1 intro game-theory

GAMUTGAMUT

WORKSHOP ON WORKSHOP ON COMPUTATIONAL COMPUTATIONAL ASPECTS OF GAME THEORYASPECTS OF GAME THEORY

J U N E 1 6 - 2 0 , 2 0 1 4

E C S U , I N D I A N S T A T I S T I C A L I N S T I T U T E K O L K A T A

PrithvirajPrithviraj (Raj) (Raj) DasguptaDasguptaAssociate Associate Professor, Computer Professor, Computer Science Science Department, Department,

University University of Nebraska, Omahaof Nebraska, Omaha

SPEAKER’S BACKGROUND

• Associate Professor, Computer Science, University of Nebraska, Omaha (2001- present)• Director, CMANTIC Lab (robotics, computational

economics)

• Ph.D. (2001) Univ. of California, Santa Barbara• Ph.D. (2001) Univ. of California, Santa Barbara• Computational economics using software agents

• B.C.S.E (1995) – Jadavpur University• 1994: Summer internship at I.S.I

June 16-20, 2014 2Game Theory Workshop - Raj Dasgupta

UNIVERSITY OF NEBRASKA, OMAHA

• Founded in 1908• Computer Science

program started in early 80s• Department since early

90s90s

• 18 full-time faculty• ~400 undergrad, 125

Masters and 15 Ph.D. students• Research areas: AI,

Database/Data mining, Networking, Systems

June 16-20, 2014 Game Theory Workshop - Raj Dasgupta 3

C-MANTIC GROUP

• http://cmantic.unomaha.edu• Research Topics:

• Multi-robot systems path and task planning• Multi-robot/agent systems coordination using game

theory-based techniques• Modular robotic systems, Information aggregation using

prediction markets, Agent-based crowd simulation, etc.prediction markets, Agent-based crowd simulation, etc.

• Established by Raj Dasgupta in 2004• Received over $3 million as PI in external funding from

DoD Navair, ONR, NASA; over 80 publications in top-tier conferences and journals

• Currently 8 members including• 2 post-doctoral researchers with Ph.D. in robotics

(electrical, control, mechanical engineering) and vision• 4 graduate students (computer science)• 1 undergraduate students (computer engineering,

computer science)• Collaborations with faculty from Mechanical engg,

Computer science (UN-Lincoln, U. Southern Mississippi), Mathematics (UNO)

6/16/2014 Raj Dasgupta, CMANTIC Lab, UNO 4

AVAILABLE ROBOT PLATFORMS

E-puck mini robot -suitable for table-top

experiments for proof-of-

Coroware Corobot (indoor

robot)• Suitable for indoor experiments in; Coroware Explorer

6/16/2014 Raj Dasgupta, CMANTIC Lab, UNO 5

experiments for proof-of-

concept• Suitable for indoor experiments in;

hardware and software compatible

with Coroware Explorer robot;

• Sensors: Laser, IR, fixed camera;

Stargazer (IR-based indoor

localization); Wifi

Coroware Explorer

(outdoor robot) – all terrain robot for outdoor experiments; customized with GPS,

compass for localization

All techniques are first verified on Webots simulator using simulated models of e-puck and Corobot robots

Turtlebot (indoor robot)• Suitable for experiments in indoor

arena within lab;

• Kinect sensor; IR

Pelican UAV

(aerial robot)• Newly acquired

robot

• Sensors: Camera;

gyro

RESEARCH PROBLEM

• How to coordinate a set of robots to perform a set of complexcomplex tasks in a collaborativecollaborative manner• Complex task: single robot does not have resources to

complete the task individually

• Coordination can be synchronous or asynchronousRobots might or might not have to perform the task at the same • Robots might or might not have to perform the task at the same time

• Performance metric(s) need to be optimized while performing tasks• Time to complete tasks, distance traveled, energy expended

• Robots are able to communicate with each other• Bluetooth, Wi-fi, IR, Camera, Laser

• Some robots can fail, but system should not stall

6/16/2014 Raj Dasgupta, CMANTIC Lab, UNO

6

APPLICATIONS

• Humanitarian de-mining (COMRADES)

• Autonomous exploration for planetary surfaces (ModRED)

• Automatic Target Recognition (ATR) for search and recon (COMSTAR)recon (COMSTAR)

• Unmanned Search and Rescue

• Civlian and domestic applications like agriculture, vaccum cleaning, etc.

6/16/2014 Raj Dasgupta, CMANTIC Lab, UNO

7

GAME THEORY WORKSHOP

DAY 1


OBJECTIVE

• Introduction to game theory from a computer science perspective

• Learn the fundamental concepts in game theory• Mathematical solution concepts

• Algorithms used to solve games• Algorithms used to solve games

• Applications of game theory in different application domains• Google Adwords

• Trading Agent Competition (Lemonade Stand Game)

• Develop programming tools for solving game theory problems and preparation for advanced graduate coursework


OUTCOMES

• Write software for algorithms for • Supply chain management – energy market, travel

booking, warehouse inventory management

• Auctions (e-bay, etc)

• Ad-placement (ad-auctions) for Internet search engines, • Ad-placement (ad-auctions) for Internet search engines, Youtube, etc.

• Applications that require coordination between people or software agents

• Social networks – information aggregation

• Robotics – distributed robot systems


ADX GAME

• An Ad Network bids for display ads opportunities• Fulfill advertising

contracts at contracts at minimum cost

• High quality targeting

• Sustain and attract advertisers

• https://sites.google.com/site/gameadx/


POWER-TAC GAME

• Agents act as retail brokers in a local power distribution region• Purchasing power from wholesale market and local sources

• Sell power to local customers and into the wholesale market. market.

• Solve a supply-chain problem

• Product is infinitely perishable

• Supply and demand must be exactly balanced at all times

• http://www.powertac.org


TOPICS TO BE COVERED

• Day 1: Introduction to game theory – normal form games

• Day 2: Solution concepts for normal form games (math)

Day 3: Bayesian games; applications of games• Day 3: Bayesian games; applications of games

• Day 4: Mechanism design and auctions

• Day 5: Coalition games and student presentations


STUDENT PRESENTATIONS

• Pick a topic on game theory that is covered in class• Research on the Internet on your topic and find at least

one problem or challenge related to this problem• Prepare a 10 minute presentation with slides (about 10-

12 slides)• Give overview of topic• Give overview of topic• Discuss the problem or challenge on the topic you have found

on the Internet• Mention the most interesting or appealing concept that you

learned from the course and why it is interesting to you

• Presentation should reflect your understanding of the topic

• Presentation schedule: reverse-alphabetical by last name, Friday after lunch


DAY 1: OUTLINE

• Introduction

• Some classic 2-player games

• Solving 2-players games• Dominated strategies and iterated dominance

• Pareto optimality and Nash equilibrium• Pareto optimality and Nash equilibrium

• Mixed Strategies

• Software packages for solving games• Generating games using GAMUT

• Solving games using Gambit

• Correlated Equilibrium


HISTORY OF GAME THEORY

• 19th century and earlier: mathematical formulations to solve taxation problems, profits, etc.

• First half of 20th century:• Von Neumann formalizes utility theory, lays down mathematical

foundations for analyzing two player games; early work starts

• 1950s onwards: Nash theoremNash theorem, analysis of different types of games, beyond two players, relaxing simplifying assumptions games, beyond two players, relaxing simplifying assumptions (e.g., complete knowledge), more complex settings

• 1970s onwards: evolutionary game theory (applying biological concepts in games), learning in games, mechanism design

• 1990s onwards: • computational implementation of game theory algorithms, complexity

results (n players), • game theory software, programming competitions,• applications to real-life domains (auctions, network bandwidth

sharing, resource allocation, etc.)


SOME NOTABLE GAME THEORISTS

• John Von Neumann – Founder of field

• John Nash – Nash equilibrium

• John Harsanyi – Incomplete Information (Bayesian Games)(Bayesian Games)

• Roger Myerson – Mechanism Design

• Many others: Morgenstern, Selten, Maynard Smith, Aumann…


A SIMPLE EXAMPLE

• Simplest case: 2 players• Each player has a set (2) of actions• Each player has to select an action• By doing action, the player gets a payoff or utility

• Rationality assumption: Each player selects action that gives it highest payoff

• A player’s decision (selected action) affects the other player’s decision (selected action)• And other player’s payoff, and in turn its own payoff• And other player’s payoff, and in turn its own payoff

Actions: Go, Stop

Payoff(Go) = 1Payoff (Stop) = -1

Actions: Go, Stop

Payoff(Go) = 1Payoff (Stop) = -1


A SIMPLE EXAMPLE




Actions: Go, Stop

Actions: Go, Stop

Payoff(Go, Go) = -2, -2Payoff (Go, Stop) = 2, -1Payoff (Stop, Go) = -1, 2

Payoff (Stop, Stop) = -1, -1

Payoff(Go, Go) = -2, -2Payoff (Go, Stop) = 2, -1Payoff (Stop, Go) = -1, 2Payoff (Stop, Stop) = -1,

-1


A SIMPLE EXAMPLE




20

Actions: Go, Stop

Actions: Go, Stop

Go Stop

Go -2, -2 2, -1

Stop -1, 2 -1, -1

Go Stop

Go -2, -2 2, -1

Stop -1, 2 -1, -1June 16-20, 2014 Game Theory Workshop - Raj Dasgupta

THE MAIN PROBLEM IN GAME THEORY…SAID SIMPLY

• Main Problem: Given that each player knows the actions and payoffs of each other, how can each player individually make a decision (select an action) that will be ‘best’ for itself

• ‘best’ is loosely defined

Go Stop

Go -2, -2 2, -1

Stop -1, 2 -1, -1

• ‘best’ is loosely defined• Minimize regret

• Maximize sum of payoffs to all (both) players

• Stable or equilibrium action – if player deviates by itself from that action, it will end up lowering its own payoff … should hold for every player

Nash Equilibrium


MULTI-AGENT INTERACTIONS

• For solving game theory computationally, players are implemented as software agents

• Main Problem (restated in terms of agents): When two agents interact, what action When two agents interact, what action should each agent take?

• Simplifying (natural) assumption: agents are self-interested

• each agent takes an action that maximizes its own benefit


PREFERENCES AND UTILITY

• Since two agents are acting, action is referred to as joint action

• Each (joint) action results in a different outcome• Selecting a joint action ≡ selecting an outcome

• Selecting the best (highest benefit) outcome is modeled as a probability distribution over the set of possible outcomes

• called the preferences of the agent over its set of possible outcomes

Go Stop

Go -2, -2 2, -1

Stop -1, 2 -1, -1

Outcome

Joint actions: (go, go) (go, stop), (stop, go) (stop, stop)June 16-20, 2014 23Game Theory Workshop - Raj Dasgupta

UTILITY FUNCTION

• Assign a numeric value to outcomes

• More preferred outcome, higher utility

• Called utility function

• Utility is also called payoff

Go Stop

Go -2, -2 2, -1

Stop -1, 2 -1, -1

• Utility is also called payoff


NOTATIONS

• A: set of agents

• Ω = (ω1, ω2, ...): set of outcomes

• U: A X Ω R : utility function→

→


PREFERENCE RELATION

• Suppose i, j ε A and ω, ω’ ε Ω• u( i, ω) >= u(i, ω’) means agent i prefers outcome ω over ω’

• also denoted as ui(ω) >= ui (ω’) or ω >=i ω‘

• Relation >= gives an ordering over Ω

• Satisfies following properties• Satisfies following properties• reflexivity

• transitivity

• comparability

• substitubility

• decomposability

• >=: weak preference

• > : strong preference• only satisfies transitivity and comparability


AGENT INTERACTION MODEL

• Consider only 2 agents: i, j ε A

• i and j take their actions simultaneously

• Outcome ω ε Ω is a result of both i and j’s actions (sometimes called joint outcome)

• Other assumptions• Other assumptions• each agent has to take action

• each agent cannot see what action other agent has taken (only knows outcome)

• We assume that each agent knows• the set of possible actions of itself and the other agent

• the utility for each possible outcome both for itself and for the other agent

• (that is, each agent knows entire game matrix)Go Stop

Go -2, -2 2, -1

Stop -1, 2 -1, -1June 16-20, 2014 27Game Theory Workshop - Raj Dasgupta

TRANSFORMATION FUNCTION

• τ: Ac X Ac Ω

• Ac: set of possible action

• τ : state transformation function

→


EXAMPLE

• Let Ac = C, D for both agents• Let τ(C,C) = ω1, τ(C,D) = ω2, τ(D,C) = ω3,

τ(D,D) = ω4

• Let ui(ω1)=4, ui(ω2)=4, ui(ω3)=1, ui(ω4)=1• Let uj(ω1)=4, uj(ω2)=1, uj(ω3)=4, uj(ω4)=1• Since each pair of joint actions gives one • Since each pair of joint actions gives one

specific joint outcome, we can write• ui(C,c)=4, ui(C,d)=4, ui(D,c)=1, ui(D,d)=1• uj(C,c)=4, uj(C,d)=1, uj(D,c)=4, uj(D,d)=1• For agent i, the preference over outcomes

is(C,c) >= i (C, d) > i (D,c) >=i (D,d)


PAYOFF MATRIX

4, 4,C

dc

1, 1,

4,

D

Agent i

• Each entry gives (utility to agent i, utility to agent j)June 16-20, 2014 30Game Theory Workshop - Raj Dasgupta

PAYOFF MATRIX

4,4 4,1C

dc

Agent j

1,4 1,1

4,1

D

Agent i

• Each entry gives (utility to agent i, utility to agent j)June 16-20, 2014 31Game Theory Workshop - Raj Dasgupta

PAYOFF MATRIX

4,4 4,1C

dc

Agent j

1,4 1,1

4,1

D

Agent i

•For agent i, selecting C is always better than selecting D

irrespective of what agent j doesJune 16-20, 2014 32Game Theory Workshop - Raj Dasgupta

PAYOFF MATRIX

4,4 4,1C

dc

Agent j

1,4 1,1

4,1

D

Agent i

•For agent j, selecting C is always better than selecting D

irrespective of what agent i doesJune 16-20, 2014 33Game Theory Workshop - Raj Dasgupta

PAYOFF MATRIX

4,4 4,1C

dc

Agent j

1,4 1,1

4,1

D

Agent i

• Finally, (C, c) remains as the only feasible (rational) outcome


DOMINANT STRATEGIES

• Let Ω=ω1, ω2, ω3, ω4: be the set of possible outcomes

• Let Ω1 and Ω2 be two proper subsets of Ωsuch that Ω1=ω1, ω2 and Ω2=ω3, ω41 1 2 2 3 4

• Suppose that for some agent k, ω1 =k ω2 >k

ω3 =k ω4

• For agent k, every outcome in Ω1 is preferred over every outcome in Ω2

• We say, for agent k, Ω1strongly dominates Ω2


STRATEGIES

• Actions of agents are also called strategies

• Outcome of playing strategy s denoted by s*

• For e.g.: (from previous example)• For e.g.: (from previous example)• For agent i, C* = ω3, ω4 and D* = ω1, ω2

• Using this notation, strategy s1 dominates strategy s2 if every outcome in s1

* dominates every outcome in s2

*

• For e.g, in previous example, C dominates D for agent i (also for agent j)


SOLVING A GAME WITH DOMINANT STRATEGIES

• Solving a game means finding the outcome of the game

• Procedure1. Inspect strategies one at a time

2. If a strategy is strongly dominated by another strategy, 2. If a strategy is strongly dominated by another strategy, remove the dominated strategy

3. If the agent ends up with only one strategy by applying step 2 repeatedly, then the remaining strategy is the dominant strategy


WEAKLY DOMINATED STRATEGY

• Let s1 and s2 be two strategies

• s1 weakly dominates s2 if every outcome in s1* is >=

every outcome in s2*


NASH EQUILIBRIUM

PRISONER’S DILEMMA GAME

• Two men are together charged with the same crime and held in separate cells. They have no way of communicating with each other, or, making an agreement beforehand. They are told thatbeforehand. They are told that• if one of them confesses the crime and the other

does not, the confessor will be freed while the other person will get a term of 3 years• if both confess the crime, each will get term of 2

years• if neither confess the crime, each will get a 1-year

term


PRISONER’S DILEMMA GAME

• Each person (player) has two strategies• confess (C)

• don’t confess (D)

• Which strategy should each player play?

Remember each player is a self-interested utility • Remember each player is a self-interested utility maximizer


PD GAME: PAYOFF MATRIX

C D

C -2, -2 0, -3Player 1

Player 2

Are there any strongly dominated strategies for any player?

C -2, -2 0, -3

D -3, 0 -1, -1

Player 1


PD GAME REASONING

• Agent i reasons:• I don’t know what j is going to play, but I know he will play

either C or D. Let me see what happens to me in either of these cases.

• If I assume that J is playing C, I will get a payoff of –2 if if I play C and a payoff of -3 if I plays Dplay C and a payoff of -3 if I plays D

• If I assume that J is playing D, I will get a payoff of 0 if if I play C and a payoff of -1 if I plays D

• Therefore, irrespective of what J plays, I’m better off by playing C

• Agent j reasons in a similar manner (since the game is symmetric)

• Both end up playing (C, C)C D

C -2, -2 0, -3

D -3, 0 -1, -1June 16-20, 2014 43Game Theory Workshop - Raj Dasgupta

EXAMPLE: PRISONER’S DILEMMA

CD

D -2,-2 -10,-1

P2

• Utility values are changed from last example, order or rows are interchanged

• Outcome of the game still remains same (C, C)

• Representation of the game does not change its outcome as long as relative utility values remain same

C -1,-10 -5,-5P1


NASH EQUILIBRIUM

• Two strategies s1 and s2 are in Nash equilibrium when• under assumption agent i plays s1, agent j can do no better

than play s2

• under assumption agent j plays s2, agent i can do no better • under assumption agent j plays s2, agent i can do no better than play s1

• Neither agent has any incentive to deviate from a Nash equilibrium


PD GAME EQUILIBRIUM

• Is (C, C) a Nash equilibrium of the game?• given agent j is playing C, agent i can do no

better than play C

• given agent i is playing C, agent j can do no better than play Cbetter than play C

• Is (D, D) a Nash equilibrium of the game?• given agent j is playing D, agent i can do better

by playing C

• given agent i is playing D, agent j can do better by playing C

C D

C -2, -2 0, -3

D -3, 0 -1, -1June 16-20, 2014 46Game Theory Workshop - Raj Dasgupta

EXAMPLE: NASH EQUILIBRIUM


FEATURES OF NASH EQUILIBRIUM

• Not every game has a Nash equilibrium in purestrategies

• Nash’s Theorem• Every game with a finite number of players and action

profiles has at least one Nash equilibrium (considering pure profiles has at least one Nash equilibrium (considering pure and mixed strategies)

• Some interaction scenarios have more than one Nash equilibrium


GAMES WITH MULTIPLE NASH EQUILIBRIUM

• Game of chicken (Earlier bridge crossing example)

• Stag-hunt

Go Stop

Go -2, -2 2, -1

Stop -1, 2 -1, -1

Stag Hare

Stag 2, 2 0,1

Hare 1, 0 1, 1


COMPETITIVE AND ZERO-SUM GAMES

• Strictly competitive game• For i, j ε A, ω, ω’ ε Ω : ω >i ω’ iff ω’ >j ω

• preferences of the players are diametrically opposite to each other

• Zero-sum game• Zero-sum game• for all ω ε Ω: ui(ω) + uj(ω) = 0

• zero sum games have no chance of cooperative behavior because positive utility for agent j means negative utility for agent i and vice-versa


ZERO-SUM GAME: MATCHING PENNIES

• Two players simultaneously flip two pennies. If both have the same side up, player 1 keeps both of them, else player 2.

Agent j

1, -1

-1, 1 1, -1

-1,1H

T

TH

Agent i

Agent j


ITERATED DOMINANCE: DISTRICT ATTORNEY’S BROTHER

• Recall: Given a game, to solve it, first remove all strictly dominated strategies

• All games might not have strictly dominated strategies

• Try iterated removal of dominated strategies • Try iterated removal of dominated strategies

• Same scenario as prisoners’ dilemma except:• one of the prisoners is the DA’s brother

• allows his brother (prisoner 1) to go free if both prisoners don’t confess


EXAMPLE: PRISONER’S DILEMMA

Prisoners’ Dilemma DA’s Brother

C D

C -2, -2 0, -3

D -3, 0 -1, -1

C D

C -2, -2 0, -3

D -3, 0 0, -1

• For players 1 and 2, D is strictly dominated

• Therefore, (C, C) is the preferred strategy (and Nash outcome)

• For player 1 D is not strictly dominated now

• But, for player 2, D is still strictly dominated

• Player 1 reasons – Player 2 will never play D

• Now, D becomes strictly dominated for player 1

• (C, C) still remains outcome of game

Strict dominance: Each player can solve the game individually

Iterated dominance: A player has to build opponent’s model to solve the gameJune 16-20, 2014 53Game Theory Workshop - Raj Dasgupta

ASSUMPTIONS IN DA’S BROTHER

• Each player is rational

• Players have common knowledge of each other’s rationality • P1 knows P2 will behave in a rational manner

• Allows iterated deletion of dominated strategies• Allows iterated deletion of dominated strategies

• However: Each iteration requires common knowledge assumption to be one level deeper


ITERATED STRATEGY DELETION:ORDER OF DELETION

• Order of deletion does not have effect on final outcome of game• if eliminated strategies are strongly dominated

• Order of deletion has effect on final outcome of gamegame• if eliminated strategies are weakly dominated


COMMON PAYOFF GAME

• Both (all) agents get equal utility in every outcome (e.g., two drivers coming from opposite sides have to choose which side of road to drive on)

1, 1

0, 0 1, 1

0, 0L

R

RL

Agent i

Agent j


BATTLE OF SEXES GAME

• Husband prefers going to a football game, which wife hates

• Wife prefers going to the opera, which husband hateshusband hates

• They like each other’s company

2, 1

0, 0 1, 2

0, 0F

O

OF

h

w


DEFINITION: NORMAL FORM GAME

• For a game with I players, the normal form representation of the game ΓN specifies for each player i, a set of strategies Si (with si Є Si) and a payoff or utility function ui(s1, s2, s3, s …s ) giving the utility associated with the s4…sI) giving the utility associated with the outcome of the game corresponding to the joint strategy profile (s1, s2, s3, s4…sI).

• Formal notation:

ΓN =[ I, Si, ui(.)]


DEFINITION: STRICTLY DOMINATED STRATEGY

• A strategy si Є Si is strictly dominated for player I in

game ΓN =[ I, Si, ui(.)] if there exists another

strategy si‘ Є Si such that for all s-i Є S-i

ui(si‘,s-i) > ui(si,s-i)ui(si‘,s-i) > ui(si,s-i)

• In this case we say that strategy si‘ strictly

dominates strategy si


EXAMPLE: STRICT DOMINANCE

RL

U 1, -1 -1, 1

P1

P2

D

M -1, 1 1, -1

-2, 5 -3, 2

P1

• For player 1, D is strictly dominated by U and M


DEFINITION: WEAKLY DOMINATED STRATEGY

• A strategy si Є Si is weakly dominated for player I in

game ΓN =[ I, Si, ui(.)] if there exists another

strategy si‘ Є Si such that for all s-i Є S-i

ui(si‘,s-i) >= ui(si,s-i)i i -i i i -i

• with strict inequality for some s-i. In this case, we say that strategy si

‘ weakly dominates strategy si

• A strategy is a weakly dominant strategy for player I in game ΓN =[ I, Si, ui(

.)] if it weakly dominates every other strategy in Si


EXAMPLE: WEAK DOMINANCE

RL

U 5,1 4,0

P1

P2

D

M 6,0 3,1

6,4 4,4

P1

• For player 1, U and M are weakly dominated by D


MIXED STRATEGIES

ZERO-SUM GAME: MATCHING PENNIES

• Is there a Nash equilibrium in pure strategies?

THAgent j

1, -1

-1, 1 1, -1

-1,1H

T

Agent i


MIXED STRATEGY NASH EQUILIBRIUM

• Objective • Each player randomizes over its own strategies in a way

such that its opponents’ choice becomes independent of it actions


SOLVING MIXED STRATEGY NASH EQUILIBRIUM (1)

• P1 tries to solve: What probability should I play H and T with so that my (expected) utility is independent of

H T

H 1, -1 -1, 1

T -1, 1 1, -1

P1

P2

p

1-p

• P1 tries to solve: What probability should I play H and T with so that my (expected) utility is independent of whether P2 plays H or T

• Suppose P1 plays H with probability ‘p’ and T with probability (1-p)• Utility to PI when P2 plays H: 1.p + (-1) (1-p) = 2p -1• Utility to PI when P2 plays T: (-1)1.p + 1.(1-p) = 1 – 2p

• To find mixed strategy P1 solves for p in:• 2p -1 = 1- 2p, or, p = 0.5


SOLVING MIXED STRATEGY NASH EQUILIBRIUM (2)

H T

H 1, -1 -1, 1

T -1, 1 1, -1

P1

P2

1-qq

0.5

0.5

0.5 0.5

• P2 solves similarly denotes q as probability of playing H and (1-q) as probability for playing T• Solving for q in the same manner as for P1 gives q = 0.5

• Mixed strategy Nash equilibrium is:

• ((0.5, 0.5) (0.5, 0.5))


EXAMPLE: MEETING IN NY GAME:MIXED STRATEGY NASH

EQUILIBRIUM


EXAMPLE: MEETING IN NY GAME:MIXED STRATEGY NASH

EQUILIBRIUM (2)• T reasons:• Let me play GC with probability σs and ES with probability (1- σs)• Then,• if S plays G it gets a payoff 100 σs + 0(1- σs)• if S plays E it gets a payoff 0 σs +1000(1- σs)

• Recall:• Each player should play its strategies with such probabilities that it • Each player should play its strategies with such probabilities that it

does not matter to its opponent what strategy that player plays.

• In other words, T should be indifferent (get the same payoff) from either strategy that S plays

• Therefore,• 100 σs + 0(1- σs) = 0 σs +1000(1- σs)• or, 1100σs =1000, or, σs = 10/11

• By similar reasoning, S plays GC with probability σT =10/11• σs = σT = 10/11 constitutes a mixed strategy nash equilibrium of

the meeting in NY game


BATTLE OF SEXES GAME

• Husband prefers going to a football game, which wife hates

• Wife prefers going to the opera, which husband hateshusband hates

• They like each other’s company

2, 1

0, 0 1, 2

0, 0F

O

OF

h

w


DEFINITION: MIXED STRATEGY

• Given player i’s (finite) pure strategy set Si, a mixed strategy for player i , σi: Si → [0,1], assigns to each pure strategy si Є Si a probability σi(si) >=0 that it will be played, where Σ si Є Si σi(si) =1where Σ si Є Si σi(si) =1• The set of all mixed strategies for player i is

denoted by ∆(Sj)=(σ1,σ2,σ3,... ,σMi) Є RM: σmi >0 for all m = 1...M and Σm=1

Μ σmi=1• ∆(Sj) is called the mixed strategy profile• si is called the support of the mixed strategy

σi(si)


DEFINITION: STRICT DOMINATION IN MIXED STRATEGIES

• A strategy σi Є ∆(Si) is strictly dominated for player i in game ΓN =[ I, ∆(Si), ui(

.)] if there exists another strategy σi

‘ Є ∆(Si) such that for all σ-i Є Π

j<>i ∆(Sj)

ui(σi‘, σ-i) > ui(σi, σ-i)ui(σi‘, σ-i) > ui(σi, σ-i)

• In this case we say that strategy σi‘ strictly

dominates strategy σi

• A strategy σi is a strictly dominant strategyfor player i in game ΓN =[ I, ∆(Si), ui(

.)] if it strictly dominates every other strategy in ∆(Sj)


PURE VS. MIXED STRATEGY

• Player i’s pure strategy is strictly dominated in game

ΓN =[ I, ∆(Si), ui(.)] if and only if there exists

another strategy σi ‘ Є ∆(Si) such that

ui(σi‘, s-i) > ui(si, s-i)ui(σi , s-i) > ui(si, s-i)

for all s-i Є S-i


EXAMPLE: PURE VS. MIXED STRATEGY

RL

U 10,1 0,4

P2

D

M 4,2 4,3

0,5 10,2

P1


BEST RESPONSE AND NASH EQUILIBRIUM

• In a game ΓN =[ I, ∆(Si), ui(.)] , strategy s*

i is a best response for player i to its opponents’ strategies s-i if

ui(s*i, s-i) >= ui(si, s-i)ui(s

*i, s-i) >= ui(si, s-i)

for all si ε ∆(Si).

• A strategy profile s = (s*1 ,s

*2 … s*

N) is a Nash equilibrium if s*

i is a best response for all players i= 1…N


DELETION OF NEVER A BEST REPONSE

• Strategy si is never a best response if there is no s-i for which si is a best response

• Strictly dominated strategy can never be a best response

• Converse is not true: A strategy that is not a strictly • Converse is not true: A strategy that is not a strictly dominated might be ‘never a best response’

• Therefore, eliminating strategies that are ‘never a best response’ eliminates• strictly dominated strategies

• possibly some more strategies


ITERATED DELETION

• Never a best response strategies can be removed in an iterated manner using rational behavior and common knowledge (similar to iterated deletion of strictly dominated strategies)dominated strategies)

• Order of deletion does not affect the strategies that remain in the end

• Strategies that remain after iterated deletion are those that a player can rationalize assuming opponents also eliminate their never a best reponse strategies


RATIONALIZABLE STRATEGIES

• In game ΓN =[ I, ∆(Si), ui(.)], the strategies in ∆(Si)

that survive the iterated removal of strategies that are never a best response are known as player i’s rationalizable strategies.


WHY WILL A GAME HAVE A NASH EQUILIBRIUM

1. Nash equilibrium as a consequence of rational inference

2. Nash equilibrium as a necessary condition if there is a unique predicted outcome to the gamethe game

3. Focal Points• e.g: restaurants around Grand Central Station

are better than those around Empire State Building. Increases the payoff of meeting at Grand Central

4. Nash equilibrium as a self-enforcing agreement

5. Nash equilibrium as a stable social conventionJune 16-20, 2014 79Game Theory Workshop - Raj Dasgupta

MAXMIN STRATEGY

• Maxmin strategy maximizes the worst payoff that player i can get

s i maxmin = arg max s_i min s_-i ui(si, s-i)

• Find the set of my strategies that give me the minimum utilities corresponding to every joint minimum utilities corresponding to every joint strategy of the opponent players (everybody except me)

• Find the strategy that gives me the highest utility from the set in the previous step


MINMAX STRATEGY FOR 2-PLAYER GAME

• Minmax strategy – player i minimizes the highest payoff that player –i (opponent) can get

s i minmax = arg min s_i max s_-i u-i(si, s-i)s i = arg min s_i max s_-i u-i(si, s-i)

• Find the set of my strategies that give my opponent the maximum utilities corresponding to each of my strategies

• Find my strategy from the set in the previous step that gives the lowest utility to my opponentJune 16-20, 2014 81Game Theory Workshop - Raj Dasgupta

MINMAX UTILITY FOR N-PLAYER GAME

• i will have one minmax strategy for each opponent

• In an n-player game, the minmax strategy for player i against player j (not equal to i) is i’s component of the mixed strategy profile s-j in the expression arg min max u (s , s ), where –j denotes the set of min s_-j max s_j uj(sj, s-j), where –j denotes the set of players other than j


NASH EQUILIBRIUM VS. MINMAX/MAXMIN STRATEGY

• In any finite, two-player, zero-sum gane, in any Nash equilibrium, each player receives a payoff that is euqal to both his maxmin value and his minmax value


REGRET

• An agent i’s regret for playing an strategy si if the other agent’s joint strategy profile is s-i is defined as

[max s’_i ε S_i ui(si’, s-i)] - ui(si, s-i)

• An agent i’s max regret is defined as

max s_-i’ ε S_-i [max s’_i ε S_i ui(si’, s-i)] - ui(si, s-i)

• An agent i’s minimax regret is defined as

arg min s_i ε S_i (max s_-i’ ε S_-i [max s’_i ε S_i ui(si’, s-i)] - ui(si, s-

i))


Education

1 intro game-theory