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.T. Coram Economics Letters 67 2000 315 –323
policies on such issues as expenditure on defence or health subsidies to producers of primary goods seems to be a common aspect of candidate behaviour. What is not clear, however, is the offers it
would be best for such parties to make. This raises the following questions. Do special interest voters get more than a proportional share of
the available resources in this type of bidding game? What will happen as the proportion of single issue voters increases? Will the resources available to these voters increase at the same rate?
Mueller 1983, 264 argues, for example, that there is distinction between outcomes in countries with and without strong interest groups. This seems to imply some sort of discontinuity. Is this
accounted for by vote bidding, or some other aspect of the political process? The purpose of this note is to consider a simple model of a multi-stage vote bidding game with a
mixture of voter types in order to throw light on these questions. To maximise the transparency of the analysis, the problem to be analysed is set up within the framework of a stripped down Hotelling
model. The model analysed here differs from Grossman and Helpman’s model of interest group payments
to influence voter’s opinions and to change the policies offered by parties in that special interests are treated as passive. On the other hand it has more complicated equilibrium properties because it is
assumed that what counts for parties is winning. Hence the payoffs switch between all and nothing. I concentrate on changes in party strategies as the proportion of single issue voters changes. I also
analyse the shares of the resources that single issue voters obtain. One interesting result is that, when single issue voters are half, or more than half, the total they get is all the resource.
2. A model of candidate competition in a pure distribution game
Suppose there is a single electorate in which there are two candidates, and a simple majority wins. Some voters vote along single interest lines while others are concerned with a range of public good
and general interest issues and behave in a probabilistic manner Mueller, 1991; 348–369; Carter and Guerette, 1992. Such voters are called moderates. The division of voters in this way captures and
adds to some of the theory of voting in Brennan and Lomasky 1997. They argue that voters tend to vote in an expressive fashion. It would be expected, however that groups with an overriding single
issue wish to express their support for that issue and to see value in their votes being counted. Voters that are not in this category might be expected to decrease their votes in some manner as the relative
support offered by a party declines.
The candidates each have a fixed pool of resources available and bid for votes by offering a distribution of these resources. The fixed pool means that the game is a pure distributional game
Coughlin, 1992, 25. It has often been argued that most elections allocating private goods are precisely elections entailing redistributional issues Aranson and Ordeshook, 1981. Offers are
simultaneous. This represents the notion that candidates have to develop policy positions and cannot change them instantaneously.
The only strategic choice that is of concern is the amount that candidates distribute to the different type of voters. No attempt will be made to analyse the particular coalition of single issue voters that is
bought. Cycling problems have been extensively explored elsewhere and are ignored in this model Ordeshook, 1986; Mueller, 1991. It is assumed that the candidate that makes the greater distribution
is able to buy the votes of single issue voters.
A .T. Coram Economics Letters 67 2000 315 –323
317
These assumptions can be summed up and the model presented as follows: 2.1. The model
1 2
The strategy sets for both parties are S 5 S 5 [0,1]. A strategy for party one is a distribution to
1
single issue voters s [ S and a distribution to the moderates 1 2 s . The strategies of party two are
1 1
s [ S and 1 2 s . A probability distribution of strategies is written s s , s s , . . . .
2 2
2 11 11
12 12
Assume that parties are only concerned with winning. It is also possible that parties are concerned with maximizing their vote. Winning gives a payoff of one unit.
w w are the payoff functions for
1, 2
parties 1 and 2. Let p be the probability of getting the votes of moderates and k of getting single issue voters. p and
k are density functions. The proportion of single issue voters is a and of moderate voters is 1 2 a.
This gives w 5 w [ p 1 2 a 1 ak ]
i i
i i
Letting f 5 p 1 2 a 1 ak
i i
i
1 if f . f
1 2
w 5 1 2 if f 5 f
1 1
2
otherwise. 2.2. Assumptions about voters
Moderate voters increase the probability of voting for party i as the amount of resources offered by i increases and decrease the probability of voting for i as the amount of resources offered decreases.
Assume that p s , s is continuous and differentiable.
1 1
2
≠p ≠s , 0, ≠p ≠s . 0, i ± j.
i i
i j
To make moderate votes worthwhile p 0, 1 5 p 1, 0 5 1. In addition the following assumption
1 2
will be used.
A1. Symmetry. A1 says that moderate voters respond to nothing but the offers made by the parties across the entire range of offers. That is a unit of goodies from party one is always treated in the same
way as a unit of goodies from party two. Hence p s , s 5 p s , s .
1 1
2 2
2 1
Single interest voters support the party that offers them the greatest payoff. Let k be the probability
1
of a vote for party 1. Then, k 5 1 if s . s , 1 2 if s 5 s , 0 if s , s .
1 1
2 1
2 1
2
Some intuition about this game is given in Fig. 1 for the case where special interest voters are less than half. This sets out the relation between the number of votes and a choice of s 5 x [ [0, 1] for
1
some given value of s 5 y.
2
What the figure shows is that s , y gives more moderate votes to party one and the single issue
1
votes to party two. s 5 y gives f 5 f 5 1 2. s 5 y 1 ´ for some ´ sufficiently small gives
1 1
2 1
f 5 p 1 2 a 1 a . 1 2.
1 1
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.T. Coram Economics Letters 67 2000 315 –323
Fig. 1. Payoffs for s for a , 1 2 and some specific p.
1
Observe that for a , 1 2 the game is discontinuous at every point s 5 s .
1 2
3. Analysis
