CSCI 3210: Computational Game Theory www.mtirfan.com/CSCI-3210 - - PDF document

9/28/20 1

CSCI 3210: Computational Game Theory

Mohammad T . Irfan Email: mirfan@bowdoin.edu Website: www.mtirfan.com

www.mtirfan.com/CSCI-3210

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Syllabus and required background

u Course website

u www.mtirfan.com/CSCI-3210

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You said you are here because:

u Be intellectually challenged in a class that is

• utside of my comfort zone!

u The game theory concept! u new algorithms! u I’ve been fascinated by game theory in the

past, so I’m excited to take a look at it through a cs lens.

u Learning something new 😂 u The difference between human decisions and

rational thinking

u Don’t know a lot about game theory, but

most curious about application to human choice

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Games and game theory: A brief introduction

Reading:

• 1. Ch 6 of Easley-Kleinberg (pdf on class

website)

• 2. [EGT] Ch 1-3

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Game Theory

u “Game”

u Ernst Zermelo (1913): In any chess game that

does not end in a draw, a player has a winning strategy u Mathematical theory of strategic

decision making

u John von Neumann (1944)

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Example: Split or Steal

u https://www.youtube.com/watch?v=yM38mR

HY150

u Rules of the game u Outcome

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One possible model

u What will happen? Payoff matrix

Split Steal Split

$33K, $33K $0+fr., $66K

Steal

$66K, $0+fr. $0, $0

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Payoff matrix

Split Steal Split

$33K, $33K $0+fr., $66K

Steal

$66K, $0+fr. $0, $0

Why did they end up with 0?

Lucy Tony

Nash Equilibrium

Everyone plays their best response to others simultaneously

John F . Nash Nobel Prize, 1994

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Nash equilibrium

Practical scenarios = Stable outcome = Nash equilibrium

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Applications

u Application: market equilibria

u Predict where the market is heading to

u Mechanism design and auctions

u Google and Yahoo apply game-theoretic techniques

u Keyword search auction

u Spectrum allocation among wireless companies

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Applications

u Understanding the Internet

u Selfish routing is a constant-factor off from

• ptimal

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Other applications

u Load balancing and resource allocation u p2p and file sharing systems u Cryptography and security u Social and economic networks, etc.

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Next: Formal discussion

(Without math)

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Game

u One-shot games (simultaneous move) u 3 components

u Players u Strategies/actions u Payoffs

Payoff matrix

Split Steal Split

$33K, $33K $0+fr., $66K

Steal

$66K, $0+fr. $0, $0

Tony Lucy

Call these pure strategies Pure-strategy NE

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Nash equilibrium (NE)

u A joint strategy (one strategy/player) s.t.

every player plays their best response to

• thers simultaneously

u (Equiv.) A joint strategy s.t. no player gains

by deviating unilaterally

u Useful for checking whether a cell is NE

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Famous example: prisoner's dilemma

u What will they do? Payoff matrix

Not Confess Confess Not Confess

• 1, -1
• 10, 0

Confess

0, -10

• 5, -5

Suspect 1 Suspect 2

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Assumptions

u Payoffs reflect player’s preference u Payoffs are known to all u Actions are known to all (different players

could have different actions– but everyone knows everyone’s actions)

u Each player wants to maximize own payoff

subject to others' actions

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Commonly Used Terms

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Best response

u Best strategy of a player, given the other

players’ strategies

u Always exists!

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(Strictly/weakly) dominant strategy

u A strategy of a player that is

(strictly/weakly) better than any of their

• ther strategies, no matter what the other

players do

u Does not always exist

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Pure-strategy Nash equilibrium (PSNE)

u Players do not use any probability in

choosing strategies as they do in "mixed- strategy" (to be covered later)

u Every player plays their best response to others

simultaneously

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Checkpoint

u What is the difference between a dominant

strategy and a best response?

u What is the difference between best

response and PSNE?

u What is the difference between weakly and

strictly dominant strategies? Will a player always have one?

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Checkpoint: “Generalization”

u Watch the following clip from the movie A

Beautiful Mind that tries to portray John Nash’s discovery of Nash equilibrium.

u Is this actually a Nash equilibrium?

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Misconceptions

u Equilibrium signifies a tie/draw/balance u Equilibrium outcome is the best possible

• utcome for all players (A Beautiful Mind)

u Self-interested players want to hurt each

• ther

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Questions

u Does NE always exist? (Answer later ...) u If it exists, is it unique?

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Games with multiple NE

• 1. Battle of the sexes (Coordination)
• 2. Hawk-dove game (anti-coordination)

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Does NE always exist? Mixed-strategy NE

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Penalty kick game

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Penalty kick game (continued)

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Penalty kick game (continued)

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Penalty kick game- first model

u E[GK plays Left] = p(1) + (1-p)(-1) = 2p – 1 u E[GK plays Right] = p(-1) + (1-p)(1) = 1 – 2p u 2p – 1 = 1 – 2p è p = ½ u Similarly, q = ½

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Left (q) Right (1- q) Left (p) -1, +1 +1, -1 Right (1- p)

+1, -1 -1, +1

Goalkeeper Shooter

Zero-sum Game

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Penalty kick game (real-world)

u “Professionals Play Minimax”- Ignacio

Palacios-Huerta

Left (0.42) Right (0.58) Left (0.38)

0.58, 0.42 0.95, 0.05

Right (0.62)

0.93, 0.07 0.70, 0.30

Shooter Goalkeeper

Equilibrium probabilities (computed by solving equations) match real-world probabilities from data! From real- world data

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What does mixed strategy mean?

u Active randomization – tennis, soccer u Proportion interpretation – evolutionary

biology

u Probabilities of player 1 are the beliefs of

player 2 about what player 1 is doing (Bob Aumann)

u Misconception

u Players just choose probabilities

u Correct

u players play pure strategies chosen according to

these probabilities

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Von Neumann’s Theorem (1928)

u Every finite 2-person zero-sum game has a

mixed equilibrium John von Neumann (1903 – 1957)

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Theorem of Nash (1950)

u Every finite game has an equilibrium in

mixed strategies

u Reading

John F . Nash (1928 – 2015) Nobel Prize, 1994

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Pure-strategy vs. mixed-strategy NE

Hawk-dove game

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Key take-away messages

u Players act simultaneously, but NE outcome

is stable in the sense that there is no incentive for unilateral deviation.

u There is always at least one mixed-strategy

• NE. A pure-strategy NE is not guaranteed.

u The concept of NE doesn't say how NE

happens.

u NE is not a balance or tie. It is often times a

socially-inefficient outcome.

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Formal (Mathematical) Definitions

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Background: discrete math

u Set theory

u Sets u Representation: list, describe properties u Belongs to u Subset u Empty set u Power set u Operations: union, intersection, difference,

product

u Sum, product

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Normal form games

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Example

u Idea: Connor Marrs u Assume: Cars don’t want to wait; all turns/S same One way St

L/R L/R/S

Photo modified from topdriver.com

Car1 Car2

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• 1. Who are the players?

u N = {1, 2}

• 2. What can each player do?

u A1 = {L, R, S}; A2 = {L, R} u Set of action profiles

A= A1 x A2 = {(L, L), (L, R), (R, L), (R, R), (S, L), (S, R)}

u Each element (e.g., (L,R)) is called an action profile

• 3. What are the payoffs?

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L R L 1, 1 0, 0 R 0, 0 1, 1 S 0, 0 1, 1

One way St

L/R L/R/S Car 1 Car 2

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Zero-sum/constant-sum game

u Is constant sum the same as zero sum?

action profile

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Example (zero-sum game)

u You and opponent flip two coins u Same parity (both heads up or tails up)=> you

win (keep both coins)

u Otherwise, opponent wins

You Opponent

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Mixed-strategy NE

u Mixed strategy of a player and mixed-

strategy profile of all players

u Expected utility u Best response u Nash equilibrium

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Mixed strategy

• A mixed strategy is an element of Si

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Expected utility

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Best response

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Nash equilibrium

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