qgames-1-intro.txt

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Quantum games are an extension of classical game
theory introducing new quantum resources and quantum
protocols.
We introduce this subject by considering
five topics, classical game theory, a quantum game known
as PQ Penny Flipover, a quantum version
of the classical Prisoner's Dilemma game,
the classical Tragedy of the Commons,
and the quantum version of known as a quantum public goods game.
Let us begin with some history, context, and definitions
from classical game theory.
Modern game theory was founded by Von Neumann and Morgenstern
in the 1940s and 1950s.
What it is is a multi-person decision theory,
an analysis of the decision-making process,
assuming participants in the game act rationally.
The idea is that each participant or person
in the game seeks to maximize the rewards.
These rewards maybe profits, incomes, subjective benefits,
or other kinds of rewards.
A classic example was provided in the 1950s by Albert Tucker.
In this example, two burglars are caught
and are separated by police so that the two may not
talk with each other.
These two burglars have the following options.
When interviewed by the police, they
may choose to cooperate with their partner in crime,
or they may defect from that partnership.
Consider the possible outcome.
These are recorded in what is known as the payoff matrix.
Let the burglars be named Al and Bob.
Al may cooperate or defect, and Bob may cooperate or defect.
There are four possibilities.
The payoffs to Bob may be three or five
if Al cooperates or zero or one if Al defects.
On the other hand, the payoffs to Al, shown here in green,
are three or zero or five or one.
These numbers may be interpreted as follows.
Let us call three the reward.
Let us call one punishment.
Let us call five the tempting payoff
and zero the sucker's payoff.
Given these payoffs which are known to both Al and Bob,
what should each one of them do not knowing
what the other one will do?
Each burglar's goal is to maximize their own payoff.
And remember, the rule is no communication
is allowed between the two burglars.
Consider what would happen if the two burglars cooperated.
Then clearly, for example, Bob could get a higher payoff
by defecting instead of cooperating.
Similarly, Al could get a higher payoff also by defecting.
But if Al defected, then Bob would do better
by also defecting.
And thus, the clear, optimum choice for each player
is to defect and both then defect their partnership.
The choices that each player makes in a game
are known as a strategy.
In this case, there is a name to the strategy that is
employed by the two burglars.
It is known as the dominant strategy.
A dominant strategy is one which earns a player a larger
payoff than any other strategy, regardless
of what other players do.
When a set of dominant strategies,
one for each player, exists, this
is known as a dominant-strategy equilibrium.
In this game, we have such a case
where the dominant-strategy equilibrium is for both players
to defect.
This example is a classic problem
in game theory known as the Prisoner's Dilemma.
Both prisoners would be better off
if they had cooperated with each other,
but because of the incentive to defect from each other,
each one of them ends up defecting, not knowing what the other player will do.