin equilibrium value is precisely the maximin (minimax) value and it coincides with the maximin strategies in two-player zero-sum games. We also show that for every Nash equilibrium that is not a maximin equilibrium there exists a maximin equilibrium that Pareto dominates it. In addition, a maximin equilibrium is never Pareto dominated by a Nash equilibrium. Finally, we discuss maximin equi- librium predictions in several games including the traveler's dilemma. JEL-Classification: C72 ∗I thank Jean-Jacques Herings for his feedback. I am particularly indebted to Ronald Peeters for his continuous comments and suggestions about the material in this paper. I am also thankful to the participants of the MLSE seminar at Maastricht University. Of course, any mistake is mine. yMaastricht University. E-mail: [email protected]. 1 Introduction In their ground-breaking book, von Neumann and Morgenstern (1944, p. 555) describe the maximin strategy1 solution for two-player games as follows: \There exists precisely one solution.