QR 38, 2/27/07Minimax and other pure

strategy equilibria

I. Minimax strategies

II. Cell-by-cell inspection

III. Three players

IV. Multiple equilibria or no equilibria

V. Cooperation, coordination, and competition in IR

I. Minimax strategies

• The minimax method of finding a Nash equilibrium applies only to zero-sum games.

• It relies on recognizing that whatever is good for one player is bad for the other.

• So, A will anticipate that B will choose the strategy that is worst for A.

Minimax strategies

• Knowing this, A should choose a strategy that provides the best of these bad outcomes.

• That is, A what to maximize the minimum payoff.

Boundary dispute game

• Defense is the column player• Offense is the row player• Choosing a military strategy• The number in each cell is row’s payoff;

only need to show this because the game is zero-sum.

• Row wants to maximize, column to minimize

Boundary dispute game

• So, for each player identify the worst payoff: this is the minimum number in each row, and the maximum number in each column.

Boundary dispute game

Ground Air Nuclear

Ground attack

2 5 13

Ground and air

6 5.6 10.5

Air 6 4.5 1

Nuclear 10 3 -2

Minimax method

• Offense knows that for each row, defense will choose the column with the lowest number in that row.

• So, offense should choose the row that gives the maximum among the minima; the maximin.

Minimax method

• Defense knows that for each column, offense will choose the row with the largest number.

• So defense should choose the column with the smallest number among the maxima; the minimax.

• If the maximin and the minimax values occur in the same cell, the outcome is a Nash equilibrium.

Minimax method

• The minimax method will arrive at the same equilibrium as using dominance criteria.

• But it is limited to 0-sum games.• This method may also fail to find an

equilibrium, either because none exists (in pure strategies) or because there is more than one.

II. Cell-by-cell inspection

• Cell-by-cell inspection always works to find all Nash equilibria in pure strategies.

• But it can be tedious.• Sometimes it is the only method that

works.• For each cell, ask whether either player

has a unilateral incentive to move.

OPEC game

• OPEC is the row player, Non-OPEC oil producers the column

• Each can choose to produce at a high, medium, or low level

• Highest profits if you produce at a higher level than others; but overproduction lowers profits.

OPEC game

High Medium Low

High 60, 60 36, 70 36, 35

Medium 70, 36 50, 50 30, 35

Low 35, 36 35, 30 25, 25

III. Three players

• Consider a 3-person PD

• Player 1 will choose row, 2 column

Three players

Player 3 chooses:

Cooperate Defect

C D

C 5, 5, 5 3, 6, 3

D 6, 3, 3 4, 4, 1

C D

C 3, 3, 6 1, 4, 4

D 4, 1, 4 2, 2, 2

IV. Multiple or no equilibria

• Multiple equilibria games are usually known in IR as coordination problems.

• If there is more than one Nash equilibrium, the players need to find a way to agree on which will prevail.

Coordination problems

• Can think of coordination problems as being differentiated by how much conflict of interest there is in the game.– Do all agree on which equilibrium is the

best?– Or do they disagree?

• Three important examples: Assurance, chicken, battle of the sexes

Assurance (or stag hunt)

Refrain Build

Refrain 4, 4 1, 3

Build 3, 1 2, 2

Assurance

• In assurance games, players should be able to get to their preferred outcome unless one is uncertain that the other really has assurance preferences.

• If this weren’t a concern, could just unilaterally disarm.

• Might solve with a focal point that leads to a convergence of expectations.

Chicken

Back down Fight

Back down 0, 0 -1, 1

Fight 1, -1 -2, -2

Chicken

• Here, the player who can make a commitment first will win – for example, throwing the steering wheel out of the window.

• But if both make commitments, leads to disaster.

Battle of the sexes (or two cultures)A B

A 2, 1 0, 0

B 0, 0 1, 2

No Nash equilibrium in pure strategies

• Sometimes no matter which outcome is chosen, one player has an incentive to move away from it

• This means there is no Nash equilibrium in pure strategies.

• The essence of such a game is that there is a benefit to being unpredictable.

No Nash equilibrium in pure strategies

A B

A 2, 3 3, 2

B 4, 1 1, 4

No Nash equilibrium in pure strategies

• Games like this might arise between a defender and an attacker; if the defender knows what the attacker is going to do, can take advantage of it.

• So the solution is to be unpredictable; to choose moves randomly.

No Nash equilibrium in pure strategies

• To play randomly, don’t pick any move with certainty, but have a probability distribution over the possible moves.

• This is called a mixed strategy.

• Schelling: in situations like this, want to keep your opponent guessing; this is more likely true in zero-sum games.

V. Cooperation, coordination, and competition in international

politics• Simple 2x2 games like these have been

used extensively in the study of IR.

• Basic problems of IR are: cooperation, coordination, and competition.

• When does each occur?

• What kind of framework allows solution of each?

Strategic form games and IR• Example of an important argument: the

type of international framework needed to deal with particular issues depends on the underlying game being played.

• In PD, players have a common interest in cooperation but incentives to defect.

• This is a typical collective-goods problem; also used for trade, arms control.

Strategic form games and IR

• So, in a PD, need a framework to provide monitoring and sanctioning.

• This can be done with international institutions; they can facilitate collaboration as the state does on the domestic level.

Strategic form games and IR

• Coordination problems mean that some outcomes are disliked by all actors.

• But there is a multiple equilibrium problem, and possibly disagreement on which equilibrium is preferred.

• Here, a regime would only need to identify a focal point; no need for enforcement.

Strategic form games and IR

• This use of games to generate predictions about the type of regime that states will create is an example of what BdM calls a structural theory.

• Domestic politics doesn’t matter much; states treated as the units of analysis.

• But not neorealist.

Strategic form games and IR

• So, this approach is more like what BdM calls liberal theory, since the focus is on maximizing wealth, the role of regimes, and collective action problems.

• But, BdM wrong to claim that liberal theory assumes hierarchy.

• Anarchy is just lack of external enforcement.

Strategic form games and IR

• If anarchy is lack of external enforcement, meets the assumptions of non-cooperative game theory.

• The problem is to search for equilibria that are self-enforcing.

• BdM brings in repetition and information; will get to these.

Strategic form games and IR

• Can also use simple games to generate predictions about likelihood for conflict or cooperation.

• What types of games are most likely to give rise to recurrent conflict? To cooperation? How do games map onto different issue-areas?

Strategic form games and IR

• Schelling introduces communication.• Seems to use strategic-form games, but

discussion is about sequential moves.• Strategic moves: the use of promises or

threats.• Major point: the capacity to

communicate (or not) is essential to how a game will be played.

Communication

• To really understand the role of communication, would have to explicitly add moves that allow for communication; analysis of this is a supergame.

• Paradox arises: players are often better off if they can’t receive messages.