QR 38

3/13/07, Spatial models

I. Unidimensional models

II. Median voter theorem

III. Multiple dimensions

I. Unidimensional models

How to represent preferences and choices in a spatial context.

• Note that this is not strictly speaking an application of noncooperative game theory, but it is a useful tool.

Social choice and cycling

• The problem of social cycling: even if individuals’ preferences are transitive, the result of a social choice process might not be.

• This results in cycling; no policy is guaranteed a majority over all alternatives.

• This could often arise unless there are restrictions on preferences or the agenda.

Cuban missile crisis

Three options for the U.S. response to Soviet missiles:

1. Diplomacy2. Blockade3. InvasionThree groups of advisors:1. Hawks: I>D>B (JCS, Acheson)2. Doves: D>B>I (Stevenson)3. Statesmen: B>I>D (RFK, McNamara)

Cuban missile crisis• Assume that all three groups have

equal weight, and that the choice of policy is determined by majority vote (2/3).

• D would defeat B; I would defeat D; and B would defeat I.

• So a cycle develops with no clear winner.

Restriction on preferences: single-peaked

• Even though individual preferences are transitive, aggregate ones are not.

• To avoid this problem, we often place a simple restriction on preferences: that they are single-peaked.

One-dimensional spatial model

In order to show this decision problem spatially, need to specify the following things:

• How policy options are arrayed on one dimension (here, the use of military force)

• The ideal point for each faction– The ideal point is the most preferred

outcome

One-dimensional spatial models

• The utility for each outcome– Curves moving away from the ideal point

indicate how utility shifts as policy moves away from this point.

Cuban missile crisis

Aggressiveness

Utility

D B I

Doves

Hawks

Statesmen

Cuban missile crisisTo avoid the problem of cycling in social

choice, we have to impose some restrictions on preferences. Here, we impose the restriction of single-peaked preferences.

The “problem” with the Cuban missile crisis example is that the Hawks’ preferences are not single-peaked.

Negotiation example

Negotiation example from BdM.

• President (P) and leader of a foreign country (F) are negotiating with one another along a single dimension.

• If they don’t reach an agreement, the status quo (Q) prevails.

Negotiation example

• P is unsure of F’s ideal point, and relies on an Agent (A) to get information about this.

• A can make a proposal.• After seeing this proposal, P can

suggest what the negotiated outcome will be.

• If F accepts, the new outcome prevails.

Negotiation example

A3FPA1 A2 Q

F*

Negotiation example

• P will accept anything left of Q• F will accept anything between Q and F* (F*

is the same distance from F’s ideal point as Q is)

• A1 will propose F*, because anything further left will be rejected by F.

• But P will reject A1’s proposal and propose P instead.– He can infer from A1’s proposal that F is between

P and Q, and therefore F will accept P

Negotiation example

• A2 will propose his ideal point.

• P will accept this, because he can’t be sure that F will accept anything further left.

• A3 will propose Q.

• This provides little information to P, and he will propose either P or Q, depending on the costs of having a proposal rejected.

II. Median voter theorem• Previous example showed one way to use

ideal points to predict outcomes.

• Another important tool is the median voter theorem: if decisions are made by simple majority rule, the median voter always wins.

• The median is the voter in the middle: with the same number of voters to the right and to the left.

Median voter theorem

• Note that the median voter is not usually the same as the mean voter.

IncomeMeanMedian

Median voter example

• C is the median voter below

• A proposal at C will beat any other proposal, even x3

A B C D E

x1 x2 x3x4

Median voter theorem

• The median voter theorem applies to policy choice, the platforms of political parties, or the location of hot dog vendors on the beach.

• The preferences of the median voter are very important.

Median voter in IR• In IR, “votes” are usually weighted (for

example, consider voting in the UN or EU).

• So we have to take the number of votes each actor has into account as well as their ideal points.

• So outcomes are a function of both preferences and power.

Strategic voting

• We have assumed strategic voting: you vote for something other than your most preferred outcome, anticipating that this will lead to a better outcome for you in the end.

• This contrasts with sincere or naïve voting.

Strategic voting

• For example, if you expect your preferred outcome (Nader) to lose, but the other two options are tied, would strategically choose one of the other two.

• The median voter theorem holds even with strategic voting, as long as there is always a head-to-head choice between the median and all other proposals.

III. Multiple dimensionsLinked issues lead to multidimensional

decisions; issues often are not decided one at a time.

• For example, if deciding environmental policy, economic growth may be a concern

• Or military and economic policies could be linked.

Multiple dimensions

• To illustrate two linked issues, show two dimensions (x and y, horizontal and vertical)

• Show each actor’s ideal point in this two-dimensional space.

• Consider indifference curves: curves that indicate sets of points among which the actor is indifferent.

Indifference curves

• If the actor gives the same weight to both dimensions, indifference curves will be circular.

• The space contains an infinite number of indifference curves.

• The indifference curves that run through the status quo are especially interesting, because they show us the set of points that the actor prefers to the status quo.

Indifference curves

Guns

Butter A

Q

Indifference curves

• The circle that runs through Q and has an actor’s ideal point at the center shows the points that the actor prefers to Q – the interior of the circle.

• This is the actor’s preferred to set.

Win sets

• If moving policy away from the status quo requires the approval of two actors, the new policy must be in the overlap of their preferred-to sets.

• This overlap is called the win set: the set of points that can beat the status quo.

End of Cold War

Economic orientation

Centralized Market

Foreignpolicy

Dovish

Hawkish

G

L

Q

Y

End of Cold War example

• If agreeing on a policy requires 2 out of the 3 factions, 3 win-sets exist.

• The G-Y win-set is substantially larger than G-L or L-Y. So there are more points that G and Y could agree on.

• Policies within the G-Y win-set will involve more dovish foreign policy, but not necessarily more market orientation.

Trade bargaining example

U.S. tariffs

Low High

Foreigntariffs

Low

High

CP

F QQ’

T*

Trade bargaining example

• Assume that the president wants to reduce tariffs, but needs congressional approval.

• All actors prefer that the other country have zero tariffs; ideal points are on the axes.

• As long as tariff reductions are unilateral, the lowest tariff the president can get is Q’.

Tariff bargaining example

• Assume the president is able to reduce tariffs to Q’ before beginning foreign negotiations.

• Now there is scope for substantial reduction in both tariff levels – to a point in the win-set. Could go as low as T*.