Game Theory

Game theory analyzes rational strategies in multi-party situations.  Some games have a single equilibrium solution; others may have no equilibrium or multiple equilibria.  A real-world “game” with very high stakes is OPEC’s pricing and production quotas: if all members adhere to their quotas, oil prices remain high and they all benefit.  But the high prices give each OPEC member a strong incentive to sneak extra oil into the market.  This increases short-run profits for the cheaters, but reduces oil prices and long-term profits for everyone. 

The best-known strategy "game" is the "prisoners’ dilemma:" The police have arrested two suspects after a bank robbery and are interrogating them separately.  If they can both keep silent, they will both go free.  If one suspect confesses and helps convict the other, he will get 1 year in prison while his accomplice will get 10 years.  If they both confess, they each get 5 years.  The optimal outcome would be for both suspects to hold out, but since neither suspect can be sure the other isn't ratting him out, there is a strong incentive to confess, so the likeliest outcome is for both of them to plead guilty and serve 5 years each.   The strategy game can be represented in a payoff matrix.  Here the cell values represent prison terms for A and B respectively:

Prisoner's Dilemma	B keeps silent	B confesses
A keeps silent	Both go free	10 years for A; 1 year for B
A confesses	1 year for A; 10 years for B	Both get 5 years

Confessing is a "max-min" strategy: assuming the other party will choose the strategy that includes the worst possible outcome for you, you should choose the strategy that yields the best of these bad outcomes for you. 

A similar problem is the "chicken game" where two idiots drive straight at each other on a road and see who swerves first:

Chicken	B swerves	B doesn't swerve
A swerves	Draw; both live	B calls A chicken
A doesn't swerve	A calls B chicken	Both die

Or a couple chooses between a football game (his preference) or a skating exhibition (her preference) when it's no fun being apart.  The payoffs are his and her utilities.

Football or Skating	she attends skating	she attends football
he attends skating	1,5	0,0
he attends football	0,0	5,1

In general, cooperation is likely to fail any time either party has sufficient incentive to defect from the cooperative arrangement.  Let $C be the reward for Cooperation; $T be the Temptation to defect; $S be the Sucker’s reward for cooperating and getting betrayed; and $N be the payoff for Non-Cooperation:

Cooperate or Defect	B cooperates	B defects
A cooperates	$C,$C	$S,$T
A defects	$T,$S	$N,$N

A “Nash equilibrium” is when, once the players have chosen their strategies, no player can benefit by changing his or her own strategy when the other players keep theirs unchanged.  In general, the non-optimal solution $N,$N will be an equilibrium when $T > $C > $N > $S. 

Games can be expanded in various dimensions: more players, more strategies per player, differential payoffs (e.g., $C_A < $C_B); multiple iterations of the game from which players can learn from each others’ strategies, etc. 

In iterative games, mutual cooperation may emerge as the dominant solution when $C + $C > $T + $S, even though it is not a single-game Nash equilibrium.  Defection can serve as retaliation for another player’s defection in a previous round.  In these cases, the typical winning iterative-game strategy is to cooperate, using defection only as retaliation against another player’s prior defection.

Here’s a game with no equilibrium:

Cooperate or Defect	B cooperates	B defects
A cooperates	$4,$3	$1,$4
A defects	$3,$2	$2,$1

Mutual cooperation is Pareto-optimal, but B has an incentive to defect, which causes A to defect, which causes B to cooperate, which causes A to cooperate, etc.: the game cycles clockwise.