Advertising competition in presidential elections

Abstract

Presidential candidates purchase advertising based on each state’s potential to tip the election. The structure of the Electoral College concentrates spending in battleground states, such that a majority of voters are ignored. We estimate an equilibrium model of multimarket advertising competition between candidates that allows for endogenously determined budgets. In a Direct Vote counterfactual, we find advertising would be spread more evenly across states, but total spending levels can either decrease or increase depending on the contestability of the popular vote. Spending would increase by 13 % in the extremely narrow 2000 election, but would decrease by 54 % in 2004. These results suggest that the Electoral College greatly increases advertising spending in typical elections.

Appendices

Appendix A: Estimation in a contest

This appendix describes a general approach to estimating contest models. The method is applicable to any problem where an agent (or agents) exerts observed effort to pass a threshold while facing some unobservables. The general objective function under a threshold goal for agent j ∈ J≥1 choosing effort allocations $A_{j}\in \mathbb {R}_{+}^{M}$ is

$$\underset{A_{j}}{\max}~\pi_{j}(A;\theta)=R_{j}\cdot\mathbb{E}\left[d_{j}\left( A,\eta;\theta\right)\right]- \omega^{\prime}A\,, $$

where $R_{j}\in \mathbb {R}_{\ge 0}$ is the agent’s prize for success, $A\in \mathbb {R}_{\geq 0}^{JM}$ is the collection of effort choices across agents, $\omega \in \mathbb {R}_{\geq 0}^{M}$ is the cost of effort, and θ ∈ Θ^k is a set of structural parameters. d _j ∈{0,1} is the success function that determines whether agent j wins the prize, such that $\mathbb {E}\left [d_{j}\right ]$ is the probability of success that integrates over the unobservable $\eta \in \mathbb {R}^{JM}$ with known CDF F _η . An important component of d _j is the scoring function $s_{jm}\left (A,\eta ;\theta \right )\in \mathbb {R}_{\geq 0}$, which determines the agent’s score given all effort choices and the unobservables. For all m, we assume s _{j m} (A,η;θ) is strictly increasing in A _{j m} and η _{j m} and strictly decreasing in A _{k m} and η _{k m} , for k≠j. These conditions ensure that s(A,η) is invertible in η, which is necessary for the computational approach. We suppress θ for the rest of the exposition.

Our paper estimates θ using GMM based on the FOCs of the agent’s problem with respect to A _j . To compute the FOC, one requires a method to estimate the marginal effect of (say) advertising $\frac {\partial \mathbb {E}\left [d_{j} \left (A,\eta ;\theta ^{v}\right )\right ]}{\partial A_{mj}}$ on the probability of winning the contest. Below we discuss ways to calculate this quantity under different conditions.

Single agent with scalar uncertainty

The simplest application involves a single agent (J = 1) who chooses a single effort level A≥0 (M = 1) with a scalar unobservable η in an attempt to exceed threshold $\overline {V}$. The success function is

$$d\left( A,\eta\right)=1\cdot\left\{ s\left( A,\eta\right)>\overline{V}\right\} \,. $$

The scoring function produces a value v = s(A,η), which is a random variable due to the presence of η. Assume the scoring function can produce a value high enough to exceed the threshold, such that $s(A,\eta )\geq \overline {V}$ for some combination of A and η. Otherwise, the agent is guaranteed to fail and she will not exert any effort. The probability of success can be expressed as

$$\begin{array}{@{}rcl@{}} \mathbb{E}\left[d\left( A,\eta\right)\right] & = & \Pr\left( s\left( A,\eta\right)>\overline{V}\right)\\ & = & 1-F_{v}\left( s\left( A,\eta\right)\leq\overline{V}\right)\,, \end{array} $$

where F _v (⋅) is the unknown CDF of the score v. Let s ⁻¹(A,v) be the inverse of s(A,η) through its second argument. Given the known distribution function for η, we use a change-of-variables to re-express the cumulative distribution F _v in terms of η,

$$\begin{array}{@{}rcl@{}} F_{v}\left( s\left( A,\eta\right)\leq\overline{V}\right) & = & F_{\eta}\left( s^{-1}\left( A,v\right)\leq s^{-1}\left( A,\overline{V}\right)\right)\,, \end{array} $$

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$$\begin{array}{@{}rcl@{}} & = & F_{\eta}\left( \eta\leq\eta^{*}\left( A,\overline{V}\right)\right)\,, \end{array} $$

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where $\eta ^{*}\equiv \eta ^{*}\left (A,\overline {V}\right )=s^{-1}\left (A,\overline {V}\right )$ is the critical value of the shock that equates the score and the threshold, such that $s\left (A,\eta ^{*}\right )=\overline {V}$. Thus, given a particular value of A , the marginal effect of A on the probability of success is:

$$\begin{array}{@{}rcl@{}} \frac{\partial\mathbb{E}\left[d\left( A,\eta\right)\right]}{\partial A} & = & \frac{\partial\left[1-F_{\eta} \left( \eta\leq\eta^{*}\left( A,\overline{V}\right)\right)\right]}{\partial A}\\ & = & -f_{\eta}\left( \eta^{*}\left( A,\overline{V}\right)\right)\frac{\partial s^{-1}\left( A, \overline{V}\right)}{\partial A}\,. \end{array} $$

If s ⁻¹(A,v) and ∂ s ⁻¹(A,v)/∂ A have closed forms, then knowledge of f _η yields an analytic expression for the marginal effect of effort.

Multiple markets and a competing agent

Our presidential election application involves an extension to a competing agent and multiple markets, where each market requires an action and has an agent-market specific unobservable. The success function in such a contest is:

$$\begin{array}{@{}rcl@{}} d_{j}\left( A,\eta\right) & = & 1\cdot\left\{ s_{j}\left( A,\eta\right)>s_{k}\left( A,\eta\right)\right\} \,. \end{array} $$

The scoring function is

$$s_{j}\left( A,\eta\right)=\underset{m}{\sum}~s_{mj}\left( A_{m},\eta_{m}\right)\,, $$

where A _m and η _m are those values relevant to market m across all agents. For a given market, we can rewrite the success function in the following way:

$$\begin{array}{@{}rcl@{}} d_{j}\left( A,\eta\right) & = & 1\cdot\left\{ s_{mj}\left( A_{m},\eta_{m}\right)-s_{mk}\left( A_{m},\eta_{m}\right)> \underset{n\neq m}{\sum}~s_{nk}\left( A_{n},\eta_{n}\right)-s_{nj}\left( A_{n},\eta_{n}\right)\right\} \\ & = & 1\cdot\left\{ IM_{mj}\left( A_{m},\eta_{m}\right)>EM_{mj}\left( A_{-m},\eta_{-m}\right) \right\} \end{array} $$

Intuitively, an agent’s score relative to his opponent’s in market m must be greater than his deficit across all other markets. We refer to these terms as the internal margin I M _{m j} and external margin E M _{m j} . The inversion to obtain a critical η ^∗ is now in terms of the internal margin instead of a single agent’s score. Note that an inverse of I M _{m j} through η _{m j} exists due to our shape assumptions on s(A,η). The probability of success can now be expressed as

$$\begin{array}{@{}rcl@{}} \mathbb{E}\left[d_{j}\left( A,\eta\right)\right] & = & \Pr\left( s_{j}\left( A,\eta\right)>s_{k}\left( A,\eta \right)\right)\\ & = & 1-F_{IM}\left( IM_{mj}\left( A_{m},\eta_{m}\right)\leq EM_{mj}\left( A_{-m},\eta_{-m}\right)\right)\\ & = & 1-F_{\eta}\left( IM_{mj}^{-1}\left( A_{m},s_{mj}-s_{mk},\eta_{mk}\right)\leq IM_{mj}^{-1}\left( A_{m},EM_{mj} \left( A_{-m},\eta_{-m}\right),\eta_{mk}\right)\right)\\ & = & 1-F_{\eta}\left( \eta_{mj}\leq IM_{mj}^{-1}\left( A_{m},\eta_{mj}^{*},\eta_{mk}\right)\right) \end{array} $$

Given values for A _−m and η _−m , E M _{m j} is just a scalar representing agent j’s deficit or surplus outside market m. Thus the critical value of agent j’s shock in market m is the $\eta _{mj}^{*}\equiv \eta _{mj}^{*}\left (A_{m},EM_{mj},\eta _{mk}\right )$ that sets

$$IM_{mj}\left( A_{m},\eta_{mj}^{*}\left( A_{m},EM_{mj},\eta_{mk}\right),\eta_{mk}\right)=EM_{mj}\left( A_{-m}, \eta_{-m}\right)\,. $$

We combine this result with Monte Carlo simulation over the opponent’s shock inside market m and the collection of shocks outside market m to produce an estimate of the marginal effect of A _{j m} . The key steps are:

• 1. Simulate r = 1,…,N S draws over $\eta _{mk}^{r}$ and $\eta _{-m}^{r}=\left \{ \eta _{-nj}^{r}, \eta _{-nk}^{r} \right \}_{n\neq m}$ using the known distributions $f_{\eta _{mj}}$. In our application, these are N(0,σ ²), which can be generalized at higher computational cost.

• 2. Use $\eta _{-m}^{r}$ to calculate the JM values of $EM_{mj}^{r}\equiv EM_{mj}^{r}\left (A_{-m}, \eta _{-m}^{r} \right )$, given some particular value A _−m .

• 3. Solve for the critical values $\eta _{mj}^{r*}\equiv \eta _{mj}^{r*}\left (A_{m},EM_{mj}^{r},\eta _{mk}^{r}\right )$ that set $IM_{mj}^{r}\left (A_{m},\eta _{mj}^{r*},\eta _{mk}^{r}\right )=EM_{mj}^{r}$ for all m and j.

• 4. The estimate of the marginal effect of A _{m j} is:

  $$ \widehat{\frac{\partial\mathbb{E}\left[d\left( A,\eta\right)\right]}{\partial A_{mj}}}=\frac{1}{NS} \underset{r}{\sum}-f_{\eta_{mj}} \left( \eta_{mj}^{r*}\right)\frac{\partial\eta\left( A_{m},EM_{mj}^{r}, \eta_{mk}^{r}\right)}{\partial A_{mj}}\,. $$

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  where $\partial \eta (\cdot )/\partial A_{mj}\equiv \partial IM_{mj}^{-1}(\cdot )/\partial A_{mj}$ and evaluated at $\eta _{mj}=\eta _{mj}^{r*}$.

The next two sections discuss the details of implementing this general form of the derivative in the Direct Vote and Electoral College, respectively.

Application to a Direct Vote

The realities of an election place certain restrictions on the empirical analogues of the score functions. Under a Direct Vote, Eq. 16 describes the derivative if it is always feasible for the internal margin of votes to exceed the external margin of votes in all markets. In practice, bounds on the internal margin are implied by the number of voters in the market, giving us the following expression for the derivative in a Direct Vote:

$$\widehat{\frac{\partial\mathbb{E}\left[d\left( A,\eta\right)\right]}{\partial A_{mj}}}=\frac{1}{NS}\underset{r}{\sum} -f_{\eta_{mj}}\left( \eta_{mj}^{r*}\right)\frac{\partial\eta\left( A_{m},EM_{mj}^{r},\eta_{mk}^{r}\right)}{\partial A_{mj}}1\cdot\left\{ N_{m}>EM_{mj}^{r}\right\} $$

where $N_{m}={\sum }_{c\in C_{m}}N_{c}$ is the total number of voters in the market. If $N_{m}<EM_{mj}^{r}$, then there does not exist a $\eta _{mj}^{r*}$ to equate the internal and external margins and the derivative at the r ^th draw is zero.

Using the notation from the body of the paper, the marginal effect of advertising on the probability of winning is:

$$\begin{array}{@{}rcl@{}} \widehat{\frac{\partial\mathbb{E}\left[\tilde{d}\left( A,\bar{\delta},\boldsymbol{\eta};\alpha\right)|\sigma\right]} {\partial A_{mj}}} & = & \frac{1}{NS}\underset{r}{\sum}-f_{\eta_{mj}}\left( \eta_{mj}^{r*}|\sigma\right) \left( \frac{\partial\eta_{mj}\left( A_{m},EM_{mj}^{r},\eta_{mk}^{r}\right)}{\partial A_{mj}}\right).\\ & = & \frac{1}{NS}\underset{r}{\sum}~f_{\eta_{mj}}\left( \eta_{mj}^{r*}|\sigma\right)\frac{\alpha}{1+A_{mj}} \end{array} $$

where $EM_{mj}^{r}\equiv EM_{mj}^{r}\left (A_{-m},\eta _{-m}^{r}\right )$ is the external margin in market m and $\eta _{mj}^{r*}=\eta _{mj}\left (A_{m},EM_{mj}^{r},\eta _{mk}^{r}\right )$ is the critical value of the shock that equates the external and internal margins. We have suppressed an indicator function that sets the derivative for the r ^th draw equal to zero when $N_{m}<E{M_{m}^{r}}$. Note that an interior solution requires us to solve a system of J M(1 + N S) equations in as many unknowns. Setting N S = 10, 000 helps ensure an accurate Monte Carlo approximation to the integral, but makes computing the equilibrium nontrivial.

Application to the Electoral College

The Electoral College introduces an added layer of complexity because of the state-level contests. We need to introduce additional notation corresponding to multiple markets intersecting a single state and markets intersecting multiple states. Let M _s(m) be the set of all markets intersecting state s(m), where m is the focal market for the derivative. Let C _{m s} denote the set of counties intersecting both state s and market m. Finally, let $\tilde {S}\left (m\right )$ denote the set of states that market m intersects.

The Electoral College involves two relevant thresholds for winning an election: (i) the electoral vote margin and (ii) the state-level popular vote margin. For a given r, the derivative is non-zero only if the electoral votes attainable through advertising in market m are greater than the electoral vote deficit implied by the shocks in all other markets:

$$\underset{s\in\tilde{S}\left( m\right)}{\sum}~E_{s}>\underset{s^{\prime}\notin\tilde{S}\left( m\right)}{\sum} E_{s^{\prime}k}\left( \eta_{s^{\prime}}^{r}\right)-E_{s^{\prime}j}\left( \eta_{s^{\prime}}^{r}\right)\,. $$

If a market is “in play” based on the electoral votes, then the state-level margin can be separated into an internal and external margin as above. If a market intersects multiple states there is a potentially relevant internal and external margin for each:

$$\begin{array}{@{}rcl@{}} IM_{mj}^{rs} & = & \underset{c\in C_{ms}}{\sum} N_{c}\left( s_{cj}(A_{m},\eta_{mj},\eta_{mk})-s_{ck}(A_{m},\eta_{mj}, \eta_{mk})\right)\\ EM_{mj}^{rs} & = & \underset{n\in M_{s\left( m\right)}\setminus m}{\sum}~~\underset{c\in C_{ns}}{\sum} N_{c} \left( s_{ck}(A_{m},\eta_{n})-s_{cj}(A_{m},\eta_{n})\right)\,. \end{array} $$

We therefore calculate an $\eta _{mj}^{*rs}$ , as described above, for each state $s\in \tilde {S}\left (m\right )$. Next, we define the relevant critical value, $\eta _{mj}^{*r}$, to be the smallest of these shocks that yields enough electoral votes to offset the electoral vote deficit implied by the r ^th set of draws.

The derivative for the r ^th draw is therefore:

$$\begin{array}{@{}rcl@{}} \frac{\partial\mathbb{E}\left[d\left( A,\eta\right)\right]}{\partial A_{mj}}^{r}=\left\{\begin{array}{ll} -f_{\eta_{mj}}\left( \eta_{mj}^{r*}\right)\frac{\partial\eta\left( A_{m},EM_{mj}^{r},\eta_{mk}^{r}\right)}{\partial A_{mj}} & \text{if } {\sum}_{c\in C_{ms}} N_{c}>EM_{mj}^{r}\\ & \text{and}\underset{s\in\tilde{S}\left( m\right)}{\sum} E_{s}>\underset{s^{\prime}\notin\tilde{S}\left( m\right)}{\sum} E_{s^{\prime}k}\left( \eta_{s^{\prime}}^{r}\right)-E_{s^{\prime}j}\left( \eta_{s^{\prime}}^{r}\right)\\ 0 & \text{otherwise} \end{array}\right. \end{array} $$

and the overall derivative is

$$\widehat{\frac{\partial\mathbb{E}\left[d\left( A,\eta\right)\right]}{\partial A_{mj}}}=\frac{1}{NS} \underset{r}{\sum}\frac{\partial\mathbb{E}\left[d\left( A,\eta\right)\right]}{\partial A_{mj}}^{r}\,. $$

Using the notation from the body of the paper, the marginal effect of advertising is equal to:

$$\begin{array}{@{}rcl@{}} \widehat{\frac{\partial\mathbb{E}\left[d_{j}\left( A,\bar{\delta},\boldsymbol{\eta};\alpha\right)|\sigma\right]} {\partial A_{mj}}} & = & \frac{1}{NS}\underset{r}{\sum}f\left( \eta_{mj}^{r*}|\sigma\right)\frac{-\partial \eta \left( A_{m},EM_{mj}^{r},\eta_{mk}^{r}\right)}{\partial A_{mj}}\\ & = & \frac{1}{NS}\underset{r}{\sum}f\left( \eta_{mj}^{r*}|\sigma\right)\frac{\alpha}{1+A_{mj}}\\ & & \quad\quad\text{if}\,\,E_{s}>\underset{\ell\neq s}{\sum}E_{\ell k}\left( A_{\ell},\eta_{\ell}^{r}\right)-E_{\ell j}\left( A_{\ell},\eta_{\ell}^{r}\right) \end{array} $$

where ∂ η(⋅,⋅,⋅)/∂ A _{m j} is evaluated at $\eta _{mj}^{r*}$. The derivative of η(⋅) with respect to A _{m j} in our application is $\frac {-\alpha }{1+A_{mj}}$ because A _{m j} and η _{m j} are perfectly substitutable within the utility function as follows: $\delta _{cj}=\bar {\delta }_{cj}+ {\alpha }\log \left (1+A_{mj}\right )+\eta _{mj}$. Intuitively, the derivative we seek equals the probability of drawing a critical value $\eta _{mj}^{*}$ times the derivative of this critical value with respect to advertising. The condition on the right requires that the state be pivotal in the election’s outcome: the number of electoral votes at stake, E _s , must be larger than the candidate’s electoral deficit outside that state, otherwise the derivative at the r ^th draw is zero. A benefit of this approach is that the Monte Carlo integration is effectively over η _{m k} and E M _{m j} , combined with an analytic expression for $\eta _{mj}^{*}$, instead of the original M×J dimensional integral.

Note that the above characterizes the marginal effect of effort in a contest with a general contest success function. One primary focus of the theoretical contest literature has been on the derivation of analytically tractable success functions (Skaperdas 1996). In practice, contests such as elections have their own specific success functions implying CDFs for the probability of success that may inherently not be analytically tractable. We show that the above approach is beneficial by compressing a large multidimensional integration problem into unidimensional external and internal margins.

Appendix B: Advertising and advertising prices

We construct a market-candidate observed aggregate advertising level and advertising price (A _{m j} and w _{m j} ) based on two observed variables. Expenditure _{m j a d} is CMAG’s estimate of the dollars spent by candidate j in market m on an advertisement a in daypart d. CPP _{m d} is SQAD’s reported advertising price for the 18 and over demographic in market m during daypart d. We use the CPP from the 3rd quarter of the election year.²⁹

Let the daypart level of advertising by candidate j in market m be:

$$GRP_{mjd}=\frac{{\sum}_{a\in\mathbb{A}_{mjd}}\,\text{Expenditure}_{mjad}}{{\sum}_{d=1}^{8}CPP_{md}} $$

where $\mathbb {A}_{tmjd}$ is the set of advertisements for a candidate in a market and daypart. Then total advertising by candidate j in market m is:

$$A_{mj}=\sum\limits_{d=1}^{8}GRP_{mjd}. $$

The market-specific advertising price for candidate j is defined as follows:

$$w_{mj}=\left\{\begin{array}{ll} CPP_{md}\frac{GRP_{mjd}}{A_{mj}} & \text{ if }A_{mj}>0\\ CPP_{m} & \text{ if } A_{mj}=0 \end{array}\right. $$

where

$$CPP_{m}=\sum\limits_{d=1}^{8}\left[CPP_{md}\frac{{\sum}_{j=1}^{J}{\sum}_{m=1}^{M}GRP_{mjd}}{{\sum}_{j=1}^{J}{\sum}_{m=1}^{M} {\sum}_{d=1}^{8}GRP_{mjd}}\right]. $$

In other words, we use a weighted average across the dayparts in which candidate j advertised in market m if the candidate did in fact advertise there, or a weighted average based on both candidates advertising in all markets within each daypart if the candidate did not advertise in the market.

The advertising price in our candidate-side estimation is ω _{m j} = w _{m j} + v _{m j} where v _{m j} is the candidate’s market-specific unobservable component of advertising. (Recall that the SQAD prices are forecasts) When we analyze the cost per marginal vote, we use C P P _m in all markets to highlight the role of diminishing marginal effectiveness and political leaning in the costs of acquiring an additional vote. Finally, when we solve the Direct Vote counterfactual, we use w _{m j} as the price of advertising. This avoids odd implications from large local residuals that likely do not relate to costs, but retains a source of local variation in advertising. We remove both the candidate and local market ad price variation in the final simulation by setting an equal price per thousand people (CPM) such that $\tilde {w}_{mj}=((\frac {1}{2M}{\sum }_{j=1}^{2}{\sum }_{m=1}^{M}CPM{}_{mj})\times Pop)/100$.

Appendix C: A comparison of voter turnout in the Electoral College and the Direct Vote in 2000

Turnout in the 2000 Direct Vote increases by 0.9 %, or about 1.8 million voters. The popular vote in four states—Iowa, New Mexico, Oregon, and Wisconsin, all with thin margins—flips from Gore to Bush. Gore, however, gains enough votes in the Democratic stronghold of California to win the election even though his national vote margin shrinks from about 543,000 to 390,000.

An important distinction between the Electoral College and a Direct Vote is a state’s relative influence in the election outcome. Under the Electoral College, a state’s influence is fixed and proportional to its fraction of the total electoral votes.³⁰ The Electoral College essentially protects states from political losses if a state implements policies that make it more difficult or disqualifies certain voters from casting their votes. Furthermore, the winner-take-all rule gives partisan members of a state’s government strong motivation to influence voter turnout to favor their own political party (as witnessed recently in the form of voter identification and anti-voter fraud laws proposed in many states).

In contrast, in a Direct Vote, a state’s relative influence in the election outcome is endogenous—it is proportional to the percent of its population that turns out to vote relative to national voter turnout. Figure 7 depicts the difference in representation of a state between each electoral mechanism and the representation that their population constitutes as percentage of the US population over age 18. States are ordered on the left axis by increasing size of their voting age population. On the top, the series of positive bars reflect the electoral college’s protection of small states. On the bottom, large states such as California, Texas and Florida are under-represented in both the electoral college and a Direct Vote. Under-representation in the Direct Vote arises from a smaller fraction of the state’s voting age population actually voting. Other states such as Georgia, Arizona and Nevada also are under-represented in a Direct Vote. Minnesota, Wisconsin, Michigan and Ohio are however over-represented in a Direct Vote. A Direct Vote therefore eliminates both the electoral college’s protection of small states and the tie in to state population size, as a state is now represented only by its voters turning out for the election.

Fig. 7

States’ election influence under the Electoral College and Direct Vote in 2000: the horizontal axis reports the difference between a state’s relative influence in the election outcome under a particular electoral system relative to the state’s voting-age population. Under the Electoral College, a state’s influence is its number of electoral votes divided by the total number of electoral votes in the country. Under a Direct Vote, a state’s influence is its voter turnout divided by the total voter turnout in the country. Bars to the left of zero indicate that a state has less influence under that system relative to its share of the total voting-age population. States are sorted from top to bottom in order of ascending population