Donald G. SaariInstitute for Mathematical Behavioral Sciences

University of California, Irvinedsaari@uci.edu

The surprising complexity of economics

Pricesp = (p1, p2)

Initial endowmentw = (w1, w2)

Demandx = (x1, x2)

Budget

afford (p,w) = p1w1+p2w2

cost (p,x) = p1x1+p2x2

Budget line; (p, x-w)=0

Rational agent:optimize utility

x*

Commodity 1

Commodity 2

Supply

DemandIndividual excess demand Ϛi( p) = Di( p) - Si( p)

Economics 101

Aggregate excess demand Ϛ( p) = Σ Ϛi( p)

Walras’ Laws

What are the properties of Ϛ( p)?

1. Ϛ( λp) = Ϛ( p)

2. Budget constraint (Ϛ( p), p) = 0

3. Ϛ( p) is continuous

What are the properties of Ϛ( p)?

Walras’ Laws 1. Ϛ( λp) = Ϛ( p)

2. Budget constraint (Ϛ( p), p) = 0

3. Ϛ( p) is continuous

“Invisible hand”

Sonnenschein

Ϛ( p) has a dynamical attractor

p1

p2 Does it?

Finding all properties of aggregate excess demand

Sonnenschein, Mantel, Debreu TheoremFor c≥2 commodities, a≥ c agents, and ε > 0, choose any f(

p) that satisfies Walras’ laws. There exists a nice pure exchange economy so that for pj ≥ ε, we have that f( p) =

Ϛ( p)

Scarf’s example

No other propertiesNot even “invisible hand”

Theory vs. reality?Charlie Plott

Why? How does this fit in with, say, voting theory?

Ϛ( p)

Ϛ( p)

x

Extensions; e.g., revealed preferences

Saari (1997) For c≥2 commodities, a≥ c agents, and ε > 0, for each subset C of two or more commodities

choose any fC( p) that satisfies Walras’ laws. There exists a nice pure exchange economy so that for pj ≥ ε, we

have that fC( p) = ϚC( p)

Idea coming from my voting theory results

3 A>C>D>B 2 C>B>D>A

6 A>D>C>B 5 C>D>B>A

3 B>C>D>A 2 D>B>C>A

5 B>D>C>A 4 D>C>B>A

X

OUTCOME: A>B>C>D

by 9: 8: 7: 6

X

Now: C>B>A

x

Now: D>C>B2 43 6

For economics, think of “substitutes”

All results from social choice, voting extend to economics

Dynamics?

* pn+1 = Ϛ( pn)

(Saari 1990?) For at least two commodities and at least as many agents as

commodities, there exists an open set of economies and an open set of initial conditions so that * not only never

converges to the price equilibrium, but it can be made to stay a distance away.

M( , …, Dk Ϛ( p),

…, Ϛ( ps), …, Dk Ϛ( ps))

n-body problem

Resolution?Help from Arrow’s Theorem!

Finite amount of market info does not work!!

Arrow

Inputs: Voter preferences are transitiveNo restrictions

Output: Societal ranking is transitive

Voting rule: Pareto: Everyone has same ranking of a pair, then that is the societal ranking

Binary independence (IIA): The societal ranking of a pair depends only on the voters’

relative ranking of pair

Conclusion: With three or more alternatives, rule

is a dictatorship

With Red wine, White wine, Beer, I prefer R>W.

Are my preferences transitive?

Cannot tell; need more information

Determining societal ranking

cannot use info thatvoters have transitive

preferences

Modify!!

You need to know my {R, B} and {W, B} rankings!

A>B, B>C implies A>C No voting rule is fair!

Borda 2, 1, 0

And transitivity

Think of this with price setting

Arrow’s dictator is a profile restriction!!

Ann Connie EllenBob David Fred

Science Soc. Science History

Vote for onefrom each

column

Three voters

Bob David Fred

2:1Representative

outcome?Ann, Connie, Ellen; Bob, Dave, FredBob, Dave, Fred

Ann, David, Ellen; Bob, Connie, Fred Bob, Dave, FredAnn, Connie, Fred; Bob, Dave, Ellen; Bob, Dave, Fred Ann, Dave, Fred; Bob, Connie,

Ellen; Bob, Dave, FredAnn, Dave, Fred; Bob, Connie, Fred; Bob, David, EllenOutlier: Pairwise vote not designed to

recognize any condition imposed among pairs

Five profiles

Wheaton College

Tommy Ratliff Public Choice

INCLUDING Transitivity!

2001, APSRwith K. Sieberg

Ethnic groups, etc., etc.

Ann Connie Ellen

Bob David Fred

Bob = A>B, Ann = B>A

B>A

A>B

Connie= C>B, Dave= B>C

C>B

B>C

Ellen = A>C, Fred = C>A

A>C

C>A

Ann, Dave, Fred; Bob, Connie, Fred; Bob, David, EllenB>C>A C>A>B A>B>C The Condorcet

triplet!

Mixed Gender =

Transitivity!!

Ann, Connie, Ellen; Bob, Dave, Fred; Bob, Dave, Fred2) A>B, B>C, C>A 1) B>A, C>B, A>CSo, “pairwise” forces certain profiles to

be treated as being cyclic!!also IIA, etc.

APSR, Sieberg, result--average of all profiles

Name change“Pairwise emphasis” severs intended connections

Lost information

Maybe a similar explanation holds for economics

Lost information, myopic emphasis!!

x*

Saari (1997) For c≥2 commodities, a≥ c agents, and ε > 0, for each subset C of two or more commodities choose any fC( p) that satisfies Walras’ laws. There

exists a nice pure exchange economy so that for pj ≥ ε, we have that fC( p) = ϚC( p)

and satisfies a bounded variation condition!

Dynamics? To a large extent remain, for reasons of local, myopic emphasis

rational agent

Reasons why economics and social sciences can be so complex can be found in

social choice and voting theory

Lost information!! Cannot see full symmetryFor a price, I will come to your department ....

10 A>B>C>D>E>F10 B>C>D>E>F>A10 C>D>E>F>A>B

D

E C B

A F

DC

BA

F

Mathematics?

16 2

5 3 4

A

F B

E C

D

Ranking Wheel

A>B>C>D>E>F

65 1

4 2 3

Rotate -60 degrees

B>C>D>E>F>A

C>D>E>F>A>B etc.

Symmetry: Z6 orbit

No candidate is favored: each is infirst, second, ... once.

All problems with pairwise comparisons due to Zn orbits

Coordinate direction!Yet, pairwise elections are cycles! 5:1

Pairwise majority voting

1 2 3

Core: Point that cannot be beaten by any other point

Core is widely used; e.g., median voter theorem

In one-dimensional setting, core always exists

Two issues or two dimensions?

Resembles an attractor from dynamics

No matter what you propose, somebody wants to “improve it.”

1

2

3

core does not exist

McKelvey: Can start anywhere and end up anywhere

Monica Tataru: Holds for q-rules; i.e., where q of the n votes are needed to win

Actual examples: MAA, Iraq

Salary

Hours

Tataru has upper and lower bounds on numberof steps needed to get from anywhere to anywhere else

Stronger rules?No matter what you propose, somebody wants to

“improve it.”

{1, 3}

Some Consequences:campaigning

negative campaigning:changing voters’ perception of

opponent

1

2

3

Positive

With McKelvey and Tataru, everything extends to any

number of voters

When does core exist?

Two natural questions

If not, what replaces the core?Generically

ˆ

McKelvey

Theorem: (Saari) A core exists generically for a q-rule if there are no more than 2q-n issues. (Actually, more general result

with utility functions, but this will suffice for today.)

Number of voters who must change their

minds to change the outcomeq=41, n=6019 on losing side, so need to persuade41-19 = 22 voters to change their votes

So this core persists up to 22 different issues

Saari, Math Monthly,

March 2004

Answered question when core exists generically.

Plottdiagram

Added stability

BanksAlways

q=6, n = 115 on losing side6-5=1 to change

vote

Proof by singularity

theory

Consequences of my theorem(All in book associated with lectures)

Single peaked conditionsfor majority rule

Essentially a single dimensional issue space

Generalization for q rules

Ideas of proofSingularity theory

Algebra: Number of equations, number of unknownsExtend to generalized inverse function theorem

Extend to “first order conditions”

Replacing the core

Core: point that cannot be beaten

Finesse point: point that minimizes what it takes to avoid being beaten

lens width, 2d, is sum of two radii minus distance between ideal points

All points on ellipse have samelens width of 2d

Define “d-finesse pt”in terms of ellipses

Ellipse: sum of distances is fixed

Predict what might happen?

d-finesse point is where all three d-ellipses meet

Generalizes to any number of voters, any number of issues and any q-rule

Minimizes what it takes torespond to any change -- d

For minimal winning coalition C, let C(d) be the Pareto Set for C and all d-ellipses for each pair of ideal points

Finesse point is a point in all C(d) sets, and d is the smallest value for which this is true.

Practical politics:incumbent advantage

The finesse point provides one practicalway to handle these problems

Most surely there are other, maybemuch better approaches

And, they are left for you to discover

But, the real message is the centrality of mathematics to understand crucial issues from

society

ArrowInputs: Voter preferences are transitive

No restrictions

Output: Societal ranking is transitive

Voting rule: Pareto: Everyone has same ranking of a pair, then that is the societal ranking

Binary independence (IIA): The societal ranking of a pair depends only on the voters’

relative ranking of pair

Conclusion: With three or more alternatives, rule

is a dictatorship

With Red wine, White wine, Beer, I prefer R>W.

Are my preferences transitive?

Cannot tell; need more information

Determining societal ranking

cannot use info thatvoters have transitive

preferences

Modify!!

You need to know my {R, B} and {W, B} rankings!

Lost information!! Cannot see full symmetryFor a price, I will come to your department ....

10 A>B>C>D>E>F10 B>C>D>E>F>A10 C>D>E>F>A>B

D

E C B

A F

DC

BA

F

Mathematics?

16 2

5 3 4

A

F B

E C

D

Ranking Wheel

A>B>C>D>E>F

65 1

4 2 3

Rotate -60 degrees

B>C>D>E>F>AC>D>E>F>A>B etc.

Symmetry: Z6 orbit

No candidate is favored: each is infirst, second, ... once. Yet, pairwiseelections are cycles! 5:1

All problems with pairwise comparisons due to Zn orbits

For a price ...I will come to your organization for your next election. You tell

me who you want to win. I will talk with everyone, and then design a “fair” election procedure. Your candidate will win.

10 A>B>C>D>E>F10 B>C>D>E>F>A10 C>D>E>F>A>B

Why??

Everyone prefers C, D, E, to

F

D

E C B

A F

DC

BA

F

F wins with 2/3 vote!!

Consensus?

Election outcomes need not represent what the voters want!