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Classical game theory. Quantum games: models. Quantum strategic games: relation to
classical equilibria. Quantum advantage against classical players Quantum advantage with quantum players.
Quantum information processing Understanding the power of quantum
Computation: quantum algorithms/complexity Communication: quantum info. theory Cryptography: quantum cryptography, quantum
algorithm breaking classical cryptosystems.
Another field: game theory
Game: Two basic forms
strategic (normal) form
players: , …, has a set of
strategies has a utility
function ℝ
Equilibrium
Equilibrium: each player has adopted an optimal strategy, provided that others keep their strategies unchanged
Classical strategic games
is an equilibrium if ,
Nash equilibrium: is product distribution. Correlated equilibrium: is general distribution.
𝑠1′
𝑠2′
𝑝
output utility
𝑢1(𝑠1′ ,𝑠2
′ )
𝑢2(𝑠1′ ,𝑠2
′ )
𝑠1
𝑠2
potentialaction
“quantum games”
Non-local games
EWL-quantization of strategic games*1
Others Meyer’s Penny Matching*2 Gutoski-Watrous framework for refereed game*3
*1. Eisert, Wilkens, Lewenstein, Phys. Rev. Lett., 1999.*2. Meyer, Phys. Rev. Lett., 1999.*3. Gutoski and Watrous, STOC, 2007.
EWL model
𝐽
⋮⋮ ⋮
𝒔𝟏
𝒔𝒏
⋮
|0
|0
What’s this classically?
𝒖𝟏(𝒔)
𝒖𝒏(𝒔)
⋮
States of concern
output utilitypotential
action
EWL model
Classically we don’t undo the sampling (or do any re-sampling) after players’ actions. If a joint operation is allowed, then a simple swapping solves
Prisoner’s Dilemma classically!
𝐽
⋮⋮ ⋮
|0
|0
𝒔𝟏
𝒔𝒏
⋮
𝒖𝟏(𝒔)
𝒖𝒏(𝒔)
⋮
output utilitypotential
action
EWL model
𝐽
⋮⋮ ⋮
|0
|0
and consider the state at this point
𝒔𝟏
𝒔𝒏
⋮
𝒖𝟏(𝒔)
𝒖𝒏(𝒔)
⋮
output utilitypotential
action
Quantum strategic games
⋮ ⋮ρ
and consider the state at this point
A simpler model, corresponding to classical
games more precisely.
CPTP
𝒔𝟏
𝒔𝒏
⋮
𝒖𝟏(𝒔)
𝒖𝒏(𝒔)
⋮
output utilitypotential
action
*1. Zhang, ITCS, 2012.
Other than the model
Main differences than previous work in quantum strategic games:
We consider general games of growing sizes. Previous: specific games, usually 2*2 or 3*3
We study quantitative questions. Previous work: advantages exist? Ours: How much can it be?
Central question: How much “advantage” can playing quantum provide?
Measure 1: Increase of payoff Measure 2: Hardness of generation
First measure: increase of payoff We will define natural correspondences
between classical distributions and quantum states.
And examine how well the equilibrium property is preserved.
Quantum equilibrium
classical quantum
Φ1
Φn
⋮ ⋮
s1
sn
⋮ρ
u1(s)
un(s)
⋮
C1
Cn
⋮
s1’
sn’⋮p → s
u1(s’)
un(s’)
⋮
classical equilibrium:
No player wants to do anything to the assigned strategy si, if others do nothing on their parts- p = p1…pn: Nash equilibrium- general p: correlated equilibrium
quantum equilibrium:
No player wants to do anything to the assigned strategy ρ|Hi, if others do nothing on their parts- ρ = ρ1… ρn: quantum Nash equilibrium- general ρ: quantum correlated equilibrium
Correspondence of classical and quantum states
classical quantum
p: p(s) = ρss
(measure in comp. basis)
ρ
p: distri. on S
ρp = ∑s p(s) |ss| (classical mixture)
|ψp = ∑s√p(s) |s (quantum superposition)
ρ s.t. p(s) = ρss (general class)
Φ1
Φn
⋮ ⋮s1
sn
⋮ρC1
Cn
⋮s1’
sn’⋮p→s
Preservation of equilibrium?
classical quantum
p: p(s) = ρss ρ
p
ρp = ∑s p(s) |ss|
|ψp = ∑s√p(s) |s
ρ s.t. p(s) = ρss
Obs: ρ is a quantum Nash/correlated equilibrium p is a (classical) Nash/correlated equilibrium
p NE CE
ρp
|ψp
gen. ρ
Question: Maximum additive and multiplicative increase of payoff (in a [0,1]-normalized game, |Si| = n)?
Maximum additive increase
classical quantum
p: p(s) = ρss ρ
p
ρp = ∑s p(s) |ss|
|ψp = ∑s√p(s) |s
ρ s.t. p(s) = ρss
p NE CE
ρp 0 0
|ψp 0 1-Õ(1/log n)
gen. ρ 1-1/n 1-1/n
max payoff increase, additive
Question: Maximum additive and multiplicative increase of payoff (in a [0,1]-normalized game, |Si| = n)?
Maximum multiplicative increase
classical quantum
p: p(s) = ρss ρ
p
ρp = ∑s p(s) |ss|
|ψp = ∑s√p(s) |s
ρ s.t. p(s) = ρss
p NE CE
ρp 1 1
|ψp 1 Ω(n0.585…)
gen. ρ n n
Question: Maximum additive and multiplicative increase of payoff (in a [0,1]-normalized game, |Si| = n)?
max payoff increase, multipliative
Next
Classical game theory. Quantum games: models. Quantum strategic games: relation to
classical equilibria. Quantum advantage against classical players Quantum advantage with quantum players.
Meyer’s game: classical
Actions : to flip or not to flip
Alice’s Goal: 0. Bob’s Goal: 1. A Nash equilibrium: flip with half prob.
Then each wins with half prob.
0 0/1?
Meyer’s game: quantum
Bob remains classical: is either or identity. Alice is quantum: can be any 1-qubit
operation. Alice’s Goal: . Bob’s Goal: . Now Alice can win for sure by applying a
Hadamard gate. . .
|0 ⟩ or ?
Meyer’s game: fairness issue
Despite the quantum advantage, there is clear a fairness issue. Alice has two actions. And the actions are in a fixed order of .
Question: Can quantum advantage still exist in a more fair setting?
For fairness: each player makes just one action, simultaneously. This is nothing but strategic games!
|0 ⟩ or ?
Quantization*1 of strategic game: Penny Matching
is an equilibrium if both players are classical, Each wins with half prob.
If Alice turns to quantum: turns into . Then she wins for sure! Message: quantum player has a huge advantage when playing
against a classical player.
|𝑎 ⟩
|𝑏 ⟩
goal:
goal:
|𝜑 ⟩
*1. Zu, Wang, Chang, Wei, Zhang, Duan, NJP, 2012.
¿¿¿
potential action classical outcome utility
Quantization of strategic game: Penny Matching
State is symmetric, so it doesn’t matter who takes which qubit.
We can also let the classical player Bob to choose the target goal. If Bob wants , then Alice applies .
¿|0 ⟩ ¿¿
|𝑎 ⟩
|𝑏 ⟩
goal:
goal:
|𝜑 ⟩¿¿¿
potential action classical outcome utility
Quantum advantage in strategic games
is an equilibrium if both players are classical, each winning with prob. = ½
If Alice uses quantum, increases her winning prob. to ¾. Question*2: Is discord necessary?
Yes, if each player’s part (of the shared state) is a qubit, No, if each player’s part (of the shared state) has dimension 3 or more.
Entangled. Necessary?
No! No entanglement. But has discord.
*2. Wei, Zhang, manuscript, 2014.
|𝑎 ⟩
|𝑏 ⟩
goal:
goal:
|𝜑 ⟩¿¿¿
potential action classical outcome utility
Games between quantum players After these examples, Bob realizes that he
should use quantum computers as well. Question: Any advantage when both players are
quantum? Previous correspondence results imply a
negative answer for complete information games.
But quantum advantage exists for Bayesian games!
Quantum Bayesian games
Each player has a private input/type . is known to Player only. The joint input is drawn from some distribution .
Each player can potentially apply some operation . A measurement in the computational basis gives output for
Player , who receives utility .
|𝑦1 ⟩
|𝑦2 ⟩
|𝜑 ⟩(𝑥1 , 𝑥2)←𝑃
𝑢1(𝑥1 , 𝑥2, 𝑦1 , 𝑦2)
𝑢2(𝑥1 ,𝑥2, 𝑦 1, 𝑦 2)
potential actionclassical outcome utility
Quantum Bayesian games
Classical state distribution . Classical strategy . Classical payoff
is equilibrium if no player can gain a higher payoff by changing her strategy unilaterally.
|𝑦1 ⟩
|𝑦2 ⟩
|𝜑 ⟩(𝑥1 , 𝑥2)←𝑃
𝑢1(𝑥1 , 𝑥2, 𝑦1 , 𝑦2)
𝑢2(𝑥1 ,𝑥2, 𝑦 1, 𝑦 2)
potential actionclassical outcome utility
Quantum Bayesian games
Quantum strategy , . Quantum payoff
is equilibrium if no player can gain a higher payoff by changing her strategy unilaterally.
|𝑦1 ⟩
|𝑦2 ⟩
|𝜑 ⟩(𝑥1 , 𝑥2)←𝑃
𝑢1(𝑥1 , 𝑥2, 𝑦1 , 𝑦2)
𝑢2(𝑥1 ,𝑥2, 𝑦 1, 𝑦 2)
potential actionclassical outcome utility
Quantum Bayesian games
*1. Pappa, Kumar, Lawson, Santha, Zhang, Diamanti, Kerenidis, PRL, 2015.
A game*1 combining Battle of the Sexes and CHSH.
The players need to coordinate like in CHSH, except when , in which case they need to anti-coordinate.
In Table I, they have conflicting interest.
|𝑦1 ⟩
|𝑦2 ⟩
|𝜑 ⟩(𝑥1 , 𝑥2)←𝑃
𝑢1(𝑥1 , 𝑥2, 𝑦1 , 𝑦2)
𝑢2(𝑥1 ,𝑥2, 𝑦 1, 𝑦 2)
potential actionclassical outcome utility
Quantum Bayesian games
*1. Pappa, Kumar, Lawson, Santha, Zhang, Diamanti, Kerenidis, PRL, 2015.
is uniform. Classical:
And a fair equilibrium with
Quantum: a fair equilibrium with
|𝑦1 ⟩
|𝑦2 ⟩
|𝜑 ⟩(𝑥1 , 𝑥2)←𝑃
𝑢1(𝑥1 , 𝑥2, 𝑦1 , 𝑦2)
𝑢2(𝑥1 ,𝑥2, 𝑦 1, 𝑦 2)
potential actionclassical outcome utility
Viewed as non-locality
Traditional quantum non-local games exhibit quantum advantages when the two players have the common goal. CHSH, GHZ, Magic Square Game, Hidden Matching
Game, Brunner-Linden game. Now the two players have conflicting interests. Quantum advantages still exist. Message: If both players play quantum strategies in
an equilibrium, they can also have advantage over both being classical.
Summary
Quantum strategic games: be careful with the model.
Quantum vs. classical: quantum has advantage.
Quantum vs. quantum: both have advantages compared to both being classical.
(Experiments for the above two results.) The field of quantum game theory: call for
more systematic studies in proper models.