Probabilistic Voting Model

Probabilistic Voting Model Advanced Political Economics Fall 2012 Riccardo Puglisi

PROBABILISTIC VOTING MODEL • Majoritarian voting model for two opportunistic candidates (or parties) • Novelty: Voters have preferences over the policy implemented by the winner but also over the identity of the candidate [ideological/sympathy component] • New concept: “Swing” voter rather than “median” voter • Methodological advancement: Nash equilibrium typically exists. This is true also for situations where it would not exist in the lack of this additional ideological component (e.g. multidimensional policy space with multidimensional conflict)

The Model: Candidates • Simple Majoritarian Election over two Candidates A & B • Each Candidate is Opportunistic: only cares about winning the election • Candidates – simultaneously but independently – Determine their Policy Platform • The Policy Platform Consists of two Issues (x, y) – for example: Welfare State and Foreign Policy

The Model: Voters • Individuals Voting Behavior Depends on: • Policy Component: How the Policy Platform Affect their Utility (ex. Welfare State and Foreign Policy) • Individual Ideology (or Sympathy) towards a Candidate (ex. Scandals or Feeling L or R) • Imperfect Information: Candidates do not know with Certainty the Voters’ Ideology (or Sympathy)

The Voters

Voters closer to A Voters closer to B FJ Neutral Voters - 1/2F J 0 1/2F J Individual Ideology • How are Voters Distributed within each Group (P, M, R) according their Ideology?  Uniform Distribution Function with Density FJ s • Notice that as the Density Increase (F J ), the Group becomes “Less Ideological”: Fewer Voters have an Ideology or Sympathy towards a Candidate

Candidates’ Average Popularity • Voters Decisions are also affected by the Candidates’ Average Popularity before the Election • Candidates cannot Control their Popularity before the Election. • The Outbreak of Scandals or other News may Reduce one Candidate Popularity, while increasing the other’s (e.g. Monica Lewinsky): • d >0 means that Candidate B is more Popular • d <0 means that Candidate A is more Popular • Candidates only know with which probability a “scandal” will take place: “Scandal” favors A “Scandal” favors B No Scandals Y d - 1/2 Y 0 1/2 Y

Individual Voting Decision • Voters Consider three Elements before Deciding who to Vote for: • Policy: the Utility Induced by the Candidate Policy Platform: UJ(XA,YA) and UJ(XB,YB) Notice this Element is Group Specific • Individual Ideology: siJ • Average Popularity: d • Voter i in Group JVote for Candidate B if: UJ(XB,YB)+ siJ+d > UJ(XA,YA)

Timing Of The Game • ELECTORAL CAMPAIGN: Candidates Announce – Independently and Simultaneously - their Policy Platform (XA,YA) and (XB,YB). • [Notice: they know the Distribution of Individual Ideology, but they do not know their Average Popularity] • Before the election, a SHOCK may occur that determines the Average Popularity of the candidates, d. • ELECTION: Voters Choose their Favorite Candidate • POLICY: After the Election, the Winner Implements her Policy Platform

Voters for A Voters for B FJ sJ - 1/2F J 0 1/2F J The “SWING” Voter • The “Swing” Voter is the Voter who – after Considering the Policy Platform and the Average Popularity – is Indifferent between Voting for Candidate A or B: (in group J) sJ = UJ(XA, YA) - UJ (XB, YB) - d • Why is this Voter Relevant? A Small Change in the Policy Platform is sufficient to Gain her Vote SWING VOTER Group J • Notice: Candidates set their Platform before the Average Popularity is known  they do not know who the Swing Voter is

Voters for A FJ Group J sJ - 1/2F J 0 1/2F J The Candidate Decision • Candidates have to set their Policy Platform before the Average Popularity is known  They maximize the Probability of being Elected – subject to “Scandal” • Who Votes for Candidate A? Voters to the left of the Swing Voter in each Group (sJ+1/2FJ) FJ = sJFJ +1/2 = 1/2 + FJ[U J(XA, YA) - U J(XB, YB)] - d FJ Voters in group J:

The Candidate Decision • Total votes for A (in all groups): PA = SaJ/2 + SaJFJ [UJ(XA,YA) - UJ(XB,YB)] - d SaJFJ PA > 1/2 • When does candidate A win the election? PA = 1/2 +SaJFJ [UJ(XA,YA) - UJ(XB,YB)] - dF > 1/2 Since SaJ = 1 and F = SaJFJ is the Average Ideology  PA > 1/2  SaJFJ [ UJ(XA,YA) - UJ(XB,YB)] - d F > 0

Candidate A wins Y d - 1/2 Y 1/2 Y The Candidate Decision • Candidate A wins the Election if PA > 1/2  d < S aJFJ/F [ UJ(XA,YA) - UJ(XB,YB)] = d • Not Surprisingly, Candidate A wins if she is not hit by a Scandal • But Candidate A does not know δ she will set the Policy Platform (XA,YA)to Maximize the Probability of Winning the Election: Pr (PA > 1/2) = Pr (d < d) Pr (d < d) = (d + 1/2 Y) Y

The Candidate Decision • Candidate A chooses (XA,YA) in order to maximize • Pr (d < d) = 1/2 + (Y/F)[SaJFJ (UJ(XA,YA) - UJ(XB,YB))] • Policy chosen to please the voters UJ(XA, YA) • More Relevance is given to the More Numerous Group (aJ) and to the “Less Ideological” Group (FJ) • Candidate B chooses (XB,YB)to maximize Pr (d > d) = 1 - Pr (d < d) Both Candidates Set the Same Policy Platform (XA, YA) = (XB, YB)

Probabilistic Voting: Novelty • Majoritarian Voting Model with Two Opportunistic Candidates • NOVELTY: • Voters have Preferences over the Policy Implemented by the Politicians and over the Identity/Ideology of the Candidates • Before the Election, a Shockmay occur that Changes the Average Popularity of the Candidates

Probabilistic Voting: Insights • POLITICAL CONVERGENCE: Both Candidates Converge on the Same Policy Platform • IDEOLOGY: Relevance of the “Less Ideological” (or “Swing”) Voters. They are easier to “Convince” through an Appropriate Policy

Probabilistic Voting Model