The surprising complexity of economics Donald G Saari

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The surprising complexity of economics Donald G. Saari Institute for Mathematical Behavioral Sciences University

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 = (p

Economics 101 Commodity 2 Rational agent: optimize utility x* Supply Prices p = (p 1, p 2) Demand x = (x 1, x 2) Initial endowment w = (w 1, w 2) Budget afford (p, w) = p 1 w 1+p 2 w 2 cost (p, x) = p 1 x 1+p 2 x 2 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) =

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 p 2 Does it? p 1 “Invisible hand”

Finding all properties of aggregate excess demand Sonnenschein, Mantel, Debreu Theorem For c≥ 2

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)

Extensions; e. g. , revealed preferences Idea coming from my voting theory results All

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 f. C( p) that satisfies Walras’ laws. There exists a nice pure exchange economy so that for pj ≥ ε, we have that f. C( p) = ϚC( p)

Dynamics? * pn+1 = M( Ϛ( pn) , …, Dk Ϛ( p), …, Ϛ(

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

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

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 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, Connie, Fred; Bob, David, Ann, Dave, Fred; Bob, Outlier: Pairwise vote not designed to Ellen 2001, APSR recognize any condition imposed among with K. Sieberg pairs. Ethnic groups, etc. INCLUDING Transitivity!

“Pairwise emphasis” severs intended connections Name change B>A Ann Connie C>B Ellen A>C Mixed

“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 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

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 f. C( p) that satisfies Walras’ laws. There exists a nice pure exchange economy so that for pj ≥ ε, we have that f. C( 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

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: Z 6 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>A C>D>E>F>A>B etc. Coordinate direction!

Pairwise majority voting 1 2 3 Core: Point that cannot be beaten by any

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 Mc. Kelvey: Can

Hours 3 {1, 3} 2 1 Salary core does not exist Mc. Kelvey: Can start anywhere and end up anywhere Actual examples: MAA, Iraq Stronger rules? No matter what you propose, somebody wants to n Monica Tataru: Holds for q-rules; i. e. , where q of the 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

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

Two natural questions Generically When does core ˆ exist? Always Plott diagram Mc. Kelvey

Two natural questions Generically When does core ˆ exist? Always Plott diagram Mc. Kelvey 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 2 q-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

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

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, 2 d, 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 2 d Define “d-finesse pt” in terms of ellipses

Minimizes what it takes to respond to any change -- d Practical politics: incumbent

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

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,

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

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: Z 6 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>A C>D>E>F>A>B etc. B C Zn orbits

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

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?