Bayesian Nash Equilibria & Bell Inequalities

Taksu Cheon(Kochi Tech)

Talk presented at KEK Workshop “Stability and Instability”, Mar. 23, 2007Copyright, T.Cheon & Associates, 2007

Plan of the Talk

“ Why should we care about Game Theory? ”

Introduction to game theory

Game strategy in joint probability formalism

Quantum strategy

Bell inequality and quantum gain in certain games

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A Game against Nature

Probabilistic Play

Payoff

Two Best Responsesdepending on Q

choice 1-Q Q

0 1 0

1 0 3

3

Strategy

1-P

P

P ! = 0, Π! = 1−Q (Q ≤ 1/4)P ! = ∗, Π! = 3/4 (Q = 1/4)

P ! = 1, Π! = 3Q (Q ≥ 1/4)

= (1−Q)− (1− 4Q)P

Π(P ) = (1− P )(1−Q) + 3PQ

A Game against Human

Human can thinkindependently

Ai thinks that Bill also wants higher payoff

Best Response to Best Response: Nash Equilibrium

Pareto Efficient N.E.

Ai\Bl 0 1

0 1 0

1 0 3

Ai\Bl 0 1

d 1-P

P

1-Q Q

(P !, Q!) = (0, 0), Π! = 1

(P !, Q!) = (1, 1), Π! = 3

4

Battle of Sexes

Women are obstinate

Rule of the game can be cruel

Two conflicting Nash E.

Two N.E. coexist in ensemble of pairs

Ai\Bl 0 1

0 1 \ 3 0

1 0 3 \ 1

1-P

P

1-Q Q

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Men

(P !, Q!) = (0, 0), (Π!Ai,Π

!Bl) = (1, 3)

(P !, Q!) = (1, 1), (Π!Ai,Π

!Bl) = (3, 1)

Rock-Scissors-Paper Game

No dominant strategy

No apparent Nash E.

Random play is bestfor both

: Mixed Nash Equilibrium

Both just break even (Stop telling trivialities...)

Ai\Bl 0 1 2

0 0 - \ + + \ -1 + \ - 0 - \ +2 - \ + + \ - 0

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P !0 = P !

1 = P !2 = 1/3

Π! = 0

Calculating Payoffs

Payoff Matrix Joint Probability Matrix

Payoff is calculated as

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MAB 0 1 2

0 M00 M01 M02

1 M10 M11 M12

2 M20 M21 M22

PAB 0 1 2

0 P0Q0 P0Q1 P0Q2

1 P1Q0 P1Q1 P1Q2

2 P2Q0 P2Q1 P2Q2

MAB PAB = PAQB

ΠAi =∑A,B

PABMAB

(Strategy)

Lizards’ R-S-P Game

Animals Play Games Uta Stansburiana:

male behavioral types Guardian Usurper Sneaker

Population ratio 1 : 1 : 1 irrespective to underlying genetics

8

Elements of Game Theory

Payoff matrix (game table)

Joint probability (strategy)

Payoff

Nash Equilibria (solutions) plus “edge solutions”

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ΠBl =∑A,B

PABLAB

ΠAi =∑A,B

PABMAB

PAB = PAQB

∂QΠBl|(P !,Q!) = 0∂P ΠAi|(P !,Q!) = 0

MAB LAB

One good choicefor all occasion:Dominant strategy

‘Bad’ Dominant Nash

Less than Pareto efficient (3,3)

Conflict between Personal Gain & Public Good

Ai\Bl 0 1

0 bd 1 \ 1 5 \ 0

1 go 0 \ 5 3 \ 3

Ai\Bl 0 1

0 bd 1 \ 1 5 \ 0

1 gd 0 \ 5 3 \ 3

Dominant Strategy & Prisoner’s Dilemma

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(P !, Q!) = (0, 0), (Π!Ai,Π

!Bl) = (1, 1)

Multisector Game of Incomplete Information PD can be made to have Pareto-Nash Equilibrium

PD with Punishers

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A \ B 0 1 0 1

0 1 5 -20 -25

1 0 3 0 3

0 -1 0 0 -5

1 0 0 0 0

b=0 90% b=1 10%

a=090%

a=110%

MAB

UndercoverPunisher[Type 1]

Multi-Sector Game

Type [a], [b] with mixtures S[a], T[b]

Payoff Matrices for Ai and Bill

Joint strategy with Type Locality assumption

Sector Payoffs

Total Payoffs

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M [ab]AB , L[ab]

AB

Π[ab]Ai =

∑A,B

P [ab]AB M [ab]

AB Π[ab]Bl =

∑A,B

P [ab]AB L[ab]

AB

Π[ab] =∑a,b

S[a]T [b]Π[ab]

P [ab]AB = P [a]

A Q[b]B

Understand System of Autonomous Agents

Solve System Design Inefficiency ... Economics Sociology Political Sciences Magnagement Robotics

Understand the Law of Unintended Consequences

Game Theory is Here to ...

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AestheticUgly math with underlying probability vector and arbitrary matrix

TechnicalHard to include “player correlation” by its construction

NanotechnologicalNeed eventually to handle quantum devices

Defects of Current Theories

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Hilbert Space Game Theory

Many-body dynamics described indirectly with Matrix and Probability distribution : reminiscentof quantum mechanics à la von Neumann

Why assume a priori that Probability Distributions to be real P0+P1+..+PN-1=1, Q0+Q1+..+QN-1=1?

Try Probability Distribution aus Unitary Vector!

Sidestep Decision-Locality (no correlation) possible?

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PAB = | 〈AB|Ψ〉 |2

Minimal Quantum Theory Measurement along z-axis of a Spin

Desired probability with proper

Independent measurements of two Spins

yield paradoxical results showing nonlocality

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|Ψ〉 = (|00〉 − |11〉)/√2

P00 = | 〈00|Ψ〉 |2 = 1/2P11 = | 〈11|Ψ〉 |2 = 1/2

P10 = | 〈10|Ψ〉 |2 = 0P01 = | 〈01|Ψ〉 |2 = 0,

,

P0 = | 〈0|α〉 |2 = cos2 α P1 = | 〈1|α〉 |2 = sin2 α,

|α〉 = Uα |0〉 = cos α |0〉 + eiξ sinα |1〉|α〉 = Uα |1〉 = −e−iξ sinα |0〉+ cos α |1〉

Player Action & Probability

Classical Strategy : Individual Probabilities

: Ai, : Bill

Quantum Strategy : Individual Unitary Actions

: Ai, : Bill

When , back to Classical w. identifications

: Play Strategy PA (QB) = Adjust ‘angle’ ()

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|Φ〉 = |00〉

P [ab]AB = P [a]

A Q[b]B

U [a]α V [b]

βP [ab]

AB = |〈AB| U [a]α V [b]

β |Φ〉|2

P [a]A Q[b]

B

P [a]A = |〈A| U [a]

α |0〉|2 and Q[b]B = |〈B| V [b]

β |0〉|2

Multisector Quantum Game Type [a], [b] with mixtures S[a], T[b]

Payoff Matrices for Ai and Bill

Joint strategy with quantum actions U and Von

Sector Payoffs

Total Payoffs

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M [ab]AB , L[ab]

AB

Π[ab]Ai =

∑A,B

P [ab]AB M [ab]

AB Π[ab]Bl =

∑A,B

P [ab]AB L[ab]

AB

Π[ab] =∑a,b

S[a]T [b]Π[ab]

P [ab]AB = |〈AB| U [a]

α V [b]β |Φγφ)〉|2

Implementation1) Pre-game calibration with =0 2) Game play with full state

Nonlocality: Results of an action of Ai seems affected by action of Bill (et vice versa)

ITC Quantum Strategy

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Aida

A

P [ab]AB

B

β[b]α[a]Bluebeard

Rigoletto

ITC Scheme

|Φ(γ,φ)〉|Φ〉 = cos

γ

2|00〉 + eiφ sin

γ

2|11〉

Cereceda Game

A two-sector Incomplete Information extension of Battle of Sexes Game

A \ B 0 1 0 1

0 1 \ 3 0 -1 \ -3 0

1 0 3 \ 1 0 -3 \ -1

0 -1 \ -3 0 -3 \ -1 0

1 0 -3 \ -1 0 -1 \ -3

b=0 50% b=1 50%

a=050%

a=150%

M\L

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Classical and Quantum PAB

Distribute to get high score

Classical strategy Quantum strategy

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P [ab]AB

0.2 0 0.1 0.1

0.8 0 0.4 0.4

0 0 0 0

1 0 0.5 0.5

Q0 Q1

P0

P1

0.43 0.07 0.07 0.43

0.07 0.43 0.43 0.07

0.07 0.43 0.07 0.43

0.43 0.07 0.43 0.07

V0 V1

U0

U1

PAB = PQ PAB = |〈UV Φ〉|2

Classical Nash Equilibria

Random play results inNegative Payoff

Eight Nash E. : examples -->

Inequitable Split in BoS sector

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1 0 1 0

0 0 0 0

0 0 0 0

1 0 1 0Π!

Ai = Π!Bl = 0

0 1 0 10 0 0 0

0 1 0 10 0 0 0

P [ab]AB

Π[00]!Ai = 3, Π[00]!

Bl = 1

Π[00]!Ai = 1, Π[00]!

Bl = 3

Π[00]!Ai = 0, Π[00]!

Bl = 0

Quantum Nash Equilibrium Maximally entangled state

Beat classical logic

Equitable Split in BoS sector

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P [ab]

AB

τ =12

cos2π

8

σ =12

sin2 π

8

=0.427=0.073

Π!Ai = Π!

Bl = 4σ√2

Π[00]!Ai = Π[00]!

Bl = 4τ

γ =π

2β!

0 − α!0 = π/8

β!1 − α!

0 = −5π/8β!

0 − α!1 = 3π/8

Gedanken experiment on dichotomic 2 x 2 system

Ai’s spin measured in settings a = 0, 1, projection A = 0, 1 (sA=(-1)A)

Bill’s spin measured in settings b = 0, 1, projection B = 0, 1 (sB=(-1)B)

With Local Realism, satisfy

Bell Inequality

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P [ab]AB

CerecedaP [00]

00 − P [10]00 − P [01]

00 − P [11]11 ≤ 0

P [00]11 − P [10]

11 − P [01]11 − P [11]

00 ≤ 0

Bell & Quantum Nash

Payoff of Cereceda Game

Positive payoffs are result of nonlocal strategy

Never achieved with classical strategies

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ΠAi =14(P [00]

00 − P [10]00 − P [01]

00 − P [11]11 )

+34(P [00]

11 − P [10]11 − P [01]

11 − P [11]00 )

+14(P [00]

11 − P [10]11 − P [01]

11 − P [11]00 )

ΠBl =34(P [00]

00 − P [10]00 − P [01]

00 − P [11]11 )

1 -1

-1

-1

1 -1

-1

-1

Anatomy of Quantum Move

Identify

1st+2nd terms: Game-Symmetrizer / Altruism

3rd term: Quantum Interference / Nonlocality

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P [ab]00 = cos2

γ

2P [a]

0 Q[b]0 + sin2 γ

2P [a]

1 Q[b]1 + cos φ sin γ

√P [a]

0 P [a]1 Q[b]

0 Q[b]1

P [ab]01 = cos2

γ

2P [a]

0 Q[b]1 + sin2 γ

2P [a]

1 Q[b]0 − cos φ sin γ

√P [a]

0 P [a]1 Q[b]

0 Q[b]1

P [ab]11 = cos2

γ

2P [a]

1 Q[b]1 + sin2 γ

2P [a]

0 Q[b]0 + cos φ sin γ

√P [a]

0 P [a]1 Q[b]

0 Q[b]1

P [ab]10 = cos2

γ

2P [a]

1 Q[b]0 + sin2 γ

2P [a]

0 Q[b]1 − cos φ sin γ

√P [a]

0 P [a]1 Q[b]

0 Q[b]1

P [a]1 = sin2 α[a], Q[b]

1 = sin2 β[b]

|Φ〉 = cosγ

2|00〉 + eiφ sin

γ

2|11〉

Altruism and Nonlocality Altruism most visible in = /2, = /2 case

A local, thus classical correlation (“cheap talk”)

Nonlocal and altruistic in = /2, = 0 case

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P [ab]AB =

12P [a]

A Q[b]B +

12P [a]

B Q[b]A (since M [ab]

AB = L[ab]BA)

Π[ab]Ai = Π[ab]

Bl =12

∑A,B

(M [ab]AB + L[ab]

AB)P [a]A Q[b]

B

Π[ab]Ai =

∑A

M [ab]AA cos2(α[a]−β[b]) +

∑A !=B

M [ab]AB sin2(α[a]−β[b])

Some Observations

In joint probability formalism, Quantum Strategy is a natural extension of Classical Strategy

Separation of control variable and probability-> Correlated and Nonlocal Strategies inclusive

Concept of Control (strategy) and Gain (payoff) to Quantum Information and Quantum Metaphysics

Mathematics mostly understood, now set for “practical” application!

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Future Directions (gen)

Do quantum game experiment!

Dynamical (evolutionary)quantum game theory

N player quantum games

Application in auction, finance?

Application in quantum information processing!(proper 2-particle control to enhance desired phenomena)

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Future Directions (pro)

More general 2 player games (more C-ineq. exist)

Other Schemes to generatequantum strategies

Inclusion of mixed state(or already included?)

General Hermitian game(or already in formalism?)

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Aida

P [ab]AB

β[b]α[a]

Bluebeard

Rigoletto

CT Scheme

J(γ1, γ2)

References

T.Cheon Homepage

http://www.mech.kochi-tech.ac.jp/cheon/

T.Cheon and A.Iqbal, “Quantum strategies and Bell inequalities”, in Proc. SPIE workshop “Fluctuations and Noise”, Firenze, May 2007.

T.Ichikawa, I.Tsutsui and T.Cheon, arXiv.org, quant-ph/0702167.

T.Cheon, Europhys. Lett. 69 (2005) 149-155.

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