taksu cheon- bayesian nash equilibria & bell inequalities

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 ...

13

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?)

30

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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