Extending Arrow’s Theorem to --- just about everything

Multi scale analysis; in particular for systems:How are activities of systems at different scales connected?

Basic issue in the social and behavioral sciences

EconomicsEngineering

Physical sciencesComplexity

History: NSF, nano-technology

Specialization, understand behavior of

individual partsuse experts,

use physical laws, or other info to find

behavior of individual parts

Don Saari, Institute for Math Behavioral SciencesUniversity of California, Irvine, CA 92697-5100

dsaari@uci.edu

Examples:

Astronomy; e.g., galaxy: find mass, rotational velocity,

luminosityNano-technology:

Find behavior of various parts

Consequences:

Serious incompatibilities:multiples of 50 to 100 off

source of dark matter

Biological systems have the first level of organization at the nanoscale. Proteins, DNA, RNA, ion channels are nanoscale systems that leverage molecular interactions to perform specific tasks. Integrated nano-bio systems have emerged as strong candidates for single molecule detection, genomic sequencing, and the harnessing of naturally occurring biomotors. Design of integrated nano-bio devices can benefit from simulation, just as the design of microfluidic devices have benefited. Currently a large stumbling block is the lack of simulation methods capable of handling nanoscale physics, device level physics, and the coupling of the two.

Complexity?or wrong methodology?

Engineering: Division of labor, experts, develop

parts, optimize

Reality:Cannot handle

connections between different levels of systems

Inefficiency, mistakesNIAAA

Arrow’s TheoremInputs: Voter preferences are transitive

No restrictionsOutput: 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

No voting rule is fair!

Notice: Emphasis on “parts,”

Simple examples of multiscale systemsGroup decisions;

voting, etc.

Also notice, “whole” exists:Plurality, Borda rankings, etc.

Alternative conclusion:

With three or more alternatives, if one

voter does not make all of the decisions,

then there must exist situations

where the “whole need not resemble the collection of the

parts”

Maybe this interpretation extends;maybe for most systems, there exist

settings where “whole need not resemble parts”

part-wholeconflict

This is the case

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

me who you want to win. After talking with everyone, I will design a “fair” election rule.

Your candidate will win.10 A>B>C>D>E>F10 B>C>D>E>F>A10 C>D>E>F>A>B

Decision for each part is “optimal”Outcome is inferior

Multiscale outcomes can be inferior when built on parts

Everyone prefers C, D, E, to F

D

E C B

A F

DC

BA

F

F wins with 2/3 vote!!A landslide victory!!

Consequences:

Evidence “strongly” supports conclusion of F!

i.e., with “parts”, expect path dependency

Developing the result

Arrow Inputs: Voter preferences are

complete & transitive. No restrictions

Arrow’s Output: Societal ranking is complete,

transitive

Macro system

Micro

three compatibility conditions:Mine: 1) All elements are needed

2) some combinations are not compatible

3) compensativeMine: Structure satisfies compatibility conditions: 1) all

elements are needed, 2) some combinations are not compatible, 3) compensative

Pareto: If all participants agree on the feature for some component, then that is the

macro outcomeIndependence: The outcome for each component depends only on each participant’s

input for that component

Conclusion: with systems having

three or more parts, there exist settings where “whole does

not resemble collection of parts”

One of severalmass velocity luminositydesign manufacturing sales

Might define outcome, but an inferior one!

(Path dependency)

Each “participant” selects compatible input, no restriction

Consequences:Nano-technology

conferenceAstronomy; dark matter:

Madrid

Resolution? To do so, first have to handle and resolve

the “easier” Arrow’s Theorem. Inputs: Voter preferences are transitive

No restrictions

Output: Societal ranking is transitive Rule: Pareto: Everyone has

same ranking of a pair, then that is the societal rankingBinary 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 rankingYou need to know my {R, B} and {W, B} rankings!

cannot use info thatvoters have transitive

preferences

Modify!! And transitivity Borda 2, 1, 0

Coordinated symmetry

Illustrate: solve voting problem

Lost information: about “connections”Here, “connections” means transitivity

A>B, B>C implies A>C

Cambridge University press

Consequences -- so far:

Engineering:Manufacturing

Conference

Statistics: NonparametricKruskal-Wallis

Voting:Understanding all voting paradoxes--

connectionsAstronomy:Partial results (paper on dark matter)

Work in progress:Approach for all areas now is

understood; must be developed

Complexity--mainly multiscale problems

International relationsevolutionary game theory

addiction? maybeetc., etc., etc.

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, Ellen

Mixed gender!

Outlier: Pairwise vote not designed to recognize any condition imposed

among pairs

Five profiles

INCLUDING Transitivity!

2001, APSRwith K. Sieberg Ethnic groups, etc.,

etc.

Ann

Bob

Connie

David

Ellen

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, Fred

2) A>B, B>C, C>A 1) B>A, C>B, A>C

So, “pairwise” forces certain profiles to be treated as being cyclic!!also IIA, etc.

APSR, Saari-Sieberg, result--average of all profiles

Name change“Pairwise emphasis” severs intended connections

Lost information

Sen, etc.

Z3