Moderating Government
Francesco De Sinopoli
Leo Ferraris
Giovanna Iannantuoni
Electing a Government
We study a society composed by policy-motivated
citizens voting in a Presidendial and Congressional
election.
Electing a Government
We study a society composed by policy-motivated
citizens voting in a Presidendial and Congressional
election.
The President is elected by majority rule in a single
national district.
Electing a Government
We study a society composed by policy-motivated
citizens voting in a Presidendial and Congressional
election.
The President is elected by majority rule in a single
national district.
The Congress is elected by multidistrict majority rule.
Electing a Government
We study a society composed by policy-motivated
citizens voting in a Presidendial and Congressional
election.
The President is elected by majority rule in a single
national district.
The Congress is elected by multidistrict majority rule.
There are two parties, L and R, with given positions.
Policy Outcome
The policy depends on who is elected President and on
the composition of the Congress.
Policy Outcome
The policy depends on who is elected President and on
the composition of the Congress.
Policy inside Congress function of seats distribution and
not of share of votes.
Policy Outcome
We assume:
given a Congress composition, the policy is more
leftist if the President is L than if the President is R;
given an elected President, the policy is decreasing in
the number of districts carried by L.
Conditional Sincerity
Given weakness of Nash in this context, we define:
Presidential Sincerity (PS): given Congress
composition, voters vote for their preferred President;
Conditional Sincerity
Given weakness of Nash in this context, we define:
Presidential Sincerity (PS): given Congress
composition, voters vote for their preferred President;
District Sincerity (DS): given a President and the
result of Congress outside their district, voters vote
for their preferred candidate in Congress.
Results
Two “pure” policy outcomes.
Government moderation.
Results
Two “pure” policy outcomes.
Government moderation.
A stumbling block: if a voter is pivotal in both
election, PS and DS pure strategy
combinations may not be equilibria.
Joint Conditional Sincerity
Joint Sincerity (JS): voters vote for their most preferred
outcome given the legislative results in the other
districts.
Joint Conditional Sincerity
Joint Sincerity (JS): (for twice pivotal) voters vote for
their most preferred outcome given the legislative
results in the other districts.
Joint Conditional sincerity (JCS): a pure strategy
combination is JCS if PS, DS and JS.
Results
- A pure strategy JCS equilibrium always exists.
Results
- A pure strategy JCS equilibrium always exists.
- Sufficient condition for Government to be divided.
Results
- A pure strategy JCS equilibrium always exists.
- Sufficient condition for Government to be divided.
- A pure strategy combinagion is JCS iff it is b-perfect.
Related Literature
• De Sinopoli, Ferraris and Iannantuoni (2010).
• Alesina and Rosenthal (1995, 1996).
• Ingberman and Rosenthal (1997).
The Model
•Policy Space: X =[0,1].
•Parties: L and R with θL< θR
•Voters: N={1,2,….,n}
Each voter i has single-peaked and symmetric
preferences represented by ui and with bliss point θi.
•Strategies: si =(si1,si2) ∈ Si={LL,LR,RL,RR}
S= S1x S2x.......xSn
The Model
• District: D={1,2,…,d,….,k}, (k odd)
• Nd set of voters in district d.
• # Nd odd, md median voter in district d.
• (w.l.o.g) m1≤m2 ≤….≤ md ≤ …≤ mk.
• m median voter for the entire polulation.
The electoral rule
The President: Party L wins iff
{#i∈N s.t. si1=L} > {#i∈N s.t. si1=R}
Given s, we define P(s) the elected President.
The electoral rule
The President. Party L wins iff
{#i∈N s.t. si1=L} > {#i∈N s.t. si1=R}
Given s, we define P(s) the elected President.
The Congress. District d is carried by L iff
{#i∈Nd s.t. si2=L} > {#i∈Nd s.t. si2=R}
Given s, we define dL(s) the number of districts
won by L.
Policy Outcome
Given s, X(P(s), dL(s)) such that
• X(R,j)>X(L,j)
∀j∈{0,1,…,k}
• X(P,j’)>X(P,j)
∀j’<j and P=L,R
Indifference Conditions
No voters’ bliss point coincides with
αj=X(R,j)+X(L,j)
2
αPj=X(P,j)+X(P,j-1)
2
Example
• Three districts: {0.21, 0.21, 0.21}, {0.49, 0.49,
0.49}, {0.79,0.79,0.79}
• ui(X)= -|X- θi|
θL=0.15
θR=0.85
• 0.15=X(L,3)<0.25=X(L,2)<0.35=X(L,1)
<0.45=X(L,0)<0.55=X(R,3)<0.65=X(R,2)
<0.75=X(R,1)<0.85=X(R,0)
Example
• Three districts: {0.21, 0.21, 0.21}, {0.49, 0.49,
0.49}, {0.79,0.79,0.79}
• ui(X)= -|X- θi|
θL=0.15
θR=0.85
• 0.15=X(L,3)<0.25=X(L,2)<0.35=X(L,1)
<0.45=X(L,0)<0.55=X(R,3)<0.65=X(R,2)
<0.75=X(R,1)<0.85=X(R,0)
The strategy combinations
(LL,LL,LL,LL,LL,LL,LL,LL,LL)
(RR,RR,RR,RR,RR,RR,RR,RR,RR)
with outcomes X(L,3) and X(R,0) are Nash equilibria.
Example
Presidential Sincerity. Given s, si is PS, given dL(s):
θi < αdL(s) iff si1=L
s is PS if si is PS ∀i.
α0=0.65 > 0.49=m
α1=0.55 > 0.49=m
α2=0.45 < 0.49=m
α3=0.35 < 0.49=m
→ X(L,0) PS outcome
→ X(L,1) PS outcome
→ X(R,2) PS outcome
→ X(R,3) PS outcome
Example
District Sincerity. Given s, si is DS, if given P(s) and
dL-d(s) :
θi<αP(s)dL-d(s)+1 iff si2=L
s is DS if si is DS ∀i.
Given P(s)=L:
m1=0.21 < αL1=0.4; m2=0.49 > αL2=0.3;
m3=0.79 > αL3=0.2
→
X(L,1) DS outcome
Given P(s)=R:
m1=0.21 < αR1=0.8; m2=0.49 < αR2=0.7;
m3=0.79 > αR3=0.6
→
X(R,2) DS outcome
Example
To conclude:
1. Two PS and DS outcomes: X(L,1)=0.35 and
X(R,2)=0.65 supported by:
(LL,LL,LL,LR,LR,LR,RR,RR,RR)
(LL,LL,LL,RL,RL,RL,RR,RR,RR)
2. Divided Government.
General Results
_
dL=max d st md<αLd if m1<αL1 (0 otherwise) if L President
_
dR=max d st md<αRd if m1<αR1 (0 otherwise) if R President
General Results
_
dL=max d st md<αLd if m1<αL1 (0 otherwise) if L President
_
dR=max d st md<αRd if m1<αR1 (0 otherwise) if R President
Proposition: There are two DS pure outcomes
_
(i) X(L, dL) if L president
_
(ii) X(R, dR) if R President
General Results
★ Moderation Results:
_ _
dR ≤ dL
★ At least one PS and DS pure outcome exists.
★ If no voter is pivotal in both elections, a PS
and DS pure strategy combination is an
Equilibrium.
Example 2
Three districts: {0.21, 0.21, 0.21}, {0.21, 0.49, 0.79},
{0.79,0.79,0.79}
As before, two PS and DS pure outcomes X(L,1)=0.35
and X(R,2)=0.65 supported by
(LL,LL,LL,LL,LR,RR,RR,RR,RR) and
(LL,LL,LL,LL,RL,RR,RR,RR,RR)
Example 2
Three districts: {0.21, 0.21, 0.21}, {0.21, 0.49, 0.79},
{0.79,0.79,0.79}
As before, two PS and DS pure outcomes X(L,1)=0.35
and X(R,2)=0.65 supported by
(LL,LL,LL,LL,LR,RR,RR,RR,RR) and
(LL,LL,LL,LL,RL,RR,RR,RR,RR)
BUT the second one is NOT an equilibrium because
voter in 0.49 prefers a leftist President and Congress
with a rightist majority to the opposite situation.
Joint Conditional Sincerity
Joint Sincerity (JS): voters vote for their most
preferred policy conditioning on the results of
the legislature results in the other districts.
Joint Conditional sincerity (JCS): a pure strategy
combination is JCS if PS, DS and JS.
Example 2
Voter located in 0.49 is pivotal in both elections
votes in JS manner for a leftist President and a
righstist Congressmen → X(L,1) the only JCS
outcome.
General Results
◆ A JCS pure strategy combination is an
Equilibrium.
◆ Let s and s’ two JCS pure strategy
combinations st P(s)=L and P(s’)=R, then
dL(s)≤dR(s’)
◆ A JCS pure strategy combination exists.
Divided Government
Given s and k odd, government is divided if:
a. when P(s)=L,
dL (s) < (k+1)/2
b. when P(s)=R,
dL(s) ≥ (k+1)/2
Divided Government
◆ In a JCS pure strategy equilibrium
Government is divided if
αL(k+1)/2 < m(k+1)/2 < αR(k+1)/2
b-Perfection
Def. A b-strategy for voter i is a function
bi:{1,2}→[0,1].
Def. A strategy combination is a b-Perfect Equilibrium
is there exists a sequence of completely mixed bstrategy combination converging to b(s) st the
original strategy combination is a best reply against
the entire sequence for all players.
b-Perfection
◆ If no voter is pivotal in both elections, every s is a
bPE iff it is PS and DS.
◆ A pure strategy combination is a bPE iff it is JCS.
Example
Three districts: {0.36, 0.36, 1}, {0, 0.66,
0.66},{0,0.84,0.84}
Parties: θL=0
θR=1
Policy: 0.15=X(L,3)<0.25=X(L,2)<0.35=X(L,1)
<0.45=X(L,0)<0.55=X(R,3)<0.65=X(R,2)
<0.75=X(R,1)<0.85=X(R,0)
Example
Three districts: {0.36, 0.36, 1}, {0, 0.66,
0.66},{0,0.84,0.84}
Parties: θL=0
θR=1
Policy: 0.15=X(L,3)<0.25=X(L,2)<0.35=X(L,1)
<0.45=X(L,0)<0.55=X(R,3)<0.65=X(R,2)
<0.75=X(R,1)<0.85=X(R,0)
The strategy (LL,LL,RR,LL,RL,RL,LL,RR,RR) is PE
and JCS with X(R,2)=0.65.
Example
Three districts: {0.36, 0.36, 1}, {0, 0.66,
0.66},{0,0.84,0.84}
Parties: θL=0
θR=1
Policy: 0.15=X(L,3)<0.25=X(L,2)<0.35=X(L,1)
<0.45=X(L,0)<0.55=X(R,3)<0.65=X(R,2)
<0.75=X(R,1)<0.85=X(R,0)
The strategy (LL,LL,RR,LL,RL,RL,LL,RR,RR) is PE
and JCS with X(R,2)=0.65.
The strategy (LR,LR,RR,LL,RL,RL,LL,RR,RR) is PE
but not JCS with X(R,1)=0.75.