1. Political Economics Riccardo Puglisi Lecture 2 Content: Probabilistic Voting Model

2. 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 then “median” voter

3. 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 Issue (x, y) – for example: Welfare State and Foreign Policy

4. The Model: Voters  Individuals Voting Behavior Depends on: 1)Policy Component: How the Policy Platform Affect their Utility (ex. Welfare State and Foreign Policy) 2)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)

5.  3 Groups of Individual – for instance – Poor, Middle- Income, Rich (P, M, R): Income: Y P < Y M < Y R Proportion in the Population: a P, a M, a R. And  a J = 1  Within each Group, Voters Differ according to their Ideology (or sympathy) toward the two Candidates:   iJ measures the Ideology of a Voter i in Group j   iJ > 0 means that Voter i is Ideologically closer to candidate B   iJ < 0 means that Voter i is Ideologically closer to candidate A The Voters

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

7. 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):   >0 means that Candidate B is more Popular   <0 means that Candidate A is more Popular  Candidates only know with which probability a “scandal” will take place: 0 - 1/2  1/2   “Scandal” favors A No Scandals “Scandal” favors B 

8. Individual Voting Decision  Voters Consider three Elements before Deciding who to Vote for: 1) Policy: the Utility Induced by the Candidate Policy Platform: U J (X A,Y A ) and U J (X B,Y B ) Notice this Element is Group Specific 2) Individual Ideology:  iJ 3) Average Popularity:   Voter i in Group J Vote for Candidate B if: U J (X B,Y B )+  iJ +  > U J (X A,Y A )

9. Timing Of The Game 1.ELECTORAL CAMPAIGN: Candidates Announce – Independently and Simultaneously -- their Policy Platform (X A,Y A ) and (X B,Y B ). [Notice: they know the Distribution of Individual Ideology, but they do not know their Average Popularity] 2.Before the election, a SHOCK may occur that determines the Average Popularity of the candidates, . 3.ELECTION: Voters Choose their Favorite Candidate 4.POLICY: After the Election, the Winner Implement the proposed Policy Platform

10. 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)  J = U J (X A, Y A ) - U J (X B, Y B ) -   Why is this Voter Relevant? A Small Change in the Policy Platform is sufficient to Gain her Vote 0 - 1/2  J 1/2  J  J J Voters for A Voters for B  J J SWING VOTER  Notice: Candidates set their Platform before the Average Popularity is known  they do not know who the Swing Voter is Group J

11. 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 0 - 1/2  J 1/2  J JJ Voters for A  J J Group J (  J +1/2  J )  J =  J  J +1/2 = 1/2 +  J [U J (X A, Y A ) - U J (X B, Y B )] -  J Voters in group J:

12.  Total votes for A (in all groups):  A =   J /2 +   J  J [U J (X A,Y A ) - U J (X B,Y B )] -  J  J  When does candidate A win the election?  A =   J  J [U J (X A,Y A ) - U J (X B,Y B )] -   A > 1/2 The Candidate Decision Since  J = 1 and  =  J  J is the Average Ideology   A > 1/2    J  J [ U J (X A,Y A ) - U J (X B,Y B )] - 

13.  Candidate A wins the Election if  A > 1/2   <  J  J /  [ U J (X A,Y A ) - U J (X B,Y B )] =   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 (X A,Y A ) to Maximize the Probability of Winning the Election: Pr (  A > 1/2) = Pr (  <  ) - 1/2  1/2   Candidate A wins  Pr (  <  ) = (  + 1/2  )  The Candidate Decision

14.  Candidate A chooses (X A,Y A ) in order to maximize Pr (  <  ) = 1/2 + (  /  )[  J  J ( U J (X A,Y A ) - U J (X B,Y B ))]  Policy chosen to please the voters U J (X A, Y A )  More Relevance is given to the More Numerous Group (  J ) and to the “Less Ideological” Group (  J )  Candidate B chooses (X B,Y B ) to maximize Pr (  >  ) = 1 - Pr (  <  ) Both Candidates Set the Same Policy Platform (X A, Y A ) = (X B, Y B ) The Candidate Decision

15. Probabilistic Voting: Novelty  Majoritarian Voting Model with Two Opportunistic Candidates  NOVELTY: 1.Voters have Preferences over the Policy Implemented by the Politicians and over the Identity/Ideology of the Candidates 2.Before the Election, a Shock may occur that Changes the Average Popularity of the Candidates

16. Probabilistic Voting: Insights 1.POLITICAL CONVERGENCE: Both Candidates Converge on the Same Policy Platform 2.IDEOLOGY: Relevance of the “Less Ideological” (or “Swing”) Voters. They are easier to “Convince” through an Appropriate Policy