The surprising complexity of economics
Donald G. Saari
Institute for Mathematical Behavioral Sciences
University of California, Irvine
dsaari@uci.edu
Economics 101
Commodity 2
Rational agent:
optimize utility
x*
Supply
Prices
p = (p1, p2)
Demand
x = (x1, x2)
Initial endowment
w = (w1, w2)
Budget
afford (p,w) = p1w1+p2w2
cost (p,x) = p1x1+p2x2
Budget line; (p, x-w)=0
Commodity 1
Demand
Individual excess demand Ϛi( p) = Di( p) - Si( p)
Aggregate excess demand Ϛ( p) = Σ Ϛi( p)
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
Sonnenschein
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
Ϛ( p) has a dynamical attractor
p2
Does it?
p1
“Invisible hand”
Finding all properties of aggregate excess demand
Sonnenschein, Mantel, Debreu Theorem
For 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)
No other properties
Not even “invisible hand”
Why? How does this fit in with, say, voting theory?
Scarf’s example
Theory vs. reality?
Charlie Plott
Ϛ( p)
Ϛ( p)
Extensions; e.g., revealed preferences
Idea coming from my voting theory results
All results from social choice,
For economics,
voting extend to economics
think of “substitutes”
3
x
A>C>D>B
2
C>B>D>A
6
A>D>C>B
x
5
C>D>B>A
3
B>C>D>A
2
X
D>B>C>A
5
B>D>C>A
4
X
D>C>B>A
Now: C>B>A
OUTCOME: A>B>C>D
Now: D>C>B
by
9: 8: 7:
2 3
4 6
6
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)
Dynamics?
* pn+1 =
M(
Ϛ( pn) , …, Dk Ϛ( p),
…, Ϛ( ps), …, Dk Ϛ( ps))
Finite amount of market info does not work!!
(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.
n-body problem
Resolution?
Help from Arrow’s Theorem!
Think of this with price setting
Arrow
A>B, B>C implies A>C
Arrow’s dictator is a profile
restriction!!
No voting rule is fair!
Inputs: Voter preferences are transitive
No restrictions
Conclusion: With three or
more alternatives, rule
is a dictatorship
Output: Societal ranking is transitive
cannot use info that
Voting rule:Pareto: Everyone has same
voters have transitive
ranking of a pair, then that is the societal ranking
preferences
Binary independence (IIA): The societal ranking of a
Borda
2,
1,
0
pair depends only on the voters’ relative ranking of
pair
And transitivity
Modify!!
With Red wine, White wine, Beer, I prefer R>W.
Are my preferences transitive?
Cannot tell; need more information
You need to know my {R, B} and {W, B} rankings!
Determining societal ranking
Science Soc. Science History
Ann
Bob
Connie
David
Ellen
Fred
Three voters
Wheaton
Vote
for one
College
from
each
Tommy Ratliff
column
Public
Choice
Bob
David
Fred
Representative
Five profiles
2:1
outcome?
Ann, Connie, Ellen; Bob, Dave, Fred Bob, Dave, Fred
Ann, David, Ellen; Bob, Connie, FredBob, Dave, Fred
Ann, Connie, Fred; Bob, Dave, Ellen; Bob, Dave,
Fred Ann, Dave, Fred; Bob, Connie, Ellen; Bob,
Dave,Connie,
Fred Fred; Bob, David,
Ann, Dave, Fred; Bob,
Outlier: Pairwise
vote not designed to
Ellen
2001, APSR
recognize any condition imposed among
with K. Sieberg
pairsEthnic groups, etc., etc.
INCLUDING Transitivity!
“Pairwise emphasis” severs intended connections
Name change
B>A
Ann
Connie
C>B
Ellen
A>C
Mixed Gender
A>B
Bob
David
B>C
Fred
C>A
=
Lost information
Transitivity!!
Bob = A>B, Ann = B>A Connie= C>B, Dave= B>C
APSR, Sieberg, result-average of all profiles
Ellen = A>C, Fred = C>A
nn, Dave, Fred; Bob, Connie, Fred; Bob, David, Ellen
B>C>A C>A>B A>B>C The Condorcet triplet!
nn, Connie, Ellen; Bob, Dave, Fred; Bob, Dave, Fred
2) A>B, B>C, C>A 1) B>A, C>B, A>C
So, “pairwise” forces certain profiles to be
also IIA, etc. treated as being cyclic!!
Maybe a similar explanation holds for economics
Lost information, myopic emphasis!!
Reasons why economics and
social sciences can be so
complex can be found in social
choice and voting theory
rational agent
x*
and satisfies a bounded variation
condition!
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)
Dynamics?
To a large extent remain, for reasons of local, myopic emphasis
Lost information!! Cannot see full symmetry
For aproblems
price, I will
come
to yourcomparisons
department ....due to Zn orbits
All
with
pairwise
Mathematics?
10 A>B>C>D>E>F
10 B>C>D>E>F>A
10 C>D>E>F>A>B
Rotate -60 degrees F 6
5
Symmetry: Z6 orbit
D
E
D
C
E
A
B
A
F
F
No candidate is favored: each is in
first, second, ... once.
Yet, pairwise elections
are cycles! 5:1
A
61
34
B
12
23
C
D
B
C
45
Ranking Wheel
A>B>C>D>E>F
B>C>D>E>F>A
C>D>E>F>A>B
etc.
Coordinate direction!
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
Resembles an attractor from dynamics
In one-dimensional setting, core always exists
Two issues or two dimensions?
No matter what you propose, somebody wants to
“improve it.”
Hours
3
{1, 3}
2
1
Salary
core does not exist
McKelvey: Can start anywhere and end up anywhere
Actual examples: MAA, Iraq
Stronger rules?
No matter
whatHolds
you propose,
somebody
wants
Monica
Tataru:
for q-rules;
i.e., where
q oftothe n
votes
are needed
“improve
it.” to win
Tataru has upper and lower bounds on number
of steps needed to get from anywhere to anywhere else
Some Consequences:
campaigning
negative campaigning:
changing voters’ perception of opponent
Positive
3
2
1
With McKelvey and Tataru,
everything extends to any
number of voters
Two natural questions
Generically
When does core ˆ exist?
Always
Plott
diagram
McKelvey
q=6, n = 11
5 on losing side
6-5=1 to change
vote
Saari, Math
Monthly, March
2004
Proof by
singularity
theory
If not, what replaces the core?
Banks
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 outcome
q=41, n=60 19 on losing side, so need to persuade
41-19 = 22 voters to change their votes
So this core persists up to
22 different issues
Added stability
Answered question when core exists generically.
Consequences of my theorem
(All in book associated with lectures)
Single peaked conditions
for majority rule
Essentially a single dimensional issue space
Generalization for q rules
Ideas of proof
Singularity theory
Algebra: Number of equations, number of unknowns
Extend to generalized inverse function theorem
Extend to “first order conditions”
Replacing the core
Core: point that cannot be beaten
Predict what
might happen?
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 same
Ellipse: sum of distances is fixed lens width of 2d
Define “d-finesse pt”
in terms of ellipses
Minimizes what it takes to
respond to any change -- d
Practical politics:
incumbent advantage
d-finesse point is where all three d-ellipses meet
Generalizes to any number of voters, any number of issues and
any q-rule
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.
The finesse point provides one practical
way to handle these problems
Most surely there are other, maybe
much better approaches
But, the real message is the centrality of mathematics to
understand crucial issues from society
And, they are left for you to discover
Arrow
Inputs: Voter preferences are transitive
No restrictions
Conclusion: With three or
more alternatives, rule
is a dictatorship
Output: Societal ranking is transitive
cannot use info that
Voting rule:Pareto: Everyone has same
voters have transitive
preferences
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
Modify!!
With Red wine, White wine, Beer, I prefer R>W.
Are my preferences transitive?
Cannot tell; need more information
You need to know my {R, B} and {W, B} rankings!
Determining societal ranking
Lost information!! Cannot see full symmetry
For a All
price,problems
I will come
to your
department
....
with
pairwise
comparisons
due to
Mathematics?
10 A>B>C>D>E>F
10 B>C>D>E>F>A
10 C>D>E>F>A>B
Rotate -60 degrees F 6
5
Symmetry: Z6 orbit
D
E
D
C
E
A
B
A
F
A
61
34
12
23
D
B
C
45
Ranking Wheel
F
No candidate is favored: each is in
first, second, ... once. Yet, pairwise
elections are cycles! 5:1
A>B>C>D>E>F
B>C>D>E>F>A
C>D>E>F>A>B
etc.
B
C
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>F
10 B>C>D>E>F>A
10 C>D>E>F>A>B
D
D
E
C
C
B
B
A
A
F
Everyone prefers
C, D, E, to F
F
F wins with 2/3 vote!!
Election outcomes need not represent what the voters want!
Why??
Consensus?