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

Games Computers
(and Computer Scientists)
Play
Avi Wigderson

Computer
Science

Games

Game
Theory

=
Information Processing by
Computers

Agents

•
•

•
•
•
•

Competing
Cooperating
Faulty
Colluding
Secretive
Adversarial

Computationally Bounded
Communicating Digitally

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

Complexity of Games

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

Extensive Form

Question: Can a winning strategy be
efficiently computed?

Rectangle Game
m=4
n=5

1
m

1

5

2
4

3
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?

Zero-Sum Games
Matching Pennies
(simultaneous play)

H

T

H1

-11 1-

T1

1-1 -1

Strategic Form

“Best” strategy for each player is to flip a fair coin. Game value is 0.
m
1
2
j

: ]Theorem [von Neumann ‘28
1

Every 0-sum game has a
2
.Min-Max) value(

v -v
Question: Can the value, i
strategies be computed? n
: ]Theorem [Khachian ‘80
.Yes – Efficient linear programming algorithm
ij

ij

Nash Equilibrium
Chicken [Aumann]

C
C

D

11 02

Strategic Form

Probabilistic strategies (Sw, Sb).
D 2 0 -3 -3

Nash Equilibrium: No player has an incentive to
.change its strategy given the opponent’s strategy
¼[here Sw=Sb = [C with prob ¾, D with prob

.Theorem [Nash]: Every (matrix) game has an equilibrium
Question: Can the players compute (any) equilibrium?
).Best known algorithm: exponential time (infeasible

Implementing Games

The Millionaires’ Problem
A

Alice

B

Bob

Both want to know who is richer
Neither gets any other information
Question: Is that possible?

Joint random decisions
3/4 1/4

Nash eq. With Independent Strategies

C

Expected value = 3/4
Prob[CC[ = 9/16

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

3/4 C

D

11 02

1/4 D 2 0 -3 -3

Nash eq. With Correlated Strategies [Aumann]
Prob[CD[ = 1/2
Prob[DC[ = 1/2
Prob[CC[ = 0
Prob[DD[ = 0

Expected value = 1

Question: How to flip a coin jointly?

Simultaneity

1/2 1/2
H

T

Expected value = 0
)if they play simultaneously( 1/2

H1

-11 1-

1/2

T1

1-1 -1

Question: How do we guarantee simultaneity
A computational representation:
outcome
Parity Function

xW xB Parity(xW, xB )

P

xW

xB

0
1
0
1

0
1
1
0

0
0
1
1

Privacy vs. Resilience
• Voting
Majority Function
M

x1 x2 x3

x1

x2

x3 Majority(x1, x2, x3 )

0
0
0
1
0
1
1
1

0
0
1
0
1
0
1
1

0
1
0
0
1
1
0
1

0
0
0
0
1
1
1
1

Q1: How to guarantee x15?
Q2: How to guarantee x1 remains private?

• Millionaire’s Problem
• Poker
• Any game

Completeness Theorem
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)
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!

Correct & Private digital implementation
Trusted party
Ideal implementation

Secrets
Preferences
Strategies

s1

s2

sn

1

2

n

Internet
Digital implementation
Internet

How to ensure Privacy
Oblivious Computation [Yao[
1

f(inputs)

P

1

1

0

M

P

1

1

0

P

M

P

1 0 0

1 0

1 0

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.

Corollary: Players can be forced to act legally,
without fear of compromising secrets.

Where is Waldo? [Naor[

Designing Games

How to minimze players’ influence
Public Information Model [Ben-Or—Linial] :
Function
Joint random coin flipping
Parity
Every good player flips, then combine Majority
majorit
parit
y y
M

M

M
P

M

Influenc
1
1/7

Iterated
Majority

1/8
M

Theorem [Kahn—Kalai—Linial] : For every function, some
player has non-proportional influence.
Theorem [Alon—Naor] : There are “multi-round” function
for which no 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
New GT Issues: Complexity, Algorithms

Coping with Uncertainty
Competing against
Clairvoyance

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

day
1

Muggle’s
action

Wizard’s
action

2

3

4

5

6

7

8

9

Profit/loss

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”
[Borodin—El-Yaniv[

Every Game? Any secrecy requirements?
Incomplete information
Game in Extensive form

Nature
Alice
Bob

...

...
...

...

Nature
Alice
...

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

Completeness Theorems
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)
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