The Theory of Moves

• Recall the definition of Nash equilibrium in a two-by-two ordinal game. It is a pair of strategies such that
  • Row will do worse by unilaterally changing her strategy if Column does not.
  • Column will also do worse by unilaterally changing her strategy.
• So if two players settle on a Nash equilibrium, neither has an incentive to cheat.
• However, we have seen that a Nash equilibrium may lead to an outcome that neither player likes.
• For example, the Nash equilibrium in Prisoner's Dilemma results in both players getting 5 years by testifying against each other, while a better outcome would be for both players to keep silent.
• Similarly, our model of the Israel-Palestine conflict (discussed on Thursday) has a Nash equilibrium which makes both sides unhappy, even though each side has a higher preference for peace.
• It seems that both sides have an incentive to change their minds once they've settled on a bad outcome.
• How can we model mathematically the notion that players would sometimes cooperate even though mutual cooperation is not a Nash equilibrium?
• Notice the following shortcoming of Nash equilibrium:
  • We allow one player to change her mind and then assume the result will be final.
  • What if the other player also wants to change her strategy in response?
  • Is it possible the two players could look at the mutual benefit and rationally decide to move toward it?
• This question can be answered using decision trees.
• We used them in Chapter 1 to model the possibilities for two players taking turns bidding in a dollar auction.
• Now let's use the same idea to model two players changing their strategies in an ordinal game.
• This idea is called the Theory of Moves (TOM).
• How does it work?
• Let's 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.

	N I
D B	(2,2) (4,1) (1,3) (3,4)

• Israel has a dominant strategy of building settlements.
• The only Nash equilibrium is that Israel keeps building settlements and the Palestinians continue the intifada.

• Suppose the two sides negotiate.
• Israel starts, the Palestinians respond, and the game continues until:
  • Either side gets its highest preference (e.g. Israel will stop if the Palestinians move so as to give Israel a 4).
  • OR either side decides to stay with its strategy with what the other has chosen.
  • OR the two sides move back to the starting strategy, in which case it's a draw. (Note: the "draw" stopping criterion is not in the textbook. So games in the textbook can theoretically continue indefinitely; see Exercise 5 in Chapter 7.)
• Suppose they start at the Nash equilibrium.
• Israel is building settlements, Palestinians are fighting an intifada.
• First choice: Israel can either stop building settlements or continue.
• Then the Palestinians can either stop the intifada or continue.
• Then Israel can either restart settlements or not...
• etc.
• But eventually all options will be exhausted.
• So we can draw the decision tree. (On the board)
• No we prune the decision tree, exactly as we would in the dollar auction. (On the board)
• Here is the result.

• The decision tree says that if the two sides are given the opportunity to negotiate, they will (according to this model) choose the compromise that Israel stops building settlements and then the Palestinians stop the intifada.
• Why doesn't this happen in the real situation? Several possibilities:
  • The model leaves out important choices.
  • The model gets the preferences wrong.
  • At least one side is not behaving rationally.
  • It did happen, but at least one side "cheated" by switching strategies.
  • The model is correct and both sides are behaving rationally, but the two sides do not know the other's preferences. (This is the possibility we'll discuss.)
• Since the Oslo agreement and other peace proposals do basically propose "land for peace" deals, the Theory of Moves does seem to predict the situation fairly well, at least at the simplest level of approximation.
• Your opinion may legitimately vary, however.
• Now suppose we change the starting point to (4,1).
• Israel is building settlements and Palestinians are at peace.
• Have the Palestinians start first instead of the Israelis.
• (On board)
• Here is the result.
• We get a different result: the ultimate outcome is (2,2).
• Settlements continue and the intifada continues.
• Does this model work?
• According to some, that's what happened before the 1999 intifada...
• Exercises:
  Put your name on it and hand it in. If you have questions, ask!
  • Work out the Theory of Moves prediction if the initial position is (1,3) and the Israelis start.
  • Work out the Theory of Moves prediction if the initial position is (2,2) and the Palestinians start.

• A different point of view:
  • Some Israelis believe Palestinians want to fight Israel even without settlements.
  • If this were true, then the above model could still work, but the preferences would change.
  • The Palestinians' preferences would be (highest to lowest):
    • 4: No settlements, intifada
    • 3: No settlements, peace
    • 2: Settlements, intifada
    • 1: Settlements, peace
  
  
  • 3 and 4 have switched.
  • Then the game becomes Prisoner's Dilemma. (Write matrix on board)
  • Theory of Moves then predicts that (2,2) is stable. (Read in Section 7.3)
  • So the war continues even if both sides have a chance to change their moves.
• Yet another point of view:
  • Some Palestinians believe that Israel cares more about the settlements than it does about peace.
  • If this were true, the preferences in the above model would change again.
  • Israel's preferences would be (highest to lowest):
    • 4: Settlements, peace
    • 3: Settlements, intifada
    • 2: No settlements, peace
    • 1: No settlements, intifada
  
  
  • Then the game changes to one we haven't seen before. (Write matrix on board)
  • If we do the Theory of Moves, we will see that the war outcome (3,2) is stable.
  • Here is the result.
• The interesting thing about this very simple model of the conflict is that it's actually pretty robust.
• By changing the preferences slightly, we can see it from different sides' point of view.
• This could explain why the original model's prediction (that both sides would move from war to peace) didn't work.
• The two sides did not have perfect knowledge about each other.
• Since they didn't know the other's true preferences, they could not plan out a rational strategy.
• So the conflict continues...
• Important note: all this is certainly subject to debate.
• Mathematics tells us what will happen given certain assumptions.
• In other words, if these are the preferences and these options are the only important ones, this is what must happen under these rules.
• One can argue about the assumptions, or about the interpretations, but not about the mathematics itself.