- A Crash Course in Game Theory
- An Analysis of a Simple Game Matrix
- The Prisoner's Dilemma Game Matrix

The Bonnie and Clyde story is an example of a situation known in mathematical game theory as the ``prisoner's dilemma.'' A prisoner's dilemma always involves two ``game players,'' and each has a choice between ``cooperating'' and ``defecting.'' If the two players cooperate, they each do moderately well; if they both defect, they each do moderately poorly. If one player cooperates and the other defects, then the defector does extremely well and the cooperator does extremely poorly. (In the case of the Bonnie and Clyde story, ``cooperating'' means cooperating with one's partner--i.e., holding out on the police--and ``defecting'' means confessing to bank robbery.) Before formalizing the prisoner's dilemma situation, we need to introduce some basic game theory notation.

In game theory, a *two-person binary-choice game* is represented
as a two-by-two matrix. Figure 1 shows a hypothetical game matrix.
The two players in this case are called **A** and **B**, and
the choices are called ``cooperate'' and ``defect.''

B cooperates | B defects | |

A cooperates | A gets 5 | A gets 2 |

B gets 5 | B gets 3 | |

A defects | A gets 3 | A gets 1 |

B gets 2 | B gets 1 |

Players **A** and **B** can play a single game by separately (and
secretly) choosing to either cooperate or defect. Once each player has
made a choice, he announces it to the other player; and the two then
look up their respective scores in the game matrix. Each entry in the
matrix is a pair of numbers indicating a score for each player,
depending on their choices. Thus, in Figure 1, if Player **A**
chooses to cooperate while Player **B** defects, then **A** gets 2
points and **B** gets 3 points. If both players defect, they each
get 1 point. Note, by the way, that the game matrix is a matter of
public knowledge; for instance, Player **A** knows before the game
even starts that if he and **B** both choose to defect, they will
each get 1 point.

In an *iterated game*, the two players play repeatedly: thus,
after finishing one game, **A** and **B** may play another.
(Admittedly, there is a little confusion in the terminology here: you
can think of each individual game as a single ``round'' of the larger,
iterated game.) There are a number of ways in which iterated games may
be played; in the simplest situation, **A** and **B** play for
some fixed number of rounds (say, 200), and before each round they are
able to look at the record of all previous rounds. For instance,
before playing the tenth round of their iterated game, both **A**
and **B** are able to study the results of the previous nine rounds.

The game depicted in Figure 1 is a particularly easy one to analyze.
Let's examine the situation from Player **A**'s point of view
(Player **B**'s point of view is identical):

``Suppose **B** cooperates. Then I do better by cooperating
myself (I receive five points instead of three). On the other
hand, suppose **B** defects. I still do better by cooperating
(since I get two points instead of one). So no matter what
**B** does, I am better off cooperating.''

Player **B** will, of course, reason the same way, and both will
choose to cooperate. In the terminology of game theory, both **A**
and **B** have a *dominant* choice--i.e., a choice that gives
a preferred outcome no matter what the other player chooses to do.
Figure 1, by the way, does *not* represent a prisoner's dilemma
situation, since when both players make their dominant choice, they
also both achieve their highest personal scores. We'll see an example
of a prisoner's dilemma game very shortly.

To recap: in any particular game using the matrix of Figure 1, we would expect both players to cooperate; and in an iterated game, we would expect both players to cooperate repeatedly, on every round.

Now consider the game matrix shown in Figure 2.

B cooperates | B defects | |

A cooperates | A gets 3 | A gets 0 |

B gets 3 | B gets 5 | |

A defects | A gets 5 | A gets 1 |

B gets 0 | B gets 1 |

In this case, Players **A** and **B** both have a dominant
choice--namely, defection. No matter what Player **B** does,
Player **A** improves his own score by defecting, and vice versa.

However, there is something odd about this game. It seems as though the two players would benefit by choosing to cooperate. Instead of winning only one point each, they could win three points each. So the ``rational'' choice of mutual defection has a puzzling self-destructive flavor.

The matrix of Figure 2 is an example of a prisoner's dilemma game
situation. Just to formalize the situation, let *CC* be the number
of points won by each player when they both cooperate; let *DD* be
the number of points won when both defect; let *CD* be the number
of points won by the cooperating party when the other defects; and let
*DC* be the number of points won by the defecting party when the
other cooperates. Then the prisoner's dilemma situation is
characterized by the following conditions:

DC > CC > DD > CD |

CC > (DC + CD) / 2 |

In the game matrix of Figure 2, we have:

DC = 5 |

CC = 3 |

DD = 1 |

CD = 0 |

so both conditions are met. In the Bonnie and Clyde story, by the way, you can verify that:

DC = 0 |

CC = -1 |

DD = -10 |

CD = -20 |

Again, these values satisfy the prisoner's dilemma conditions.