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 Economics Engineering Physical sciences Complexity History: NSF, nano-technology Specialization, understand behavior of individual parts use experts, use physical laws, or other info to find behavior of individual parts Don Saari, Institute for Math Behavioral Sciences University of California, Irvine, CA

Examples: Astronomy; e.g., galaxy: find mass, rotational velocity, luminosity Nano-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, mistakes NIAAA

Arrow’s Theorem Inputs: 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 No voting rule is fair! Notice: Emphasis on “parts,” Simple examples of multiscale systems Group 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-whole conflict 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>F 10 B>C>D>E>F>A 10 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 D C B A 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) compensative Message: Beware; evidence may appear to provide overwhelming support about the existence of a connection, a result, etc., yet it can be wrong Positive results are being developed Mine: 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 outcome Independence: 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 several mass velocity luminosity design manufacturing sales Might define outcome, but an inferior one! (Path dependency) Each “participant” selects compatible input, no restriction

Consequences: Nano-technology conference Astronomy; 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 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 You need to know my {R, B} and {W, B} rankings! cannot use info that voters 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: Nonparametric Kruskal-Wallis Voting: Understanding all voting paradoxes--connections Astronomy: Partial results (paper on dark matter) Work in progress: Approach for all areas now is understood; must be developed Complexity--mainly multiscale problems International relations evolutionary game theory addiction? maybe etc., etc., etc.

AnnConnieEllen BobDavidFred ScienceSoc. ScienceHistory Vote for one from each column Three voters Bob David Fred 2:1 Representative outcome? Ann, Connie, Ellen; Bob, Dave, Fred Bob, Dave, Fred Ann, David, Ellen; Bob, Connie, Fred Bob, Dave, Fred Ann, Connie, Fred; Bob, Dave, Ellen; Bob, Dave, Fred Ann, Dave, Fred; Bob, Connie, Ellen; Bob, Dave, Fred Ann, 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, APSR with 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, Ellen B>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. Z3Z3