Probabilistic Voting Model
Advanced Political Economics
Fall 2013
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:
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)
The Voters
Individual Ideology
 How are Voters Distributed within each Group (P, M, R)
according their Ideology?
 Uniform Distribution Function with Density FJ
Voters
closer to A
- 1/2F J
Neutral
Voters
0
Voters
closer to B
1/2F J
FJ
s
that as the Density Increase (F J ), the
Group becomes “Less Ideological”: Fewer Voters have
an Ideology or Sympathy towards a Candidate
 Notice
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
- 1/2 Y
No
Scandals
0
“Scandal”
favors B
1/2 Y
Y
d
Individual Voting Decision
 Voters Consider three Elements before Deciding who
to Vote for:
1)
Policy: the Utility Induced by the Candidate Policy Platform:
UJ(XA,YA) and UJ(XB,YB)
Notice this Element is Group Specific
2)
Individual Ideology: siJ
3)
Average Popularity: d
 Voter i in Group J Vote for Candidate B if:
UJ(XB,YB)+ s iJ+d > UJ(XA,YA)
Timing Of The Game
1. 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]
2. Before the election, a SHOCK may occur that determines
the Average Popularity of the candidates, d.
3. ELECTION: Voters Choose their Favorite Candidate
4. POLICY: After the Election, the Winner Implements her
Policy Platform
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
Group J
Voters SWING
for A VOTER
- 1/2F J
sJ 0
Voters
for B
FJ
1/2F J
 Notice: Candidates set their Platform before the Average
Popularity is known  they do not know who the Swing Voter is
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
Voters
for A
Group J
- 1/2F J
Voters in group J:
FJ
sJ 0
1/2F 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
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
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)
Candidate A
wins
- 1/2 Y
d
Y
1/2 Y
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:
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
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