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Quantum Games
Quantum Strategies in Classical Games
Presented by Yaniv Carmeli
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Talk Outline
Introduction
PQ Games
Game Theory
Why quantum games?
PQ penny flip
2x2 Games
Quantum strategies
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Game Theory
The study of decision making of competing
agents is conflict situations.
Economic problems
Diplomatic relations
Social sciences
Biology
Engineering
John von Neumann
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Why Quantum Games?
Attempt to understand the source of the advantages of
quantum computation.
Quantum algorithms as games.
Quantum communication as a game.
Which problems are solvable more efficiently using quantum
algorithms?
The objective is to maximize effective communication
Quantum cryptography as a game
Eve’s objective is to learn the contents of the conversation.
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Game Theory - Terminology
Players
Moves
Strategy
Instructions to the player how to react to all
scenarios of the game.
Pure strategy – Always play a given move.
Mixed strategy – Probabilistic choice of moves.
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Game Theory – Terminology (Cont.)
Utility
Numerical measure of the desirability of an outcome.
Payoff Matrix
Gives the utility for all the players and for all game
outcomes.
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Game Theory – Terminology (Cont.)
A Nash Equilibrium (NE)
A combination of strategies from which no player can
improve his payoff by unilateral change of strategy.
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PQ Coin Flip
P
Q
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Coin Flip (Cont.)
The Game:
A penny is placed in a box head up.
Q can choose to flip or not to flip.
P can choose to flip or not to flip.
Q can choose to flip or not to flip
At the end: If the coin is head up, Q wins,
else P wins.
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Coin Flip (Cont.)
No deterministic solution.
Best mixed strategy: Flip with probability ½, don’t flip with
probability ½.
Expected payoff: 0.
General probabilistic strategy: Flip with probability p, don’t
flip with probability 1-p.
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PQ Coin Flip
P
Q
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Coin Flip – Quantum Representation
The coin is represented by a qubit, where
represents head up, and
represents tail.
Initial state:
Flipping the coin:
Not flipping:
Probabilistic strategy:
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Coin Flip – A Quantum Player
A quantum player is allowed any unitary strategy.
Q’s first operation is
After Picard’s mixed strategy:
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Coin Flip (Cont.)
What if the game was to end here?
If Q were to employ a strategy for which
Picard could get an expected payoff of
by selecting p=0 (or p=1).
If Picard were to choose
Q could get an expected payoff of
by selecting a=1 (or b=1)
NE is:
Where
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Coin Flip (Cont.)
NE is:
where
This represent the same results as in a classic
game.
A quantum player has no advantage
if he has only one move, and there
is no entanglement involved.
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Coin Flip (Cont.)
Q has a winning strategy:
After Q’s first move:
After P’s move:
After Q’s second move:
The mixed/quantum equilibria:
with exp. payoff of 1 to Q.
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Coin Flip (Cont.)
What about a game with two quantum players?
Consider an arbitrary pair of quantum strategies
If
Q can improve his expected payoff
by choosing
If
P can improve his
expected payoff by choosing
No Equilibrium!
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Coin Flip – Bad Example?
Can be implemented classically – not an example for
superiority of a quantum player (S.J. van Enk, 2000).
Classical implementations are not scalable – quantum
implementations are (D.A. Meyer, 2000).
It’s like losing a game of chess and
saying: “If we would have played on
a larger board, I would have won”.
(S.J. van Enk, 2000).
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Game Theory – Terminology (Cont.)
A Dominant Strategy
Does at least as well as any other strategy against
any possible moves by the other players.
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Game Theory – Terminology (Cont.)
A Pareto optimal outcome
An outcome from which no player can increase his
utility without reducing the utility of another player.
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2x2 Games
2 players
Each has the choice between 2 pure strategies.
The choice has to be made without communication,
and before knowing the opponent’s chosen move.
The payoff matrix is known.
Assumptions on the players:
Rationality - Players aim to maximize their payoffs
Each player knows that other players are rational
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The Prisoners’ Dilemma
Two suspects held in separate cells are charged with a
major crime. However, there is not enough evidence.
Both suspects are told the following policy:
If neither confesses - both will be convicted of a minor
offense and get one year in jail.
If both confess - both will be
sentenced to six years.
If one confesses but the other
does not, then the confessor will
be released, but the other will be
sentenced to jail for nine years.
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The Prisoners’ Dilemma
Each player has the choice between 2 strategies:
C (Cooperate)
D (Defect)
Nash Equilibrium: [D,D]
Pareto Optimal Strategy: [C,C]
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PD – Another Version
Two firms, Reynolds and Philip, share a market.
Each firm earns $60M from its customers if neither
do advertising.
Advertising costs a firm $20M.
Advertising captures $30M from the competitor.
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PD – Third Version
Two cyclists halfway in a race, with the rest of the
cyclists far behind them.
Each has two options: Taking the lead, where there is
no shelter from the wind (C), or staying behind and
riding in the other’s slipstream (D).
If they both make no effort to stay ahead, the rest of
the cyclists will catch up. If one takes the lead, he
works much harder and the other cyclist is likely to win.
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PD – Yet Another Version
Two states engaged in an arms race.
They both have two options, either to increase
military expenditure (D) or to make an agreement
to reduce weapons (C).
Neither state can be certain that the other one will
keep to such an agreement; therefore, they both
incline towards military expansion.
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Chicken
Each player has the choice between 2 strategies:
C (Cooperate \ Swerve)
D (Defect \ Don’t Swerve)
Nash Equilibria: [C,D],[D,C]
Pareto Optimal Strategy: [C,C]
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Battle of The Sexes
Each player has the choice between 2 strategies:
O (Opera)
T (Television)
Nash Equilibria: [O,O],[T,T]
Pareto Optimal Strategy: [O,O],[T,T]
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The Dating Problem
Alice may want a date with Bob, but if he
doesn’t want a date with her, she doesn’t want
him to know that she was interested.
Bob may want a date with Alice, but if she
doesn’t want a date with him, he doesn’t want
her to know that he was interested.
Is there still hope for the shy Alice
and Bob?
Is this a 2x2 game?
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Quantum 2x2 Games
The players receive two qubits (one for each) in a
known initial state.
Each player manipulates his qubit according to his
chosen move.
At the end, both qubits are measured using a
predetermined known basis.
The expected payoff is determined according
to the payoff matrix:
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Quantum 2x2 Games (Cont.)
Observation: If the qubits’ initial state is not entangled,
there is no advantage over a classical player utilizing a
mixed strategy.
What if the initial state was the maximally entangled
state
?
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Quantum Prisoners’ Dilemma
For the rest of this section we consider the Prisoners’
Dilemma with the initial state:
and the basis for measurement:
The payoff of final state σ:
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One Parameter Set of Strategies
When Alice and Bob select their strategies from S(CL)
(local rotations with one parameter):
Note that:
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One Parameter Set of Strategies
The expected payoff:
The same as mixed strategies in the classical version.
(Like choosing to cooperate with probability
).
Equilibrium is still [D,D].
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Two Parameter Set of Strategies
Now Alice and Bob select their strategies from S(TP)
(the two-parameter set of operators):
Note that:
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Two Parameter Set of Strategies
The expected payoff:
There is a new unique equilibrium [Q,Q], where:
Since
, it is also the optimal
solution (Given without proof).
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Two Parameter Set of Strategies
Proof that [Q,Q] is an equilibrium:
If Bob uses Q, then for every strategy of Alice U(θ,φ)
the same argument holds for every strategy of Bob,
when Alice uses Q.
Q is unique – given without proof.
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General Unitary Operations as Strategies
If Alice and Bob select their strategies from S(GU)
(the set of general local unitary operations):
For every move of Bob
(where a,b,c,d are
appropriate complex numbers), there exists a move for
Alice
, s.t.
.
The same argument is true for Bob, so there is no
Nash Equilibrium in pure strategies.
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General Unitary Operations as Strategies
There exists a NE in mixed strategies:
Expected payoff for both is 2.5.
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General Unitary Operations as Strategies
Problem: It is not the only equilibrium.
For example: The following is also an equilibrium
It has the same property as the previous one.
And there are more…
Which of them will be chosen?
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General Unitary Operations as Strategies
There exist probabilities
operators
s.t.:
and unitary
If Alice will choose “R”, Bob’s actions will not
change the state of the quantum system anymore.
[R,R] is the single NE which is the only one that gives
an expected payoff of 2.25 for both players
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General Unitary Operations as Strategies
Focal points are outcomes which are distinguished
from others on the basis of some characteristics which
are not included in the formalism of the model. Those
characteristics may distinguish an outcome as a result
of some psychological or social process and may even
seem trivial, such as the names of the actions.
If there are more that one NEs and one of them can be
considered a focal equilibrium, then it is the one that
will be chosen.
The NE is [R,R]
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Is It Really That Good?
Can we say we have cracked the prisoners’ dilemma?
The solution [D,D] is not a NE anymore!
The effect of Q can be described classically:
If Alice chooses Q and Bob chooses C or D then his choice is
changed to the one he didn’t pick (and vice versa).
If both choose Q – the payout is as if both cooperated.
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Is It Really That Good?
In non-cooperative games players are not allowed to
communicate, and the use of correlated random
variables is not allowed.
Letting Alice and Bob use an entangled state, means
they are using the correlations present in such a state.
This goes against the spirit of the prisoners’ dilemma.
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2x2 Games With Entanglement
Before each player makes his move we apply an
entangling operation to the qubits.
We would like the classic game to be a sub-game of
the quantum one. Hence:
After the players make their moves, we will apply
.
commutes with any direct product of any pair of
classical strategies.
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2x2 Games With Entanglement
With the entangling operator:
we can control the level of entanglement.
- No entanglement.
- Maximally entangled state.
Is there a critical level of entanglement, below which
the quantum player has no advantage over the
classical player?
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