For any player i, a strategy weakly dominates another strategy if (With at least one S -i that gives a strict inequality) strictly dominates if where.

Slides:



Advertisements
Similar presentations
Chapter 17: Making Complex Decisions April 1, 2004.
Advertisements

Pondering more Problems. Enriching the Alice-Bob story Go to AGo to B Go to A Alice Go to B Go to A Go to B Go shoot pool Alice.
Game Theory Assignment For all of these games, P1 chooses between the columns, and P2 chooses between the rows.
Mixed Strategies For Managers
Chapter Twenty-Eight Game Theory. u Game theory models strategic behavior by agents who understand that their actions affect the actions of other agents.
Simultaneous- Move Games with Mixed Strategies Zero-sum Games.
Non-Cooperative Game Theory To define a game, you need to know three things: –The set of players –The strategy sets of the players (i.e., the actions they.
ECO290E: Game Theory Lecture 5 Mixed Strategy Equilibrium.
MIT and James Orlin © Game Theory 2-person 0-sum (or constant sum) game theory 2-person game theory (e.g., prisoner’s dilemma)
EC3224 Autumn Lecture #04 Mixed-Strategy Equilibrium
EC941 - Game Theory Lecture 7 Prof. Francesco Squintani
Part 3: The Minimax Theorem
Working Some Problems. Telephone Game How about xexed strategies? Let Winnie call with probability p and wait with probability 1-p. For what values of.
An Introduction to Game Theory Part I: Strategic Games
2008/02/06Lecture 21 ECO290E: Game Theory Lecture 2 Static Games and Nash Equilibrium.
Chapter 6 © 2006 Thomson Learning/South-Western Game Theory.
EC941 - Game Theory Prof. Francesco Squintani Lecture 8 1.
Introduction to Nash Equilibrium Presenter: Guanrao Chen Nov. 20, 2002.
by Vincent Conitzer of Duke
EC102: Class 9 Christina Ammon.
A camper awakens to the growl of a hungry bear and sees his friend putting on a pair of running shoes, “You can’t outrun a bear,” scoffs the camper. His.
Game Theory Game theory models strategic behavior by agents who understand that their actions affect the actions of other agents. Been used to study –Company.
An Introduction to Game Theory Part II: Mixed and Correlated Strategies Bernhard Nebel.
Lecture 1 - Introduction 1.  Introduction to Game Theory  Basic Game Theory Examples  Strategic Games  More Game Theory Examples  Equilibrium  Mixed.
Games of pure conflict two person constant sum. Two-person constant sum game Sometimes called zero-sum game. The sum of the players’ payoffs is the same,
Chapter Twenty-Eight Game Theory. u Game theory models strategic behavior by agents who understand that their actions affect the actions of other agents.
6.1 Consider a simultaneous game in which player A chooses one of two actions (Up or Down), and B chooses one of two actions (Left or Right). The game.
An Introduction to Game Theory Part III: Strictly Competitive Games Bernhard Nebel.
An introduction to game theory Today: The fundamentals of game theory, including Nash equilibrium.
Static Games of Complete Information: Subgame Perfection
Extensive Game with Imperfect Information Part I: Strategy and Nash equilibrium.
QR 38, 2/22/07 Strategic form: dominant strategies I.Strategic form II.Finding Nash equilibria III.Strategic form games in IR.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 1 of 68 Chapter 9 The Theory of Games.
Nash Equilibrium - definition A mixed-strategy profile σ * is a Nash equilibrium (NE) if for every player i we have u i (σ * i, σ * -i ) ≥ u i (s i, σ.
EC941 - Game Theory Francesco Squintani Lecture 3 1.
UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Bargaining and Negotiation Review.
An introduction to game theory Today: The fundamentals of game theory, including Nash equilibrium.
Minimax strategies, Nash equilibria, correlated equilibria Vincent Conitzer
CPS 170: Artificial Intelligence Game Theory Instructor: Vincent Conitzer.
Chapter 9 Games with Imperfect Information Bayesian Games.
1. 2 Non-Cooperative games Player I Player II I want the maximum payoff to Player I I want the maximum payoff to Player II.
Microeconomics Course E John Hey. Examinations Go to Read.
Introduction to Game Theory application to networks Joy Ghosh CSE 716, 25 th April, 2003.
Game Theory: introduction and applications to computer networks Game Theory: introduction and applications to computer networks Lecture 2: two-person non.
Game Theory: introduction and applications to computer networks Game Theory: introduction and applications to computer networks Introduction Giovanni Neglia.
Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.
Lecture 5 Introduction to Game theory. What is game theory? Game theory studies situations where players have strategic interactions; the payoff that.
1 What is Game Theory About? r Analysis of situations where conflict of interests is present r Goal is to prescribe how conflicts can be resolved 2 2 r.
Strategic Behavior in Business and Econ Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies.
Normal Form Games, Normal Form Games, Rationality and Iterated Rationality and Iterated Deletion of Dominated Strategies Deletion of Dominated Strategies.
1 a1a1 A1A1 a2a2 a3a3 A2A Mixed Strategies When there is no saddle point: We’ll think of playing the game repeatedly. We continue to assume that.
Empirical Aspects of Plurality Elections David R. M. Thompson, Omer Lev, Kevin Leyton-Brown & Jeffrey S. Rosenschein COMSOC 2012 Kraków, Poland.
5.1.Static Games of Incomplete Information
1 The Volunteer’s Dilemma (Mixed Strategies). 2 The Volunteer Dilemma Game Simultaneously and independently, players have to decide if they wish to volunteer.
Games of pure conflict two-person constant sum games.
Market Design and Analysis Lecture 2 Lecturer: Ning Chen ( 陈宁 )
Pondering more Problems. Enriching the Alice-Bob story Go to AGo to B Go to A Alice Go to B Go to A Go to B Go shoot pool Alice.
Microeconomics Course E John Hey. Examinations Go to Read.
Day 9 GAME THEORY. 3 Solution Methods for Non-Zero Sum Games Dominant Strategy Iterated Dominant Strategy Nash Equilibrium NON- ZERO SUM GAMES HOW TO.
Working Some Problems.
Game theory basics A Game describes situations of strategic interaction, where the payoff for one agent depends on its own actions as well as on the actions.
Mixed Strategies Keep ‘em guessing.
Games of pure conflict two person constant sum
Communication Complexity as a Lower Bound for Learning in Games
نسیم لحیم‌گرزاده استاد درس: دکتر فرزاد توحیدخواه
Chapter 14 & 15 Repeated Games.
Chapter 14 & 15 Repeated Games.
Molly W. Dahl Georgetown University Econ 101 – Spring 2009
Lecture Game Theory.
Presentation transcript:

For any player i, a strategy weakly dominates another strategy if (With at least one S -i that gives a strict inequality) strictly dominates if where S -i represents the product of all strategy sets other than player's i

In simple terms, the matrix looks like this: It is then possible to make general the point values: DefectCooperate Defectlose-loselose more-win more Cooperatewin more-lose morewin-win

Given an initial allocation of goods among a set of individuals, a change to a different allocation that makes at least one individual better off without making any other individual worse off is called a Pareto improvement. An allocation is defined as Pareto optimal (or Pareto efficient) when no further Pareto improvements can be made.

Option AOption BOption C Option A0, 025, 405, 10 Option B40, 250, 05, 15 Option C10, 515, 510, 10 A Payoff Matrix - Nash Equlibria in bold There is an easy numerical way to identify Nash equilibria on a payoff matrix. It is especially helpful in two-person games where players have more than two strategies. If the first payoff number, in the duplet of the cell, is the maximum of the column of the cell and if the second number is the maximum of the row of the cell - then the cell represents a Nash equilibrium. We can apply this rule to a 3×3 matrix: Using the rule, we can very quickly (much faster than with formal analysis) see that the Nash Equlibria cells are (B,A), (A,B), and (C,C). Indeed, for cell (B,A) 40 is the maximum of the first column and 25 is the maximum of the second row. For (A,B) 25 is the maximum of the second column and 40 is the maximum of the first row. Same for cell (C,C). For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns. This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and check if the second member of the pair is the maximum of the row. If these conditions are met, the cell represents a Nash Equilibrium. Check all columns this way to find all NE cells.

Informal definition Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks himself or herself: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?“ If any player would answer "Yes", then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the set of strategies is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium. The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because it may happen that a Nash equilibrium is not Pareto optimal.

Let (S, f) be a game with n players, where S i is the strategy set for player i, S=S 1 X S 2... X S n is the set of strategy profiles and f=(f 1 (x),..., f n (x)) is the payoff function for x S. Let x i be a strategy profile of player i and x -i be a strategy profile of all players except for player i. When each player i {1,..., n} chooses strategy x i resulting in strategy profile x = (x 1,..., x n ) then player i obtains payoff f i (x). Note that the payoff depends on the strategy profile chosen, i.e., on the strategy chosen by player i as well as the strategies chosen by all the other players. A strategy profile x * S is a Nash equilibrium if no unilateral deviation in strategy by any single player is profitable for that player, that isstrategy profiles Nash proved that if we allow mixed strategies, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium.mixed strategies When the inequality above holds strictly (with > instead of ) for all players and all feasible alternative strategies, then the equilibrium is classified as a strict Nash equilibrium. If instead, for some player, there is exact equality between and some other strategy, then the equilibrium is classified as a weak Nash equilibrium.

Finger Morra Game

The utility U E is the expected value for E.

Utility:

p k is the prob. that the kicker kicks right p j is the prob. that the goalie jumps right