Games &Computers.ppt 2564KB Jun 23 2011 12:12:22 PM

Games Computers

(

and Computer Scientists

)

Play

Games Computer

Science

Game Theory

=

Information Processing by

Computers _Agents

• Competing • Cooperating • Faulty

• Colluding • Secretive • Adversarial

Plan

• Complexity of Games

• Implementation of Games

• Design of Games

Theorem

[Zermelo]

: In every finite

win/lose perfect information

2-player game, White or Black can

force a win.

Extensive Form

Rectangle Game

m

n m=4

n=5 1

2

3

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?

4

5

Zero-Sum Games

Matching Pennies

(simultaneous play) 1 -11 1

-1

1 1 -1

H

H T

T

Strategic Form

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

1

1 2

2

m

n

v_ij -v_ij i

j

Theorem

[von Neumann ‘28

]

:

Every 0-sum game has a

(

Min-Max)

value

.

Question: Can the value,

strategies be computed?

Theorem

[Khachian ‘80

]

:

Nash Equilibrium

Chicken

[Aumann]

1 1 0 2 0

2 -3 -3 C

C D

D

Strategic Form

Probabilistic strategies (S_w, S_b).

Nash Equilibrium: No player has an incentive to

change its strategy given the opponent’s strategy .

here S_w=S_b = [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

Alice

Bob

B A

Both want to know who is richer

Neither gets any other information

Joint random decisions

1 1 0 2 0

2 -3 -3

C D

C

D

Nash eq. With Independent Strategies

Nash eq. With Correlated Strategies [Aumann]

3/4

1/4

3/4 1/4

Expected value = 3/4

Prob[CC[ = 9/16 Prob[CD[ = 3/16 Prob[DC[ = 3/16 Prob[DD[ = 1/16

Prob[CD[ = 1/2 Prob[DC[ = 1/2 Prob[CC[ = 0 Prob[DD[ = 0

Expected value = 1

Simultaneity

1 -11 1

-1

1 1 -1

H T H T 1/2 1/2 1/2 1/2

Expected value = 0

( if they play simultaneously )

Question: How do we guarantee simultaneity?

x_W x_B

A computational representation:

outcome

Parity Function xW xB Parity(xW, xB )

0 0 0

1 1 0

0 1 1

1 0 1

Privacy vs. Resilience

Q

₁

: How to guarantee x

₁



5?

Q

₂

: How to guarantee x

₁

remains private?

Majority Function

x₁ x₃

x1 x2 x3 Majority(x1, x2, x3)

0 0 0 0

0 0 1 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

• Voting

M

x₂

• Millionaire’s Problem

• Poker

Completeness Theorem

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 [Yao, Goldreich –Micali –Wigderson[: 1. More than 1/2 of the players are honest 2. Players computationally bounded

3. Trap-door functions exist (e.g. factoring integers is hard)

Correct & Private digital implementation

Secrets Preferences

Strategies

Trusted party

Ideal implementation

1 2 n

s₁ s₂ s_n

Internet

Internet

How to ensure Privacy

Oblivious Computation [Yao[

1 0 0 1 0 1 0

1 1 0

1 0

1

f(inputs)

P M P

M P

P

How to ensure Correctness

Definition [Goldwasser-Micali-Rackof[:

zero-knowledge proofs:

• Convincing

• Reveal no information

Theorem [Goldreich-Micali-Wigderson[:

Every provable mathematical statement has a zero-knowledge proof.

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

P

parit

y M

majorit y

M M M

M Iterated

Majority 1/8

Theorem [Kahn—Kalai—Linial] : For every function, some player has non-proportional influence.

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

Coping with Uncertainty

On-line Problems

Investor’s Problem (One-way trading)

day price

1 2 3 4 5 6 7 8 9

Profit/loss Muggle’s

action

On-line problems are everywhere:

• Computer operating systems

• Taxi dispatchers

• Investors’ decisions

• Battle decisions

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”

... ...

Nature

... ...

Alice

Nature

...

Alice

Bob

Information Sets

•

Player’s action depends

only on its information set

Every Game? Any secrecy requirements?

Completeness Theorems

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 [Yao, Goldreich –Micali –Wigderson[: 1. More than 1/2 are honest

2. Players computationally bounded

3. Trap-door functions exist (e.g. factoring integers is hard)

Theorem [Ben-Or –Goldwasser –Wigderson[: 1’.