CS182 Lecture 5: Games and Adversarial Search
|
|
- Myrtle Anderson
- 7 years ago
- Views:
Transcription
1 CS182 Lecture 5: Games and Adversarial Search Agenda EOLQ: uninformed and informed search Games The difference an adversary makes Minimax Alpha-beta pruning Static evaluation functions (for depth-limited search) Games with chance (briefly: Expectimax) Announcements EOLQ CS 182: Intelligent Systems: Reasoning, Actions, & Plans Fall 2013
2 Announcements AIMA 3e Reading: 5 through 5.5 and 5.7 (skip 5.6) Assignment 1: (1) due Thursday; (2) extra credit problem: Optional. Can earn up to 2 points, but can t get more than 100 total points on assignment. Good job answering each other in the forum!!! Depth-first search in PacMan: impact of graph search, saving states to check for repetitions. Lecture of interest: Dave Ferrucci (leader of IBM Watson effort), Wednesday September 18th, 5-6pm in MD 119. See course website for more information. CS 182: Intelligent Systems: Reasoning, Actions, & Plans Fall 2013
3 Search Summary Systematic search to find sequences of actions (aka paths ) from initial state to a goal state: Uninformed search: BFS, DFS, UCS, IDS Heuristic functions estimate costs of shortest paths and can dramatically reduce search cost. A* search expands lowest g+h: Complete and optimal. IDA* Greedy best-first search expands lowest h: Incomplete and not always optimal Local search: for optimization Don t need path (seq. of acts) so can ignore history Do need an objective (evaluation) function Hill-climbing (many variants), simulated annealing, beam search, genetic algorithms CS 182: Intelligent Systems: Reasoning, Actions, & Plans Fall 2013
4 Why/When Use Local Search? Solution, not path, is what matters. Objective function Better is better than best Large/infinite complex state spaces Gain: little memory and ability to operate in large landscapes. Lose: optimality/completeness
5 An Additional Species of Search Problem So far Systematic search for action sequences: least-cost path to a goal at unknown depth Searching for a goal state: Local/Optimization New: Choosing an action in the presence of an adversary: Games! Examples: tic-tac-toe, chess, backgammon, Adversary might prevent path to best goal Want best assured outcome
6 Game Playing Characteristics of games Unpredictable opponent, so solution is a strategy, specifying a move for every possible opponent reply. Time limits: need to approximate rather than do complete search. Also, course will consider only 2-player game, zerosum (win, lose, draw games or fast/anytime valued). Search goal: given state of the game, choose a next move, by Evaluating the future, i.e., how the game might proceed from this state. Choosing as the best move the move that leads to best payoff in the future. what s best?
7 Why games? One of the first tasks undertaken by AI Easy to represent, precise rules Games and strategy coincide with our intuitive notions of intelligence (humans as symbol manipulators, Newell & Simon) Better than people in Othello and Checkers, defeated world champions in chess, backgammon... but not Go Games are often too hard to solve optimally: Chess has a branching factor of 35, nodes Drives bounded-rationality research They re fun!
8 Types of Games
9 Research Program Advice We may hope that machines will eventually compete with men in all purely intellectual fields. But which are the best ones to start with? Even this is a difficult decision. Many people think that a very abstract activity, like the playing of chess, would be best. It can also be maintained that it is best to provide the machine with the best sense organs that money can buy, and then teach it to understand and speak English... Again I do not know what the right answer is, but I think both approaches should be tried. Turing, Computing Machinery and Intelligence, Mind, p. 460
10 Representing Games as Trees S Agent Moves Opponent Moves Agent Moves G Possible Goal State (winning situation for agent)
11 Game Tree for Tic-Tac Toe
12 Tic-tac-toe: first move Start Node X X X
13 Tic-tac-toe: second move Start Node O X O X X X O X X O X X O X
14 Tic-tac-toe: third move Start Node X O X O X X O X O X O X O X
15 The Rational Opponent The Game tree gives us the possible ways the game could proceed, but how do we decide what s the best move? Especially since we can t plan or control the whole path Minimax principle: [Von Neumann/Morgenstern, 1944] Assume that both players always play optimally You try to maximize your winnings They try to minimize your winnings
16 Winning Strategy? I choose I choose You choose You choose I win I lose I win I lose I win I win I win I lose I don t have a winning strategy strategy. I do have a winning strategy.
17 Evaluating Moves 1 me 1 0 you me value (me-node) = value (you-node) = max of children min of children
18 Generalize to Different Leaf-node Values 2 me = max 2 1 you = min me = max
19 A Max vs Min Game Tree Suppose there are two players: MAX (that s you!) and MIN (your opponent) After one turn each, say the game is over. What move should MAX pick? 3 max(3,2,2) MAX has 3 possible moves: a1, a2, a3 3 min(3,12,8) 2 min(2,4,6) 2 min(14,5,2) MIN has 3 possible responses in each case Utility (winnings) of the final state Conceptually, Construct game tree (root to leaves) Propagate utility upwards (leaves to root) Minimax Principle: Maximize your winnings, given that your opponent will try to minimize them
20 Minimax Approach Basic Algorithm Construct search tree down to the leaves Determine minimax values at leaves, Propagate minimax values up the tree Eventually, read off the minimax decision at root. Formally Let n = node, n = children of node n MINIMAX_VALUE (n) = Utility(n), if n is terminal node (leaf) = max ( MINIMAX(n ) ) if n is MAX node (maximize the value) = min ( MINIMAX(n ) ) if n is MIN node (minimize the value)
21 Minimax algorithm Uses depth-first search (to save space) 21
22 Exercise: Minimax Search
23 Properties of minimax Complete? Only if tree is finite; note that a finite strategy can exist even in an infinite tree Optimal? Yes, against an optimal opponent; otherwise? Time complexity? O(b m ) (b = legal moves, m = maximum depth of tree) Space complexity? O(bm) (depth-first exploration Yikes!! 23
24 Time Complexity Too High to Play the Games Suppose the game tree depth is at most m, with b possible moves Time = O(b m ) is still exponential What is the size for Chess? Tic-tac-toe? Go? Backgammon?
25 Some Sizes of Game Trees Chess: b ~ 35 (average branching factor) d ~ 100 (depth of game tree for typical game) b d ~ (~10 154) nodes!! Tic-Tac-Toe At most 9 half-moves, <=9 choices 9! = 362,880 (Computer goes first) 8! = 40,320 (Computer goes second) Backgammon: b ~20 x 20 (because of chance nodes) Go: Branching factor starts at 361 (19x19 board)
26 Can We Avoid Exploring Every Path? In games, time is everything! Two strategies that help: Prune the tree (alpha-beta pruning) Don t waste time looking at situations that don t improve anything Look a limited way into the future (heuristic function) Use a heuristic to estimate the future goodness of this choice (ala A*?) without exploring all the way
27 Pruning the Search 2 max ( me ) =2 min ( you ) max
28 and values ( value) 2 max ( me ) ( value) =2 1 min ( you ) max No point expanding this node further.
29 and Values Going Deeper =2 max ( me ) =2 min ( you ) =2 =7 max No point expanding this node further.
30 α-β pruning example 30
31 α-β pruning example 31
32 α-β pruning example 32
33 α-β pruning example 33
34 α-β pruning example 34
35 - Pruning Approach: cutoff at a MIN node Consider a MAX node, and the second MIN node below it. If the MAX player will not choose the action leading to that node then can prune a MAX b c At MAX = the best (highest) choice we have found so far at any choice point along path for MAX 3 Eventually, your minimax value is Anything less than 3 on b branch, prune this effort!
36 - Pruning Approach: cutoff at a MAX node Now consider a MIN node, and the second MAX mode below it. Again, if the MIN player will not choose the action leading to that node then can prune a MIN b c At MIN = the best (lowest) choice we have found so far at any choice point along path for MIN 100 Eventually, your minimax value is Anything more than a 100 on b branch, prune this effort!
37 Putting it Together = 3 α = 3 MIN a MAX b = 2 c At MAX = the best (highest) choice we have found so far at any choice point along path for MAX Eventually, your minimax value is a b c At MIN = the best (lowest) choice we have found so far at any choice point along path for MIN α = 100 α = 150 α = 3 Eventually, your minimax value is
38 and Cut-off Values Provisional backed up values; become final when done on path below them. PBV( me -node) = max(value of successors so far) = value value can never decrease PBV( you -node) = min(value of successors so far) = value value can never increase
39 Cutting Off Useless Search - procedure: stop searching below any min ( you ) node s.t. value value of any max ancestor [ - cutoff] max ( me ) node s.t. value value of any min ancestor [ - cutoff]
40 The α-β algorithm 40
41 MAX 2 MIN 2 a = - b = MAX MIN The entire subtree is pruned
42 Effectiveness of - pruning The effectiveness relies on having a good move ordering heuristic If we are considering children left to right, then we don t prune last tree even though we could have
43 Pruning does not affect final result Good move ordering improves effectiveness of pruning With perfect ordering time complexity=o(b m/2 ) doubles depth of search (In worst case, there is no improvement) A simple example of the value of reasoning about which computations are relevant (a form of metareasoning) Unfortunately, is still impossible Properties of α-β 43
44 Try It Out: α-β Search
45 The Importance of Move Ordering (Knuth & Moore 75) [best-case] If successors are ordered best-first Only examine O(b d/2 ) nodes instead of O(b d ) Can look twice as far ahead in same amount of time! [avg-case] If successors are examined in random order Then nodes will be O(b 3d/4 ) for moderate b [worst case] No improvement over exhaustive search For chess, a fairly simple ordering function (e.g. captures, then threats, then forward, then backward moves) gets within about a factor of 2 to theoretical limit
46 What more can we do? Minimax search with alpha-beta pruning Has to search all the way to terminal states at least once Still much too expensive in typical games such as Chess. What else can we do? Bound the depth of search (limit) Treat bottom nodes as terminal nodes Use evaluation function (aka heuristic evaluation function) to estimate the utility of these nodes (whether they will win) What makes a good heuristic?
47 Heuristic Evaluation Functions Examples Othello: Number of white pieces - Number of black pieces Chess: Value of all white pieces - Value of all black pieces Heuristics are where domain knowledge comes in Not as theoretically nice as with A*/admissibility f Instead you ask domain 3 = f 1 /f expert 2 for features Express as linear weighted sum Maybe learn the weights? Problems?? Features for chess: f 1 = number of white pieces f 2 = number of black pieces f 4 = number of white bishops f 5 = estimate of threat to white king Eval(s)=w 1 f 1 (s)+w 2 f 2 (s)+ +w 5 f 5 (s)
48 Other Important Techniques Iterative Deepening Instead of a fixed depth, keep solutions at different depths. When you run out of time, choose the last solution. Transposition Table Different permutations of move, can result in previously seen games. Store and remove repeated states (essential!)
49 More Important Techniques Quiescent Search Use deeper search on positions where there might be wild swings in value (non-quiescent positions) e.g. in chess, if there is the potential for a capture (domain knowledge v. important here) Endgame/Opening Databases Precompute choices for smaller games, and store them Checkers: 400 billion positions with less than 9 pieces Chess: Ken Thompson (UNIX) and Stiller solved all 5-piece endgames (these games can be quite long!!!)
50 Games with chance Some games have chance events Rolling a dice Flipping a coin Drawing a card These are called chance nodes How to consider chance nodes? Evaluate the expected value 50
51 Example Game with Chance: Backgammon In backgammon, the goal is to get your pieces (say white) all the way off the board white must travel from 0 to 25 to do this. You can move to a position so long as 1 opponent, if 1 opponent you capture them (reset them to zero) CHANCE comes in because you roll dice to determine what moves you can take
52 With Chance, Need to Compute Expected Value Assume the dice has n outcomes d 1,d 2,d 3,..,d n Every outcome has a value v(d 1 ),v(d 2 ),..,v(d n ) Every outcome has a probability p(d 1 ),p(d 2 ),..,p(d n ) The expected value is p(d 1 )v(d 1 )+p(d 2 )v(d 2 )+ +p(d n )v(d n ) Also called the weighted average 52
53 Computing the Optimal Move When Chance Nodes Include chance nodes in addition to MAX and MIN nodes, Calculate expected minimax values Explodes branching-factor O(b d n d ), for n distinct rolls backgammon, n=21, b 20 so around 3-ply is possible
54 54
55 Search tree with probabilities MAX MIN
56 Search tree with probabilities MAX MIN
57 Game Tree with Chance Nodes MAX move (for MIN) (for MAX) MIN move MAX move
58 Summary of Adversarial Search Algorithms Game Playing Programs The goal is to choose a next move Evaluate potential futures, then choose the best move Model the Opponent as being just as smart Two Key Questions: How do we compute best? How do we compute it fast We saw Minimax principle ( best ) Alpha-Beta pruning ( fast ) Heuristic Evaluation Functions (fastest, but at the expense of best) Chance games are a lot harder.
59 Games have a long history in AI Famous early computer scientists thought a lot about games, competition, and economics John McCarthy, Allen Newell, Herb Simon: chess Arthur Samuel wrote a checkers program that could learn. Hans Berliner, Backgammon Von Neumann (cf. Game Theory) Chess Ratings for Chess programs 1950s, Shannon, Turing 1958s, NSS Chess (Newell et al, McCarthy) 1997, Deep Blue 1987, HITECH (Berliner, et al) First to defeat human grandmaster 1982, Belle (Condon & Thompson) Rating 2250, first master-level) 1974, Kaissa, (Moscow ITEP, 1900) 1967, MacHack 6 (Greenblatt et al, 1400) Below novice
60 State-of-the art game programs Optimal strategies for Othello (Logistello, 1997) Backgammon champion is a computer 60
61 Deep blue: Chess Deep Blue defeated human world champion Garry Kasparov in a six-game match in Other strong programs: Deep Junior, and Fritz. Two camps: emulation camp and engineering camp. Mostly engineering + lots of knowledge Complex evaluation function: material value, positional value, pawn structure, obstructed lines, center situation, mobility, etc. Vast library of openings and endgames compiled by a team of Chess Masters. No learning!! 62
62 Chinook: optimal checkers player In 2007, the entire search tree of Checkers was spanned and the optimal strategy was calculated How Chinook works? Parallel iterative alpha-beta, deep searches (17-30) Heuristic function: weighted sum of 25 components; 4 parameter sets for different phases of the game Perfect knowledge of all endgames with <= 7 and 40% of 8 piece positions (about 40x10 9 ) in 2GB Openings library of 6,000 positions + antibook : library of bad positions 64
63 Checkers is Solved Originally published in Science Express on 19 July 2007 Science 14 September 2007: Vol. 317, no. 5844, pp Jonathan Schaeffer (University of Alberta), Neil Burch, Yngvi Björnsson, Akihiro Kishimoto, Martin Müller, Robert Lake, Paul Lu, Steve Sutphen 65
64 What do most programs do? Combination of Brute force search Heuristics Game database Some programs attempt to learn Improve the heuristic function by comparing its estimates with actual outcomes Attempt to discover rules and heuristic functions 66
65 Game of Go A very popular game in Japan More than 3,000 years old Branching factor is ~360 Search depth is also ~360 No good evaluation function Minimax approach not effective Open challenge! Very recent approach achieved master level in a small version of Go, using a different (not minimax) approach 67
66 Games Summary Games are fun to work on! They illustrate several important points about AI Perfection is unattainable must approximate Good idea to think about what to think about Uncertainty constrains the assignment of values to states Optimal decisions depend on information state, not real state Games are to AI as grand prix racing is to automobile design 68
67 Games: Some Final Project Ideas Scrabble Dictionary gives the computer an edge finding words But what is a good strategy? ( chance ) Connect 4 Tutor Game trees allow you to model games. Instead of using this to defeat an opponent, can you train the opponent Can you help them learn to be better? Simulated Soccer Soccer is a *very* different game distributed and continuous space/time How would you design a team? (robocup simulation league) Technical Game Playing Algorithms Local search (over moves); Monte Carlo Evaluation, etc
68 Summary: Classes of Search Algorithms Systematic search for action sequence: want least-cost path to goal, which typically is at unknown depth. Optimization/Local search to find state that maximizes/minimizes objective function (which captures important/relevant properties of states). Decisions with an adversary (games) search to find best assured outcome in presence of an adversary who might prevent path to best goal
69 Upcoming lectures and sections Sept. 19: Constraint satisfaction Thursday/Friday sections: Games and Constraint Satisfaction Next week: Representation and modeling wider (than search) worlds CS 182: Intelligent Systems: Reasoning, Actions, & Plans Fall
70 Announcements For Thursday: AIMA 3e Chapter 6 (skim 6.5) Keep up the good work answering each other in the forum!!! Assignment 1: (1) due Thursday; (2) extra credit problem: Optional. Can earn up to 2 points, but can t get more than 100 total points on assignment. Lecture of interest: Dave Ferrucci (leader of IBM Watson effort), Wednesday September 18th, 5-6pm in MD 119. See course website for more information. CS 182: Intelligent Systems: Reasoning, Actions, & Plans Fall 2013
71 EOLQ What question didn t you get to ask today? What s still puzzling or not clear? What idea would you like to hear a bit more about? CS 182: Intelligent Systems: Reasoning, Actions, & Plans Fall 2013
Game playing. Chapter 6. Chapter 6 1
Game playing Chapter 6 Chapter 6 1 Outline Games Perfect play minimax decisions α β pruning Resource limits and approximate evaluation Games of chance Games of imperfect information Chapter 6 2 Games vs.
More informationChess Algorithms Theory and Practice. Rune Djurhuus Chess Grandmaster runed@ifi.uio.no / runedj@microsoft.com October 3, 2012
Chess Algorithms Theory and Practice Rune Djurhuus Chess Grandmaster runed@ifi.uio.no / runedj@microsoft.com October 3, 2012 1 Content Complexity of a chess game History of computer chess Search trees
More informationChess Algorithms Theory and Practice. Rune Djurhuus Chess Grandmaster runed@ifi.uio.no / runedj@microsoft.com September 23, 2014
Chess Algorithms Theory and Practice Rune Djurhuus Chess Grandmaster runed@ifi.uio.no / runedj@microsoft.com September 23, 2014 1 Content Complexity of a chess game Solving chess, is it a myth? History
More informationGame Playing in the Real World. Next time: Knowledge Representation Reading: Chapter 7.1-7.3
Game Playing in the Real World Next time: Knowledge Representation Reading: Chapter 7.1-7.3 1 What matters? Speed? Knowledge? Intelligence? (And what counts as intelligence?) Human vs. machine characteristics
More informationCS91.543 MidTerm Exam 4/1/2004 Name: KEY. Page Max Score 1 18 2 11 3 30 4 15 5 45 6 20 Total 139
CS91.543 MidTerm Exam 4/1/2004 Name: KEY Page Max Score 1 18 2 11 3 30 4 15 5 45 6 20 Total 139 % INTRODUCTION, AI HISTORY AND AGENTS 1. [4 pts. ea.] Briefly describe the following important AI programs.
More informationAI: A Modern Approach, Chpts. 3-4 Russell and Norvig
AI: A Modern Approach, Chpts. 3-4 Russell and Norvig Sequential Decision Making in Robotics CS 599 Geoffrey Hollinger and Gaurav Sukhatme (Some slide content from Stuart Russell and HweeTou Ng) Spring,
More informationMeasuring the Performance of an Agent
25 Measuring the Performance of an Agent The rational agent that we are aiming at should be successful in the task it is performing To assess the success we need to have a performance measure What is rational
More information10. Machine Learning in Games
Machine Learning and Data Mining 10. Machine Learning in Games Luc De Raedt Thanks to Johannes Fuernkranz for his slides Contents Game playing What can machine learning do? What is (still) hard? Various
More informationDecision Making under Uncertainty
6.825 Techniques in Artificial Intelligence Decision Making under Uncertainty How to make one decision in the face of uncertainty Lecture 19 1 In the next two lectures, we ll look at the question of how
More informationMinimax Strategies. Minimax Strategies. Zero Sum Games. Why Zero Sum Games? An Example. An Example
Everyone who has studied a game like poker knows the importance of mixing strategies With a bad hand, you often fold But you must bluff sometimes Lectures in Microeconomics-Charles W Upton Zero Sum Games
More informationFIRST EXPERIMENTAL RESULTS OF PROBCUT APPLIED TO CHESS
FIRST EXPERIMENTAL RESULTS OF PROBCUT APPLIED TO CHESS A.X. Jiang Department of Computer Science, University of British Columbia, Vancouver, Canada albertjiang@yahoo.com M. Buro Department of Computing
More informationGame Theory and Algorithms Lecture 10: Extensive Games: Critiques and Extensions
Game Theory and Algorithms Lecture 0: Extensive Games: Critiques and Extensions March 3, 0 Summary: We discuss a game called the centipede game, a simple extensive game where the prediction made by backwards
More informationSearch methods motivation 1
Suppose you are an independent software developer, and your software package Windows Defeater R, widely available on sourceforge under a GNU GPL license, is getting an international attention and acclaim.
More informationIntroduction Solvability Rules Computer Solution Implementation. Connect Four. March 9, 2010. Connect Four
March 9, 2010 is a tic-tac-toe like game in which two players drop discs into a 7x6 board. The first player to get four in a row (either vertically, horizontally, or diagonally) wins. The game was first
More informationLaboratory work in AI: First steps in Poker Playing Agents and Opponent Modeling
Laboratory work in AI: First steps in Poker Playing Agents and Opponent Modeling Avram Golbert 01574669 agolbert@gmail.com Abstract: While Artificial Intelligence research has shown great success in deterministic
More informationThe Taxman Game. Robert K. Moniot September 5, 2003
The Taxman Game Robert K. Moniot September 5, 2003 1 Introduction Want to know how to beat the taxman? Legally, that is? Read on, and we will explore this cute little mathematical game. The taxman game
More informationAcknowledgements I would like to thank both Dr. Carsten Furhmann, and Dr. Daniel Richardson for providing me with wonderful supervision throughout my
Abstract Game orientated application programming interfaces (API), built using Java, are intended to ease the development of Java games, on multiple platforms. Existing Java game APIs, tend to focus on
More informationTEACHER S GUIDE TO RUSH HOUR
Using Puzzles to Teach Problem Solving TEACHER S GUIDE TO RUSH HOUR Includes Rush Hour 2, 3, 4, Rush Hour Jr., Railroad Rush Hour and Safari Rush Hour BENEFITS Rush Hour is a sliding piece puzzle that
More informationMastering Quoridor. Lisa Glendenning THESIS. Submitted in Partial Fulfillment of the Requirements for the Degree of
Mastering Quoridor by Lisa Glendenning THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science Computer Science The University of New Mexico Albuquerque, New Mexico
More informationLab 11. Simulations. The Concept
Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that
More informationCOMP 590: Artificial Intelligence
COMP 590: Artificial Intelligence Today Course overview What is AI? Examples of AI today Who is this course for? An introductory survey of AI techniques for students who have not previously had an exposure
More informationBinary Search Trees. A Generic Tree. Binary Trees. Nodes in a binary search tree ( B-S-T) are of the form. P parent. Key. Satellite data L R
Binary Search Trees A Generic Tree Nodes in a binary search tree ( B-S-T) are of the form P parent Key A Satellite data L R B C D E F G H I J The B-S-T has a root node which is the only node whose parent
More informationSequential lmove Games. Using Backward Induction (Rollback) to Find Equilibrium
Sequential lmove Games Using Backward Induction (Rollback) to Find Equilibrium Sequential Move Class Game: Century Mark Played by fixed pairs of players taking turns. At each turn, each player chooses
More informationTD-Gammon, A Self-Teaching Backgammon Program, Achieves Master-Level Play
TD-Gammon, A Self-Teaching Backgammon Program, Achieves Master-Level Play Gerald Tesauro IBM Thomas J. Watson Research Center P. O. Box 704 Yorktown Heights, NY 10598 (tesauro@watson.ibm.com) Abstract.
More informationMonte Carlo Tree Search and Opponent Modeling through Player Clustering in no-limit Texas Hold em Poker
Monte Carlo Tree Search and Opponent Modeling through Player Clustering in no-limit Texas Hold em Poker A.A.J. van der Kleij August 2010 Master thesis Artificial Intelligence University of Groningen, The
More information20-30 minutes, can be used within a longer activity
Locked-in 1 Age group successfully used with: Abilities assumed: Time: Size of group: 11 adult None 20-30 minutes, can be used within a longer activity anything from 2 to hundreds Focus What is an algorithm?
More informationKeywords-Chess gameregistration, Profile management, Rational rose, Activities management.
Volume 5, Issue 2, February 2015 ISSN: 2277 128X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: www.ijarcsse.com Online Chess
More informationRoulette Wheel Selection Game Player
Macalester College DigitalCommons@Macalester College Honors Projects Mathematics, Statistics, and Computer Science 5-1-2013 Roulette Wheel Selection Game Player Scott Tong Macalester College, stong101@gmail.com
More information6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitely-repeated prisoner s dilemma
More informationImproving Depth-first PN-Search: 1 + ε Trick
Improving Depth-first PN-Search: 1 + ε Trick Jakub Pawlewicz, Łukasz Lew {pan,lew}@mimuw.edu.pl Institute of Informatics, Warsaw University Banacha 2, 02-097 Warsaw, Poland Abstract. Various efficient
More informationMonte-Carlo Methods. Timo Nolle
Monte-Carlo Methods Timo Nolle Outline Minimax Expected Outcome Monte-Carlo Monte-Carlo Simulation Monte-Carlo Tree Search UCT AMAF RAVE MC-RAVE UCT-RAVE Playing Games Go, Bridge, Scrabble Problems with
More informationA Knowledge-based Approach of Connect-Four
A Knowledge-based Approach of Connect-Four The Game is Solved: White Wins Victor Allis Department of Mathematics and Computer Science Vrije Universiteit Amsterdam, The Netherlands Masters Thesis, October
More informationSIMS 255 Foundations of Software Design. Complexity and NP-completeness
SIMS 255 Foundations of Software Design Complexity and NP-completeness Matt Welsh November 29, 2001 mdw@cs.berkeley.edu 1 Outline Complexity of algorithms Space and time complexity ``Big O'' notation Complexity
More informationClock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system
CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical
More informationCSE 517A MACHINE LEARNING INTRODUCTION
CSE 517A MACHINE LEARNING INTRODUCTION Spring 2016 Marion Neumann Contents in these slides may be subject to copyright. Some materials are adopted from Killian Weinberger. Thanks, Killian! Machine Learning
More informationDynamic programming formulation
1.24 Lecture 14 Dynamic programming: Job scheduling Dynamic programming formulation To formulate a problem as a dynamic program: Sort by a criterion that will allow infeasible combinations to be eli minated
More informationLoad Balancing. Load Balancing 1 / 24
Load Balancing Backtracking, branch & bound and alpha-beta pruning: how to assign work to idle processes without much communication? Additionally for alpha-beta pruning: implementing the young-brothers-wait
More informationBest-First and Depth-First Minimax Search in Practice
Best-First and Depth-First Minimax Search in Practice Aske Plaat, Erasmus University, plaat@theory.lcs.mit.edu Jonathan Schaeffer, University of Alberta, jonathan@cs.ualberta.ca Wim Pijls, Erasmus University,
More information6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games
6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4, 2009 1 Introduction Outline Decisions, utility maximization Strategic form games Best responses
More informationUNIVERSALITY IS UBIQUITOUS
UNIVERSALITY IS UBIQUITOUS Martin Davis Professor Emeritus Courant Institute, NYU Visiting Scholar UC Berkeley Q 3 a 0 q 5 1 Turing machine operation: Replace symbol ( print ) Move left or right one square,
More informationBayesian Nash Equilibrium
. Bayesian Nash Equilibrium . In the final two weeks: Goals Understand what a game of incomplete information (Bayesian game) is Understand how to model static Bayesian games Be able to apply Bayes Nash
More informationSYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis
SYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October 17, 2015 Outline
More informationUniversity of Alberta. Library Release Form
University of Alberta Library Release Form Name of Author: Michael Bradley Johanson Title of Thesis: Robust Strategies and Counter-Strategies: Building a Champion Level Computer Poker Player Degree: Master
More informationLearning Example. Machine learning and our focus. Another Example. An example: data (loan application) The data and the goal
Learning Example Chapter 18: Learning from Examples 22c:145 An emergency room in a hospital measures 17 variables (e.g., blood pressure, age, etc) of newly admitted patients. A decision is needed: whether
More information10/13/11 Solution: Minimax with Alpha-Beta Pruning and Progressive Deepening
10/1/11 Solution: Minimax with Alpha-Beta Pruning and Progressive Deepening When answering the question in Parts C.1 and C. below, assume you have already applied minimax with alpha-beta pruning and progressive
More informationA Systematic Approach to Model-Guided Empirical Search for Memory Hierarchy Optimization
A Systematic Approach to Model-Guided Empirical Search for Memory Hierarchy Optimization Chun Chen, Jacqueline Chame, Mary Hall, and Kristina Lerman University of Southern California/Information Sciences
More informationINTRODUCTION 4 1 OVERALL LOOK ON CHESS PROGRAMMING 8
Table of contents INTRODUCTION 4 AIM OF THE WORK 4 LAYOUT 4 RANGE OF THE WORK 5 RESEARCH PROBLEM 5 RESEARCH QUESTIONS 6 METHODS AND TOOLS 6 1 OVERALL LOOK ON CHESS PROGRAMMING 8 1.1 THE VERY BEGINNING
More information6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationOne pile, two pile, three piles
CHAPTER 4 One pile, two pile, three piles 1. One pile Rules: One pile is a two-player game. Place a small handful of stones in the middle. At every turn, the player decided whether to take one, two, or
More informationCourse Outline Department of Computing Science Faculty of Science. COMP 3710-3 Applied Artificial Intelligence (3,1,0) Fall 2015
Course Outline Department of Computing Science Faculty of Science COMP 710 - Applied Artificial Intelligence (,1,0) Fall 2015 Instructor: Office: Phone/Voice Mail: E-Mail: Course Description : Students
More informationMath Games For Skills and Concepts
Math Games p.1 Math Games For Skills and Concepts Original material 2001-2006, John Golden, GVSU permission granted for educational use Other material copyright: Investigations in Number, Data and Space,
More informationStandard 12: The student will explain and evaluate the financial impact and consequences of gambling.
TEACHER GUIDE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Priority Academic Student Skills Personal Financial
More informationHistory of Artificial Intelligence. Introduction to Intelligent Systems
History of Artificial Intelligence Introduction to Intelligent Systems What is An Intelligent System? A more difficult question is: What is intelligence? This question has puzzled philosophers, biologists
More information15-466 Computer Game Programming Intelligence I: Basic Decision-Making Mechanisms
15-466 Computer Game Programming Intelligence I: Basic Decision-Making Mechanisms Maxim Likhachev Robotics Institute Carnegie Mellon University AI Architecture from Artificial Intelligence for Games by
More informationCSE 326: Data Structures B-Trees and B+ Trees
Announcements (4//08) CSE 26: Data Structures B-Trees and B+ Trees Brian Curless Spring 2008 Midterm on Friday Special office hour: 4:-5: Thursday in Jaech Gallery (6 th floor of CSE building) This is
More informationNear Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
More informationCheckers Is Solved. *To whom correspondence should be addressed. E-mail: jonathan@cs.ualberta.ca
Checkers Is Solved Jonathan Schaeffer,* Neil Burch, Yngvi Björnsson, Akihiro Kishimoto, Martin Müller, Robert Lake, Paul Lu, Steve Sutphen Department of Computing Science, University of Alberta, Edmonton,
More informationGamesman: A Graphical Game Analysis System
Gamesman: A Graphical Game Analysis System Dan Garcia Abstract We present Gamesman, a graphical system for implementing, learning, analyzing and playing small finite two-person
More informationFrom Last Time: Remove (Delete) Operation
CSE 32 Lecture : More on Search Trees Today s Topics: Lazy Operations Run Time Analysis of Binary Search Tree Operations Balanced Search Trees AVL Trees and Rotations Covered in Chapter of the text From
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1
More informationOutline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits
Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique
More informationGame theory and AI: a unified approach to poker games
Game theory and AI: a unified approach to poker games Thesis for graduation as Master of Artificial Intelligence University of Amsterdam Frans Oliehoek 2 September 2005 ii Abstract This thesis focuses
More information(Refer Slide Time: 2:03)
Control Engineering Prof. Madan Gopal Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 11 Models of Industrial Control Devices and Systems (Contd.) Last time we were
More informationWatson. An analytical computing system that specializes in natural human language and provides specific answers to complex questions at rapid speeds
Watson An analytical computing system that specializes in natural human language and provides specific answers to complex questions at rapid speeds I.B.M. OHJ-2556 Artificial Intelligence Guest lecturing
More informationCPSC 211 Data Structures & Implementations (c) Texas A&M University [ 313]
CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 313] File Structures A file is a collection of data stored on mass storage (e.g., disk or tape) Why on mass storage? too big to fit
More informationOptimization in ICT and Physical Systems
27. OKTOBER 2010 in ICT and Physical Systems @ Aarhus University, Course outline, formal stuff Prerequisite Lectures Homework Textbook, Homepage and CampusNet, http://kurser.iha.dk/ee-ict-master/tiopti/
More informationCMPSCI611: Approximating MAX-CUT Lecture 20
CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to
More informationCSC384 Intro to Artificial Intelligence
CSC384 Intro to Artificial Intelligence What is Artificial Intelligence? What is Intelligence? Are these Intelligent? CSC384, University of Toronto 3 What is Intelligence? Webster says: The capacity to
More informationConditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of conditional probability and independence
More informationGeneralized Widening
Generalized Widening Tristan Cazenave Abstract. We present a new threat based search algorithm that outperforms other threat based search algorithms and selective knowledge-based for open life and death
More informationRafael Witten Yuze Huang Haithem Turki. Playing Strong Poker. 1. Why Poker?
Rafael Witten Yuze Huang Haithem Turki Playing Strong Poker 1. Why Poker? Chess, checkers and Othello have been conquered by machine learning - chess computers are vastly superior to humans and checkers
More informationCost Model: Work, Span and Parallelism. 1 The RAM model for sequential computation:
CSE341T 08/31/2015 Lecture 3 Cost Model: Work, Span and Parallelism In this lecture, we will look at how one analyze a parallel program written using Cilk Plus. When we analyze the cost of an algorithm
More informationCS104: Data Structures and Object-Oriented Design (Fall 2013) October 24, 2013: Priority Queues Scribes: CS 104 Teaching Team
CS104: Data Structures and Object-Oriented Design (Fall 2013) October 24, 2013: Priority Queues Scribes: CS 104 Teaching Team Lecture Summary In this lecture, we learned about the ADT Priority Queue. A
More informationGuessing Game: NP-Complete?
Guessing Game: NP-Complete? 1. LONGEST-PATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES 2. SHORTEST-PATH: Given a graph G = (V, E), does there exists a simple
More informationSmart Graphics: Methoden 3 Suche, Constraints
Smart Graphics: Methoden 3 Suche, Constraints Vorlesung Smart Graphics LMU München Medieninformatik Butz/Boring Smart Graphics SS2007 Methoden: Suche 2 Folie 1 Themen heute Suchverfahren Hillclimbing Simulated
More informationUnderstanding Proactive vs. Reactive Methods for Fighting Spam. June 2003
Understanding Proactive vs. Reactive Methods for Fighting Spam June 2003 Introduction Intent-Based Filtering represents a true technological breakthrough in the proper identification of unwanted junk email,
More informationMath 202-0 Quizzes Winter 2009
Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile
More informationMetaGame: An Animation Tool for Model-Checking Games
MetaGame: An Animation Tool for Model-Checking Games Markus Müller-Olm 1 and Haiseung Yoo 2 1 FernUniversität in Hagen, Fachbereich Informatik, LG PI 5 Universitätsstr. 1, 58097 Hagen, Germany mmo@ls5.informatik.uni-dortmund.de
More information1 Representation of Games. Kerschbamer: Commitment and Information in Games
1 epresentation of Games Kerschbamer: Commitment and Information in Games Game-Theoretic Description of Interactive Decision Situations This lecture deals with the process of translating an informal description
More information6.042/18.062J Mathematics for Computer Science. Expected Value I
6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you
More informationInternational Journal of Advanced Research in Computer Science and Software Engineering
Volume 3, Issue 7, July 23 ISSN: 2277 28X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: www.ijarcsse.com Greedy Algorithm:
More informationLecture 10: Regression Trees
Lecture 10: Regression Trees 36-350: Data Mining October 11, 2006 Reading: Textbook, sections 5.2 and 10.5. The next three lectures are going to be about a particular kind of nonlinear predictive model,
More informationHow I won the Chess Ratings: Elo vs the rest of the world Competition
How I won the Chess Ratings: Elo vs the rest of the world Competition Yannis Sismanis November 2010 Abstract This article discusses in detail the rating system that won the kaggle competition Chess Ratings:
More information6.080/6.089 GITCS Feb 12, 2008. Lecture 3
6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my
More informationBASIC RULES OF CHESS
BASIC RULES OF CHESS Introduction Chess is a game of strategy believed to have been invented more then 00 years ago in India. It is a game for two players, one with the light pieces and one with the dark
More informationFinal Exam. Route Computation: One reason why link state routing is preferable to distance vector style routing.
UCSD CSE CS 123 Final Exam Computer Networks Directions: Write your name on the exam. Write something for every question. You will get some points if you attempt a solution but nothing for a blank sheet
More informationLearning Agents: Introduction
Learning Agents: Introduction S Luz luzs@cs.tcd.ie October 22, 2013 Learning in agent architectures Performance standard representation Critic Agent perception rewards/ instruction Perception Learner Goals
More informationClassification/Decision Trees (II)
Classification/Decision Trees (II) Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Right Sized Trees Let the expected misclassification rate of a tree T be R (T ).
More informationBPM: Chess vs. Checkers
BPM: Chess vs. Checkers Jonathon Struthers Introducing the Games Business relies upon IT systems to perform many of its tasks. While many times systems don t really do what the business wants them to do,
More informationHow to Win Texas Hold em Poker
How to Win Texas Hold em Poker Richard Mealing Machine Learning and Optimisation Group School of Computer Science University of Manchester / 44 How to Play Texas Hold em Poker Deal private cards per player
More informationDecision Theory. 36.1 Rational prospecting
36 Decision Theory Decision theory is trivial, apart from computational details (just like playing chess!). You have a choice of various actions, a. The world may be in one of many states x; which one
More informationA Working Knowledge of Computational Complexity for an Optimizer
A Working Knowledge of Computational Complexity for an Optimizer ORF 363/COS 323 Instructor: Amir Ali Ahmadi TAs: Y. Chen, G. Hall, J. Ye Fall 2014 1 Why computational complexity? What is computational
More informationAP Stats - Probability Review
AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose
More informationCHAPTER 15: IS ARTIFICIAL INTELLIGENCE REAL?
CHAPTER 15: IS ARTIFICIAL INTELLIGENCE REAL? Multiple Choice: 1. During Word World II, used Colossus, an electronic digital computer to crack German military codes. A. Alan Kay B. Grace Murray Hopper C.
More informationThe Turing Test! and What Computer Science Offers to Cognitive Science "
The Turing Test and What Computer Science Offers to Cognitive Science " Profs. Rob Rupert and Mike Eisenberg T/R 11-12:15 Muenzinger D430 http://l3d.cs.colorado.edu/~ctg/classes/cogsci12/ The Imitation
More informationThe Effects of Start Prices on the Performance of the Certainty Equivalent Pricing Policy
BMI Paper The Effects of Start Prices on the Performance of the Certainty Equivalent Pricing Policy Faculty of Sciences VU University Amsterdam De Boelelaan 1081 1081 HV Amsterdam Netherlands Author: R.D.R.
More informationThe UCT Algorithm Applied to Games with Imperfect Information
Otto-von-Guericke-Universität Magdeburg Fakultät für Informatik Institut für Simulation und Graphik Diploma Thesis The UCT Algorithm Applied to Games with Imperfect Information Author: Jan Schäfer July
More informationDynamic Programming. Lecture 11. 11.1 Overview. 11.2 Introduction
Lecture 11 Dynamic Programming 11.1 Overview Dynamic Programming is a powerful technique that allows one to solve many different types of problems in time O(n 2 ) or O(n 3 ) for which a naive approach
More informationHow To Understand The Relation Between Simplicity And Probability In Computer Science
Chapter 6 Computation 6.1 Introduction In the last two chapters we saw that both the logical and the cognitive models of scientific discovery include a condition to prefer simple or minimal explanations.
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More information