Turnout ABMs & Social Networks

1. Turnout ABMs &Social Networks James Fowler University of California, San Diego

2. Habitual Voting and Behavioral Turnout • Turnout is the “paradox that ate rational choice theory” (Fiorina 1990) • Bendor, Diermeier, and Ting (2003) develop behavioral ABM • Advantages • Innovative • High turnout, other realistic aggregate features • Disadvantages • Behavioral assumption biases result towards high turnout • Causes individuals to engage in casual voting instead of habitual voting (Miller and Shanks 1996; Plutzer 2002; Verba and Nie 1972) • “Moderating feedback” in the behavioral mechanism affects the BDT model • I develop an alternative model (JOP 2005) without feedback • yields both high turnout and habitual voting

3. BDT Behavioral Model of Turnout • Finite electorate with nD>0 Democrats, nR>0 Republicans who always vote for their own party • Each period t an election is held in which each citizen i chooses to vote (V) or abstain (A), given a propensity to vote • Election winner is party with highest turnout • Payoffs (πi,t)

4. BDT Behavioral Model of Turnout • Voters also have aspirationsai,t • Propensity adjustment (Bush and Mosteller 1955) • If πi,t ≥ai,t then • If πi,t <ai,t then where • Aspiration adjustment (Cyert and March 1963) • where

5. Moderating Feedbackin the BDT Model of Turnout • Expected propensity: • Stable only if which is true iff • 50% success rate → 50% turnout! • Adaptive aspirations + monotonicity = bias towards high aggregate turnout

6. Voting is Habitual, Not Casual • Validated Turnout in the 1972, ‘74, ‘76 NES Panel Survey • South Bend (1976-1984)

7. Distribution of Individual Turnout Frequency in South Bend (1976-1984) vs. Turnout Frequency Predicted by BDT Model of Turnout

8. An Alternative Behavioral Model of Turnout • New propensity adjustment parameter • If πi,t ≥ ai,t then • If πi,t < ai,t then • BDT computational model is a special case when  = 1 • Proposition 1. If the speed of adjustment () is not too fast then there exists a range of propensities such that for  > 0 there is moderating feedback and for  = 0 there is no feedback • Corollary 1.1 (BDT computational model). If  = 1, then all propensities are subject to moderating feedback • Corollary 1.2 (model without feedback). If  = 0, then propensities in the range are not subject to moderating feedback

9. An Alternative Behavioral Model of Turnout • Expected propensity: • Notice that if  = 0 ,then →E[pi,t+1] = pi,tregardless of the value of the prior propensity • No bias!

10. Moderating Feedback in Both Models

11. Distribution of Individual Turnout Frequency in South Bend (1976-1984) vs. Turnout Frequency Predicted by Behavioral Models of Turnout

12. Aggregate Turnout • Remarkably, 1/3 of the BDT voters continue to vote even when c>b!

13. The Limits of Closed-Form Reason • Bendor argue that their propositions cover both the BDT and alternative model, so differences must be a mistake • However, key propositions based on assumption all voters have low (or all high) aspirations • These conditions never observed in 100,000 simulations with randomly drawn parameters

14. Lesson about Convergence • Bendor also refused to believe results at first because they had “played with” a step-adjustment rule • I used their own C code to show them that if they waited long enough, it would generate my results • Need a way to assess convergence! • Fortunately, we know this process is ergodic

15. CODA library for Markov Chains • Brooks-Gelman (1997) • start more than one chain at divergent starting points • check within variance vs. between variance • when ratio is near one (<1.1), you’ve reached convergence • Geweke (1992) • Test for equality of the means of the first and last part of a Markov chain

16. CODA library for Markov Chains • Raftery and Lewis (1992) • Run on a pilot chain • Takes into account autocorrelation to suggest how long to run iteration • q - quantile to be estimated • r - desired margin of error of the estimate • s - probability of obtaining an estimate in interval (q-r,q+r) • Heidelberger and Welch (1982) • Tests the null hypothesis that the sampled values come from a stationary distribution using Cramer von Mises statistic

17. Summary and Conclusion • BDT model • Feedback biases it towards high turnout • Feedback yields casual voting • Alternative model • generates high turnout (albeit at a lower cost) • yields habitual voting • Warning for future work in “formal behavioralism” • 1950s and 1960s psychologists studied stochastic learning rules • 1970s rules abandoned because they could not explain individual-level behavior • Lesson: look at both population and individual levels!

18. Computational vs. Analytical Results • Argument appears in two places • Parties, Mandates, and Voters: How Elections Shape the Future (with Oleg Smirnov) 2007 • “Policy-Motivated Parties in Dynamic Political Competition,” JTP 2007 • Errors occur in both proofs and programs • e.g. Roemer 1997 corrects errors in Wittman 1983 • Computer forces consistency in programs • program may not run • Humans must catch mistakes in proofs

19. Numerical Comparative Statics • Given no errors in proof, comparative statics for a given parameter space are certain • Claim: f(a,b) is always increasing in a. • Proof: df(a,b)/da > 0 • Given no errors in program, comparative statics for a given parameter space are uncertain • But we can estimate the uncertainty by sampling the parameter space

20. Estimating Uncertainty of Computational Claims • For one set of parameters • Claim: f(a,b) is always increasing in a • Test: if f(a + ε,b) ≤ f(a,b) then claim is contradicted • For n i.i.d. sets of parameters • Let p be the portion of the space that contradicts the claim • Probability of not contradicting claim is (1 – p)n • To be 95% confident of our estimate of p, let (1 – p)n=0.05, • Implies p = 1 – 0.051/n or approximately 3/n • No observed failures means we can be 95% confident that 3/n part of the space (or less) contradicts the claim

21. Numerical Comparative Statics • Draw n = 100,000 sets of parameters • If a claim is not falsified, we can be 95% confident that only 0.003% (or less) of the parameter space contradicts the results • We use this method to characterize numerically propositions in a dynamic model of party competition with policy-motivated parties

22. Network Theory

23. Some Network Terminology • Each case can be thought of as a vertex or node • An arc i  j = case i cites case j in its majority opinion (directed or two-mode network) • An arc from case i to case j represents • an outward citation for case i • an inward citation for case j • A tie i  j = nodes are connected to one another (bilateral or symmetric network) • Total arcs/ties leading to and from each vertex is the degree • in degree = total inward citations • out degree = total outward citations

24. Clustering Coefficient • What is the probability that your friends are friends with each other? • Network level • Count total number of transitive triples in a network and divide by total possible number • Ego level • For ego-centered measure, divide total ties between friends by total possible number

25. Degree Centrality • Degree centrality = number of inward citations(Proctor and Loomis 1951; Freeman 1979) • InfoSynthesis uses this to choose cases for its CD-ROM containing the 1000 “most important” cases decided by the Supreme Court • However, treats all inward citations the same • Suppose case a is authoritative and case z is not • Suppose case a  i and case z  j • Implies i is more important than j

26. Eigenvector Centrality:An Improvement • Eigenvector centrality estimates simultaneously the importance of all cases in a network (Bonacich 1972) • Let A be an n x n adjacency matrix representing all citations in a network such that aij = 1 if the ith case cites the jth case and 0 otherwise • Self-citation is not permitted, so main diagonal contains all zeros

27. Eigenvector Centrality:An Improvement • Let x be a vector of importance measures so that each case’s importance is the sum of the importance of the cases that cite it:xi = a1i x1 + a2i x2 + … + ani xnorx = ATx • Probably no nonzero solution, so we assume proportionality instead of equality:λxi = a1i x1 + a2i x2 + … + ani xnor λx = ATx • Vector of importance scores x can now be computed since it is an eigenvector of the eigenvalue λ

28. Problems with Eigenvector Centrality • Technical • many court cases not cited so importance scores are 0 • 0 score cases add nothing to importance of cases they cite • citation is time dependent, so measure inherently biases downward importance of recent cases • Substantive • assumes only inward citations contain information about importance • some cases cite only important precedents while others cast the net wider, relying on less important decisions

29. Well-Grounded Cases • How well-grounded a case is in past precedent contains information about the cases it cites • Suppose case h is well-grounded in authoritative precedents and case z is not • Suppose case h  i and case z  j • Implies i is more authoritative than j

30. Hubs and Authorities • Recent improvements in internet search engines (Kleinberg 1998) have generated an alternative method • A hub cites many important decisions • Helps define which decisions are important • An authority is cited by many well-grounded decisions • Helps define which cases are well-grounded in past precedent • Two-way relation • well-grounded cases cite influential decisions and influential cases are cited by decisions that are well-grounded

31. Hub and Authority Scores • Let x be a vector of authority scores and y a vector of hub scores • each case’s inward importance score is proportional to the sum of the outward importance scores of the cases that cite it:λx xi = a1i y1 + a2i y2 + … + ani ynorx = ATy • each case’s outward importance score is proportional to the sum of the outward impmortance scores of the cases that it cites:λy yi = ai1x1 + ai2x2 + … + ain xnory = Ax • Equations imply λx x = ATAxand λy y = AATy • Importance scores computed using eigenvectors of principal eigenvalues λx andλy

32. Closeness Centrality • Sabidussi 1966 • inverse of the average distance from one legislator to all other legislators • let ij denote the shortest distance from i to j • Closeness is

33. Closeness Centrality • Rep. Cunningham 1.04 • Rep. Rogers 3.25

34. Betweeness Centrality • Freeman 1977 • identifies individuals critical for passing support/information from one individual to another in the network • let ik represent the number of paths from legislator i to legislator k • let ijk represent the number of paths from legislator i to legislator k that pass through legislator j • Betweenness is

35. Large Scale Social Networks • Sparse • Average degree << size of the network • Clustered • High probability that one person’s acquaintances are acquainted with one another (clustering coefficient) • Small world • Short average path length “Six degrees of separation” (Milgram 1967)

36. Large Scale Social Network Data

37. Citations in High Energy Physics

38. Judicial Citations

39. Scientific and Judicial Citations • Unifying property is the degree distribution • P(k) = probability paper has exactly k citations • Degree distributions exhibit power-law tail • Common to many large scale networks • Albert and Barabasi 2001 • Common to scientific citation networks • Redner 1998; Vazquez 2001 • Suggests similar processes • Academics may be as strategic as judges!

40. The Watts-Strogatz (WS) Model(Nature 1998) Order Chaos “Real”Social Network

41. Barabasi and Albert, Science 1999 Add new nodes to a network one by one, allow them to “attach” to existing nodes with a probability proportional to their degree Yields scale-free degree distribution Preferential Attachmentand the Scale Free Model

42. Ravasz and Barabasi 2003 Hierarchical Networks

43. Identifying Networks

44. Turnout in a Small World Social Logic of Politics 2005, ed. Alan Zuckerman • Why do people vote? • How does a single vote affect the outcome of an election? • How does a single turnout decision affect the turnout decisions of one’s acquaintances?

45. Pivotal Voting Literature • Most models assume independence between voters • Decision-theoretic modelsDowns 1957; Tullock 1967; Riker and Ordeshook 1968; Beck 1974; Ferejohn and Fiorina 1974; Fischer 1999 • Empirical modelsGelman, King, Boscardin 1998; Mulligan and Hunter 2001 • Game theoretic models imply negative dependence between votersLedyard 1982,1984; Palfrey and Rosenthal 1983, 1985; Meyerson 1998; Sandroni and Feddersen 2006

46. Social Voting Literature • Turnout is positively dependent • between spouses (Glaser 1959; Straits 1990) • between friends, family, and co-workers Lazarsfeld et al 1944; Berelson et al 1954; Campbell et al 1954; Huckfeldt and Sprague 1995; Kenny 1992; Mutz and Mondak 1998; Beck et al 2002 • Influence matters • many say they vote because their friends and relatives vote (Knack 1992) • Mobilization increases turnout • Organizational (Wielhouwer and Lockerbie 1994; Gerber and Green 1999, 2000a, 2000b) • Individual -- 34% try to influence peers (ISLES 1996)

47. Turnout Cascades • If turnout is positively dependent thenchanging a single turnout decision may cascade to many voters’ decisions, affecting aggregate turnout • If political preferences are highly correlated between acquaintances, this will affect electoral outcomes • This may affect the incentive to vote • Voting to “set an example”

48. Small World Model of Turnout • Assign each citizen an ideological preference and initial turnout behavior • Place citizens in a WS network • Randomly choose citizens to interact with their “neighbors” with a small chance of influence • Hold an election • Give one citizen “free will” to measure cascade

49. Simplifying Assumptions • Social ties are • Equal • Bilateral • Static • Citizens are • Non-strategic • Sincere in their discussions

50. Model Analysis • Analytic--to a point: • Create Simulation • Analyze Model Using: • A Single Network Tuned to Empirical Data • Several Networks for Comparative Analysis