taksu cheon- bayesian nash equilibria & bell inequalities
TRANSCRIPT
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
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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
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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
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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
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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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