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)

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 → X(L,0) PS outcome α1=0.55 > 0.49=m → X(L,1) PS outcome α2=0.45 < 0.49=m → X(R,2) PS outcome α3=0.35 < 0.49=m → 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 b-strategy 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.