qgames-9-qprisoner.txt

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This is an example of a quantum version
of the prisoner's dilemma game.
Perhaps the most important rule to the prisoner's dilemma
setting is that no communication is allowed between the players.
However, what if the players were
allowed to hold an entangled quantum state?
Suppose we define a single qubit Hilbert
space with basis state's C and D.
And allow each player to hold one qubit.
We may model a quantum version of the prisoner's dilemma game
as this quantum circuit.
The players are assumed to initially hold
qubits in the C state.
A two qubit operation J is applied to these qubits.
J will be defined in a moment.
The two player's actions are represented
by U sub A and U sub B-- two single qubit operations.
The two qubits are then passed through the inverse operator J
dagger and then measured.
We define J as being the two qubit entangling
operator II plus xx divided by square root of 2.
This is an entangling operation, which
is a pi over 4 rotation about the two Pauli xx operation.
Now, it is a little unclear exactly who
is doing J and J dagger.
But it should be clear that U sub A and U sub B
are the prisoner's actions.
Where the unitary operations are either
I-- representing cooperation with a partner,
or x-- representing defection from the partner.
Note that the two qubit operation J commutes
with all four possible classical actions which may be chosen
as strategies by the players.
Thus, assuming that the measurement results give
the probabilities of the prisoners deciding on CC, CD,
DC, or DD as their actions, then the payoffs
represented by these two formulas
reduce to exactly the same as expected
from the classical version of the prisoner's dilemma game.
In these formulas R, P, T, and S are, as before,
the reward, punishment, tempt payoff, and the sucker's payoff.
What is different now is that the payoffs involve
probabilities P sub CC, P sub DD, and so forth, given
by the measurement results.
So these are now random variables.
In the case where the player's action is cooperate or defect
this circuit behaves classically and the output
is deterministic.
Thus, the payoff matrix is exactly the same
as in the classical case.
With this quantum setting, however, we
may now introduce a new quantum strategy represented
by the operator Z chosen for either U sub A or U sub B,
or both.
Let us call this the quantum action versus C or D
being the classical actions.
What payoffs may result if a quantum action is chosen?
Well, suppose Al chooses Q-- the quantum action,
and Bob chooses to cooperate C. Then in this case, the output
state before measurement, psi, is given by this formula
J dagger Z tensored I times J acting on CC- the input state.
Working through the algebra for this
using the known Pauli matrix commutation relationships
we find that the output state is up to an overall phase
DD for this action QC.
Thus, the output payoff is-- for each one of the players--
the punish payoff.
What if both players choose Q?
What is the payoff?
In this case, with use of A and use of B both being Z,
the output state is found to be up to an overall phase CC.
That is because J commutes with Z tensor Z. Thus,
the payoff is 3 for both players.
3 being the reward payoff.
A similar computation can be done for the case of Q
and defect.
Putting these new results with quantum strategies
into the payoff matrix we find these results.
Note now that this payoff matrix no longer
has a dominant strategy in contrast
with the original payoff matrix.
Here now we have a Nash equilibrium strategy
that is both players choosing the quantum action.
And in this case both prisoners receive the reward payoff.
However, there are issues with this setting.
For example, what happens if the prisoners
choose other unitary operators than I, x, and Z.
How does one enforce J and J dagger?
In fact, who performs them?
Are they performed by the police?
Are they performed by a judiciary?
Perhaps they are performed by a trusted third party.
Perhaps they are performed by a trusted physical apparatus.
But then again, why does one want
to help the prisoners in this setting?
What is the proper interpretation
of the game now that we have a quantum set of strategies?
Nevertheless, this example demonstrates
some of the principal ideas used in introducing
quantum strategies into classical games.