1 Advances in Political Economy - Department of Political Science

EDITOR’S PROOF

294 K. McAlister et al.

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vote function and the Hessian matrix of second derivatives is non-positive, meaning

that the eigenvalues are all non positive. More simply put, a vector,z∗,isaLNE

if each party locates itself at a local maximum in its respective vote function. This

means, that given the opportunity to make moves in the policy space and relocate

its platform, no vote-maximizing party would choose to move. We assume that par-

ties can estimate how their vote shares would change if they marginally move their

policy position. The local Nash equilibrium is that vector z of party positions so

that no party may shift position by a small amount to increase its vote share. More

formally a LNE is a vectorz=(z 1 ,...,zj,...,zp)such that eachVj(z)is weakly

locally maximized at the positionzj. To avoid problems with zero eigenvalues we

also define a strict local Nash equilibrium (SLNE) to be a vector that strictly lo-

cally maximizesVj(z). We typically denote an LNE byz(K)whereKrefers to

the model we consider. Using the estimated MNL coefficients we simulate these

models and then relate any vector of party positions,z, to a vector of vote share

functionsV(z)=(V 1 (z),... , Vp(z)), predicted by the particular model with p par-

ties.

Given that we have defined the errors as cumulatively coming from a Type-I ex-

treme value distribution, the probabilityρij(z)has a multinomial logit specification

and can be estimated. For each voteriand partyjthe probability thativotes forj

givenzis given by:

ρij(z)=

exp(u∗ij(xi,zj))

∑p

k= 1 exp(u

∗

il(xi,zk))

=

[

1 +

∑p

k=j

exp(fk)

]− 1

wherefk=

∑p

k= 1

(

u∗il(xi,zk)

)

−

(

u∗ij(xi,zj)

)

.

Thus

dρj(z)

dzj

= 2 β(zj−xi)

[

1 ×

[

1 +

∑p

k=j

exp(fk)

]]− 2 [p

∑

k=j

exp(fk)

]]

= 2 β(zj−xi)×

[

ρij(z)

][

1 −ρij(z)

]

in regionk, with population,Nk,ofsizenkthe first order condition becomes

dVjk(zk)

dzj

∣

∣

∣

z−j=z

=

1

nk

2 βk

∑

i∈Nk

ρij k( 1 −ρij k)(zj−xi)= 0 , (1)

sozj=

∑

i∈Nk

wijxi, (2)

wherewij=

ρij k( 1 −ρij k)

∑

k= 1 ρij k(^1 −ρij k)

. (3)