Modelling the Effect of Campaign Advertising on US Presidential Elections

Abstract

We provide a stochastic electoral model of the US Presidential election. The availability of smart phone and social media coupled with large data set on voters’ personal characteristics and information has given candidates the ability to send messages directly to voters. In our model, candidates directly communicate with voters. Prior to the election candidates announce their policies and advertising campaigns. Voters care about candidates’ policies relative to their ideal policy and about the messages candidates sent relative to their ideal message frequency, called the campaign tolerance level. The electoral mean is a strick (weak) local Nash equilibrium (LNE) of the election if the expected vote shares of all candidates are greater than the sufficient (necessary) pivotal vote shares. The sufficient pivotal vote share rises when voters give greater weight to the policy or advertising dimensions. The necessary pivotal vote share may increase or decrease in the importance votes give to the policy or advertising dimensions. If the expected vote share of at least one candidate is lower than the necessary pivotal vote share, then the electoral mean is not a LNE of the election.

Appendix

1.1 Second Order Conditions (SOC)

Section 5 derived candidates’ best response policies and ad campaign functions and the critical values satisfying the first order conditions for a maximum of the vote share functions. Lemma 1 proved that if candidates adopt the same electoral campaign, they adopt the electoral mean, (z _m , a _m ), as their campaign strategy. To evaluate if candidates’ vote share functions are at a maximum at (z _m , a _m ) we use the Hessian matrix of second order partial derivatives of the vote share functions of each candidate evaluated at (z _m , a _m ).

Before proceeding recall from Sect. 6 that when all candidates adopt the electoral mean as their campaign strategy, the probability that i votes for j, ρ _ij in (4) when evaluated at (z _m , a _m ), is independent of voters’ ideal policies and campaign tolerance levels as well as independent of candidates’ policies and ad campaigns and gives j’s expected vote share so that from (16),

$$\displaystyle{ \rho _{ij}^{m} =\rho _{ j}^{m} = V _{ j}^{m}\mbox{ for all }i \in \mathcal{N}. }$$

(31)

1.1.1 Proof of Proposition 1

We use the Hessian matrix of second order partial derivatives of candidate j to determine whether j’s vote share function, V _j in (5), is at a maximum at (z _m , a _m ) holding the electoral campaigns of all other candidates at (z _m , a _m ). To find this Hessian we need the second order partial derivatives of the probability that i votes for j, D ² ρ _ij evaluated at (z _m , a _m ), D ² ρ _ij ^m, using ρ _ij in (4). Then using D ² ρ _ij ^m we find the Hessian of j’s vote share, D ² V _j at (z _m , a _m ).

Suppose all candidates adopt the electoral mean, (z _m , a _m ) as their campaign strategy. Using (31), the partial derivatives of (7) wrt z _j and a _j evaluated at (z _m , a _m ), i.e., D ² ρ _ij ^m, is given by

$$\displaystyle\begin{array}{rcl} \mathbf{D}^{2}\rho _{ ij}^{m}& \equiv & \mathbf{D}^{2}\rho _{ ij}(\mathbf{z}_{m},\mathbf{a}_{m}) \equiv \left (\begin{array}{cc} \frac{\partial ^{2}\rho _{ ij}} {\partial (z_{j})^{2}} & \frac{\partial ^{2}\rho _{ ij}} {\partial z_{j}\partial a_{j}} \\ \frac{\partial ^{2}\rho _{ ij}} {\partial z_{j}\partial a_{j}} & \frac{\partial ^{2}\rho _{ ij}} {\partial (a_{j})^{2}} \end{array} \right ) \\ & =& 2\rho _{j}^{m}(1 -\rho _{ j}^{m})\left [2(1 - 2\rho _{ j}^{m})\mathbf{BE}_{ (\mathbf{z}_{m},\mathbf{a}_{m})}^{ij\mathbf{T}}\mathbf{E}_{ (\mathbf{z}_{m},\mathbf{a}_{m})}^{ij}\mathbf{B} -\mathbf{B}\right ] \\ & =& 2\rho _{j}^{m}(1 -\rho _{ j}^{m})\left [2(1 - 2\rho _{ j}^{m})\boldsymbol{\nabla }_{ i}^{m} -\mathbf{B}\right ] {}\end{array}$$

(32)

$$\displaystyle{ \text{where }\mathbf{B} \equiv \left (\begin{array}{cc} \beta &0\\ 0 & e \end{array} \right ),\quad \mathbf{E}_{(\mathbf{z}_{m},\mathbf{a}_{m})}^{ij} \equiv \left [\begin{array}{cc} (x_{ i} - z_{m})&(t_{i} - a_{m}) \end{array} \right ]. }$$

(33)

and where a vector with a T denotes the transpose of that vector. The diagonal of B shows the importance voters give the policy and advertising dimensions and $\mathbf{E}_{(\mathbf{z}_{m},\mathbf{a}_{m})}^{ij}$ the differences between i’s ideals (x _i , t _i ) and j’s electoral campaign at the electoral mean (z _m , a _m ). The matrix $\boldsymbol{\nabla }_{i}^{m}$ in (32) defined by

$$\displaystyle{ \boldsymbol{\nabla }_{i}^{m} \equiv \mathbf{BE}_{ (\mathbf{z}_{m},\mathbf{a}_{m})}^{ij\mathbf{T}}\mathbf{E}_{ (\mathbf{z}_{m},\mathbf{a}_{m})}^{ij}\mathbf{B} \equiv \left (\begin{array}{cc} \beta (x_{i} - z_{m})^{2}\beta & \beta (x_{ i} - z_{m})(t_{i} - a_{m})e \\ \beta (x_{i} - z_{m})(t_{i} - a_{m})e& e(t_{i} - a_{m})^{2}e \end{array} \right ) }$$

(34)

gives the weighted variance-covariance matrix of i ’s ideals around the electoral mean (z _m , a _m ).

Using D ² ρ _ij ^m in (32), the Hessian of second order partial derivatives of V _j in (5) wrt z _j and a _j , evaluated at (z _m , a _m ) is given by

$$\displaystyle\begin{array}{rcl} \mathbf{D}^{2}V _{ j}^{m}& \equiv & \mathbf{D}^{2}V _{ j}(\mathbf{z}_{m},\mathbf{a}_{m}) \equiv \frac{1} {n}\sum \nolimits _{i\in \mathcal{N}}\mathbf{D}^{2}\rho _{ ij}^{m} \\ & =& \frac{1} {n}\sum \nolimits _{i\in \mathcal{N}}2\rho _{j}^{m}(1 -\rho _{ j}^{m})\left [2(1 - 2\rho _{ j}^{m})\boldsymbol{\nabla }_{ i}^{m} -\mathbf{B}\right ] \\ & =& 2\rho _{j}^{m}(1 -\rho _{ j}^{m})\left \{2(1 - 2\rho _{ j}^{m})\left ( \frac{1} {n}\sum \nolimits _{i\in \mathcal{N}}\boldsymbol{\nabla }_{i}^{m}\right ) -\mathbf{B}\right \}.{}\end{array}$$

(35)

Note that averaging $\boldsymbol{\nabla }_{i}^{m}$ in (34) across all voters, we obtain the matrix

$$\displaystyle{ \boldsymbol{\nabla }^{m} \equiv \frac{1} {n}\sum \nolimits _{i\in \mathcal{N}}\boldsymbol{\nabla }_{i}^{m}, }$$

(36)

i.e., the weighted variance-covariance matrix of the distribution of voters’ ideal policies and campaign tolerance levels around the electoral mean where the weights are given by B in (33).

After substituting $\boldsymbol{\nabla }^{m}$ from (36) into (35), we get that j’s Hessian matrix is given by

$$\displaystyle{ \mathbf{D}^{2}V _{ j}^{m} = 2\rho _{ j}^{m}(1 -\rho _{ j}^{m})\left \{2(1 - 2\rho _{ j}^{m})\boldsymbol{\nabla }^{m} -\mathbf{B}\right \} }$$

(37)

Candidate j adopts the electoral mean, (z _m , a _m ), as its campaign strategy, iff j’s vote share is at a maximum at (z _m , a _m ). Hence, to show that (z _m , a _m ) for all $j \in \mathcal{C}$ is a strick (weak) local Nash equilibrium [S(W)LNE] of the election we must find the conditions under which j’s Hessian matrix, D ² V _j in (35) when evaluated at (z _m , a _m ), D ² V _j ^m, is negative definite (semi-definite) for all $j \in \mathcal{C}$ which happens only when its eigenvalues are negative (non-positive) for all $j \in \mathcal{C}$, that is, iff the trace and the determinant of D ² V _j ^m are respectively negative and positive as in this case the eigenvalues of D ² V _j ^m are both negative.

Before deriving the trace and the determinant of D ² V _j ^m and to simplify the calculations, we give the following definitions.

Definition 5

Let C _j ^m ≡ C _j (z _m ,a _m ) be the characteristic matrix of candidate j when all candidates adopt the electoral mean as their campaign strategy (z _m ,a _m ), i.e.,

$$\displaystyle\begin{array}{rcl} \mathbf{C}_{j}^{m}& \equiv & \mathbf{C}_{ j}(\mathbf{z}_{m}\mathbf{,a}_{m}) \equiv A_{j}^{m}\boldsymbol{\nabla }^{m} -\mathbf{B}{}\end{array}$$

(38)

$$\displaystyle\begin{array}{rcl} \text{where }A_{j}^{m}& \equiv & A_{ j}^{m}(\mathbf{z}_{ m}\mathbf{,a}_{m}) \equiv 2(1 - 2\rho _{j}^{m}){}\end{array}$$

(39)

is the characteristic factor of C _j ^m, ∇ ^m is given by (36), ρ _j ^m by (31) and B by (33).

Let q _j ^m be the joint probability that voters vote for j and for any other candidate at (z _m ,a _m ),

$$\displaystyle{ q_{j}^{m} \equiv q_{ j}(\mathbf{z}_{m}\mathbf{,a}_{m}) \equiv \rho _{j}^{m}(1 -\rho _{ j}^{m}). }$$

(40)

Using (39) and (40), D ² V _j ^m in (37) can be expressed as a function of C _j ^m in (38), i.e.,

$$\displaystyle{ \mathbf{D}^{2}V _{ j}^{m} = 2q_{ j}^{m}\left \{A_{ j}^{m}\boldsymbol{\nabla }^{m} -\mathbf{B}\right \} \equiv 2q_{ j}^{m}\mathbf{C}_{ j}^{m}. }$$

(41)

So that the trace of D ² V _j ^m in (38) is then given by

$$\displaystyle{ Tr\left (\mathbf{D}^{2}V _{ j}^{m}\right ) = 2q_{ j}^{m}Tr(\mathbf{C}_{ j}^{m}). }$$

(42)

Since q _j ^m in (40) is always positive, $Tr\left (\mathbf{D}^{2}V _{j}^{m}\right ) < 0$ iff Tr(C _j ^m) < 0.

The trace of C _j ^m in (38) is given by

$$\displaystyle{ Tr(\mathbf{C}_{j}^{m}) \equiv Tr\left [A_{ j}^{m}\boldsymbol{\nabla }^{m} -\mathbf{B}\right ] = A_{ j}^{m}Tr\left (\boldsymbol{\nabla }^{m}\right ) - Tr(\mathbf{B}) }$$

(43)

where $Tr(\boldsymbol{\nabla }^{m})$ is given by (27), Tr(B) by (23) and A _j ^m by (39). From (43), Tr(C _j ^m) < 0 when

$$\displaystyle{ A_{j}^{m} < \frac{Tr(\mathbf{B})} {Tr\left (\boldsymbol{\nabla }^{m}\right )}. }$$

(44)

After substituting A _j ^m from (39) into (44) and some manipulation, we obtain

$$\displaystyle{ \frac{1} {2} -\frac{1} {4} \frac{Tr(\mathbf{B})} {Tr\left (\boldsymbol{\nabla }^{m}\right )} <\rho _{ j}^{m} }$$

(45)

The left hand side (LHS) of (45) is independent of candidates’ policies and ad campaigns and is thus the same for all candidates. Define the LHS of (45) as the necessary pivotal (np) vote share at the electoral mean, (z _m , a _m ), i.e.,

$$\displaystyle{ \mathcal{V}_{np}^{m} \equiv \frac{1} {2} -\frac{1} {4} \frac{Tr(\mathbf{B})} {Tr\left (\boldsymbol{\nabla }^{m}\right )} = \frac{1} {2} -\frac{1} {4} \frac{\beta +e} {\frac{1} {n}\sum \nolimits _{i\in \mathcal{N}}\beta (x_{i} - z_{m})^{2}\beta + \frac{1} {n}\sum \nolimits _{i\in \mathcal{N}}e(t_{i} - a_{m})^{2}e} }$$

(46)

where the numerator in the last term follows from (23) and the denominator from (27).

Since ρ _j ^m in (31) is independent of candidates’ policies and ad campaigns and is the same for all voters then ρ _j ^m also gives j’s expected vote share at the electoral mean, i.e., V _j ^m = ρ _j ^m. Therefore, condition (45) can be re-written as

$$\displaystyle{ \mathcal{V}_{np}^{m} \equiv \frac{1} {2} -\frac{1} {4} \frac{Tr(\mathbf{B})} {Tr\left (\boldsymbol{\nabla }^{m}\right )} <\rho _{ j}^{m} = V _{ j}^{m} }$$

(47)

This condition says that if j’s expected vote share at (z _m , a _m ), V _j ^m, is higher than the necessary pivotal vote share, $\mathcal{V}_{np}^{m}$, then Tr(C _j ^m) < 0. If for some candidate $j \in \mathcal{C}$, $\mathcal{V}_{np}^{m} >\rho _{ j}^{m} = V _{j}^{m}$, then for this candidate Tr(C _j ^m) > 0.

Therefore, the necessary condition for j to adopt (z _m , a _m ) as its electoral campaign, when all other candidates adopt the electoral mean as their electoral campaign, is

$$\displaystyle{ \mathcal{V}_{np}^{m} <\rho _{ j}^{m} = V _{ j}^{m}. }$$

(48)

That is, candidate j must expect a high enough vote share at (z _m , a _m ) for j to adopt the electoral mean as its campaign strategy.

The sufficient SOC for j to converge to (z _m , a _m ) is that the eigenvalues of D ² V _j ^m in (37) be both negative implying that the determinant of D ² V _j ^m must be positive at (z _m , a _m ), i.e., $\det \left (\mathbf{D}^{2}V _{j}^{m}\right ) > 0$. From (37), the determinant of D ² V _j when evaluated at (z _m ,a _m ) is given by

$$\displaystyle{ \det \left (\mathbf{D}^{2}V _{ j}^{m}\right ) =\det \left [2q_{ j}^{m}\left \{A_{ j}^{m}\boldsymbol{\nabla }^{m} -\mathbf{B}\right \}\right ] =\det \left [2q_{ j}^{m}A_{ j}^{m}\boldsymbol{\nabla }^{m} - 2q_{ j}^{m}\mathbf{B}\right ] }$$

After some manipulation and using (23) and (27), det(D ² V _j ^m) becomes

$$\displaystyle\begin{array}{rcl} \det (\mathbf{D}^{2}V _{ j}^{m})& =& 4\left [q_{ j}^{m}A_{ j}^{m}\right ]^{2}\det (\boldsymbol{\nabla }^{m}) \\ & & +4\left [q_{j}^{m}\right ]^{2}\det (\mathbf{B}) \times \left \{1 - A_{ j}^{m}Tr\left ((\mathbf{B})^{-1}\boldsymbol{\nabla }^{m}\right )\right \}{}\end{array}$$

(49)

From the triangle inequality, the determinant of the variance-covariance matrix of voters’ policies and campaign tolerance levels around the electoral mean, $\det (\boldsymbol{\nabla }^{m})$ is always non-negative. The determinant of D ² V _j ^m is positive iff the last term in (49) is positive. Since det(B) = β e > 0 and since from (40) q _j ^m > 0, the determinant of D ² V _j ^m is positive iff the term in curly brackets is positive, i.e., iff

$$\displaystyle{ A_{j}^{m} < \frac{1} {Tr\left [(\mathbf{B})^{-1}\boldsymbol{\nabla }^{m}\right ]}. }$$

(50)

After substituting A _j ^m from (39) into (50) and some manipulation, we obtain

$$\displaystyle{ \frac{1} {2} -\frac{1} {4} \frac{1} {Tr\left [(\mathbf{B})^{-1}\boldsymbol{\nabla }^{m}\right ]} <\rho _{ j}^{m}. }$$

(51)

Note that the LHS of (51) is the same for all candidates. Define the LHS of (51) as the sufficient pivotal (sp) vote share at (z _m , a _m ), i.e.,

$$\displaystyle{ \mathcal{V}_{sp}^{m}\,\equiv \,\frac{1} {2} -\frac{1} {4} \frac{1} {Tr\left [(\mathbf{B})^{-1}\boldsymbol{\nabla }^{m}\right ]}\,=\,\frac{1} {2} -\frac{1} {4} \frac{1} {\frac{1} {\beta } \frac{1} {n}\sum \nolimits _{i\in \mathcal{N}}\beta (x_{i} - z_{m})^{2}\beta + \frac{1} {e} \frac{1} {n}\sum \nolimits _{i\in \mathcal{N}}e(t_{i} - a_{m})^{2}e} }$$

(52)

where the denominator in the last term follows from (28).

As before recall that j’s expected vote share at the electoral mean is V _j ^m = ρ _j ^m. Therefore, using (51) we have that the sufficient condition for j to adopt (z _m , a _m ) as electoral campaign—when all other candidates adopt (z _m , a _m ) as their electoral campaign—is that

$$\displaystyle{ \mathcal{V}_{sp}^{m} <\rho _{ j}^{m} = V _{ j}^{m}, }$$

(53)

i.e., V _j ^m must be high enough at (z _m ,a _m ) for j to adopt to the electoral mean as its electoral campaign. ■

For the necessary part of the proof, assume (z _m ,a _m ) is a weak local Nash equilibrium of the election. Then for all $j \in \mathcal{C}$, j’s Hessian matrix evaluated at (z _m , a _m ) must be negative semi-definite. This implies that Tr(D ² V _j ^m) ≤ 0 which is true if and only if $Tr(\mathcal{C}_{j}^{m}) \leq 0$ for all $j \in \mathcal{C}$. If $\mathcal{V}_{np}^{m} > V _{j}^{m}$ for some $j \in \mathcal{C}$ then it must be the case that $Tr(\mathcal{C}_{j}^{m})$ must be strictly positive violating the weak Nash equilibrium condition. This completes the proof of necessity. ■

1.2 Proof of Lemma 2

We only need the proof for candidate j. Using (52) and (46), $\mathcal{V}_{sp}^{m} > \mathcal{V}_{np}^{m}$ when

$$\displaystyle{ \frac{1} {2} -\frac{1} {4} \frac{1} {Tr\left [(\mathbf{B})^{-1}\boldsymbol{\nabla }^{m}\right ]} > \frac{1} {2} -\frac{1} {4} \frac{Tr(\mathbf{B})} {Tr\left (\mathbf{\nabla }^{m}\right )} }$$

This translates into

$$\displaystyle{ Tr(\mathbf{B})Tr\left [(\mathbf{B})^{-1}\boldsymbol{\nabla }^{m}\right ] > Tr\left (\boldsymbol{\nabla }^{m}\right ) }$$

which using (23), (28) and (27) and after some manipulation translates into

$$\displaystyle{ \left (1 + \frac{e} {\beta } \right )var(x) + \left (1 + \frac{\beta } {e}\right )var(t) > var(x) + var(t). }$$

After multiplying this inequality by $\left [\beta e\right ]^{-1} > 0$ and some manipulation, we get

$$\displaystyle{ var(x) + var(t) > 0, }$$

so that $\mathcal{V}_{sp}^{m} > \mathcal{V}_{np}^{m}$ always as required and this holds for all $j \in \mathcal{C}$. ■

1.3 Comparative Statics

To find the effect of β on the sufficient pivotal vote share, when all candidates adopt the electoral mean (z _m ,a _m ) as their campaign strategy, $\mathcal{V}_{sp}^{m}$ in (22), take the derivative of $\mathcal{V}_{sp}^{m}$ wrt β, ceteris paribus, i.e.,

$$\displaystyle{ \frac{\partial \mathcal{V}_{sp}^{m}} {\partial \beta } = \frac{1} {4} \frac{\frac{1} {\beta ^{2}} var(x)} {\left [\frac{1} {\beta } var(x) + \frac{1} {e}var(t)\right ]^{2}} > 0, }$$

so that $\mathcal{V}_{sp}^{m}$ increases as β increases. A similar analysis would show that $\mathcal{V}_{sp}^{m}$ increases when e increases.

The effect of β on the necessary pivotal vote share, $\mathcal{V}_{sp}^{m}$ in (22), i.e., the derivative of $\mathcal{V}_{np}^{m}$ wrt β, ceteris paribus, is given by

$$\displaystyle\begin{array}{rcl} \frac{\partial \mathcal{V}_{np}^{m}} {\partial \beta } & =& -\frac{1} {4} \frac{var(x) + var(t) - (\beta +e) \frac{1} {n}\sum \nolimits _{i\in \mathcal{N}}2\beta (x_{i} - z_{m})^{2}} {\left [var(x) + var(t)\right ]^{2}} {}\\ & =& -\frac{1} {4} \frac{var(x) + var(t) - 2var(x) - 2 \frac{1} {n}\sum \nolimits _{i\in \mathcal{N}}\beta (x_{i} - z_{m})^{2}e} {\left [var(x) + var(t)\right ]^{2}} {}\\ & =& -\frac{1} {4} \frac{-var(x) + var(t) - 2\frac{e} {\beta } var(x)} {\left [var(x) + var(t)\right ]^{2}} {}\\ & =& \frac{1} {4} \frac{(1 + 2\frac{e} {\beta } )var(x) - var(t)} {\left [var(x) + var(t)\right ]^{2}} {}\\ \end{array}$$

where the fourth line follows after multiplying by $\frac{\beta } {\beta }$. The necessary pivotal vote share increases when β increases iff the numerator in the last line of the above equation is positive, i.e., iff

$$\displaystyle{ (1 + 2\frac{e} {\beta } )var(x) > var(t) }$$

Adding var(x) on both sides of the above inequality gives

$$\displaystyle{ 2\frac{\beta +e} {\beta } var(x) > var(x) + var(t) }$$

The RHS of this inequality equals $Tr\left (\boldsymbol{\nabla }^{m}\right )$, given in (27). Thus, when β increases the necessary pivotal vote share increases when

$$\displaystyle{ \frac{\beta +e} {\beta } var(x) > \frac{1} {2}Tr\left (\boldsymbol{\nabla }^{m}\right ), }$$

i.e., when the variance of voters’ ideal policies weighted by $\frac{\beta +e} {\beta }$ is larger than half the aggregate variance of voters’ preferences about the electoral mean.

A similar analysis shows that $\mathcal{V}_{np}^{m}$ rises as e increases iff

$$\displaystyle{ \frac{\beta +e} {e} var(x) > \frac{1} {2}Tr\left (\boldsymbol{\nabla }^{m}\right ). }$$

Suppose voter’s preferences become more dispersed along the policy dimension while maintaining the electoral mean (z _m , a _m ) constant, so that the distribution of voters’ ideal policies undergoes a mean preserving spread. This means that var(x) increases but the electoral mean (z _m , a _m ), var(t), β and e remain unchanged.

We now examine what happens to the pivotal vote shares as var(x) or var(t) increase. The effect of higher var(x) or var(t) on the sufficient pivotal vote share is given by

$$\displaystyle\begin{array}{rcl} \frac{\partial \mathcal{V}_{sp}^{m}} {\partial var(x)}& =& \frac{1} {4} \frac{1} {\beta } \frac{1} {\left [\frac{1} {\beta } var(x) + \frac{1} {e}var(t)\right ]^{2}} {}\\ \frac{\partial \mathcal{V}_{sp}^{m}} {\partial var(t)}& =& \frac{1} {4} \frac{1} {e} \frac{1} {\left [\frac{1} {\beta } var(x) + \frac{1} {e}var(t)\right ]^{2}} {}\\ \end{array}$$

which are both always positive. That is, a mean preserving increase in var(x) or var(t) increases the sufficient pivotal vote share. Thus, making it more difficult for the electoral mean to be a strick LNE of the election.

The effect of a mean preserving increase in var(x) or var(t) on the necessary pivotal vote in (22) is given by

$$\displaystyle{ \frac{\partial \mathcal{V}_{np}^{m}} {\partial var(x)} = \frac{\partial \mathcal{V}_{np}^{m}} {\partial var(t)} = \frac{1} {4} \frac{(\beta +e)} {\left [var(x) + var(t)\right ]^{2}} > 0. }$$

The necessary pivotal vote share also increases in the variance of voters’ preferences along any dimension. The necessary condition for convergence to the electoral mean is then harder to meet.