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.


4 Responses to “Minor party entry barrier”

  1. Harald Korneliussen Says:

    Early in the morning trying to read through this, but it seems there is an assumption that a party either wins power or does not. Even for first-past-the-post systems, this is not entirely true, and in PR systems it is far from true.

    That’s the one issue, one which the model could possibly be extended to take into account? A more serious issue is the actor’s imperfect knowledge.

  2. Yoram Gat Says:


    I don’t think that an assumption of all-or-nothing is inherent in the model. One could assume an all-or-nothing competition within districts and have the decision of whether to pursue a campaign in each district as being made independently. In fact, it is also possible to see the model as describing a PR system, interpreting pk as the proportion of seats won by party k (and thus as the proportion of the value of controlling government – G – won by the party). I think, however, that the model is more credibly seen as describing a FPTP system, where the parties tend to be devoid of inherent identities, and thus their appeal can be considered as being constant across all voters. Minor parties in a PR tend to have more defined identities and therefore appeal in varying degrees to different voters.

    As for the imperfect knowledge issue – how do you see this issue as relevant to this model?

    BTW, just to be clear, I am not really suggesting that the Tullock model is to any substantial degree a good description of any electoral system. I am suspicious of any quantitative model of human behavior. I was just looking at the way in which the narrowing-the-field function of elections is reflected in the model.

  3. Yoram Gat Says:

    BTW, Harald, are you familiar with the book Elections as instruments of democracy: majoritarian and proportional visions by one G. Bingham Powell? It may be useful for the FPTP vs. PR comparison you are interested in.

  4. Harald Korneliussen Says:

    Not that particular book, no, though I’ve read a good deal about it (mostly arguments, rather than statistics). Right now, I’m interested in The Comparative Study of Electoral Systems, a huge effort to build and analyze datasets on these issues. I found their first publication at the local university library, but it’s a bit hard to make sense of for me.

    I’ll look for Bingham Powell’s book.

    One thing I have read about, though, and found absolutely fascinating, is some of the research on power indices (it’s also much easier for me to follow the mathematics). From that, I think that what you suggest:

    “In fact, it is also possible to see the model as describing a PR system, interpreting p_k as the proportion of seats won by party k (and thus as the proportion of the value of controlling government – G – won by the party). “

    is not correct. Winning a given share of seats in a PR system does not mean winning the same share of power, or proportion of the value of controlling government.

    I suppose that, and the imperfect knowledge issue isn’t relevant to this model as such – but it’s a reason this model doesn’t tell us much. In particular, it doesn’t tell us how many parties we’re likely to get, or how small or large interest groups are favored by the system.

    One book I recommend that has a lot of useful information on power incdices is Power, Freedom and Voting. Since it’s an interdisciplinary collection, it’s very readable – the article authors don’t assume so much about what you know.

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