qgames-11-qpublic-goods.txt

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The potential for quantum strategies in game theory
is illustrated by the following example
of a quantum public goods game-- a proportional multiplayer
game.
Suppose a group of coworkers collects
a pool of funds for a birthday celebration.
Assume each individual starts with y dollars
and is asked to contribute.
So say John, Ann, Kay, and Chloe each
have the option of contributing some amount.
However, contributions are anonymous,
and everyone gets to share equally
in the birthday party, independent of
whether or not he or she contributed.
Let n be the total number of individuals-- players
in this game.
And let y be known as the initial individual endowment.
Let C sub k be the contribution from player k.
Another variable, a, defines what
is known as the public gain-- the fact
that the whole may be a times greater than the sum
of the contributions.
We may model each individual share of their birthday party
as a payoff given the individual endowment
minus the contribution plus an equal share
of the total contribution weighted by the public gain.
Note, again, that contributions are optional.
Thus, it is apparent from this formula
that individual payoffs are maximized by contributing
nothing to the pool.
Thus, the Nash equilibrium for the game is for nobody at all
to contribute, or in other words, everybody defects.
Let us look specifically at the case
of two players, Alice and Bob, each
with an initial endowment of $1.
suppose each either contributes their dollar, or defects
and contributes nothing.
In this case, the payoff may be written as a two
by two matrix shown here.
And in fact, this instance of the game
is just like the prisoner's dilemma,
when a is bounded between 1 and 2.
Both players defecting is the dominant strategy.
Now turn to the case of three players.
We may employ the payoff formula to tabulate
the payoffs and the total birthday pool contribution
as follows.
When A, B, and C each contribute $0,
then their payoffs are each just their initial $1 endowments.
When one player contributes $1, that dollar
multiplied by the public gain A is
shared between all the players.
Shown here are the payoffs when C contributed.
Note that C gets less than A and B.
Similarly, when two players contribute--
say, B and C-- then A has the highest payoff.
If all three contribute, then the total payout
is the highest.
What is the Nash equilibrium?
Since an individual gets a higher payoff
by not contributing, the Nash equilibrium
is for nobody to contribute.
This is unfortunate, given the birthday
party is a public good.
This is another example of the tragedy of the commons.
How might quantum strategies be allowed
to change the Nash equilibrium?
Here is one proposal from Kay-Yut Chen, Tad Hogg,
and Ray Beausoleil.
The idea is that the players employ a quantum circuit, shown
here, for three players.
Each player is given a qubit with Hilbert space spanned
by c and d.
The qubits start in state c and are
entangled using a multiqubit operation,
J. Each player then makes their choice
as a single qubit unitary operator.
The result, fed through J dagger,
is then measured to give the payoff.
J is performed by a third party or trusted apparatus.
Each player's unitary operation may be identity for cooperate,
a Pauli-x for defect, or a Pauli-z as a new quantum
strategy.
J is an entangling operation defined
as an equal sum of identity and a tensor
product of x in all qubits.
This protocol gives an average payoff of 1
plus a quantity divided by 2 when players employ the Nash
equilibrium strategy, which is for everyone to choose
the quantum strategy.
Noting that a is between 1 and 2,
we find that the quantum outcome is
more optimal than the classical equilibrium
payoff, which is $1.
And instead of trusting in a third party,
with this procedure, one essentially
puts trust into quantum mechanics operating properly.