1. Advanced Political Economics
Probabilistic Voting Model
Advanced Political Economics
Fall 2013
Riccardo Puglisi

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 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)

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 Issues (x, y) – for example: Welfare State and Foreign Policy

4. 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)

5. The Voters

6. Individual Ideology
How are Voters Distributed within each Group (P, M, R) according their Ideology?
 Uniform Distribution Function with Density FJ
- 1/2F J
1/2F J
F J
Voters
closer to A
closer to B
Neutral Voters s
Notice that as the Density Increase (F 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):
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
- 1/2 Y
1/2 Y d

8. 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 J Vote for Candidate B if:
UJ(XB,YB)+ s iJ+d > UJ(XA,YA)

9. 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

10. (in group J) sJ = UJ(XA, YA) - UJ (XB, YB) - d
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
- 1/2F J
1/2F J
F J
Voters
for A
for B
s J
SWING VOTER
Group J
Notice: Candidates set their Platform before the Average Popularity is known  they do not know who the Swing Voter is

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
- 1/2F J
1/2F J FJ
Voters
for A
s J
Group J
(s J+1/2F J) F J = s JF J +1/2 =
1/2 + F J[U J(XA, YA) - U J(XB, YB)] - d F J
Voters in group J:

12. The Candidate Decision
Total votes for A (in all groups):
PA = S aJ/2 + S aJFJ [UJ(XA,YA) - UJ(XB,YB)] - d SaJFJ
When does candidate A win the election?
PA > 1/2
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  S aJFJ [ UJ(XA,YA) - UJ(XB,YB)] - d F > 0

13. 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)
- 1/2 Y
1/2 Y Y
Candidate A
wins d
Pr (d < d) = (d + 1/2 Y) Y

14. 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)

15. 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 Shock may occur that Changes the Average Popularity of the Candidates

16. 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