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## Turnout ABMs & Social Networks

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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

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)

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

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

Voting is Habitual, Not Casual

- Validated Turnout in the 1972, ‘74, ‘76 NES Panel Survey
- South Bend (1976-1984)

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

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

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!

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

Aggregate Turnout

- Remarkably, 1/3 of the BDT voters continue to vote even when c>b!

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

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

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

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

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!

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Closeness Centrality

- Rep. Cunningham 1.04
- Rep. Rogers 3.25

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

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)

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!

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 ModelTurnout 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?

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

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)

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”

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

Simplifying Assumptions

- Social ties are
- Equal
- Bilateral
- Static
- Citizens are
- Non-strategic
- Sincere in their discussions

Model Analysis

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

Political Discussion Network Data

- 1986 South Bend Election Study (SBES) 1996 Indianapolis-St. Louis Election Study (ISLES)(Huckfeldt and Sprague)
- “Snowball survey” of “respondents” and “discussants”

Discussant’s Discussant

Discussant

Discussant’s Discussant

Respondent

Discussant

Discussant’s Discussant

Discussant’s Discussant

Discussant

Discussant’s Discussant

Features of a Political Discussion Network Like the ISLES

- Size:186 million, but limited to 100,000-1 million
- Degree:3.15 (but truncated sample)
- Clustering:0.47 for “talk” 0.61 for “know”
- Interactions:20 (3/week, 1/3 political, 20 weeks in campaign)
- Influence Rate:0.05 (consistent w/ 1st,2nd order turnout corr.)
- Preference Correlation: 0.66 for lib/cons, 0.47 for Dem/Rep

Turnout CascadesMagnify the Effect of a Single Vote

- A single turnout decision
- changes the turnout decision of at least 3 other people
- increases the vote margin of one’s favorite candidate by at least 2 to 3 votes
- Turnout cascades increase the incentive to vote by increasing the
- pivotal motivation (Downs 1957)
- signaling motivation (Fowler & Smirnov 2007)
- duty motivation (Riker & Ordeshook 1967)
- Consistent with people who say they vote to “set an example”

Do Turnout Cascades Exist?

- Cascades increase with number of discussants
- But this correlates strongly with interest
- How does individual-level clustering affect the size of turnout cascades?
- Social capital literature suggests monotonic and increasing

Intention toInfluence and Turnout

Individual NetworkCharacteristics

TurnoutCascades

Prediction: How Individual-Level Clustering Affects Simulated Turnout

What’s Going On?

- Clustering increases the number of paths of influence both within and beyond the group
- With a fixed number of acquaintances, clustering decreases the number of connections to the rest of the network

A

B

A

B

A

B

E

E

E

D

D

D

C

C

C

F

G

F

F

G

G

The Strength of Mixed Ties

- “Weak” ties may be more influential than “strong” ties because they permit influence between cliques (Grannovetter 1973)
- Evidence here suggests that a mixture of strong and weak ties maximizes the individual incentive to set an example by participating

Stylized Facts for Aggregate Turnout

- Turnout increases in:
- Number of contactsWielhouwer and Lockerbie 1994;Ansolabehere and Snyder 2000; Gerber and Green 1999, 2000
- Clustering of social tiesCox, Rosenbluth, and Thies 1998; Monroe 1977
- Concentration of shared interestsBusch and Reinhardt 2000; Brown, Jackson, and Wright 1999; Gray and Caul 2000; Radcliff 2001

Implications

- Turnout Cascades & Rational Voting
- Turnout cascades magnify the incentive to vote by a factor of 3-10
- Even so, not sufficient
- Explaining the Civic Duty Norm
- Establishing a norm of voting with one’s acquaintances can influence them to go to the polls
- People who do not assert such a duty miss a chance to influence people who share similar views, leading to worse outcomes for their favorite candidates

Implications

- Over-Reporting Turnout
- Strategic people may tell others they vote to increase the margin for their favorite candidates
- It is rational to do this without knowing anything about the candidates in the election!
- May explain over-reporting of turnout(Granberg and Holmberg 1991)
- Paradox: why would people ever say they don’t vote?
- Social Capital
- Bowling together is better for participation than bowling alone (Putnam 2000)
- BUT, who we bowl with is also important
- People concerned about participation should be careful to encourage a mix of strong and weak ties (Granovetter 1973)

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