• Nash equilibrium
• Chicken
• Arms races (Prisoner's Dilemma)
• Nuclear war (Chicken)

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.
• Thus a Nash equilibrium is "stable." If two players agree to that outcome, neither has an incentive to cheat.
• 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: Find all the Nash equilibria in Arbitrary Game, whose matrix is

  	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 Nash equilibria, if any.

The game of Chicken

• In Chicken (as seen in "Rebel without a Cause"), two players drive cars off a cliff.
• The first one to get out is the loser.
• The second one to get out is the winner...
• Unless he doesn't get out, in which case he's the big loser.
• [Video clip]
• We can model this as a two-by-two ordinal game if we make the following assumptions:
  • The options available to each player are: get out early (E), or stay in the car (S).
  • The rankings for each player are the same.
  • e.g. James Dean prefers most: he stays in the car and Buzz gets out early. (Then he still has time to get out and win.)
  • Next James Dean prefers both players getting out early. (It's a draw.)
  • Next James Dean prefers himself getting out early and Buzz staying. (Yes, he loses, and it's humiliating, but at least he survives.)
  • James Dean's last preference is if both players stay in their cars. (Both die.)
• A more common alternative is if both players drive their cars into each other, but this is equivalent in our model.
• Here is the matrix for Chicken, in this model.

  	E S
  E S	(3,3) (2,4) (4,2) (1,1)

• Question: is there a dominant strategy for either player?
• Question: is there a Nash equilibrium?
• Answers: No and yes. (On blackboard.)
• Nash equilibria are if one player stays and the other escapes.
• Compare to Prisoner's Dilemma, where Nash equilibrium is mutual noncooperation.
• The difference between Chicken and Prisoner's Dilemma:
  • Similarity: noncooperating (staying in the car) against a cooperating player is best.
  • Similarity: mutual cooperating is second best.
  • The difference:
    • In Prisoner's Dilemma, mutual noncooperating is third best (both get five years instead of one player getting ten years).
    • In Chicken, mutual noncooperating is worst (both die instead of both humiliated).
• Both games are used to represent political situations in which it might be in both countries' interests to cooperate, but individual greed makes them noncooperate (for a worse overall result).

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.

Example: nuclear war

• Now we change the question.
• Suppose there is a dispute in Kashmir.
• India threatens to go to war if Pakistan doesn't stop funding Kashmiri separatists.
• Pakistan threatens to go to war if India keeps repressing Kashmiris.
• The options for each country are to continue its policy (C) or yield (Y).
• Both sides know that nuclear war will result if neither backs down.
• What are the preferences?
  • India prefers continuing its policy while Pakistan yields (victory).
  • Next highest is both sides yielding (peace).
  • Next highest is Pakistan continuing and India yielding (defeat).
  • Last is both sides continuing (nuclear war).
• Pakistan's preferences are the same.
• Draw the matrix:

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

• This is the same game as Chicken.
• We know there are no dominant strategies.
• But the Nash equilibrium is for one side to continue its policy and for the other side to yield.
• So crises tend to get defused...?

More examples of game theory in politics

• Consider the current situation, U.S. vs. Iraq.
• U.S. has the option of either going to war (WAR) or not (PEACE).
• Iraq has the option of building weapons of mass destruction (WMD) or not (DISARM).
• Let's suppose: The U.S. prefers war over peace. If there is a war, the U.S. prefers Iraq to have weapons of mass destruction (to provide justification). If there is no war, the U.S. prefers Iraq not to have WMD.
• Let's also suppose: Iraq prefers peace over war. Without the threat of war, Iraq prefers not to have weapons of mass destruction (cheaper that way); but with the threat of war, Iraq does prefer WMD.
• Model this as a two-by-two ordinal game.
• Does either side have a dominant strategy?
• What are the Nash equilibria?
• (On board)
• U.S. has dominant strategy of war. Iraq has no dominant strategy.
• Nash equilibrium is that the U.S. fights a war and Iraq has weapons of mass destruction.
• Consider the Israel-Palestine conflict.
• Let's say Israel can choose to either build settlements (B) or dismantle settlements (D) in the West Bank and Gaza.
• The Palestinians can choose to fight the intifada (I) or negotiate peacefully (N).
• Suppose Israel prefers that the Palestinians negotiate rather than fighting the intifada, but otherwise prefers building settlements over dismantling.
• Suppose the Palestinians prefer that Israel dismantle settlements. If Israel dismantles the settlements, Palestinians would certainly prefer peaceful negotiations. If Israel builds settlements, Palestinians would prefer to fight.
• Model this situation as a two-by-two ordinal game.
• Are there any dominant strategies?
• What are the Nash equilibria?
• (On board)
• Israel has a dominant strategy of building settlements.
• The only Nash equilibrium is that Israel keeps building settlements and the Palestinians continue the intifada.

Larger ordinal games

Consider the following three-by-three ordinal game. Row and Column both have three options: "Yes," "No," or "Maybe."

	Yes No Maybe
Yes No Maybe	(1,6) (7,5) (5,3) (9,2) (3,8) (6,9) (2,1) (8,7) (4,4)

Find the dominant strategies, if any, and find the Nash equilibria, if any.

There are no dominant strategies. The Nash equilbria are at (No, Maybe) and (Maybe, No).