Northeastern University - MTHG131 – Final ProjectExploration of a Complex System Using Differential Equations

Bryan Licciardi 12/6/2006

Model based on paper by:Pluemper, Thomas and Martin, Christian W., Multi-Party Competition: A Computational Model with Abstention and Biased Voters (2006).

A Computational Model with Abstention and Biased Voters

What is that? Computer simulation to examine how political parties adjust

there stance on various policies Where does it come from?

Political Science – Political policy modeling Economic Theory of Democracy (1957)

Downsian Proximity model What are major features of cited paper model?

Multi-party Multi-policy (multi-dimensional) Abstention Maximizing vote share

What did I do to study this? Developed MATLAB simulation to mimic research paper Developed set of differential equations to model simulation

Feature Representation- Details of the policy space

2D Voter distribution Normal Distribution

Mean = 0,0 Std dev = 1

Voter Positions are Fixed

Voter Positions are Known (no probabilities/ estimation (deterministic))

Feature Representaion – Party Movement

Parties randomly assigned initial position

Parties scan policy space and look for the location where vote share will be maximized

Move toward that ideal locations

Numerical Solution – MATLAB Simulation

Program Architecture (how it works): Variable: # of voters to represent Generate Voter Positions w/ NormDist Variable: # of Parties to simulate Randomly generate Initial Party Positions Variable: # of time steps to run simulation Variable: Abstention distance Randomly generate party move order per time step Select first party in order and find ideal position

Calculate voter distribution with selected party at everyone position in space

Select location with best results as “ideal position” Repeat for remain parties

Move each party toward its ideal location at given speed Repeat for remain amount of time

<200 lines of code>

Analytical Solution –Differential Equations

Simplifications made 2 policy dimensions Limited to 2 party system (Party A/B)

Process to get equations Breakdown one dimension and repeat

“Took along of time, but the answer was pretty simple”

Policy Space – Voter distribution, Vote share, Ideal position

Voter Distribution (assume large pop.) Party share of voters

Policy Space – Voter distribution, Vote share, Ideal position (cont.)

Find Maximum Position for A given B Case 1: B > 0, (Party A - Party B<s)

Lower Bound = A - s Upper Bound = Mid-point

Case 2: B < 0, (Party A - Party B<s) Lower Bound = Mid-point Upper Bound = A + s

Case 3: B, (Party A - Party B>s) Lower Bound = A - s Upper Bound = A + s

Derivation of Differential Equations

f(B) = CDF(UB) – CDF(LB), derivative = 0 for max Quadratic formula solutions Final Equations

Analysis of party movement

Ideal position eq. dissection

“Not perfect but not bad”

222241212__0333441212___03334xnxnxnxmxnxnxnsPssPPfornidealsPssPPforn⎧⎛⎞⎛⎞++++<⎪⎜⎟⎜⎟⎪⎝⎠⎝⎠=⎨⎛⎞⎛⎞⎪−+−++>⎜⎟⎜⎟⎪⎝⎠⎝⎠⎩

Comparison of results

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Comparison of results

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Comparison with original

Question / Comments

Easy questions only Only positive comments