On - line Appendix for In fl uencing Connected Legislators

In this appendix we present the proofs omitted in “Influencing Connected Legislators.” Marco Battaglini Cornell University and EIEF Ithaca, NY 14853 [email protected] Eleonora Patacchini Cornell University Ithaca, NY 14853 [email protected] 1 Proof of Lemma 1 Let φ be the  dimensional column vector of voting probabilities with th element equal to . Define η :  →  as η(φ s) = φ−F(φ s), where s =(sA sB) and F(φ s) is a column vector with th element equal to 12 +Ψ 3 ()− () + (φ) +  P   (2 − 1) ́ , as defined in (5) in Section 3.1. The equilibrium probabilities φ(s) are defined as the solution of η(φ∗ s) = 0. The fact that the solution of this system exists follows from the Brower fixed-point theorem as argued in Section 3. Since, by Assumption 1, Ψ ( + + (2 ))  12, the solution is interior in (0 1). To show uniqueness of an equilibrium of the voting stage with policy motivated legislators for Ψ sufficiently small, let kk be the norm kk = X  || for any  ∈ . We have: °°F(φ s)−F(φ0 s)°° ≤ ΨÃ ·X  ̄̄̄̄ ̄ Z   X   (x) ̄̄̄̄ ̄ + 2X 3X  ́ ̄̄ −  ̄̄ ! ≤ Ψ 3  · X  (| − |) + 2 X  ̄̄  −  ̄̄ ́ ≤ Ψ ( + 2) kφ−φ0k where we use the fact that ̄̄  (x) ̄̄  1 for any x, and X   ≤ 1 for any . For any   1, we therefore have °°F(φ s)−F(φ0 s)°° ≤  · kφ−φ0k for Ψ sufficiently small. We can therefore conclude that there is a Ψ1 such that F(φ s) is a contraction in [0 1] with a unique fixed-point in (0 1) for Ψ ≤ Ψ1. We now turn to the derivatives of the voting probabilities. The implicit function theorem implies that the solution  is differentiable in    at s s if (η) is invertible in a neighborhood of (s sφ(sA sB)), where φ(sA sB) solves η(φ s s) = 0 (the expression (η) represents the Jacobian of η with respect to φ). It is easy to verify that (η) = h  − Ψ2 e i, where e  is a × matrix with   element equal to e  =  ¡ +  (φ)¢. Let ∗ be the largest eigenvalue of e  achieved for some φ (this is well defined and bounded since ( e ) is continuos in φ in and the space of feasible φ is compact). Theorem III* of Debreu and Herstein [1953] implies that h  − Ψ2 e i−1 exists and is nonegative for Ψ ≤ (2∗)−1 = Ψ2. The Jacobian of  is then  [] = Ψ · () h  − Ψ2 e i−1 1  where 1 is a -dimensional vector equal to zero except at the th dimension in which it is equal to one. Since h  − Ψ2 e i−1 is nonnegative with at least one strictly positive element for Ψ ≤ Ψ2, it follows that X     =  []  · 1  0 for  large enough. To verify concavity with respect to s, let   be the Hessian of . Consider first its diagonal entries      for any . We can write:     = Ψ ⎢⎢⎣  (  )    + 2 P       + P  3P    (φ) 3   ́3   ́ +  (φ)     ́ ⎥⎥⎦ (1)