Presidential Nominating Process

S.J. Brams, M.D. Davis / Optimal resource allocation in presidental primaries

375

1978a), and more formally with Straffin (Brams and Straffin, 1982), strategic aspects of candidates' positioning themselves on issues in multicandidate elections, in which winnowing of the less serious candidates occurs over an extended series of elections.

In the case of the second factor, with the notable exception of recent work by Aldrich (1979, 1980a-c), who has developed game-theoretic and differenceequation models of the nomination process, we know of no models of resource commitment or allocation in presidential primaries. Although the optimization model we shall propose differs structurally from Aldrich's models, we believe the models complement each other, and results derived from each are entirely consistent.

2. The basic model

In a previous model of resource allocation in the general election for the presidency, we (Brams and Davis, 1973, 1974) assumed that the probability that a voter in state i votes for the Democratic Party candidate is

P1 = di/(d;+r),

where d; and r¡ are, respectively, the resources that the Democratic and Republican candidates allocate to state i.2 In the present model, we provisionally assume that there are just two candidates, A and B, in one party's primary (or caucus) race, and the probability that a voter in primary 1 votes for candidate A is

P1 = a1/(a1+b1),

where a and b, are, respectively, the resources that candidates A and B allocate to primary 1. Thus, in primary 1, the assumption of the general-election model relating resource allocation to the probability of voting for a particular candidate is retained. Also, it is assumed that voting for each of the two candidates by concerned (nonindifferent) voters are mutually exclusive and exhaustive events: they vote either for A or B; there are no other candidates, and concerned voters do not abstain.

Assume a strict sequence of primaries, or set of primaries. Let the primaries, or set of primaries, be indexed by i, i = 1, 2, . n, where i= 1 refers to all primaries on the first primary date, i=2 refers to all primaries on the date the next one or more primaries are held, and so on for the n distinct primary dates. For convenience, we shall henceforth refer to primary 1, primary 2, and so on, with the understanding that primary i refers to all primaries held on the ith primary date in the sequence of state primaries.

The next step is to specify the general relationship between the probability,

2 For a generalization, see Shane (1977). See also Brams (1978b) for a discussion of related models.

376

S.J. Brems, M.D. Davis / Optimal resource allocation in presidental primaries

Pi=2, 3, ..., n, that a voter in primary i>1 votes for candidate A and the resources candidate A allocates to primary i. Unlike primary 1, however, we assume that the probability that a voter in any primary after the first votes for candidate A does not depend only on the resources A, versus B, allocates to that primary. Instead, we assume that P, where i>1, depends also on p¡-1, the probability that a voter in primary i− 1 voted for A. That is, we assume a kind of momentum – either positive or negative — is transferred from each primary to affect performance in the primary immediately following it.

In this manner, the resources A allocates to primary i, a¡, are discounted by p¡-1, the probability that a voter in the preceding primary voted for A; similarly, the resources B allocates, b¡, are discounted by the probability, (1 − P¡-1), that a voter voted for B in primary i-1. Necessarily,

Pi + (1 - Pi) = 1,

that is, a concerned voter in primary i votes for either candidate A or B.

To summarize, in the basic model we assume that a candidate's (say, A's) performance as gauged by Pi, the probability that a voter in primary i votes for him depends in primary 1 only on the resources he, versus his opponent (B), allocates to that primary; but in all subsequent primaries, A's performance depends on both the resources he allocates and his performance in the immediately preceding primary. The greater A's Pi-1 in that preceding primary, the less is B's complementary (1-P-1), so the more A's resource allocation in primary i, a¡, 'counts' as compared to B's. This sequential-dependence assumption means that A's p; in any primary is boosted the greater his pi-1 was in the preceding primary. In fact, if the latter probability exceeds 1/2, it is appropriate to say A's momentum going into primary i is positive; otherwise, B's is, unless p¡- 1 = 1/2.

This interdependence, however, holds not just between adjacent primaries. Because Pi-1, which affects p1, depends on Pi-2, there is an indirect transfer of momentum from primary i-2 to i. This indirect transfer of momentum extends backward to primary 1, thus rendering p; dependent on A's performance in all previous primaries. So although the model formally incorporates a direct dependence only between adjacent primaries in the sequence, there is an indirect dependence that makes performance in primary i a function of performance in all preceding primaries.

There are a couple of obvious implications of this basic model. For candidate A, if any a;=0, i = 1, 2, ..., n, this will render p=0, given b;>0. This implies (1 − p;) = 1, which means B wins the votes of all concerned voters in primary i. But then this implies that in subsequent primary i + 1, Pi + 1 = 0, because the discounting factor pi on the right-hand side of

S.J. Brams, M.D. Davis / Optimal resource allocation in presidental primaries

377

is 0. Hence, A's failure to allocate some resources to any primary i means that he not only wins no votes in that primary but also in the subsequent primary – and all primaries subsequent to that, too. Thus, the chain is broken if any primary is ignored.

This implication seems to fly in the face of some conventional wisdom, which says that a presidential candidate can on occasion afford to write off some primaries if he does sufficiently well in others that he chooses to contest.3 We shall deal with this and other issues later when we describe possible modifications in the basic model in Section 4. For now, the salient question is whether the basic model has a solution, and if so what it is.

3. A solution to the basic model

We shall proceed by showing that, when viewed as a constrained maximization problem, the basic model has a closed-form (analytic) equilibrium solution in that, if either player unilaterally deviates a small amount, he decreases his payoff. Our argument will depend on an induction hypothesis that postulates the solution for all primaries i, i = 1, 2,..., n, from which we shall derive the solution for a (n+1)st (prior) Oth primary. By demonstrating that the locally optimal allocation for the Oth primary takes the same form as that for all other primaries i, given a symmetry assumption, we shall prove that the hypothesized set of allocations for all primaries is indeed locally optimal.

The constraint assumptions will be specified in detail below, but we point out here that the presidential campaign-finance law, which took effect for the first time in the 1976 campaign, limits all presidential contenders not only to a certain total amount of spending in all state primaries/caususes but also to certain amounts in each primary/caucus - if they accept matching funds from the government. In fact, all candidates in 1976 and 1980 subscribed to these limits, with the exception of John Connally in the 1980 Republican race, whose case we shall discuss in Section 4. Since the sum of the state spending limits far exceeds the overall limit, the latter is the only serious constraint (except possibly in New Hampshire, which we shall say something about later). Accordingly, we do not incorporate state spending limits as formal constraints in our model, though we shall assume that the locally optimal allocations to each state are, by symmetry, the same for leading contenders A and B who can spend up to the overall limit.

378

S.J. Brams, M.D. Davis / Optimal resource allocation in presidental primaries

Assume candidate A seeks to maximize his expected delegate vote, across all n primaries,

where = the number of delegates of a party in primary k and

(Note that if we assume p≈ 1/2, (1) defines p for k = 1.)

Assume variables a and b are the resources that candidates A and B, respectively, allocate to each primary k, subject to the constraints

(Note that we use 'A' and 'B' both to name the two candidates and to indicate their total resources.)

To find the optimal allocation, a*, candidate A should make to primary i to maximize the objective function, V, make the following induction hypothesis:

Now consider n+1 primaries, which include the 0th primary and ʼn subsequent primaries. Our object is to find two symmetric strategies for A and B which are in local equilibrium.

Assume that A and B make allocations, a and b, respectively, to the Oth primary, and that a = b and allocations to all other primaries are the same, so A = B. Assume also that A and B make their allocations to the remaining primaries in accordance with the induction hypothesis.

Now assume that A unilaterally perturbs his allocation to the 0th primary, either increasing or decreasing it. If = a=b=b are the allocations to the Oth primary, we shall show that such a perturbation from the a calculated to be the optimal allocation in the Oth primary – according to the induction hypothesis - leads to a lowering of the expected delegate vote for A (assuming B's allocations to the remaining n primaries remain unchanged, and A's perturbed allocations remain in proportion to his initial allocations).

Since a = b,

a = b = 1/2

(2)

S.J. Brams, M.D. Davis / Optimal resource allocation in presidental primaries

379

but the distinction between a and b is maintained because when a is perturbed, b remains constant. The effect, we shall show, is to decrease the expected delegate vote for A.