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Games Computers ( and Computer Scientists ) Play Avi Wigderson Computer Science Game Theory = Information Processing by Computers Agents Games Competing Cooperating Faulty Colluding Secretive Adversarial Computationally Bounded Communicating Digitally Plan

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slide2

Computer

Science

Game

Theory

=

Information Processing by

Computers

Agents

Games

  • Competing
  • Cooperating
  • Faulty
  • Colluding
  • Secretive
  • Adversarial

Computationally Bounded

Communicating Digitally

slide3
Plan
  • Complexity of Games
  • Implementation of Games
  • Design of Games
  • Games against Clairvoyance
slide5

Extensive Form

Theorem [Zermelo]: In every finite win/lose perfect information 2-player game, White or Black can force a win.

Question: Can a winning strategy be

efficiently computed?

slide6

1

5

3

2

4

Rectangle Game

1

m=4

n=5

m

n

Theorem: White has a winning strategy.

Proof: Assume Black has a winning strategy.

Then White can mimic it and win. Contradiction!

Question: What is the winning strategy?

slide7

1 -1

1 -1

-1 1

-1 1

m

1

2

j

1

2

i

vij-vij

n

Zero-Sum Games

Matching Pennies

(simultaneous play)

H

T

Strategic Form

H

T

“Best” strategy for each player is to flip a fair coin. Game value is 0.

Theorem [von Neumann ‘28]:

Every 0-sum game has a

(Min-Max) value.

Question: Can the value,

strategies be computed?

Theorem [Khachian ‘80]:

Yes – Efficient linear programming algorithm.

slide8

-3 -3

1 1

2 0

0 2

Nash Equilibrium

Chicken [Aumann]

C

D

Strategic Form

C

Probabilistic strategies (Sw, Sb).

D

Nash Equilibrium: No player has an incentive to

change its strategy given the opponent’s strategy.

here Sw=Sb = [C with prob ¾, D with prob ¼]

Theorem [Nash]: Every (matrix) game has an equilibrium.

Question: Can the players compute (any) equilibrium?

Best known algorithm: exponential time (infeasible).

the millionaires problem

A

B

Alice

Bob

The Millionaires’ Problem

Both want to know who is richer

Neither gets any other information

Question: Is that possible?

slide11

-3 -3

1 1

2 0

0 2

C

D

3/4

1/4

3/4

C

Expected value = 3/4

Prob[CC] = 9/16

Prob[CD] = 3/16

Prob[DC] = 3/16

Prob[DD] = 1/16

1/4

D

Prob[CD] = 1/2

Prob[DC] = 1/2

Prob[CC] = 0

Prob[DD] = 0

Expected value = 1

Joint random decisions

Nash eq. With Independent Strategies

Nash eq. With Correlated Strategies [Aumann]

Question: How to flip a coin jointly?

slide12

1/2

H

1/2

T

Expected value = 0

(if they play simultaneously)

1 -1

1 -1

-1 1

-1 1

1/2

H

1/2

T

A computational representation:

outcome

xW xB Parity(xW, xB )

00 0

1 1 0

01 1

10 1

Parity Function

P

xW

xB

Simultaneity

Question: How do we guarantee simultaneity?

slide13

Privacy vs. Resilience

x1 x2 x3 Majority(x1, x2, x3)

00 0 0

0 01 0

0 1 0 0

1 0 0 0

0 1 1 1

1 0 1 1

1 1 0 1

1 1 1 1

Majority Function

  • Voting

M

x1

x2

x3

Q1: How to guarantee x15?

Q2: How to guarantee x1 remains private?

  • Millionaire’s Problem
  • Poker
  • Any game
completeness theorem
Completeness Theorem
  • Theorem [Yao, Goldreich –Micali –Wigderson]:
  • More than 1/2 of the players are honest
  • Players computationally bounded
  • Trap-door functions exist (e.g. factoring integers is hard)

Every game,

with any secrecy requirements,

can be digitally implemented

s.t. no collusion of the bad players can affect:

*correctness (rules, outcome)

* privacy (no information leaks)

Hard problems can be useful!

slide15

Trusted party

Ideal implementation

Secrets

Preferences

Strategies

s1

s2

sn

1

2

n

Internet

Digital implementation

Internet

Correct & Private digital implementation

slide16

1

M

M

P

P

P

P

1 0 0

1 0

1 0

How to ensure Privacy

Oblivious Computation [Yao]

f(inputs)

1

1

0

1

1

0

slide17

How to ensure Correctness

  • Definition [Goldwasser-Micali-Rackoff]:
  • zero-knowledge proofs:
  • Convincing
  • Reveal no information

Theorem [Goldreich-Micali-Wigderson]:

Every provable mathematical statement has a

zero-knowledge proof.

Corollary: Players can be forced to act legally,

without fear of compromising secrets.

slide20

parity

M

P

M

M

M

M

majority

How to minimze players’influence

Public Information Model [Ben-Or—Linial]:

Joint random coin flipping

Every good player flips, then combine

Function Influence

Parity 1

Majority 1/7

Iterated

Majority 1/8

Theorem [Kahn—Kalai—Linial]: For every function, some

player has non-proportional influence.

Theorem [Alon—Naor]: There are “multi-round” functions

for which no player has non-proportional influence.

slide21

How to achieve cooperation, efficiency, truthfulness

Players (agents) are selfish

  • Auction
    • Question: How to get players to bid their true values?
    • Theorem [Clarke—Groves—Vickery]:
    • 2nd price auction achieves truthfulness.
  • Internet Games
    • Question: How to get players to cooperate?
    • [Nisan]: Distributed algorithmic mechanism design.
    • [Papadimitriou]: Algorithms, Games & the Internet

New CS Issues: Pricing, incentives

New GT Issues: Complexity, Algorithms

slide23

price

day

Wizard’s

action

  • On-line Problems
    • Investor’s Problem (One-way trading)

1

2

3

4

5

6

7

8

9

Profit/loss

Muggle’s

action

slide24

On-line problems are everywhere:

  • Computer operating systems
  • Taxi dispatchers
  • Investors’ decisions
  • Battle decisions
slide25

Competitive Analysis [Tarjan—Slator]:

For every sequence of events,

Bound the competitive ratio:

muggle-cost(sequence)

wizard-cost(sequence)

Can be achieved in many settings.

Huge, successful theory.

“Online Computation and Competitive Analysis”

[Borodin—El-Yaniv]

slide26

Nature

Alice

...

...

...

...

Bob

Nature

  • Information Sets
  • Player’s action depends
  • only on its information set

Alice

...

Every Game? Any secrecy requirements?

Incomplete information

Game in Extensive form

completeness theorems
Completeness Theorems
  • Theorem [Yao, Goldreich –Micali –Wigderson]:
  • More than 1/2 are honest
  • Players computationally bounded
  • Trap-door functions exist (e.g. factoring integers is hard)

Every game, with any secrecy requirements, can be

digitally implemented s.t. no collusion of the bad players can affect:

*correctness (rules, outcome)

* privacy (no information leaks)

Theorem [Ben-Or –Goldwasser –Wigderson]:

1’.

2’. At least 3 players, more than 2/3 are honest

3’. Private pairwise communication