Analyzing Games: Mixed Strategies
|
|
- Joleen Morrison
- 5 years ago
- Views:
Transcription
1 Analyzing Games: Mixed Strategies CPSC 532A Lecture 5 September 26, 2006 Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 1
2 Lecture Overview Recap Mixed Strategies Fun Game Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 2
3 Pareto Optimality Idea: sometimes, one outcome o is at least as good for every agent as another outcome o, and there some agent who strictly prefers o to o in this case, it seems reasonable to say that o is better than o we say that o Pareto-dominates o. An outcome o is Pareto-optimal if there is no other outcome that Pareto-dominates it. Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 3
4 Best Response If you knew what everyone else was going to do, it would be easy to pick your own action Let a i = a 1,..., a i 1, a i+1,..., a n. now a = (a i, a i ) Best response: a i BR(a i) iff a i A i, u i (a i, a i) u i (a i, a i ) Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 4
5 Nash Equilibrium Now let s return to the setting where no agent knows anything about what the others will do What can we say about which actions will occur? Idea: look for stable action profiles. a = a 1,..., a n is a Nash equilibrium iff i, a i BR(a i ). Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 5
6 options what should you adopt, C or D? Does it depend on what you league will do? Furthermore, from the perspective of the network operaof behavior can he expect from the two users? Will any two users behave presented with this scenario? Will the behavior change if the network s the users to communicate with each other before making a decision? anges to the delays would the users decisions still be the same? How s behave if they have the opportunity to face this same decision with the art multiple times? Do answers to the above questions depend on how ents are and how they view each other s rationality? y Analyzing gives answers Games: Mixed to many Strategies of these questions. It tells us that any rational CPSC 532A Lecture 5, Slide 6 e s options are the columns. In each cell, the first number represents r, minus your delay), and the second number represents your colleague s Nash Equilibria of Example Games C D C 1, 1 4,0 D 0, 4 3, 3 Figure 3.1 The TCP user s (aka the Prisoner s) Dilemma.
7 e s options are the columns. In each cell, the first number represents r, minus your delay), right. and If the second players number choose the represents same side your (left colleague s or right) they have some hig otherwise they have a low utility. The game matrix is shown in Figure 3.4 Nash Equilibria of Example Games C D Left Right C 1, 1 4,0 D 0, 4 3, 3 Left 1 0 Right 0 1 Figure 3.1 The TCP user s (aka the Prisoner s) Figure Dilemma. 3.4 Coordination game. ro-sum game options what should At you the other adopt, end C or of the D? spectrum Does it depend from pure on what coordination you games lie zero league will do? Furthermore, which (bearing from in the mind perspective the comment of the we network made earlier operaof behavior can he about positive affine nstant-sum tions) expect are from more the properly two users? called Will constant-sum any two users games. behave Unlike common-pa mes presented with this scenario? Will the behavior change if the network s the users to communicate with each other c Shoham before making and Leyton-Brown, a decision? 2006 anges to the delays would the users decisions still be the same? How s behave if they have the opportunity to face this same decision with the art multiple times? Do answers to the above questions depend on how ents are and how they view each other s rationality? y Analyzing gives answers Games: Mixed to many Strategies of these questions. It tells us that any rational CPSC 532A Lecture 5, Slide 6
8 e s options are the columns. In each cell, the first number represents r, minus your delay), right. Rock and If the second players Papernumber choose Scissors the represents same side your (left colleague s or right) they have some hig otherwise they have a low utility. The game matrix is shown in Figure 3.4 Nash Equilibria of Example Games Rock C D Left Right Paper C 1, 1 4,0 Left 1 0 Scissors D 0, 4 3, 3 Right 0 1 Figure 3.6 Rock, Paper, Scissors game. Figure 3.1 The TCP user s (aka the Prisoner s) Figure Dilemma. 3.4 Coordination game. B F ro-sum game options what should At you the other adopt, end C or of the D? spectrum Does it depend from pure on what coordination you games lie zero league will do? Furthermore, which (bearing from in the mind perspective the comment of the we network made earlier operaof behavior can he about positive affine nstant-sum tions) B expect are2,1 from more the properly 0,0 two users? called Will constant-sum any two users games. behave Unlike common-pa mes presented with this scenario? Will the behavior change if the network s the users to communicate F 0,0with 1,2 each other c Shoham before making and Leyton-Brown, a decision? 2006 anges to the delays would the users decisions still be the same? How s behave if they have the opportunity to face this same decision with the Figure 3.7 Battle of the Sexes game. art multiple times? Do answers to the above questions depend on how ents are and how they view each other s rationality? y Analyzing gives answers Games: Mixed to many Strategies of these questions. It tells us that any rational CPSC 532A Lecture 5, Slide 6
9 e s options are the columns. As in the case In each of common-payoff cell, the first number games, represents we can use an abbreviated m r, minus your delay), right. and If the second players number choose the represents same side your (left colleague s or right) they have some hig represent Rock zero-sum Paper games, Scissors in which we write only one payoff value in ea otherwise they have a low utility. The game matrix is shown in Figure 3.4 Nash Equilibria value represents of Example the payoff Games of player 1, and thus the negative of the payof Rock Note, 0 though, that 1 whereas the 1 full matrix representation is unambiguous, C D Left Right the abbreviation we must explicit state whether this matrix represents a com game or a zero-sum one. Paper C A 1 1, classical 1 example 0 4,0 of a 1 zero-sum Left game 1 is the0game of matching pen game, each of the two players has a penny, and independently chooses to d Scissors heads 1or tails. The 1 two players 0 then compare their pennies. If they are th D 0, 4 3, 3 Right 0 1 player 1 pockets both, and otherwise player 2 pockets them. The pay shown in Figure 3.5. Figure 3.6 Rock, Paper, Scissors game. Figure 3.1 The TCP user s (aka the Prisoner s) Figure Dilemma. 3.4 Coordination game. B F Heads Tails ro-sum game options what should At you the other adopt, end C or of the D? spectrum Does it depend from pure on what coordination you games lie zero league will do? Furthermore, which (bearing from in the mind perspective the comment of the we network made earlier operaof behavior can he about positive affine nstant-sum tions) B expect are2,1 from more the properly 0,0 two users? called Heads Will constant-sum 1 any two users games. 1 behave Unlike common-pa mes presented with this scenario? Will the behavior change if the network s the users to communicate F 0,0with 1,2 each other c Shoham before Tails making and Leyton-Brown, 1a decision? anges to the delays would the users decisions still be the same? How s behave if they have the opportunity to face this same decision with the Figure 3.7 Battle of the Sexes game. Figure 3.5 Matching Pennies game. art multiple times? Do answers to the above questions depend on how ents are and how they view each other s rationality? y Analyzing gives answers Games: Mixed to many Strategies The popular of thesechildren s questions. game It tells of Rock, us thatpaper, any rational Scissors, CPSC 532Aalso Lecture known 5, Slideas 6 R
10 e s options are the columns. As in the case In each of common-payoff cell, the first number games, represents we can use an abbreviated m r, minus your delay), right. and If the second players number choose the represents same side your (left colleague s or right) they have some hig represent Rock zero-sum Paper games, Scissors in which we write only one payoff value in ea otherwise they have a low utility. The game matrix is shown in Figure 3.4 Nash Equilibria value represents of Example the payoff Games of player 1, and thus the negative of the payof Rock Note, 0 though, that 1 whereas the 1 full matrix representation is unambiguous, C D Left Right the abbreviation we must explicit state whether this matrix represents a com game or a zero-sum one. Paper C A 1 1, classical 1 example 0 4,0 of a 1 zero-sum Left game 1 is the0game of matching pen game, each of the two players has a penny, and independently chooses to d Scissors heads 1or tails. The 1 two players 0 then compare their pennies. If they are th D 0, 4 3, 3 Right 0 1 player 1 pockets both, and otherwise player 2 pockets them. The pay shown in Figure 3.5. Figure 3.6 Rock, Paper, Scissors game. Figure 3.1 The TCP user s (aka the Prisoner s) Figure Dilemma. 3.4 Coordination game. B F Heads Tails ro-sum game options what should At you the other adopt, end C or of the D? spectrum Does it depend from pure on what coordination you games lie zero league will do? Furthermore, which (bearing from in the mind perspective the comment of the we network made earlier operaof behavior can he about positive affine nstant-sum tions) B expect are2,1 from more the properly 0,0 two users? called Heads Will constant-sum 1 any two users games. 1 behave Unlike common-pa mes presented with this scenario? Will the behavior change if the network s the users to communicate F 0,0with 1,2 each other c Shoham before Tails making and Leyton-Brown, 1a decision? anges to the delays would the users decisions still be the same? How s behave The if they have the opportunity to face this same decision with the Figure paradox 3.7 of Battle Prisoner s of the Sexes dilemma: game. Figure the3.5 Nash Matching equilibrium Pennies is game. the only art multiple times? Do answers non-pareto-optimal the above questions outcome! depend on how ents are and how they view each other s rationality? y Analyzing gives answers Games: Mixed to many Strategies The popular of thesechildren s questions. game It tells of Rock, us thatpaper, any rational Scissors, CPSC 532Aalso Lecture known 5, Slideas 6 R
11 Lecture Overview Recap Mixed Strategies Fun Game Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 7
12 Mixed Strategies It would be a pretty bad idea to play any deterministic strategy in matching pennies Idea: confuse the opponent by playing randomly Define a strategy s i for agent i as any probability distribution over the actions A i. pure strategy: only one action is played with positive probability mixed strategy: more than one action is played with positive probability these actions are called the support of the mixed strategy Let the set of all strategies for i be S i Let the set of all strategy profiles be S = S 1... S n. Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 8
13 Utility under Mixed Strategies What is your payoff if all the players follow mixed strategy profile s S? We can t just read this number from the game matrix anymore: we won t always end up in the same cell Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 9
14 Utility under Mixed Strategies What is your payoff if all the players follow mixed strategy profile s S? We can t just read this number from the game matrix anymore: we won t always end up in the same cell Instead, use the idea of expected utility from decision theory: u i (s) = a A u i (a)p r(a s) P r(a s) = j N s j (a j ) Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 9
15 Best Response and Nash Equilibrium Our definitions of best response and Nash equilibrium generalize from actions to strategies. Best response: s i BR(s i ) iff s i S i, u i (s i, s i) u i (s i, s i ) Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 10
16 Best Response and Nash Equilibrium Our definitions of best response and Nash equilibrium generalize from actions to strategies. Best response: s i BR(s i ) iff s i S i, u i (s i, s i) u i (s i, s i ) Nash equilibrium: s = s1,..., s n is a Nash equilibrium iff i, s i BR(s i ) Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 10
17 Best Response and Nash Equilibrium Our definitions of best response and Nash equilibrium generalize from actions to strategies. Best response: s i BR(s i ) iff s i S i, u i (s i, s i) u i (s i, s i ) Nash equilibrium: s = s1,..., s n is a Nash equilibrium iff i, s i BR(s i ) Every finite game has a Nash equilibrium! [Nash, 1950] e.g., matching pennies: both players play heads/tails 50%/50% Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 10
18 Rock Paper Computing Mixed Nash Equilibria: Battle of the Sexes Scissors Figure 3.6 Rock, Paper, Scissors game. B F B 2,1 0,0 F 0,0 1,2 Figure 3.7 Battle of the Sexes game. It s hard in general to compute Nash equilibria, but it s easy when you can guess the support Strategies in normal-form games We have so far defined the actions available to each player in a game, but not yet his For set ofbos, strategies, let s or his look available for an choices. equilibrium Certainly onewhere kind of strategy all actions is to select are ure strategy a single action and play it; we call such a strategy a pure strategy, and we will use part the notation of the we support have already developed for actions to represent it. There is, however, another, less obvious type of strategy; a player can choose to randomize over the set of available actions according to some probability distribution; such a strategy is called ixed strategy a mixed strategy. Although it may not be immediately obvious why a player should introduce randomness into his choice of action, in fact in a multi-agent setting the role of mixed strategies is critical. We will return to this when we discuss solution concepts Analyzing Games: for Mixed games Strategies in the next section. CPSC 532A Lecture 5, Slide 11
19 Scissors Computing Mixed Nash Equilibria: Battle of the Sexes Figure 3.6 Rock, Paper, Scissors game. B F B 2,1 0,0 F 0,0 1,2 ure strategy ixed strategy Figure Strategies in normal-form games Battle of the Sexes game. Let player 2 play B with p, F with 1 p. If player 1 best-responds with a mixed strategy, player 2 must We have so far defined the actions available to each player in a game, but not yet his make set of strategies, him indifferent or his available between choices. Certainly F andone B kind (why?) of strategy is to select a single action and play it; we call such a strategy a pure strategy, and we will use the notation we have already developed for actions to represent it. There is, however, another, less obvious type of strategy; a player can choose to randomize over the set of available actions according to some probability distribution; such a strategy is called a mixed strategy. Although it may not be immediately obvious why a player should introduce randomness into his choice of action, in fact in a multi-agent setting the role of mixed strategies is critical. We will return to this when we discuss solution concepts for games in the next section. We define a mixed strategy for a normal form game as follows. Analyzing Games: Definition Mixed Strategies Let (N,(A 1,...,A n ),O,µ,u) be a normal form game, CPSC and532a for any Lecture 5, Slide 11
20 Scissors Computing Mixed Nash Equilibria: Battle of the Sexes Figure 3.6 Rock, Paper, Scissors game. B F B 2,1 0,0 F 0,0 1,2 ure strategy ixed strategy Figure Strategies in normal-form games Battle of the Sexes game. Let player 2 play B with p, F with 1 p. If player 1 best-responds with a mixed strategy, player 2 must We have so far defined the actions available to each player in a game, but not yet his make set of strategies, him indifferent or his available between choices. Certainly F andone B kind (why?) of strategy is to select a single action and play it; we call such a strategy a pure strategy, and we will use the notation we have already developed u 1 (B) for actions = u 1 to (F represent ) it. There is, however, another, less obvious type of strategy; a player can choose to randomize over the set of available actions according 2p + to some 0(1 probability p) = distribution; 0p + 1(1such p) a strategy is called a mixed strategy. Although it may not be immediately obvious why a player should introduce randomness into his choice of action, in fact in a multi-agent setting the role p = 1 of mixed strategies is critical. We will return to this 3 when we discuss solution concepts for games in the next section. We define a mixed strategy for a normal form game as follows. Analyzing Games: Definition Mixed Strategies Let (N,(A 1,...,A n ),O,µ,u) be a normal form game, CPSC and532a for any Lecture 5, Slide 11
21 Scissors Computing Mixed Nash Equilibria: Battle of the Sexes Figure 3.6 Rock, Paper, Scissors game. B F B 2,1 0,0 F 0,0 1,2 Figure 3.7 Battle of the Sexes game. Likewise, player 1 must randomize to make player 2 indifferent Strategies in normal-form games Why is player 1 willing to randomize? We have so far defined the actions available to each player in a game, but not yet his set of strategies, or his available choices. Certainly one kind of strategy is to select ure strategy a single action and play it; we call such a strategy a pure strategy, and we will use the notation we have already developed for actions to represent it. There is, however, another, less obvious type of strategy; a player can choose to randomize over the set of available actions according to some probability distribution; such a strategy is called ixed strategy a mixed strategy. Although it may not be immediately obvious why a player should introduce randomness into his choice of action, in fact in a multi-agent setting the role of mixed strategies is critical. We will return to this when we discuss solution concepts for games in the next section. We define a mixed strategy for a normal form game as follows. Definition Let (N,(A 1,...,A n ),O,µ,u) be a normal form game, and for any Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 11
22 Scissors Computing Mixed Nash Equilibria: Battle of the Sexes Figure 3.6 Rock, Paper, Scissors game. B F B 2,1 0,0 F 0,0 1,2 Figure 3.7 Battle of the Sexes game. Likewise, player 1 must randomize to make player 2 indifferent Strategies in normal-form games Why is player 1 willing to randomize? We have so far defined the actions available to each player in a game, but not yet his Let set of player strategies, 1 or play his available B with choices. q, FCertainly with 1one kind q. of strategy is to select ure strategy a single action and play it; we call such a strategy a pure strategy, and we will use the notation we have already developed u 2 (B) for = actions u 2 (F to represent ) it. There is, however, another, less obvious type of strategy; a player can choose to randomize over the set of available actions according q + to0(1 some probability q) = 0q distribution; + 2(1 such q) a strategy is called ixed strategy a mixed strategy. Although it may not be immediately obvious why a player should introduce randomness into his choice of action, q = 2 in fact in a multi-agent setting the role of mixed strategies is critical. We will return to this 3 when we discuss solution concepts for games in the next section. We define a mixed strategy for a normal form game as follows. ) are a Nash Thus the mixed strategies ( 2 3, 1 3 ), ( 1 3, 2 3 equilibrium. Definition Let (N,(A 1,...,A n ),O,µ,u) be a normal form game, and for any Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 11
23 Interpreting Mixed Strategy Equilibria What does it mean to play a mixed strategy? Different interpretations: Randomize to confuse your opponent consider the matching pennies example Players randomize when they are uncertain about the other s action consider battle of the sexes Mixed strategies are a concise description of what might happen in repeated play: count of pure strategies in the limit Mixed strategies describe population dynamics: 2 agents chosen from a population, all having deterministic strategies. MS is the probability of getting an agent who will play one PS or another. Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 12
24 Lecture Overview Recap Mixed Strategies Fun Game Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 13
25 Fun Game! L R T 80, 40 40, 80 B 40, 80 80, 40 Play once as each player, recording the strategy you follow. Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 14
26 Fun Game! L R T 320, 40 40, 80 B 40, 80 80, 40 Play once as each player, recording the strategy you follow. Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 14
27 Fun Game! L R T 44, 40 40, 80 B 40, 80 80, 40 Play once as each player, recording the strategy you follow. Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 14
28 Fun Game! L R T 80, 40; 320, 40; 44, 40 40, 80 B 40, 80 80, 40 Play once as each player, recording the strategy you follow. What does row player do in equilibrium of this game? Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 14
29 Fun Game! L R T 80, 40; 320, 40; 44, 40 40, 80 B 40, 80 80, 40 Play once as each player, recording the strategy you follow. What does row player do in equilibrium of this game? row player randomizes all the time that s what it takes to make column player indifferent Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 14
30 Fun Game! L R T 80, 40; 320, 40; 44, 40 40, 80 B 40, 80 80, 40 Play once as each player, recording the strategy you follow. What does row player do in equilibrium of this game? row player randomizes all the time that s what it takes to make column player indifferent What happens when people play this game? Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 14
31 Fun Game! L R T 80, 40; 320, 40; 44, 40 40, 80 B 40, 80 80, 40 Play once as each player, recording the strategy you follow. What does row player do in equilibrium of this game? row player randomizes all the time that s what it takes to make column player indifferent What happens when people play this game? with payoff of 320, row player goes up essentially all the time with payoff of 44, row player goes down essentially all the time Analyzing Games: Mixed Strategies CPSC 532A Lecture 5, Slide 14
Mixed Strategies; Maxmin
Mixed Strategies; Maxmin CPSC 532A Lecture 4 January 28, 2008 Mixed Strategies; Maxmin CPSC 532A Lecture 4, Slide 1 Lecture Overview 1 Recap 2 Mixed Strategies 3 Fun Game 4 Maxmin and Minmax Mixed Strategies;
More informationComputing Nash Equilibrium; Maxmin
Computing Nash Equilibrium; Maxmin Lecture 5 Computing Nash Equilibrium; Maxmin Lecture 5, Slide 1 Lecture Overview 1 Recap 2 Computing Mixed Nash Equilibria 3 Fun Game 4 Maxmin and Minmax Computing Nash
More informationGame Theory Week 1. Game Theory Course: Jackson, Leyton-Brown & Shoham. Game Theory Course: Jackson, Leyton-Brown & Shoham Game Theory Week 1
Game Theory Week 1 Game Theory Course: Jackson, Leyton-Brown & Shoham A Flipped Classroom Course Before Tuesday class: Watch the week s videos, on Coursera or locally at UBC Hand in the previous week s
More informationGame Theory Intro. Lecture 3. Game Theory Intro Lecture 3, Slide 1
Game Theory Intro Lecture 3 Game Theory Intro Lecture 3, Slide 1 Lecture Overview 1 What is Game Theory? 2 Game Theory Intro Lecture 3, Slide 2 Non-Cooperative Game Theory What is it? Game Theory Intro
More informationSelf-interested agents What is Game Theory? Example Matrix Games. Game Theory Intro. Lecture 3. Game Theory Intro Lecture 3, Slide 1
Game Theory Intro Lecture 3 Game Theory Intro Lecture 3, Slide 1 Lecture Overview 1 Self-interested agents 2 What is Game Theory? 3 Example Matrix Games Game Theory Intro Lecture 3, Slide 2 Self-interested
More informationGame Theory: Normal Form Games
Game Theory: Normal Form Games CPSC 322 Lecture 34 April 3, 2006 Reading: excerpt from Multiagent Systems, chapter 3. Game Theory: Normal Form Games CPSC 322 Lecture 34, Slide 1 Lecture Overview Recap
More informationMinmax and Dominance
Minmax and Dominance CPSC 532A Lecture 6 September 28, 2006 Minmax and Dominance CPSC 532A Lecture 6, Slide 1 Lecture Overview Recap Maxmin and Minmax Linear Programming Computing Fun Game Domination Minmax
More informationESSENTIALS OF GAME THEORY
ESSENTIALS OF GAME THEORY 1 CHAPTER 1 Games in Normal Form Game theory studies what happens when self-interested agents interact. What does it mean to say that agents are self-interested? It does not necessarily
More informationNoncooperative Games COMP4418 Knowledge Representation and Reasoning
Noncooperative Games COMP4418 Knowledge Representation and Reasoning Abdallah Saffidine 1 1 abdallah.saffidine@gmail.com slides design: Haris Aziz Semester 2, 2017 Abdallah Saffidine (UNSW) Noncooperative
More informationDomination Rationalizability Correlated Equilibrium Computing CE Computational problems in domination. Game Theory Week 3. Kevin Leyton-Brown
Game Theory Week 3 Kevin Leyton-Brown Game Theory Week 3 Kevin Leyton-Brown, Slide 1 Lecture Overview 1 Domination 2 Rationalizability 3 Correlated Equilibrium 4 Computing CE 5 Computational problems in
More informationUPenn NETS 412: Algorithmic Game Theory Game Theory Practice. Clyde Silent Confess Silent 1, 1 10, 0 Confess 0, 10 5, 5
Problem 1 UPenn NETS 412: Algorithmic Game Theory Game Theory Practice Bonnie Clyde Silent Confess Silent 1, 1 10, 0 Confess 0, 10 5, 5 This game is called Prisoner s Dilemma. Bonnie and Clyde have been
More informationNORMAL FORM (SIMULTANEOUS MOVE) GAMES
NORMAL FORM (SIMULTANEOUS MOVE) GAMES 1 For These Games Choices are simultaneous made independently and without observing the other players actions Players have complete information, which means they know
More informationIntroduction to (Networked) Game Theory. Networked Life NETS 112 Fall 2016 Prof. Michael Kearns
Introduction to (Networked) Game Theory Networked Life NETS 112 Fall 2016 Prof. Michael Kearns Game Theory for Fun and Profit The Beauty Contest Game Write your name and an integer between 0 and 100 Let
More informationECO 220 Game Theory. Objectives. Agenda. Simultaneous Move Games. Be able to structure a game in normal form Be able to identify a Nash equilibrium
ECO 220 Game Theory Simultaneous Move Games Objectives Be able to structure a game in normal form Be able to identify a Nash equilibrium Agenda Definitions Equilibrium Concepts Dominance Coordination Games
More informationAdvanced Microeconomics (Economics 104) Spring 2011 Strategic games I
Advanced Microeconomics (Economics 104) Spring 2011 Strategic games I Topics The required readings for this part is O chapter 2 and further readings are OR 2.1-2.3. The prerequisites are the Introduction
More informationGame Theory Lecturer: Ji Liu Thanks for Jerry Zhu's slides
Game Theory ecturer: Ji iu Thanks for Jerry Zhu's slides [based on slides from Andrew Moore http://www.cs.cmu.edu/~awm/tutorials] slide 1 Overview Matrix normal form Chance games Games with hidden information
More informationFIRST PART: (Nash) Equilibria
FIRST PART: (Nash) Equilibria (Some) Types of games Cooperative/Non-cooperative Symmetric/Asymmetric (for 2-player games) Zero sum/non-zero sum Simultaneous/Sequential Perfect information/imperfect information
More informationCS510 \ Lecture Ariel Stolerman
CS510 \ Lecture04 2012-10-15 1 Ariel Stolerman Administration Assignment 2: just a programming assignment. Midterm: posted by next week (5), will cover: o Lectures o Readings A midterm review sheet will
More informationEconS Game Theory - Part 1
EconS 305 - Game Theory - Part 1 Eric Dunaway Washington State University eric.dunaway@wsu.edu November 8, 2015 Eric Dunaway (WSU) EconS 305 - Lecture 28 November 8, 2015 1 / 60 Introduction Today, we
More informationIntroduction to (Networked) Game Theory. Networked Life NETS 112 Fall 2014 Prof. Michael Kearns
Introduction to (Networked) Game Theory Networked Life NETS 112 Fall 2014 Prof. Michael Kearns percent who will actually attend 100% Attendance Dynamics: Concave equilibrium: 100% percent expected to attend
More informationLecture #3: Networks. Kyumars Sheykh Esmaili
Lecture #3: Game Theory and Social Networks Kyumars Sheykh Esmaili Outline Games Modeling Network Traffic Using Game Theory Games Exam or Presentation Game You need to choose between exam or presentation:
More informationCSCI 699: Topics in Learning and Game Theory Fall 2017 Lecture 3: Intro to Game Theory. Instructor: Shaddin Dughmi
CSCI 699: Topics in Learning and Game Theory Fall 217 Lecture 3: Intro to Game Theory Instructor: Shaddin Dughmi Outline 1 Introduction 2 Games of Complete Information 3 Games of Incomplete Information
More informationGame Theory and Randomized Algorithms
Game Theory and Randomized Algorithms Guy Aridor Game theory is a set of tools that allow us to understand how decisionmakers interact with each other. It has practical applications in economics, international
More informationLecture 6: Basics of Game Theory
0368.4170: Cryptography and Game Theory Ran Canetti and Alon Rosen Lecture 6: Basics of Game Theory 25 November 2009 Fall 2009 Scribes: D. Teshler Lecture Overview 1. What is a Game? 2. Solution Concepts:
More informationCMU Lecture 22: Game Theory I. Teachers: Gianni A. Di Caro
CMU 15-781 Lecture 22: Game Theory I Teachers: Gianni A. Di Caro GAME THEORY Game theory is the formal study of conflict and cooperation in (rational) multi-agent systems Decision-making where several
More information1. Introduction to Game Theory
1. Introduction to Game Theory What is game theory? Important branch of applied mathematics / economics Eight game theorists have won the Nobel prize, most notably John Nash (subject of Beautiful mind
More informationGame Theory. Vincent Kubala
Game Theory Vincent Kubala Goals Define game Link games to AI Introduce basic terminology of game theory Overall: give you a new way to think about some problems What Is Game Theory? Field of work involving
More informationReading Robert Gibbons, A Primer in Game Theory, Harvester Wheatsheaf 1992.
Reading Robert Gibbons, A Primer in Game Theory, Harvester Wheatsheaf 1992. Additional readings could be assigned from time to time. They are an integral part of the class and you are expected to read
More informationCSC304 Lecture 2. Game Theory (Basic Concepts) CSC304 - Nisarg Shah 1
CSC304 Lecture 2 Game Theory (Basic Concepts) CSC304 - Nisarg Shah 1 Game Theory How do rational, self-interested agents act? Each agent has a set of possible actions Rules of the game: Rewards for the
More information1\2 L m R M 2, 2 1, 1 0, 0 B 1, 0 0, 0 1, 1
Chapter 1 Introduction Game Theory is a misnomer for Multiperson Decision Theory. It develops tools, methods, and language that allow a coherent analysis of the decision-making processes when there are
More informationGame Theory. Vincent Kubala
Game Theory Vincent Kubala vkubala@cs.brown.edu Goals efine game Link games to AI Introduce basic terminology of game theory Overall: give you a new way to think about some problems What Is Game Theory?
More informationExtensive Form Games: Backward Induction and Imperfect Information Games
Extensive Form Games: Backward Induction and Imperfect Information Games CPSC 532A Lecture 10 October 12, 2006 Extensive Form Games: Backward Induction and Imperfect Information Games CPSC 532A Lecture
More informationECON 2100 Principles of Microeconomics (Summer 2016) Game Theory and Oligopoly
ECON 2100 Principles of Microeconomics (Summer 2016) Game Theory and Oligopoly Relevant readings from the textbook: Mankiw, Ch. 17 Oligopoly Suggested problems from the textbook: Chapter 17 Questions for
More informationChapter 3 Learning in Two-Player Matrix Games
Chapter 3 Learning in Two-Player Matrix Games 3.1 Matrix Games In this chapter, we will examine the two-player stage game or the matrix game problem. Now, we have two players each learning how to play
More informationMultiagent Systems: Intro to Game Theory. CS 486/686: Introduction to Artificial Intelligence
Multiagent Systems: Intro to Game Theory CS 486/686: Introduction to Artificial Intelligence 1 1 Introduction So far almost everything we have looked at has been in a single-agent setting Today - Multiagent
More informationGames. Episode 6 Part III: Dynamics. Baochun Li Professor Department of Electrical and Computer Engineering University of Toronto
Games Episode 6 Part III: Dynamics Baochun Li Professor Department of Electrical and Computer Engineering University of Toronto Dynamics Motivation for a new chapter 2 Dynamics Motivation for a new chapter
More information(a) Left Right (b) Left Right. Up Up 5-4. Row Down 0-5 Row Down 1 2. (c) B1 B2 (d) B1 B2 A1 4, 2-5, 6 A1 3, 2 0, 1
Economics 109 Practice Problems 2, Vincent Crawford, Spring 2002 In addition to these problems and those in Practice Problems 1 and the midterm, you may find the problems in Dixit and Skeath, Games of
More informationComputational Methods for Non-Cooperative Game Theory
Computational Methods for Non-Cooperative Game Theory What is a game? Introduction A game is a decision problem in which there a multiple decision makers, each with pay-off interdependence Each decisions
More informationImperfect Information Extensive Form Games
Imperfect Information Extensive Form Games ISCI 330 Lecture 15 March 6, 2007 Imperfect Information Extensive Form Games ISCI 330 Lecture 15, Slide 1 Lecture Overview 1 Recap 2 Imperfect Information Extensive
More informationEC3224 Autumn Lecture #02 Nash Equilibrium
Reading EC3224 Autumn Lecture #02 Nash Equilibrium Osborne Chapters 2.6-2.10, (12) By the end of this week you should be able to: define Nash equilibrium and explain several different motivations for it.
More informationECON 282 Final Practice Problems
ECON 282 Final Practice Problems S. Lu Multiple Choice Questions Note: The presence of these practice questions does not imply that there will be any multiple choice questions on the final exam. 1. How
More informationLECTURE 26: GAME THEORY 1
15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 26: GAME THEORY 1 INSTRUCTOR: GIANNI A. DI CARO ICE-CREAM WARS http://youtu.be/jilgxenbk_8 2 GAME THEORY Game theory is the formal study of conflict and cooperation
More informationIntroduction to Game Theory
Introduction to Game Theory Managing with Game Theory Hongying FEI Feihy@i.shu.edu.cn Poker Game ( 2 players) Each player is dealt randomly 3 cards Both of them order their cards as they want Cards at
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India August 2012
Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India August 01 Rationalizable Strategies Note: This is a only a draft version,
More informationGAME THEORY: STRATEGY AND EQUILIBRIUM
Prerequisites Almost essential Game Theory: Basics GAME THEORY: STRATEGY AND EQUILIBRIUM MICROECONOMICS Principles and Analysis Frank Cowell Note: the detail in slides marked * can only be seen if you
More informationGame Theory: Introduction. Game Theory. Game Theory: Applications. Game Theory: Overview
Game Theory: Introduction Game Theory Game theory A means of modeling strategic behavior Agents act to maximize own welfare Agents understand their actions affect actions of other agents ECON 370: Microeconomic
More informationMultiagent Systems: Intro to Game Theory. CS 486/686: Introduction to Artificial Intelligence
Multiagent Systems: Intro to Game Theory CS 486/686: Introduction to Artificial Intelligence 1 Introduction So far almost everything we have looked at has been in a single-agent setting Today - Multiagent
More informationDominance and Best Response. player 2
Dominance and Best Response Consider the following game, Figure 6.1(a) from the text. player 2 L R player 1 U 2, 3 5, 0 D 1, 0 4, 3 Suppose you are player 1. The strategy U yields higher payoff than any
More informationCMU-Q Lecture 20:
CMU-Q 15-381 Lecture 20: Game Theory I Teacher: Gianni A. Di Caro ICE-CREAM WARS http://youtu.be/jilgxenbk_8 2 GAME THEORY Game theory is the formal study of conflict and cooperation in (rational) multi-agent
More informationIntroduction to Game Theory
Introduction to Game Theory Part 1. Static games of complete information Chapter 1. Normal form games and Nash equilibrium Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas V. Filipe
More informationNormal Form Games: A Brief Introduction
Normal Form Games: A Brief Introduction Arup Daripa TOF1: Market Microstructure Birkbeck College Autumn 2005 1. Games in strategic form. 2. Dominance and iterated dominance. 3. Weak dominance. 4. Nash
More informationSolution Concepts 4 Nash equilibrium in mixed strategies
Solution Concepts 4 Nash equilibrium in mixed strategies Watson 11, pages 123-128 Bruno Salcedo The Pennsylvania State University Econ 402 Summer 2012 Mixing strategies In a strictly competitive situation
More informationNote: A player has, at most, one strictly dominant strategy. When a player has a dominant strategy, that strategy is a compelling choice.
Game Theoretic Solutions Def: A strategy s i 2 S i is strictly dominated for player i if there exists another strategy, s 0 i 2 S i such that, for all s i 2 S i,wehave ¼ i (s 0 i ;s i) >¼ i (s i ;s i ):
More informationMultiagent Systems: Intro to Game Theory. CS 486/686: Introduction to Artificial Intelligence
Multiagent Systems: Intro to Game Theory CS 486/686: Introduction to Artificial Intelligence 1 Introduction So far almost everything we have looked at has been in a single-agent setting Today - Multiagent
More informationDominant Strategies (From Last Time)
Dominant Strategies (From Last Time) Continue eliminating dominated strategies for B and A until you narrow down how the game is actually played. What strategies should A and B choose? How are these the
More informationECO 5341 Strategic Behavior Lecture Notes 3
ECO 5341 Strategic Behavior Lecture Notes 3 Saltuk Ozerturk SMU Spring 2016 (SMU) Lecture Notes 3 Spring 2016 1 / 20 Lecture Outline Review: Dominance and Iterated Elimination of Strictly Dominated Strategies
More informationCPS 570: Artificial Intelligence Game Theory
CPS 570: Artificial Intelligence Game Theory Instructor: Vincent Conitzer What is game theory? Game theory studies settings where multiple parties (agents) each have different preferences (utility functions),
More informationResource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Game Theory
Resource Allocation and Decision Analysis (ECON 8) Spring 4 Foundations of Game Theory Reading: Game Theory (ECON 8 Coursepak, Page 95) Definitions and Concepts: Game Theory study of decision making settings
More informationMixed strategy Nash equilibrium
Mixed strategy Nash equilibrium Felix Munoz-Garcia Strategy and Game Theory - Washington State University Looking back... So far we have been able to nd the NE of a relatively large class of games with
More informationRepeated Games. ISCI 330 Lecture 16. March 13, Repeated Games ISCI 330 Lecture 16, Slide 1
Repeated Games ISCI 330 Lecture 16 March 13, 2007 Repeated Games ISCI 330 Lecture 16, Slide 1 Lecture Overview Repeated Games ISCI 330 Lecture 16, Slide 2 Intro Up to this point, in our discussion of extensive-form
More informationn-person Games in Normal Form
Chapter 5 n-person Games in rmal Form 1 Fundamental Differences with 3 Players: the Spoilers Counterexamples The theorem for games like Chess does not generalize The solution theorem for 0-sum, 2-player
More informationMicroeconomics of Banking: Lecture 4
Microeconomics of Banking: Lecture 4 Prof. Ronaldo CARPIO Oct. 16, 2015 Administrative Stuff Homework 1 is due today at the end of class. I will upload the solutions and Homework 2 (due in two weeks) later
More informationTHEORY: NASH EQUILIBRIUM
THEORY: NASH EQUILIBRIUM 1 The Story Prisoner s Dilemma Two prisoners held in separate rooms. Authorities offer a reduced sentence to each prisoner if he rats out his friend. If a prisoner is ratted out
More informationGame theory Computational Models of Cognition
Game theory Taxonomy Rational behavior Definitions Common games Nash equilibria Mixed strategies Properties of Nash equilibria What do NE mean? Mutually Assured Destruction 6 rik@cogsci.ucsd.edu Taxonomy
More informationDesign of intelligent surveillance systems: a game theoretic case. Nicola Basilico Department of Computer Science University of Milan
Design of intelligent surveillance systems: a game theoretic case Nicola Basilico Department of Computer Science University of Milan Outline Introduction to Game Theory and solution concepts Game definition
More information1. Simultaneous games All players move at same time. Represent with a game table. We ll stick to 2 players, generally A and B or Row and Col.
I. Game Theory: Basic Concepts 1. Simultaneous games All players move at same time. Represent with a game table. We ll stick to 2 players, generally A and B or Row and Col. Representation of utilities/preferences
More informationGame Theory. Wolfgang Frimmel. Dominance
Game Theory Wolfgang Frimmel Dominance 1 / 13 Example: Prisoners dilemma Consider the following game in normal-form: There are two players who both have the options cooperate (C) and defect (D) Both players
More informationU strictly dominates D for player A, and L strictly dominates R for player B. This leaves (U, L) as a Strict Dominant Strategy Equilibrium.
Problem Set 3 (Game Theory) Do five of nine. 1. Games in Strategic Form Underline all best responses, then perform iterated deletion of strictly dominated strategies. In each case, do you get a unique
More informationSection Notes 6. Game Theory. Applied Math 121. Week of March 22, understand the difference between pure and mixed strategies.
Section Notes 6 Game Theory Applied Math 121 Week of March 22, 2010 Goals for the week be comfortable with the elements of game theory. understand the difference between pure and mixed strategies. be able
More informationEcon 302: Microeconomics II - Strategic Behavior. Problem Set #5 June13, 2016
Econ 302: Microeconomics II - Strategic Behavior Problem Set #5 June13, 2016 1. T/F/U? Explain and give an example of a game to illustrate your answer. A Nash equilibrium requires that all players are
More informationAdversarial Search and Game Theory. CS 510 Lecture 5 October 26, 2017
Adversarial Search and Game Theory CS 510 Lecture 5 October 26, 2017 Reminders Proposals due today Midterm next week past midterms online Midterm online BBLearn Available Thurs-Sun, ~2 hours Overview Game
More informationNash Equilibrium. Felix Munoz-Garcia School of Economic Sciences Washington State University. EconS 503
Nash Equilibrium Felix Munoz-Garcia School of Economic Sciences Washington State University EconS 503 est Response Given the previous three problems when we apply dominated strategies, let s examine another
More informationExtensive Form Games: Backward Induction and Imperfect Information Games
Extensive Form Games: Backward Induction and Imperfect Information Games CPSC 532A Lecture 10 Extensive Form Games: Backward Induction and Imperfect Information Games CPSC 532A Lecture 10, Slide 1 Lecture
More informationGenetic Algorithms in MATLAB A Selection of Classic Repeated Games from Chicken to the Battle of the Sexes
ECON 7 Final Project Monica Mow (V7698) B Genetic Algorithms in MATLAB A Selection of Classic Repeated Games from Chicken to the Battle of the Sexes Introduction In this project, I apply genetic algorithms
More informationGame theory attempts to mathematically. capture behavior in strategic situations, or. games, in which an individual s success in
Game Theory Game theory attempts to mathematically capture behavior in strategic situations, or games, in which an individual s success in making choices depends on the choices of others. A game Γ consists
More informationGame theory. Logic and Decision Making Unit 2
Game theory Logic and Decision Making Unit 2 Introduction Game theory studies decisions in which the outcome depends (at least partly) on what other people do All decision makers are assumed to possess
More informationFinite games: finite number of players, finite number of possible actions, finite number of moves. Canusegametreetodepicttheextensiveform.
A game is a formal representation of a situation in which individuals interact in a setting of strategic interdependence. Strategic interdependence each individual s utility depends not only on his own
More informationNormal Form Games. Here is the definition of a strategy: A strategy is a complete contingent plan for a player in the game.
Normal Form Games Here is the definition of a strategy: A strategy is a complete contingent plan for a player in the game. For extensive form games, this means that a strategy must specify the action that
More informationThe extensive form representation of a game
The extensive form representation of a game Nodes, information sets Perfect and imperfect information Addition of random moves of nature (to model uncertainty not related with decisions of other players).
More informationLecture 10: September 2
SC 63: Games and Information Autumn 24 Lecture : September 2 Instructor: Ankur A. Kulkarni Scribes: Arjun N, Arun, Rakesh, Vishal, Subir Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More informationGame Theory Refresher. Muriel Niederle. February 3, A set of players (here for simplicity only 2 players, all generalized to N players).
Game Theory Refresher Muriel Niederle February 3, 2009 1. Definition of a Game We start by rst de ning what a game is. A game consists of: A set of players (here for simplicity only 2 players, all generalized
More informationGame Theory. 4: Nash equilibrium in different games and mixed strategies
Game Theory 4: Nash equilibrium in different games and mixed strategies Review of lecture three A game with no dominated strategy: The battle of the sexes The concept of Nash equilibrium The formal definition
More informationGame Theory ( nd term) Dr. S. Farshad Fatemi. Graduate School of Management and Economics Sharif University of Technology.
Game Theory 44812 (1393-94 2 nd term) Dr. S. Farshad Fatemi Graduate School of Management and Economics Sharif University of Technology Spring 2015 Dr. S. Farshad Fatemi (GSME) Game Theory Spring 2015
More informationBackward Induction. ISCI 330 Lecture 14. March 1, Backward Induction ISCI 330 Lecture 14, Slide 1
ISCI 330 Lecture 4 March, 007 ISCI 330 Lecture 4, Slide Lecture Overview Recap ISCI 330 Lecture 4, Slide Subgame Perfection Notice that the definition contains a subtlety. n agent s strategy requires a
More informationLecture 23. Offense vs. Defense & Dynamic Games
Lecture 3. Offense vs. Defense & Dynamic Games EC DD & EE / Manove Offense vs Defense p EC DD & EE / Manove Clicker Question p Using Game Theory to Analyze Offense versus Defense In many competitive situations
More informationPARALLEL NASH EQUILIBRIA IN BIMATRIX GAMES ISAAC ELBAZ CSE633 FALL 2012 INSTRUCTOR: DR. RUSS MILLER
PARALLEL NASH EQUILIBRIA IN BIMATRIX GAMES ISAAC ELBAZ CSE633 FALL 2012 INSTRUCTOR: DR. RUSS MILLER WHAT IS GAME THEORY? Branch of mathematics that deals with the analysis of situations involving parties
More informationLecture 13(ii) Announcements. Lecture on Game Theory. None. 1. The Simple Version of the Battle of the Sexes
Lecture 13(ii) Announcements None Lecture on Game Theory 1. The Simple Version of the Battle of the Sexes 2. The Battle of the Sexes with Some Strategic Moves 3. Rock Paper Scissors 4. Chicken 5. Duopoly
More informationJapanese. Sail North. Search Search Search Search
COMP9514, 1998 Game Theory Lecture 1 1 Slide 1 Maurice Pagnucco Knowledge Systems Group Department of Articial Intelligence School of Computer Science and Engineering The University of New South Wales
More informationRobustness against Longer Memory Strategies in Evolutionary Games.
Robustness against Longer Memory Strategies in Evolutionary Games. Eizo Akiyama 1 Players as finite state automata In our daily life, we have to make our decisions with our restricted abilities (bounded
More informationDECISION MAKING GAME THEORY
DECISION MAKING GAME THEORY THE PROBLEM Two suspected felons are caught by the police and interrogated in separate rooms. Three cases were presented to them. THE PROBLEM CASE A: If only one of you confesses,
More informationGame Theory and Economics Prof. Dr. Debarshi Das Humanities and Social Sciences Indian Institute of Technology, Guwahati
Game Theory and Economics Prof. Dr. Debarshi Das Humanities and Social Sciences Indian Institute of Technology, Guwahati Module No. # 05 Extensive Games and Nash Equilibrium Lecture No. # 03 Nash Equilibrium
More informationSimultaneous-Move Games: Mixed Strategies. Games Of Strategy Chapter 7 Dixit, Skeath, and Reiley
Simultaneous-Move Games: Mixed Strategies Games Of Strategy Chapter 7 Dixit, Skeath, and Reiley Terms to Know Expected Payoff Opponent s Indifference Property Introductory Game The professor will assign
More informationECO 199 B GAMES OF STRATEGY Spring Term 2004 B February 24 SEQUENTIAL AND SIMULTANEOUS GAMES. Representation Tree Matrix Equilibrium concept
CLASSIFICATION ECO 199 B GAMES OF STRATEGY Spring Term 2004 B February 24 SEQUENTIAL AND SIMULTANEOUS GAMES Sequential Games Simultaneous Representation Tree Matrix Equilibrium concept Rollback (subgame
More informationStudent Name. Student ID
Final Exam CMPT 882: Computational Game Theory Simon Fraser University Spring 2010 Instructor: Oliver Schulte Student Name Student ID Instructions. This exam is worth 30% of your final mark in this course.
More informationMath 464: Linear Optimization and Game
Math 464: Linear Optimization and Game Haijun Li Department of Mathematics Washington State University Spring 2013 Game Theory Game theory (GT) is a theory of rational behavior of people with nonidentical
More informationDominant and Dominated Strategies
Dominant and Dominated Strategies Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Junel 8th, 2016 C. Hurtado (UIUC - Economics) Game Theory On the
More informationDistributed Optimization and Games
Distributed Optimization and Games Introduction to Game Theory Giovanni Neglia INRIA EPI Maestro 18 January 2017 What is Game Theory About? Mathematical/Logical analysis of situations of conflict and cooperation
More informationDistributed Optimization and Games
Distributed Optimization and Games Introduction to Game Theory Giovanni Neglia INRIA EPI Maestro 18 January 2017 What is Game Theory About? Mathematical/Logical analysis of situations of conflict and cooperation
More informationEconomics 201A - Section 5
UC Berkeley Fall 2007 Economics 201A - Section 5 Marina Halac 1 What we learnt this week Basics: subgame, continuation strategy Classes of games: finitely repeated games Solution concepts: subgame perfect
More informationCSC304 Lecture 3. Game Theory (More examples, PoA, PoS) CSC304 - Nisarg Shah 1
CSC304 Lecture 3 Game Theory (More examples, PoA, PoS) CSC304 - Nisarg Shah 1 Recap Normal form games Domination among strategies Weak/strict domination Hope 1: Find a weakly/strictly dominant strategy
More information