Minor party entry barrier

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 p_k = x_k [ t_k ] [ x₁ / t₁ + … + x_{n /} t_n ], where x_i is the amount of resources spent by party i, and t_i is the (in)effectiveness of the party’s campaign efforts.

Each party is characterized by its campaign effectiveness and its fund raising effectiveness f_i. Given the benefits of power, G, the expected return for party k is:

U_k = p_k G – x_k [ f_k ] = x_k [ t_k ] [ x₁ / t₁ + ^… + x_{n /} t_n ] G – x_k [ f_k ].

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 z_i = x_i [ t_i ] [ G ], Z = z₁ + ^… + z_n, and r_i = t_i [ G f_i ], i = 1, …, n. Then,

U_k = G ( z_k [ Z ] – r_k z_k ),

And

d U_k [ d z_k ] = G ( (Z – z_k) [ Z² ] – r_k ).

An equilibrium with n parties 1, …, n exists if there exists a solution to the n simultaneous equations,

d Ri [d z_i] = 0, i = 1, …, n,

such that z_i > 0, i = 1, …, n.

If the solution exists, then, summing the n equations:

Z = (n – 1) [ R ],

where R = r₁ + ^… + r_n.

Then,

z_k = Z (1 – r_k Z).

Then z_k > 0 if r_k > [ Z ] = R [ n – 1 ], which is true if and only if

r_k > (R – r_k) [ n – 2 ].

Thus, a solution exists if

min_k r_k = 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.

Advertisement