Minor party entry barrier
February 8, 2011
Elections are first and foremost a mechanism for eliminating non-elite competitors for power – their role as an inter-elite arbitration mechanism is merely a derivative of their primary function. The filtering function is most severe in first-past-the-post systems, where there are often just two credible competitors, and rarely more than 3.
[ Due to typesetting constraints, I am using [ … ] as shorthand for the reciprocal of the expression in the square brackets. That is [ x ] = 1 / x. ]
Within the Tullock contest model, the entry barrier for minor parties can be calculated. The Tullock contest model specifies the chance of party k winning power as pk = xk [ tk ] [ x1 / t1 + … + xn / tn ], where xi is the amount of resources spent by party i, and ti is the (in)effectiveness of the party’s campaign efforts.
Each party is characterized by its campaign effectiveness and its fund raising effectiveness fi. Given the benefits of power, G, the expected return for party k is:
Uk = pk G – xk [ fk ] = xk [ tk ] [ x1 / t1 + … + xn / tn ] G – xk [ fk ].
The model assumes that parties spend the amount that maximizes their expected returns. Under the model assumptions there will always be at least two parties with non-zero expenditures, since a party that finds itself without a spending opponent reduces its spending, until the derivative of the expected return for a competitor becomes positive at 0.
Let zi = xi [ ti ] [ G ], Z = z1 + … + zn, and ri = ti [ G fi ], i = 1, …, n. Then,
Uk = G ( zk [ Z ] – rk zk ),
d Uk [ d zk ] = G ( (Z – zk) [ Z2 ] – rk ).
An equilibrium with n parties 1, …, n exists if there exists a solution to the n simultaneous equations,
d Ri [d zi] = 0, i = 1, …, n,
such that zi > 0, i = 1, …, n.
If the solution exists, then, summing the n equations:
Z = (n – 1) [ R ],
where R = r1 + … + rn.
zk = Z (1 – rk Z).
Then zk > 0 if rk > [ Z ] = R [ n – 1 ], which is true if and only if
rk > (R – rk) [ n – 2 ].
Thus, a solution exists if
mink rk = r(n) > (r(1) + … + r(n – 1)) [ n – 2 ],
and the number of parties at equilibrium will be the maximum number n such that
r(n) ≥ (r(1) + … + r(n – 1)) [ n – 2 ].
In particular, this implies that any party which has r > r(1) + r(2) will have negative benefit from engaging in an election campaign.