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Donald G. Saari Institute for Mathematical Behavioral Sciences University of California, Irvine

The surprising complexity of economics. Donald G. Saari Institute for Mathematical Behavioral Sciences University of California, Irvine dsaari@uci.edu. x*. Supply. Budget line; (p, x-w)=0. Demand. Economics 101. Commodity 2. Prices p = (p 1 , p 2 ). Initial endowment w = (w 1 , w 2 ).

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Donald G. Saari Institute for Mathematical Behavioral Sciences University of California, Irvine

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  1. The surprising complexity of economics Donald G. Saari Institute for Mathematical Behavioral Sciences University of California, Irvine dsaari@uci.edu

  2. x* Supply Budget line; (p, x-w)=0 Demand Economics 101 Commodity 2 Prices p = (p1, p2) Initial endowment w = (w1, w2) Rational agent: optimize utility Budget afford (p,w) = p1w1+p2w2 cost (p,x) = p1x1+p2x2 Demand x = (x1, x2) Commodity 1 Individual excess demand Ϛi( p) = Di( p) - Si( p) Aggregate excess demand Ϛ( p) = Σ Ϛi( p) What are the properties of Ϛ( p)? 1. Ϛ( λp) = Ϛ( p) Walras’ Laws 2. Budget constraint (Ϛ( p), p) = 0 3. Ϛ( p) is continuous

  3. p2 1. Ϛ( λp) = Ϛ( p) Walras’ Laws Sonnenschein 2. Budget constraint (Ϛ( p), p) = 0 3. Ϛ( p) is continuous p1 “Invisible hand” What are the properties of Ϛ( p)? Ϛ( p) has a dynamical attractor Does it?

  4. Ϛ( p) Ϛ( p) Scarf’s example 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? Theory vs. reality? Charlie Plott

  5. Extensions; e.g., revealed preferences Idea coming from my voting theory results All results from social choice, voting extend to economics For economics, think of “substitutes” x x X X Now: C>B>A OUTCOME: A>B>C>D by 9: 8: 7: 6 Now: D>C>B 3 6 2 4 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)

  6. n-body problem Dynamics? * pn+1 = Ϛ( pn) , …, Dk Ϛ( p), M( …, 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. Resolution? Help from Arrow’s Theorem!

  7. Arrow’s dictator is a profile restriction!! Arrow Think of this with price setting A>B, B>C implies A>C 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 voters have transitive preferences Pareto: Everyone has same ranking of a pair, then that is the societal ranking Voting rule: Borda 2, 1, 0 Binary independence (IIA): The societal ranking of a 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 Dictator = EX profile restriction

  8. INCLUDING Transitivity! 2001, APSR with K. Sieberg Science Soc. Science History Three voters Wheaton College Tommy Ratliff Public Choice Vote for one from each column Bob David Fred Representative outcome? Five profiles 2:1 Bob, Dave, Fred Ann, Connie, Ellen; Bob, Dave, Fred Bob, Dave, Fred Ann, David, Ellen; Bob, Connie, 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 Outlier: Pairwise vote not designed to recognize any condition imposed among pairs Ethnic groups, etc., etc. Mixed gender!

  9. A>C C>B B>A C>A A>B B>C Lost information “Pairwise emphasis” severs intended connections Name change Ann Connie Ellen Mixed Gender = Transitivity!! Bob David Fred 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 Ann, Dave, Fred; Bob, Connie, Fred; Bob, David, Ellen B>C>A C>A>B A>B>C The Condorcet triplet! 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.

  10. x* and satisfies a bounded variation condition! 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 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

  11. Lost information!! Cannot see full symmetry Allproblems with pairwise comparisons due to Zn orbits For a price, I will come to your department .... Mathematics? Ranking Wheel 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B A F B E C D 1 6 2 5 3 4 6 5 1 4 2 3 Rotate -60 degrees Symmetry: Z6 orbit D E C B A F D C B A A>B>C>D>E>F F B>C>D>E>F>A No candidate is favored: each is in first, second, ... once. C>D>E>F>A>B etc. Yet, pairwise elections are cycles! 5:1 Coordinate direction!

  12. Pairwise majority voting 2 3 1 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.”

  13. {1, 3} Hours 3 2 1 Salary core does not exist McKelvey: Can start anywhere and end up anywhere Stronger rules? Actual examples: MAA, Iraq No matter what you propose, somebody wants to “improve it.” Monica Tataru: Holds for q-rules; i.e., where q of the n votes are needed to win Tataru has upper and lower bounds on number of steps needed to get from anywhere to anywhere else

  14. Some Consequences: campaigning 3 2 1 negative campaigning: changing voters’ perception of opponent Positive With McKelvey and Tataru, everything extends to any number of voters

  15. Two natural questions Generically ˆ When does core exist? If not, what replaces the core? Always McKelvey Banks Plott diagram 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.) q=6, n = 11 5 on losing side 6-5=1 to change vote Number of voters who must change their minds to change the outcome Saari, Math Monthly, March 2004 q=41, n=60 19 on losing side, so need to persuade 41-19 = 22 voters to change their votes Proof by singularity theory So this core persists up to 22 different issues Added stability Answered question when core exists generically.

  16. 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”

  17. Replacing the core Predict what might happen? 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 same lens width of 2d Ellipse: sum of distances is fixed Define “d-finesse pt” in terms of ellipses

  18. 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.

  19. 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

  20. 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 voters have transitive preferences 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 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

  21. Lost information!! Cannot see full symmetry Allproblems with pairwise comparisons due to Zn orbits For a price, I will come to your department .... Mathematics? Ranking Wheel 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B A F B E C D 1 6 2 5 3 4 6 5 1 4 2 3 Rotate -60 degrees Symmetry: Z6 orbit D E C B A F D C B A A>B>C>D>E>F F B>C>D>E>F>A No candidate is favored: each is in first, second, ... once. Yet, pairwise elections are cycles! 5:1 C>D>E>F>A>B etc.

  22. 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. D E C B A F 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B D C B A F Everyone prefers C, D, E, to F F wins with 2/3 vote!! Election outcomes need not represent what the voters want! Why?? Consensus?

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