Shengyu Zhang The Chinese University of Hong Kong.

Shengyu Zhang

The Chinese University of Hong Kong

Content

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 extensive form

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)

⋮


action

EWL model

𝐽

⋮⋮ ⋮

|0

|0

and consider the state at this point

𝒔𝟏

𝒔𝒏

⋮

𝒖𝟏(𝒔)

𝒖𝒏(𝒔)

⋮


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)

⋮


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


|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


Maximum multiplicative increase

classical quantum

p: p(s) = ρss ρ

p


|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


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 ⟩¿¿¿


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 ⟩¿¿¿


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


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)



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)



*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)



*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)


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.

Shengyu Zhang The Chinese University of Hong Kong.

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