Review of two-by-two ordinal games

• Definition: A two-by-two ordinal game is a game where two players each have two options.
• All that matters is the preferences of each player.
• Any two-by-two ordinal game can be represented as a matrix.
• Example: the prisoner's dilemma as a two-by-two game.
• `C' denotes "cooperate," i.e. keep silent, while `N' denotes "non-cooperate," i.e. testify.

	C N
C N	(3,3) (1,4) (4,1) (2,2)

• A strategy for a two-by-two ordinal game is a choice of whether to cooperate or not.
• A dominant strategy for a player is a strategy which is better for that player than the other strategy, no matter what the opponent does.
• Example:
• In the Prisoner's Dilemma, prove that the strategy of testifying is dominant for both players (on the board).
• Example:
• In Arbitrary Game, prove that the row player has a dominant strategy of N, but that the column player has no dominant strategy (on the board). The matrix of Arbitrary Game is below.

  	C N
  C N	(1,3) (2,2) (3,1) (4,4)

• More examples:
  Generate four random two-by-two ordinal games and find the dominant strategies, if any.

Nash Equilibrium

• In a two-by-two ordinal game, a Nash equilibrium is a pair of strategies for both Row and Column such that:
  • If Row changes her strategy and Column keeps her strategy, Row will do worse.
  • If Column changes her strategy and Row keeps her strategy, Column will do worse.
• In a Nash equilibrium, neither player has any incentive to unilaterally change her strategy.
• However, the players might have an incentive to simultaneously change their strategies.
• Example: Show that in Prisoner's Dilemma, the only Nash equilibrium is (N,N), where both players testify.
• Notice that if both players switch from (N,N) to (C,C), it's better for both. But they would not do this sequentially.

Example: arms races

• Consider nuclear weapons development by India and Pakistan.
• Each has a choice: build nuclear weapons (B) or don't build nuclear weapons (D).
• Neither knows what the other is choosing.
• India's preferences (highest to lowest):
  • Have nukes, and Pakistan doesn't have nukes (B,D); India "wins"
  • Neither side has nukes (D,D); no threat => peace
  • Both sides have nukes (B,B); mutual assured destruction => "peace"
  • Not having nukes, with Pakistan having nukes (D,B); India "loses"
• Pakistan's preferences are obviously the same.
• Draw the matrix, with India as Row and Pakistan as Column.

• 	D B
  D B	(3,3) (1,4) (4,1) (2,2)

• Notice: this game is exactly the same as Prisoner's Dilemma.
• (Mathematical abstraction makes things easier!)
• So what happens?
• Both India and Pakistan have dominant strategies: noncooperation.
• So both countries will build nuclear weapons.
• So this model predicts the actual outcome.