Game Theory
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Get out your nerd glasses and pocket protectors!  It's time to talk game theory and "The Proposition".

Game theory, a blend of statistics and human psychology, examines how people behave in situations where they're given a choice between making a small personal sacrifice to improve the outcome for the group ("cooperating") and grabbing a larger individual gain at the expense of the group ("defecting").  Often, individually rational decisions lead to a collectively irrational outcome, such as bank panics, global warming, overfishing, and the Prisoner's Dilemma, a famous thought experiment devised by Merrill Flood and Alvin Dresher.

In the Prisoner's Dilemma, two men who have jointly committed a crime are taken into custody.  They are held separately, so they can't talk to each other.  The police don't have enough evidence to convict them, and they must rely on testimony from the men.  The prisoners are given a choice between turning state's evidence and confessing, or remaining silent:

If both remain silent, both will get 1 year.

If Prisoner A confesses and Prisoner B remains silent, Prisoner A will go free, while his partner will get the full 10 year sentence.  (The reverse is also true.)

If both confess, both will get a reduced 5 year sentence.

Obviously, the best outcome would be for both to "cooperate" and remain silent.  However, since they are in separate cells, they can't confer with each other and agree to do this.  Neither man knows for sure what the other one will do.  So each man's reasoning would go something like this: "If my partner stays quiet, my best move is to betray, as I would then go free instead of getting a 1 year sentence.  If my partner betrays, then my best move is still to betray, since I would get a lighter sentence of 5 years instead of 10.  No matter what my colleague does, I'm better off confessing."  Both men, reasoning in this manner, are likely to confess, and the result is a 5-year prison term for each.

So, you can see the similarities between this dilemma, and the dilemma at the heart of "The Proposition".  Charlie is told that if he kills his older brother Arthur, his younger brother Mikey will be spared.  Should Charlie cooperate, or defect from the deal?

Game theory allows us to look at the possible outcomes:

1. Charlie cooperates and Captain Stanley cooperates:  Arthur dies, and Mikey goes free.

2. Charlie defects, and Captain Stanley cooperates: Arthur lives, and Mikey goes free

3. Charlie cooperates, and Captain Stanley defects: Arthur dies, and Mikey is hanged

4. Charlie defects, and Captain Stanley defects: Arthur lives, and Mikey is hanged

From Charlie's point of view (comparing the outcomes where Charlie defects to the outcomes where he cooperates), outcome 2 is preferable to outcome 1, and outcome 4 is preferable to outcome 3.  Either way, he's better off defecting.

From Captain Stanley's point of view (comparing the outcomes where Captain Stanley defects to the outcomes where he cooperates), outcome 4 is preferable to outcome 2, and outcome 3 is preferable to outcome 1.  Either way, he's also better off defecting.

Game theory predicts that both will betray the deal, and that is in fact what happens.  Charlie, instead of killing Arthur, makes a pact with him to ride back and break out Mikey.  Eden Fletcher defects from the proposition over Captain Stanley's protests: he orders Mikey to be flogged, and Mikey dies of his injuries shortly thereafter.

The Prisoner's Dilemma is a frequently-used plot device throughout art and literature.  Puccini was very fond of it.  In Tosca, the corrupt police chief Scarpia offers Tosca this deal: if she sleeps with him, he will see that her lover Cavaradossi is spared from execution by ordering his soldiers to use blank bullets in their rifles.   Both end up defecting from the deal.  Tosca pretends to acquiesce, but stabs Scarpia to death as they lie in bed together.  Unbeknownst to Tosca, Scarpia has secretly double-crossed her and ordered his soldiers to use real bullets.  When she realizes this, Tosca jumps to her death off the ramparts of the castle.  (Amusing side note:  Sometimes, in staging this opera, a trampoline is positioned behind the set to break Tosca's fall.  It's not unusual to see Tosca come bouncing back up after jumping off the wall.)

If you've seen the movie A Beautiful Mind, you're probably somewhat familiar with game theory.  John Nash (played by Russell Crowe) made important contributions to game theory and mathematics. He is best known for "Nash equilibrium points": zero-sum game outcomes where no player, knowing with 20-20 hindsight which strategy the other person chose, would change the strategy they chose.  Nash equilibrium points represent an outcome towards which all players gravitate (even though, for the players, the individual payoffs may be less than desirable).

The Proposition beautifully illustrates a Nash equilibrium point.  Had Charlie known that Eden Fletcher was going to scotch the deal and flog Mikey to death, he still would have made the same choice to save Arthur.  Equally, Eden, had he known that Charlie was going to recruit his brother and ride back into town, would not have changed his behavior towards Mikey.  So the plot unfolds dharmically, like a Greek tragedy, along a foreordained path.  Mikey was doomed from the start.