qgames-8-pqpenny.txt

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We now consider a very simple quantum
game called PQ Penny Flip Over, introduced
by David Meyer in 1998.
This game involves two players, Picard and Q.
They flip a coin.
The coin is known to start out in tails.
Q gets to flip first, if he so desires.
P then flips, then Q. Q wins if the penny ends up heads
and Q loses if the penny ends up tails.
We may model this game with the following payoff matrix.
Q is given two moves, each of which
is either a flip or a no flip.
P gets one move.
There are eight possible payoff values.
This game has no Nash equilibrium.
The best strategy for Q turns out
to be a mixed strategy, meaning that it
is a statistical mixture of doing no flip
or a flip with probability 1/2 On average, the payoff for Q
will be zero.
On the other hand, because Q is clearly a quantum being,
Q can put the coin in a superposition state.
Instead of merely flipping the coin,
Q can apply a Hadamard operator to put the coin
in a superposition of tail and head.
P is classical, so he can only perform a flip,
modeled as an X operator, or no flip, an identity operator.
The winning strategy for Q is thus
to perform two Hadamard operations,
such that it becomes irrelevant whether P flipped the coin
or not, thus leaving the coin always in the head
state, in the end, once it is measured, giving a payoff of +1 for Q no matter what P does.