Allocating a campaign across battleground states

Six statements about the model; the mapping to the Electoral College is a reconstruction R, and no statement is an estimate of any campaign's allocation.

• One contest, two objectives. With a tied state decided by a coin, the expected weight $A$ wins is exactly the chapter's score, and the probability that it wins a majority is a second payoff on the same contest. The two objectives are different games: at equal budgets and without a safe state both have value $0$, but their optimal sets differ (Table 2.1 (p. 88)a). The comparison of the two goals is classical in a model of legislative elections (Snyder 1989); the computation here is the book's derivation within Model E1 D [Proposition 2.15 (p. 100)].

• Safe states are invisible to one objective and decisive for the other. Under captured weight a safe state only shifts the payoff by a constant, so the optimal allocation is the same whatever safe states exist; one row of Table 2.1 (p. 88)a covers them all. Under the majority objective a safe state changes the quota and with it the winning sets, and the optimal allocation moves D [Proposition 2.15 (p. 100)(a)].

• Under a majority, size is not pivotality. Without a safe state the minimal winning sets are any two of fields 1–3: field 3, of weight $2$, is exactly as decisive as fields of weight $3$, and field 4 can never change the outcome. The three fields are interchangeable: over the set of optimal strategies each receives the same range of expected units, from $1$ to $\frac{3}{2}$, and every optimal strategy leaves field 4 empty. A single optimal strategy need not treat the three alike; one of them places $\frac{3}{2}$, $1$ and $\frac{3}{2}$ units on them in expectation. Under captured weight the same contest allocates by weight: field 3 receives exactly one unit in expectation, fields 1 and 2 between $\frac{5}{4}$ and $\frac{7}{4}$ D [Proposition 2.15 (p. 100)(b), (c); Figure 2.7 (p. 87)].

• A safe state for either side can make a small state pivotal. If $B$ holds a safe state of weight $2$, the minimal winning sets become $\{1,2\}$, $\{1,3,4\}$ and $\{2,3,4\}$; field 4 is now needed whenever $A$ does not win both heavy fields. Every optimal strategy of $A$ places $\frac{3}{5}$ units on it in expectation, exactly as much as on field 3, and $\frac{7}{5}$ on each of fields 1 and 2; $A$'s value falls to $-\frac{37}{200}$, a win probability of $\frac{63}{200}$. If $A$ holds the safe state the value is $+\frac{37}{200}$, and field 4 is again contested (Table 2.1 (p. 88)a). Which small state matters therefore depends on the states that are not contested at all D [Proposition 2.15 (p. 100)(c)].

• A majority magnifies a budget edge. With five units against four, the richer side's expected share of the weight under captured weight is $\frac{97}{168}$; its probability of a majority is $\frac{235}{336}$. With six against four the figures are $\frac{23}{36}$ and $\frac{20}{23}$. The richer side begins to contest the light state under captured weight at $6:4$, between $\frac{1}{2}$ and $\frac{11}{14}$ units in expectation, but never under the majority objective at these budgets D [Proposition 2.15 (p. 100)(d); Table 2.1 (p. 88)b].

• Concentration is consistent with the model, not explained by it. The model reproduces one mechanism behind the documented pattern: under a majority objective, states that cannot change the outcome receive nothing, and states that one side is sure to win reshape the value of the others. It does not reproduce the main one. In Model E1 every contested state is equally close, so the model is silent on why resources follow the expected closeness of the vote; models of the probability of being pivotal, such as Strömberg's, address that directly. Kovenock and Roberson (2009) argue, in a dynamic model, that neither a battleground-only nor a fifty-state strategy is likely to arise in equilibrium; Model E1 is one-shot and cannot speak to that R.

States are not independent fields: their outcomes move together with a national environment, so that winning one makes winning another more likely, and the model's independent coins have no such correlation. A state is not won by whoever spends more; resources shift probabilities, and the contest function is unknown. Resources are not one budget: candidate time, advertising and field organisation have different costs, limits and returns, and media markets cross state lines. The players are not two: campaigns, parties and outside groups allocate separately. Allocation is sequential and responds to polls, while the model is one shot. Safe states are not given: in the journalistic account above advertising all but leaves a state between two elections, so which states are safe is itself an outcome of strategy and of voters. Maine and Nebraska split their electors, so not every field is winner-take-all. And the instance's sharpest effects (a field that is a dummy, a field that becomes pivotal) depend on the exact stylised weights; with other weights other fields play those roles.

The deepest break is one the campaign case does not show, because a state cannot bargain with the campaigns competing for it: in most allocation contests the field is itself a decision-maker. The episode below is read for that one line, and it is a deliberately imperfect analogy.