The Predictioneer’s Game A two-dimensional expected utility model.

The Predictioneer’s Game

A two-dimensional expected utility model

I. Preliminary Concepts

A. Terms1. Strategy: A decision to adopt a consistent set of

actions

2. Outcomes: What result from our strategies, the state of the world, and others’ strategies

3. Preferences: A comparative statement about the value of at least two outcomes. Players do not “prefer” strategies.

4. Preference notationa. Weak relation: O1 ≥ O2, or O1 R O2 (O1 is at least as good

as O2)

b. Strong relation: O1 > O2, or O1 P O2 (O1 is better than O2)

c. Indifference: O1 = O2, or O1 I O2 (O1 is as good as O2)

B. Rational Choice Theory

1. Assumes people are rational. Two conditions:

1. Preferences are connected (sometimes called “complete:” all alternative outcomes can be / have been compared to each other)

2. Preferences are transitive: If O1 > O2 and O2 > O3 then O1 > O3

2. Assumes that people choose the strategy that they believe will result in the best outcome (from their perspective)

C. Expected Utility TheoryRational choice theory, plus other assumptions1. Assumes that all preferences can be translated into subjective

utility (a measure of “attractiveness” as expressed by preference intensity)

2. Transforms ordinal preference information into interval preference information (cardinal utility) by imagining “lotteries” between the least-preferred and most-preferred outcomes.

3. Assumes that rational actors behave as if they are calculating the probabilities of different outcomes, assigning utilities to those outcomes, and then choosing the strategy that maximizes this “expected utility”

4. Does not assume that people actually calculate utility, or that utility “causes” preferences. Instead, utility is a way for researchers to transform existing preferences into interval data.

D. Expected Utility Notation

Sample equation:

Read: “The expected utility (EU) of action/strategy A is equal to the sum (∑), for each opposing strategy or state of the world that might obtain (S), of: The probability that S obtains (p) times the utility

of the outcome, where The outcome is the consequence (C) of both the

opponent’s strategy or state of the world (S) and the action (A)”

Another example

II. Game Theory FoundationsA. Game theory = formal way to represent strategic

interaction

B. Assumes rational choice, and some games assume expected utility theory

C. Two forms: 1. Normal (also known as strategic) form -- assumes simultaneous

choice.

2. Extensive form -- allows representation of sequence of choices, but can also represent simultaneous form

3. Therefore, extensive form always contains at least as much information as normal form

4. Nevertheless, when extensive and normal form offer the same information (single-shot, simultaneous choice under uncertainty about opponent’s choice) then normal is easier to use.

II. The Normal (Strategic Form)

A. Nash Equilibrium: How to make a prediction from a normal form game

1. Goal = Find an equilibrium (stable behavior, unlikely to change without change in conditions) outcome

2. Simplest tool = Nash Equilibrium Neither player could do any better by unilaterally changing its strategy choice

3. Equilibrium = Outcomes and associated Strategies played by each side (both behavior and its consequences are predicted)

B. Elements of a game in Normal (Strategic) Form1. Players – Note that these might be individuals or

organizations (like states)2. Strategies – The choices players have3. Outcomes – The results of the players’ choices4. Payoffs – How much each player values each

Outcome (can be ordinal or interval)

Player 2

Player1

Strategy A Strategy B

Strategy A

Outcome 1Player 1 Payoff, Player 2 Payoff

Outcome 2Player 1 Payoff, Player 2 Payoff

Strategy B

Outcome 3Player 1 Payoff, Player 2 Payoff

Outcome 4Player 1 Payoff, Player 2 Payoff

C. Algorithm (solution process) for Normal Form Games

1. Find Player 1’s best response to each of Player 2’s strategies

a. Assume column player (Player 2) plays its first strategy (leftmost column)

b. Examine row player’s payoffs given Player 2’s strategy (i.e. only the ones in the selected column). These are the first number in each cell.

c. Place a mark (+, for example) next to the maximum payoff(s) for Player 1 in that column.

d. If you have finished the rightmost column then proceed to (2). Otherwise, assume that Player 2 plays its next strategy (one column to the right of the one before) and follow steps (b) and (c) again.

2. Find Player 2’s best response to each of Player 1’s strategies

a. Assume row player (Player 1) plays its first strategy (topmost row)

b. Examine Player 2’s payoffs given Player 1’s strategy (i.e. only the ones in the selected row). These are the second number in each cell.

c. Place a mark (-, for example) next to the maximum payoff(s) for Player 2 in that row.

d. If you have finished the bottom row then proceed to (3). Otherwise, assume that Player 1 plays its next strategy (one row down from the one before) and follow steps (b) and (c) again.

3. Find the best responses to best responses

Look at each cell. Where you see both + and – in the same cell, that indicates that each player is doing the best it can, given what the other player is doing. This is a Nash equilibrium – each player playing its best response to the other’s strategy.

There may be zero, one, or multiple Nash equilibria, so check each cell.

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 15, -2 5, -1 -3,5 3, 3

Strategy B 3, 5 -2,0 10, 4 15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 15, -2 5, -1 -3,5 3, 3

Strategy B 3, 5 -2,0 10, 4 15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 15, -2 5, -1 -3,5 3, 3

Strategy B +3, 5 -2,0 10, 4 15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 15, -2 5, -1 -3,5 3, 3

Strategy B +3, 5 -2,0 10, 4 15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5 3, 3

Strategy B +3, 5 -2,0 10, 4 15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5 3, 3

Strategy B +3, 5 -2,0 10, 4 15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5 3, 3

Strategy B +3, 5 -2,0 +10, 4 15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5 3, 3

Strategy B +3, 5 -2,0 +10, 4 15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5 3, 3

Strategy B +3, 5 -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5 3, 3

Strategy B +3, 5 -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5 3, 3

Strategy B +3, 5 -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5 3, 3

Strategy B +3, 5 -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5- 3, 3

Strategy B +3, 5 -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5- 3, 3

Strategy B +3, 5 -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5- 3, 3

Strategy B +3, 5- -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5- 3, 3

Strategy B +3, 5- -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5- 3, 3

Strategy B +3, 5- -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10- 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5- 3, 3

Strategy B +3, 5- -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10- 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5- 3, 3

Strategy B +3, 5- -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10- 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5- 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5- 3, 3

Strategy B +3, 5- -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10- 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5- 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5- 3, 3

Strategy B +3, 5- -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10- 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5- 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8- 2, 3 4, 5

Strategy F 1, 5 2, -1 7, 0 0, 0 3, 1

Example

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5- 3, 3

Strategy B +3, 5- -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10- 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5- 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8- 2, 3 4, 5

Strategy F 1, 5- 2, -1 7, 0 0, 0 3, 1

Example: Unique Nash Equilibrium

Strategy A’ Strategy B’ Strategy C’ Strategy D’ Strategy E’

Strategy A 1, 1 +15, -2 5, -1 -3,5- 3, 3

Strategy B +3, 5- -2,0 +10, 4 +15, -4 2, 3

Strategy C 1, 10- 10, 2 6, 3 10, 5 0, 8

Strategy D 0, 3 1, 1 5, 5- 4, 2 +6, 4

Strategy E 2, 3 3, 7 8, 8- 2, 3 4, 5

Strategy F 1, 5- 2, -1 7, 0 0, 0 3, 1

IV. Games in Extensive Form: The TreeA. Extensive form adds information:

1. What is the order of moves? Example: “If you do this, then I will do that.”

2. What prior information does each player have when it makes its decision?

B. Elements1. Nodes – Points at which a player faces a choice2. Branches – Decision paths connecting a player’s

choices to the outcomes3. Information Sets – When a player doesn’t know

which node it is at4. Outcomes – Terminal nodes

C. Solving an Extensive Form Game

1. Subgame Perfect Equilibrium – A subset of Nash Equilibria -- eliminates “non-credible” threats from consideration

2. Process = Backwards induction – “If they think that we think…”

3. Algorithma. Begin with a terminal node

b. Identify the player that has the move at that terminal node.

c. Look at the player’s payoff (listed sequentially: Player 1, Player 2, Player 3, etc.) for each choice.

d. Select the one with the highest utility

e. Repeat for each such node

f. Proceed to the preceding branches leading to now-solved terminal nodes. Repeat (b) through (e). Continue the process until you reach the first node of the game, which can now be solved.

Example: A Simple Game of Deterrence

War

CapB

FSB

SQ

Preferences

A: CapB

SQ

War

FSB

B: SQ

FSB

War

CapB

Attack

Don’t

Attack

Nuke

Don’t

Nuke

Nuke

Don’t

Nuke

Deterrence Success!!!

Subgame Perfect

Equilibrium

What if preferences change? Credible Threat But No Restraint

War

CapB

FSB

SQ

Attack

Don’t

Attack

Nuke

Don’t

Nuke

Nuke

Don’t

Nuke

Deterrence Fails!!!

Preferences

A: CapB

SQ

War

FSB

B: FSB

SQ

War

CapB

Subgame Perfect

Equilibrium

War

CapB

FSB

SQ

Attack

Don’t

Attack

Nuke

Don’t

Nuke

Nuke

Don’t

Nuke

Deterrence Fails!!!

Preferences

A: CapB

SQ

War

FSB

B: SQ

FSB

CapB

War

Subgame Perfect

Equilibrium

What if preferences change? Restraint But No Credible Threat

V. A non-strategic model of choice

A. Black’s Median Voter Theorem (1948)1. Assumptions

a. Single Dimension (important one!)

b. “Single-peaked” preferences

Single-peaked preferences?

Red: Yes Blue: Yes Green: No!

When are single-peaked preferences an appropriate assumption?

I. A non-strategic model of choice

A. Black’s Median Voter Theorem (1948)1. Assumptions

a. Single Dimension (important one!)

b. “Single-peaked” preferences

c. Majority rule (more on this later…)

d. Odd number of decision-makers (trivial)

2. Conclusion: No other position can beat that of the median voter

Example

Who is the median voter?

What is the median voter’s position?

Don’t confuse median voter with moderate policy!

Actor Position

A 0

B 10

C 20

D 70

E 70

F 80

G 100

B. Expanding the model

1. We can eliminate assumptions (c) and (d) by adding power (aka potential influence) to the model. Majority rule just becomes a special case when

power is equal (one person, one vote). But how do we find the “median voter” now?

Example Who is the weighted

median voter? Sum power, then

divide by 2: 20+100+100+50+50+

100+10 = 430 Divide by 2 = 215

From either end, where does cumulative power reach 215?

Actor Position Power

A 0 20

B 10 100

C 20 100

D 70 50

E 70 50

F 80 100

G 100 10

2. Salience

We can further refine the prediction by taking into account that some actors have other items on their agendas

What proportion of their power (potential influence) are actors willing to spend on this issue?

General finding in politics: small, narrowly-focused groups often outperform large ones with broad goals

Example

Who is the weighted median voter now?

Need to find the median point of power * salience: Multiply, sum, then divide by 2:

Actor Pos. Power Salience

A 0 20 60%

B 10 100 25%

C 20 100 25%

D 70 50 20%

E 70 50 20%

F 80 100 50%

G 100 10 100%

Actual Influence = Potential Influence * Salience

Actor Pos. Power Salience EFFECT

A 0 20 60% 12

B 10 100 25% 25

C 20 100 25% 25

D 70 50 20% 10

E 70 50 20% 10

F 80 100 50% 50

G 100 10 100% 10

Actual Influence = Potential Influence * SalienceActor Pos. Power Salience EFFECT

A 0 20 60% 12

B 10 100 25% 25

C 20 100 25% 25

D 70 50 20% 10

E 70 50 20% 10

F 80 100 50% 50

G 100 10 100% 10

SUM 142

142 / 2 = 71. Who’s the WMV?Actor Pos. Power Salience EFFECT

A 0 20 60% 12

B 10 100 25% 25

C 20 100 25% 25

D 70 50 20% 10

E 70 50 20% 10

F 80 100 50% 40

G 100 10 100% 10

SUM 142

142 / 2 = 71. Who’s the WMV?Actor Pos. Power Salience EFFECT

A 0 20 60% 12

B 10 100 25% 25

C 20 100 25% 25

D 70 50 20% 10

E 70 50 20% 10

F 80 100 50% 40

G 100 10 100% 10

SUM 142

C. Limitations of the Weighted Median Voter model

1. Limited to one dimension (by assumption)

2. Assumes sincere voting – but what if being a winner is its own reward? “If you can’t beat ‘em, join ‘em.” This condition violates the single-issue assumption of Black…

3. Neglects coercion – What if actors are able to bully others into taking insincere positions?

4. Outcome is ambiguous -- The “vote” is an outcome, but does it result from acceptance by the players or war between them?

VI. Mean Voters and Multiple DimensionsA. What’s a “mean voter”?

1. Imagine that each actor “pushes” on the issue(s) (like a tug of war) in proportion to his/her power. How far does/do the issue(s) move?

2. Weighted Mean Voter Position (single issue) = Sum (Pos * Pow * Sal) / Sum (Pow * Sal)

Actor Pos. Pow Sal PPS

A 0 20 60% 0

B 10 100 25% 250

C 20 100 25% 500

D 70 50 20% 700

E 70 50 20% 700

F 80 100 50% 4000

G 100 10 100% 1000

SUM 7150

Weighted Mean Voter Position = 7150 / Sum (Pow * Sal)

Actor Pos. Pow Sal PS

A 0 20 60% 12

B 10 100 25% 25

C 20 100 25% 25

D 70 50 20% 10

E 70 50 20% 10

F 80 100 50% 50

G 100 10 100% 10

SUM 142

Weighted Mean Voter Position = 7150 / 142 = 50

Actor Pos. Pow Sal

A 0 20 60%

B 10 100 25%

C 20 100 25%

D 70 50 20%

E 70 50 20%

F 80 100 50%

G 100 10 100%

SUM

Note the large gap between the median voter’s position (70) and the mean voter’s position (50.4)

B. The Mean Voter Theorem

Caplin and Nalebuff, 1989: IF preferences single-peaked, concave, and linear

in parameters (Euclidian distance rule), THEN Voting rule needed to stabilize outcome is no more

than 1-[n/(n+1)]n

Two issues: Majority needed is no more than 1-[2/3]2 1-4/9 5/9 (55.6%)

Infinite issues: 64% rule creates unbeatable outcome

Caplin and Nalebuff, 1991: Relaxes concavity requirement

Ma and Weiss, 1993: Slightly relaxes linear utility requirement

C. Limits of the mean voter theorem

a. Much more stringent assumptions than the median voter theorem (not as generalizable) implies many exceptions to the “rule” (most work from late 1990s and 2000s)

b. While a 64% rule is always sufficient given the specified conditions, it has not been shown to be necessary (i.e. cannot assume supermajority rule is required for stability)

c. Still missing bargaining, the willingness to engage in risk-taking and potential conflict

VII. Adding Game Theory

A. Rational choice: preferences are connected and transitive

B. Expected utility: decision-makers base decisions on expected (rather than actual) payoffs

C. Rational expected utility maximizers interacting = a game. Each player plays best by anticipating behavior of opponent. Equilibrium = all are playing best given others’ play

D. Strategies: Two basic choices

1. What position to take (sincere or some alternative position)

2. Whether to threaten conflict against some or all participants by making demands/offers to them

E. Information

1. Information is imperfect: actors make their offers simultaneously (like in the PD)

2. Information is also incomplete: actors don’t know each other’s “true” preferences, so they have to have some rule for estimating them

F. Summary Actors believe that if everyone acts sincerely

and refrains from coercion, the weighted mean voter’s position (hereafter: WMV) will prevail

Actors do not know if others are making sincere proposals or will refrain from coercion (threats may or may not be credible)

Actors therefore make offers to each other designed to shift the expected outcome in their favor and/or to ensure they are part of the winning coalition

VIII. The Predictioneer’s Game

A. Dealing with incomplete information (private knowledge) – players use Bayes’ Rule (a probability function) to estimate opponents’ true positions from their previous behavior

B. Banks monotonicity theorem

1.Theorem: When information is incomplete (players have private information about their true expectations regarding conflict) then players who privately desire conflict (expect its utility to be high) will take more extreme bargaining positions

2. Implications

Together with Bayes’ Rule, this means we expect players who have taken positions far from the weighted mean to be willing to fight

Bueno de Mesquita calls these players “risk-acceptant” since they are obviously willing to gamble on getting “all or nothing” -- the conflict lottery of p(win = 1)+(1-p)(lose = 0)-c -- rather than taking a compromise position likely to get them something

C. Time

1. Why it is needed: Bayes’ Rule is about probabilities. But real preferences are 100% true. This means mistakes can be made. How can we allow players to correct others’

misconceptions?

Example Actor Pos.

A 0

B 10

C 20

D 70

E 70

F 80

G 100

SUM

Weighted Median Voter = D

What if A is not truly conflict-seeking?

What if we are wrong about…A?

There is a simple solution: A can shift positions to match the winning position (currently 70), thereby accepting the balance of power

Other players see this and conclude A is conflict-averse

But B and C are now more isolated, changing the expected utility of conflict for them.

Shows the need for multiple rounds, so players have the chance to avoid conflicts they believe they cannot win

ExampleActor Round 1 Round 2 Round 3

A 0 70 70

B 10 10 10

C 20 20 50

D 70 70 50

E 70 70 50

F 80 80 80

G 100 100 100

2. Discounting All else being equal, I would rather get my

way sooner rather than later This means that if we are very close to a

deal, I will probably just give in or offer to split the difference to avoid dragging things out

Discounting is important to prevent the model from continuing to infinity (the value of getting your way an infinite number of periods later is zero)

D. The Second Issue: Valence (Attractiveness)

1. Flexibility (or Resolve): This is essentially another dimension of preferences: how much do people care about a deal being made (claiming credit)? Mediators are usually at 100, successful politicians are usually 40-60, politicians with few constraints are often lower, and true ideologues are at zero.

2. Central insight = actors converge around the mean proposal, once its attractive force is accounted for

But what makes a proposal attractive, besides its policy content? People want to be winners – on the inside, not the outside

Bueno de Mesquita (April 12):

“What I tell my students and my consulting clients is to think of flexibility -- though not correct mathematically -- as the proportion of the issue scale a player is willing to contemplate moving in a single round. It is NOT the amount they will move but rather they will ignore proposals for greater movement as asking too much while giving added weight to proposals that ask them to move less, all else being equal.

Bueno de Mesquita (April 12)(Continued)

“Here is a useful (though asymmetric) analogy. Suppose you put a house on the market for $1 million. If someone offers you $900,000 you might respond to the offer with a counter (if you have sufficient flexibility) or you might blow them off, not responding at all (if you are sufficiently resolved currently about your asking price). If you have higher flexibility then you might consider a proposal as low as say $800,000. Still that is not saying you will accept it but rather that you will entertain it enough to make a counter-proposal.”

D. The Second Issue: Valence (Attractiveness)

1. Flexibility (or Resolve): This is essentially another dimension of preferences: how much do people care about a deal being made (claiming credit)? Mediators are usually at 100, successful politicians are usually 40-60, politicians with few constraints are often lower, and true ideologues are at zero.

2. Central insight = actors converge around the mean proposal, once its attractive force is accounted for

But what makes a proposal attractive, besides its policy content? People want to be winners – on the inside, not the outside

E. Sample Two-Dimensional Utility Function (Cobb-Douglas)

Z = Utility

X = Closeness to preferred policy

Y = Closeness to circle of winners

* Note that the surface is curved – could be used to represent risk aversion between issues)

F. Solution Process1. Start with data on (public) position, estimated

salience, power (influence), and flexibility/resolve

2. First round: Everyone sees the weighted mean as being the

outcome if nothing changes and everyone fights for his/her position

Each player makes its best offer (or no offer) to each player: choosing proposals that make the other players indifferent between imposing costs on the demander and a negotiated compromise. A negotiated compromise is always welfare-enhancing from the demander’s perspective relative to having costs imposed on it by the rival

3. Subsequent Rounds

Players shift position according to the weighted mean of credible proposals they receive

Note that the winning position can shift each round, since any shift alters the weighted mean

Many players start to converge on a few positions: the current winning position and a few alternatives where coalitions can form

Conflict results from failures of credibility (mis-estimating opponent’s costs or “types”)

During the Subsequent Rounds

Players’ positions change (as already described) Player’s influence can change (exercising power

or finding oneself gaining relative power) Players’ salience may change (can alter amount

of effort used based on expected utility of using more/less effort)

Players’ flexibility may change (if it is causing them to miss too much utility on the first dimension in exchange for gaining too little on the second dimension, or vice versa)

Bueno de Mesquita (April 12):“Each round, based on one's experience in the previous round, player flexibilities change within the model's logic. A house seller asking $1,000,000 might start off thinking they wouldn't take less than $950,000 (roughly, a flexibility of 5) but if they get no offers for sufficiently long (enough rounds) they might lower their price or be open to lower bids -- they have become more flexible.”

4. When does it end?

Sum of players’ payoffs at the end of an iteration is greater than the projected sum of those payoffs in the next iteration, indicating that the average player’s welfare is expected to decline in the sense of accumulated payoffs; OR

Sum of players’ utility, (including all games, even those which don’t involve them), is greater in the current round than the projected sum of utilities in the next iteration, indicating that the average player’s welfare is expected to decline in the sense of total utility.

G. Problems with the Predictioneer’s Game

Some elements are arbitrary: Priors set at .5 for all beliefs (hawk vs. dove, retaliatory vs.

pacific) Shape parameters of utility function are arbitrary (theta and beta

are .5 each) Forecast at any one round is “smoothed as the average of the

weighted means including the adjacent rounds just before and after the round in question”

Cannot represent risk-acceptance with Cobb-Douglas functions (although degree of concavity could represent risk-aversion)

Cost terms α, τ, γ, φ are governed by unpublished “heuristic rules”

Input variables shift according to unpublished “heuristic rules”

IX. Using the model to forecastA. Pick an issue. Note the importance of reducing

the issue to a one-dimensional scale. B. Identify the stakeholders (players)C. Create the issue scale, taking into account the

space between each positionD. Find players’ positions, power (influence),

salience, and flexibilityE. Note veto playersF. Type the data in Excel and save as tab-delimited

text (.txt) – be sure column names match the order and naming of the Zimbabwe example in the manual!

G. Input the data and run the modelH. Interpret the results

The US Role in Afghanistan

A Sample Policy Forecast

Data: Note that new model disallows Salience = 100 and Flexibility = 0

Player Influence Position Salience FlexibilityalQaeda 8 0 90 5Taliban 26 0 95 10Cdoves 20 10 40 30Pakistan 40 60 80 75Afghanistan 100 75 85 50France 15 80 40 70Germany 16 85 30 65Cmods 24 90 30 60UK 24 90 35 55Obama 92 95 65 60McChrystal 56 100 90 50Chawks 30 100 50 30

Rd 1 Rd 2 Rd 3 Rd 4 Rd 5 Rd 6 Rd 7 Rd 8

alQaedaTaliban

CdovesPakistan

AfghanistanFrance

GermanyCmods

UKObama

McChrystalChawks

Issue Forecast

0

10

20

30

40

50

60

70

80

90

100

Policy Forecast

alQaeda

Taliban

Cdoves

Pakistan

Afghanistan

France

Germany

Cmods

UK

Obama

McChrystal

Chawks

Issue Forecast

Policy Forecast: Issue Positions

Bueno de Mesquita answers questions

Daily Show appearance Interview with questions

X. Interpreting the OutputA. Issue Forecast: The obvious oneB. Level of conflict. Given as % of dyads each

round (equilibria from IIG). Could be used to forecast attacks or overall level of violence in military situations.

1. No Conflict: Players share same position2. Status Quo: Players leave each other alone3. Compromise: Players both move toward each other4. Coerce: One side gives in after suffering costs from

the other, or in anticipation of suffering such costs (Acquiescence or Capitulation)

5. Clash: The equivalent of war, wherein each side imposes costs on the other

Afghanistan Example: Conflict Becomes One-Sided

Rd 1 Rd 2 Rd 3 Rd 4 Rd 5 Rd 6 Rd 7 Rd 8

No_DisputeStatus_Quo

CompromiseCoerceClash

0

5

10

15

20

25

30

35

40

45

No_Dispute

Status_Quo

Compromise

Coerce

Clash

C. Policy positions by round

1. Use to make predictions by player, i.e. what position player X will adopt with respect to issue Y.

2. Use to identify players who remain opposed to the outcome

D. Coalition analysis

1. Look at pivotal players each round – these are the ones whose support is needed for the outcome to be an equilibrium of the game

2. See also the policy positions

E. Secondary forecasts (implications of issue forecast)

Parliamentary systems: If policy proposed by head of government fails, then government falls (look for final outcome far from leader’s policy)

Winners win: If a politician is forecasted to be on the winning side in issue after issue, can forecast that he/she is likely to gain higher office

F. Power, Salience, Flexibility

1. Model estimates remaining resources of each player, allowing one to identify winners and losers

2. Model estimates players who back away from dealing with the issue (lower salience) or choose being a winner over being right (higher flexibility)

Power by Round

0

5

10

15

20

25

30

35

alQae

da

Taliba

n

Cdove

s

Pakist

an

Afgha

nista

n

Franc

e

Germ

any

Cmod

sUK

Obam

a

McC

hrys

tal

Chawks

Player

Po

wer

Rd 1

Rd 2

Rd 3

Rd 4

Rd 5

Rd 6

Rd 7

Rd 8

XI. Another example: NATO Enlargement

A. The actors: All NATO members, Ukraine, Georgia, and Russia. Players with similar situations aggregated together (neighbors)

B. Issue Scale: Enlarge now to both Ukraine and Georgia (100), enlarge gradually or only to Ukraine (50), don’t enlarge (0)

C. Positions: Publicly available for all but four countries, whom analysts call “neutral” on the issue

D. Influence

1. Related to five factors: Votes in NATO EU membership Military spending Troop strength “Fudge factor” – country-specific weaknesses or

strengths (i.e. geography)

2. An Index of Influence

Each number normalized to a % of total (i.e. % of votes, % of spending) except EU status (which is yes/no)

Four components (all 0-1) weighted at 20% votes, 10% EU membership, 35% spending, 35% troops (whole numbers used to give a 0-100 index)

Then the raw index is adjusted:

Player Votes Prop EU 2008Mil$ Prop 2008Troops Prop RoughEst Notes FinalEstBaltics 3 0.11 1.00 1.4 0.00 20 0.00 12.42 Border Russia 6

Benelux 3 0.11 1.00 17.6 0.02 83 0.02 13.47 13.5Canada 1 0.04 1.00 19.4 0.02 55 0.01 11.85 11.9

Denmark 1 0.04 1.00 4.4 0.00 18 0.00 11.03 11EastEurope 7 0.26 0.71 15.0 0.02 300 0.06 15.06 15

France 1 0.04 1.00 66.2 0.07 347 0.07 15.70 15.7Georgia 0 0.00 0.00 2.5 0.00 17.5 0.00 0.22 0.2Germany 1 0.04 1.00 46.2 0.05 252 0.05 14.27 14.3Greece 1 0.04 1.00 10.0 0.01 134 0.03 12.08 12.1Hungary 1 0.04 1.00 1.9 0.00 19 0.00 10.95 10.9Iceland 1 0.04 0.00 0.0 0.00 0 0.00 0.74 Debts to EU members 0.3

Italy 1 0.04 1.00 30.4 0.03 195 0.04 13.28 13.3Norway 1 0.04 0.00 5.9 0.01 20 0.00 1.10 1.1Portugal 1 0.04 1.00 3.6 0.00 38 0.01 11.15 11.1Russia 0 0.00 0.00 58.6 0.06 1037 0.22 9.70 Influence from credibility 19.4Spain 1 0.04 1.00 19.0 0.02 129 0.03 12.38 12.4Turkey 1 0.04 0.00 13.3 0.01 496 0.10 4.84 4.8

UK 1 0.04 1.00 60.5 0.06 173 0.04 14.22 14Ukraine 0 0.00 0.00 2.0 0.00 148 0.03 1.15 Russian minority 0.7

USA 1 0.04 0.00 575.0 0.60 1326 0.28 31.51 31.5

E. Other variables

1. Salience defaults to 50 (one of many important issues), but is then increased or decreased to account for public statements or analysts’ perceptions of urgency (example: Georgia = 95, Turkey = 25)

2. Flexibility defaults to 50, but is lowered for Russia and some European powers, raised for some extremely vulnerable countries.

3. Veto players – Every NATO member has a veto. Unanimity rule controls admissions.

Player Influence Position Salience Flexibility VetoBaltics 6 100 60 50 1Benelux 13.5 0 50 60 1Canada 11.9 100 50 50 1Denmark 11 100 50 50 1EastEurope 15 100 60 50 1France 15.7 0 50 25 1Georgia 0.2 100 95 25 0Germany 14.3 0 75 40 1Greece 12.1 0 40 60 1Hungary 10.9 50 25 75 1Iceland 0.3 50 25 90 1Italy 13.3 0 50 50 1Norway 1.1 50 25 50 1Portugal 11.1 0 50 50 1Russia 19.4 0 85 5 0Spain 12.4 0 50 50 1Turkey 4.8 50 25 60 1UK 14 100 50 50 1Ukraine 0.7 75 90 50 0USA 31.5 100 50 50 1

F. Forecast

Your task: Interpret the results (HANDOUT)

XII. Other scholars’ use of the EUM: Did it work for them?

A. James and Lusztig (1996 using 1995 data): Forecasting Quebec-Canada relations

1. Constitutional bargaining will fail to meet Quebec’s demands. Quebec’s constitutional status is still unresolved.

2. If Quebec becomes independent, it will be admitted to NAFTA (against the opposition of Canada). Not yet testable.

XIII. Other scholars’ use of the EUMB. James and Lusztig (2000 using 1999 data):

Forecasting elements of an FTAA1. Congress will grant the President fast-track

authority, but with more limits than those Clinton requested in 1997. 2002: Congress grants even greater authority to Bush (> 100).

2. The FTAA will be formed de novo, rather than simply extended southward from NAFTA – but will still be dominated by the US. FTAA never formed, largely being replaced by agreements with individual countries or groups (CAFTA-DR).

C. Stokman and Thomson (1998 using 1997 data)

Forecasts Labour rejection of EMU even after election victory and previous statements of support

Labour rejected the EMU after winning elections

XIV. Assessing Model Accuracy

A. Must distinguish between different versions of the model: bargaining-free (weighted median voter), EUM (single issue with risk profiles), and PG (two issues with risk neutrality)

B. CIA study (in Feder 2002): 80 issues, more than 20 countries 90% accuracy for WMV model

From Feder (2002)

“During my government career, I used Bueno de Mesquita’s voting model on more than 1200 issues in more than 75 countries. Between 1982 and 1986, issues forecasted included the following (Feder 1995, p. 283): What policy is Egypt likely to adopt toward Israel? How fully will France participate in the Strategic

Defense Initiative? What is the Philippines likely to do about US bases? What policy will Beijing adopt toward Taiwan’s role in

the Asian Development Bank?”

C. Red Flag Over Hong Kong (1995)

11 of 12 forecasts were accurate (exception was land valuation)

Key predictions were about social and political rights (namely, that China would not respect Hong Kong’s special status)

D. Ray and Russett (1996): Largely focuses on EUM 1988 Prediction of defeat for Ortega in 1990

Nicaraguan elections (unclear if correct successor identified)

Feb 1989: Prediction of hard-line crackdown in China (pre-Tiananmen Square)

1989: Predicted “key features” of 1991 Cambodia Peace Accords

1991: Predicted admission of two Koreas to UN, failure of anti-Gorbachev coup

Other forecasts: oil prices, conditions of trade agreements, funding for family planning programs

E. European Union analysis

Compared “conflict model” (EUM) to “exchange model” (decision-makers trade positions when expected utility is positive) and other models

Model was accurate 97% of the time (but other models based on expert inputs also did quite well – especially the exchange model)

F. Labor Negotiations in the Netherlands (Rojer 1999)

XV. “What If?” analysis: Engineering Outcomes

A. Which proposals are credible? Check the credible proposals (read by row – row player proposes X to column player and column player believes that row player will back up the offer/threat). May show that some actors simply aren’t believed.

B. Where are the missed opportunities? Again, read by row – row player COULD have gained agreement with column player by making the offer and both sides would have gained something. Uses…

1. Counterfactual History Did players miss opportunities to end the

1914 crisis short of war? What if Chamberlain had stood up to Hitler

earlier? Would the threat have been credible?

What if Germany and Russia had renewed their alliance instead of following different banners?

What if the US had accepted disarmament proposals before the nuclear arms race?

2. Advice

Can advise actors of proposals that would achieve a less-bad outcome, perhaps even an ideal one.

Example: Environmentalists have their allies introduce a bill every year – only to see it die every year. What if…they make a proposal for half of what they want to the opposition (talking to enemies instead of allies)?

C. Scenario Analysis

Compares “base case” to alternatives Example: Compare “base case” outcome to

outcome that could be reached if one of the factions changes its policy, salience, or flexibility.

Example: Besançon (2003)

Used the model to predict results of 2002-2003 Round Table talks in Northern Ireland. Key issue = disarmament of militias

Advised the Northern Ireland Women’s Coalition (NIWC) on ways to make progress on the disarmament issue (NIWC wanted complete disarmament)

Base Case: 75 (not bad, but not 100 either)

NIWC takes a different position and devotes more of its resources to the issue

NIWC forms a coalition with PUP