Political Economics
Tommaso Nannicini
Topic 3
Content:
Majoritarian Elections
Median Voter
Logrolling
Swing Voter
References:
Mueller (2003), ch. 5 (5.1, 5.2, 5.3, 5.9, 5.13.1)
Hindriks & Myles (2013), ch. 11 (11.8)
Some normative thoughts
(before going positive)
• Intuitively, unanimity is the only voting rule that can
guarantee Pareto optimality (Wicksell criterion)
• BUT it is associated with high internal costs (e.g.,
transaction costs to reach an agreement)
• Any deviation from unanimity reduces internal costs at
the price of higher external costs (imposed on minority)
• Optimal voting rule minimizes total costs (internal +
external). No guarantee that simple majority does this
Majority rule and redistribution
Majority outcome doesn’t need to be Pareto improving (EYZ)
Majoritarian elections
Main features:
 Direct democracy: voters choose a one-dimensional
economy policy (e.g. the size of the welfare state,
degree of flexibility in the labor market)
 Each voter has single-peaked preferences over the
economic policy
 Bliss point: most preferred economic policy
 Median voter divide distribution of voters in half
 Commitment over economic policy
Meet the median voter
If preferences are single-peaked along a one-dimensional
economic policy, the median voter’s bliss point represents
the equilibrium outcome of the majoritarian voting game
 Intuition: All voters to the left of the median voter (MV) –
i.e., whose bliss point is lower than the median voter’s – prefer
the MV’s bliss point to any point chosen by voters to the right of
the MV and vice versa
 Hence, 50% of the voters prefer the MV’s bliss point to any
lower level [the voters to the right of the MV]. While 50%
prefer the MV’s bliss point to any higher level [voters to the left
of the MV]
An example
U
A
B
xa
X’
C
X*
xc
x
• B is the median voter, whose bliss point is x*
• x* is the equilibrium outcome of the voting game (i.e., majority
always prefers x* to any other possible alternative)
Simple proof of median voter theorem
• Define single-peaked preferences:
– x bliss point
– y,z>x or y,z<x  U(y)>U(z) iff |y-x|<|z-x|
• Meet the median voter:
– Bliss points of n voters  {x1,…,xn}
– NR voters have xi ≥ xm; NL voters have xi ≤ xm
– If NR ≥ N/2 and NL ≥ N/2  xm is the MV’s bliss point
• Median voter theorem:
– xm cannot lose under majority rule
– In fact:
• if z < xm at least NR/N ≥ 1/2 vote share in favor of xm
• if z > xm at least NL/N ≥ 1/2 vote share in favor of xm
How plausible are the assumptions?
• We discussed single-peaked preferences…
• What about one-dimensional issues?
• It may be seen as fairly good approximation:
– NOMINATE score by Poole and Rosenthal (1985) along liberal vs.
conservative scale
– Winning compromises (close to median) vs. ideological statements
– 81% of winning proposals in the US Senate closer to median than
status quo
– 62% of losing proposals farther away from the median than the
status quo
Applications of the median voter theorem
• Election: majority voting for political candidates (or
parties)
• Two opportunistic candidates who chose political
platform (or ideology)
• Voters care about the ideology/political platform
• Political outcome: both candidates select as their
platform the ideology of the median voter
Downs-Hotelling model
Proportion
of voters
left
Im Ib Ia
right
Ideology
 Result: Party A & B converge towards Im - the ideology of the MV
 Implication: “Policy moderation” - both parties move toward
moderate positions (ideology) and away from the extremes
Evidence: In two-party systems, “similar” positions
Logrolling
• Different intensity of preferences may create incentive for
vote exchange (banned in elections, but not in Parliament)
• Example of mutual gains from vote exchange:
–
–
–


Voter A gets -2 from X and -2 from Y
Voter B gets 5 from X and -2 from Y
Voter C gets -2 from X and 5 from Y
B and C vote to pass both X and Y
Negative externality on A, but Pareto improvement (compensation)
 The same incentive holds if:
–
–
–



Voter A gets -2 from X and -2 from Y
Voter B gets 3 from X and -2 from Y
Voter C gets -2 from X and 3 from Y
BUT the agreement is no longer Pareto improving (because 2<4)
Examples: pork barrel or local goods vs. general-interest issues
Problems: bluffing, cheating in sequential votes
Probabilistic voting model
 Majoritarian voting model with two opportunistic
candidates (or parties), A and B
 Novelty: voters have preferences over the policy
implemented by the winner but also over the identity of
the candidate [ideology + sympathy]
 Distributive politics game: set of transfers between
voters (zero-sum)
 New concept: swing voter rather then median voter
The voters
 Assume: n voters – voter i has income wi
 Voting behavior depends on:
1) Policy component
2) Ideology (or sympathy) toward a certain candidate (ex.
scandals or being leftist vs. rightist)
 Define: ui(wi+tij)+ij  utility of voter i if candidate j wins
(with u’>0 and u’’<0)
The candidates
 Simple majoritarian election over two candidates A and B
 Each candidate is opportunistic: only cares about winning the
election
 Candidates – simultaneously but independently – determine
their policy platform
 The policy platform consists of a set of transfers t=(t1,…,tn).
Set of feasible policies: T={t: ti=0, wi+ti>0}
Political competition
 Imperfect information: candidates do not know with
certainty the voters’ ideology (or sympathy), just its
distribution ex ante
 Voter i votes for candidate j iff:
ui(wi+tiA) – ui(wi+tiB) > iB - iA
 Fi and fi are the cumulative distribution function and the
density of (iB - iA), respectively
Equilibrium
 Probability that i votes for A:
i(tA,tB)=F[ui(wi+tiA) – ui(wi+tiB)]
 A will maximize i(tA,tB) w.r.t tA keeping tB fixed
 B will maximize [1-i(tA,tB)] w.r.t tB keeping tA fixed
 In equilibrium: tA = tB = t*
 ui’(wi+ti*)f(0)=
(where  is the Lagrange multiplier)
Take-home implications
 Assume identical utility functions: ui=u
 But different distribution functions fi(0)fk(0) ik
 In equilibrium: ui’(wi+ti*) =/fi(0)
 The higher the density, the lower marginal utility, and the
larger transfers
 Swing voters (i.e., those with higher density at zero) receive
larger transfers