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Transcript of Donald G. Saari Institute for Mathematical Behavioral Sciences University of California, Irvine...
Donald G. SaariInstitute for Mathematical Behavioral Sciences
University of California, [email protected]
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!