Push and pull: Modeling mobile app promotions and consumer responses

Abstract

How effective are push promotions through mobile apps for brick-and-mortar retailers and what strategies can improve the performance of targeted push marketing? To address these questions, we develop a multivariate event history model to evaluate the effects of behavior and location-based push promotions on shoppers’ app usage and offline shopping activities. Our study generates new insights into mobile app promotions and offline shopping. We find that behavior-based pushes have a higher impact on consumer responses before a shopping trip than during a trip, and their effects vary significantly across different types of retailers. The effects of pushes are positively correlated with shoppers’ propensities of app pulls and mall visits, which suggests that timing the delivery of pushes can make them more effective. Furthermore, location-based pushes exhibit stronger positive effects on app pulls and coupon outclicks during a shopping trip than behavior-based pushes, even after shoppers receive the latter before the trip, which shows that behavior and location-based pushes are not substitutable. We demonstrate through simulations that our model enables marketers to design more effective mobile targeting strategies by exploiting heterogeneous consumer responses. Addressing potential endogeneity by controlling for the information used for customer selection in the customer’s response functions, our proposed model can be applied to many empirical problems involving event history data.

Appendices

Appendix A. MCMC algorithm

We derive from the variance-covariance matrix $\Psi$ the correlation coefficient $\rho =\Psi _{12}\Psi _{11}^{-\frac{1}{2}}$ and variance $\sigma _{\mu }^{2}=\Psi _{11}$.

The following steps show the details of the MCMC algorithm used in this study:

Step 1: Sample $\alpha ^{i}$. We consider the prior distribution of $\alpha ^{i}$ following $MVN_{4}(\theta _{\alpha }, \Sigma _{\alpha })$. We use the Metropolis-Hastings algorithm to sample $\alpha ^{i}$. The accepting probability of the proposed $\alpha ^{i*}$, drawn from a $MVN_{4}$ distribution, is given as

$$ min\{\frac{L_{i}(\alpha ^{i*},\gamma ^{i},\phi ,\delta ^{i},\pi ^{i},\vartheta ^{i},\mu ^{i}|Data^{i}).MVN_{4}(\alpha ^{i*}|\theta _{\alpha },\Sigma _{\alpha })}{L_{i}(\alpha ^{i},\gamma ^{i},\phi ,\delta ^{i},\pi ^{i},\vartheta ^{i},\mu ^{i}|Data^{i}).MVN_{4}(\alpha ^{i}|\theta _{\alpha },\Sigma _{\alpha })},1\}. $$

Step 2: Sample $\gamma ^{i}$. We consider the prior distribution of $\gamma ^{i}$ following $MVN_{14}(\theta _{\gamma }, \Sigma _{\gamma })$. We use the Metropolis-Hastings algorithm to sample $\gamma ^{i}$. The accepting probability of the proposed $\gamma ^{i*}$, drawn from a $MVN_{14}$ distribution, is given as

$$ min\{\frac{L_{i}(\alpha ^{i},\gamma ^{i*},\phi ,\delta ^{i},\pi ^{i},\vartheta ^{i}|Data^{i}).MVN_{14}(\gamma ^{i*}|\theta _{\gamma },\Sigma _{\gamma })}{L_{i}(\alpha ^{i},\gamma ^{i},\phi ,\delta ^{i},\pi ^{i},\vartheta ^{i}|Data^{i}).MVN_{14}(\gamma ^{i}|\theta _{\gamma },\Sigma _{\gamma })},1\}. $$

Step 3: Sample $\phi$. We consider the prior distribution of $\phi _{j}$ following $inverse-Gamma(\bar{a}_{\phi },\bar{b}_{\phi })$. We use the Metropolis-Hastings algorithm to sample $\phi _{j}$. The accepting probability of the proposed $\phi _{j}^{*}$, drawn from a log-normal distribution, is given as

$$ min\{\frac{\prod _{i}L_{i}(\alpha ^{i},\gamma ^{i},\phi ^{*},\delta ^{i},\pi ^{i},\vartheta ^{i}|Data^{i}).\prod Gamma(\phi _{j}^{*}|\bar{a}_{\phi },\bar{b}_{\phi })\prod \phi _{j}^{*}}{\prod _{i}L_{i}(\alpha ^{i},\gamma ^{i},\phi ,\delta ^{i},\pi ^{i},\vartheta ^{i}|Data^{i}).\prod Gamma(\phi _{j}|\bar{a}_{\phi },\bar{b}_{\phi })\prod \phi _{j}},1\}. $$

Step 4: Sample $\delta ^{i}$. We consider the prior distribution of $\delta ^{i}$ following $MVN_{3}(\theta _{\delta },\Sigma _{\delta })$. The accepting probability of the proposed $\delta ^{i*}$, drawn from a $MVN_{3}$ distribution, is given as

$$ min\{\frac{L_{i}(\alpha ^{i},\gamma ^{i},\phi ,\delta ^{i*},\pi ^{i},\vartheta ^{i}|Data^{i}).MVN_{3}(\delta ^{i*}|\theta _{\delta },\Sigma _{\delta })}{L_{i}(\alpha ^{i},\gamma ^{i},\phi ,\delta ^{i},\pi ^{i},\vartheta ^{i}|Data^{i}).MVN_{3}(\delta ^{i}|\theta _{\delta },\Sigma _{\delta })},1\}. $$

Step 5: Sample $\pi ^{i}$. We consider the prior distribution of $\pi ^{i}$ following $MVN_{2}(\theta _{\pi }, \Sigma _{\pi })$. The accepting probability of the proposed $\pi ^{i*}$, drawn from a $MVN_{2}$ distribution, is given as

$$ min\{\frac{L_{i}(\alpha ^{i},\gamma ^{i},\phi ,\delta ^{i},\pi ^{i*},\vartheta ^{i}|Data^{i}).MVN_{2}(\pi ^{i*}|\theta _{\pi },\Sigma _{\pi })}{L_{i}(\alpha ^{i},\gamma ^{i},\phi ,\delta ^{i},\pi ^{i},\vartheta ^{i}|Data^{i}).MVN_{2}(\pi ^{i}|\theta _{\pi },\Sigma _{\pi })},1\}. $$

Step 6: Sample $\vartheta ^{i}$. We consider the prior distribution of $\vartheta ^{i}$ following $log-MVN_{2}(\bar{\theta }_{\vartheta },\bar{\Sigma }_{\vartheta })$. The accepting probability of the proposed $\vartheta ^{i*}$ drawn from an $log-MVN_{2}$ distribution, is given as

$$ min\{\frac{L(\alpha ^{i},\gamma ^{i},\phi ,\delta ^{i},\pi ^{i},\vartheta ^{i*}|Data).log-MVN_{2}(\vartheta ^{i*}|\bar{\theta }_{\vartheta },\bar{\Sigma }_{\vartheta })\prod _{k}\vartheta _{k}^{i*}}{L(\alpha ^{i},\gamma ^{i},\phi ,\delta ^{i},\pi ^{i},\vartheta ^{i}|Data).log-MVN_{2}(\vartheta ^{i}|\bar{\theta }_{\vartheta },\bar{\Sigma }_{\vartheta })\prod _{k}\vartheta _{k}^{i}},1\}. $$

Step 7: Sample $\theta _{n}$. We consider the prior distribution of $\theta _{n}$ following $MVN_{Q}(\bar{\theta }_{\theta _{n}}, \bar{\Sigma }_{\theta _{n}})$, where $\bar{\theta }_{\theta _{n}}=0$ and $\bar{\Sigma }_{\theta _{n}}=10^{6}I_{Q}$. The next draw, $\theta _{n}^{*}$, is drawn from a multivariate normal distribution

$$ \theta _{n}^{*}\sim MVN_{Q}(M,\,N), $$

where $M=N^{'}((\sum _{i=1}^{I}n^{i})^{'}\Sigma _{n}^{-1}+\bar{\theta }_{\theta _{n}}^{'}\bar{\Sigma }_{\theta _{n}}^{-1})^{'}$, $N=(I\Sigma _{n}^{-1}+\bar{\Sigma }_{\theta _{n}}^{-1})^{-1}$ , $n=\alpha ,\gamma ,\delta ,\pi ,$and $\vartheta$, and Q is the corresponding dimension of $\theta _{n}^{*}$.

Step 8: Sample $\Sigma _{n}$. We consider the prior distribution of $\Sigma _{n}$ following $IW(\bar{S}^{-1},\bar{\nu })$, where $\bar{S}=I_{Q}$ and $\bar{\nu }=1$. The next draw, $\Sigma _{n}^{*}$, is drawn from an inverse Wishart distribution

$$ \Sigma _{n}^{*}\sim IW((\sum _{i=1}^{I}(n^{i}-\theta _{n})(n^{i}-\theta _{n})^{'}+\bar{S})^{-1},I+\bar{\nu }), $$

where Q is the corresponding dimension of $\theta _{n}^{*}$.

Appendix B. Addressing endogeneity of behavior-based push

To test the robustness of our results and whether our approach can sufficiently address the endogeneity issue, especially for the effects of behavior-based pushes, we have fitted three versions of our model with (a) the linear functions of $S_{t}^{i}$, (b) the third-degree polynomials of $S_{t}^{i}$, (c) a control function approach (using the linear functions of $S_{t}^{i}$), which assumes a probit model for the push selection $W_{t,1}^{i}\in \{0,1\}$ and includes an additional random shock $\varepsilon _{t}^{i}$ in the app pull intensity:

$$\begin{aligned} W_{t,1}^{i}= & {\left\{ \begin{array}{ll} 1 & W_{t,1}^{i^{*}}>0\\ 0 & W_{t,1}^{i^{*}}\le 0 \end{array}\right. }\end{aligned}$$

(12)

$$\begin{aligned} W_{t,1}^{i^{*}}= & \omega _{0}^{j}+\omega _{1}^{j}\delta _{1}^{i}+\omega _{2}^{j}\gamma _{11}^{i}+\omega _{3}S_{t-3}^{i,r}+\omega _{4}^{j}X_{t,1}^{i}+v_{t}^{i},\end{aligned}$$

(13)

$$\begin{aligned} \lambda _{1}^{i}(t|H_{t}^{i})= & \exp \left( \delta _{1}^{i}+\pi _{1}^{i}X_{t,1}^{i}+\sum _{t'<t}\gamma _{1}^{i}W_{t',1}^{i}Z_{t',1}^{i}g(t-t';\phi _{01})+q(S_{t}^{i},\eta _{1}^{i})+\varepsilon _{t}^{i}\right) , \end{aligned}$$

(14)

where j indexes the promotion category;$X_{t,1}^{i}$ represents the weekend and holiday indicators; $S_{t-3}^{i,r}$ is shopper $i's$ affinity score for retailer r three days prior to the date of the push message. We assume that $\varepsilon _{t}^{i}$ in (14) and $v_{t}^{i}$ in (13) follow a joint bivariate normal distribution $\left[ \varepsilon _{t}^{i},v_{t}^{i}\right] \sim N\left( 0,\Psi \right)$, where $\Psi$ is a covariance matrix whose diagonal element (variance) for $v_{t}^{i}$ is fixed at 1. The push selection modeled is jointly estimated using Bayesian inference with the models for app pulls, mall visits and coupon outclicks.¹⁶ After fitting the models above, we find the correlation between $\varepsilon _{t}^{i}$ and $v_{t}^{i}$ is very small (the Bayesian posterior mean of the correlation in $\Psi$ is 0.006 with the 95% posterior interval = (-0.015, 0.028)), which is consistent with the argument in Section 4.2 that flexibly controlling for $S_{t}^{i}$ in the app pull and mall visit intensity functions indeed blocks the direct causal paths from $S_{t-3}^{i,r}$ to the customers’ responses. Therefore, it is no longer necessary to apply the control function approach which uses $S_{t-3}^{i,r}$ as an excluded variable. We have also fitted our model using a selected subsample of 2,375 customers (out of 5,000) whose affinity scores $S_{t-3}^{i,r}$ are within a specified interval for each push promotion, ensuring overlap in the affinity scores between push recipients and non-recipients.

We obtained the following results for the parameters in our models. The table below shows that various forms of the flexible functions of $S_{t}^{i}$ and the control function aproach yield very similar estimates, which demonstrates the robustness of our findings.

Table 13 Robustness Checks for the Effects of Behavior-based Push on App Pulls Outside Malls and Mall Visits

Appendix C. Posterior predictive checks and model comparison criterion

Based on the posterior estimates of the model parameters, we predict shoppers’ app pulls and mall visits in the 30 days following the estimation period. Following the recommendations in Gabry et al. (2019), we show in Fig. 1 the kernel density estimates of app pulls and mall visits in the holdout sample (the thick dark curve) vs. the predicated app pulls and mall visits (thin light curves), respectively, computed from 500 posterior draws of the parameter estimates. From the two plots in Fig. 1 it is evident that our proposed model fits the data very well.

We propose a metric based on the Brier score (Hernández-Orallo et al., 2012; Zhang and Zhou, 2018) to compare the models. Note that, for mall visits, we have

$$\begin{aligned} \lambda _{2}^{i}(t|H_{t}^{i})= & \underset{\triangle t\rightarrow 0}{lim}\frac{Pr\{N_{2}^{i}(t+\triangle t)-N_{2}^{i}(t)=1|H_{t}^{i}\}}{\triangle t}\\= & \frac{d\left( 1-F_{N_{2}^{i}\left( t\right) }\left( t|H_{t}^{i}\right) \right) }{dt}, \end{aligned}$$

Hence, the cdf of the kth mall visit given the previous observations including the $(k-1)$th mall visit is $F_{N_{2}^{i}\left( t\right) }\left( t|H_{t}^{i}\right) =1-\exp \left( -\int _{t_{k}^{i}}^{t}\lambda \left( t|H_{t}^{i}\right) \right) dt$. After $\lambda \left( t|H_{t}^{i}\right)$ is computed from the estimated parameters in the model, we calculate the modified Brier score as

$$ BrierScore=\frac{1}{n}\sum _{i=1}^{n}\left\{ \frac{1}{K_{i}}\sum _{k=1}^{K_{i}}\left[ I\left( t_{k}^{i}|H_{t_{k}}^{i}\right) -F_{N_{2}^{i}\left( t\right) }\left( t_{k}^{i}|H_{t_{k}}^{i}\right) \right] ^{2}\right\} $$

where $I\left( t_{k}^{i}|H_{t_{k}}^{i}\right)$ is an indicator function (empirical cdf ) which is equal to 1 for $t\ge t_{k}^{i}$.

Fig. 8

Posterior Predictive Checks on App Pulls and Mall Visits

Appendix D. Simulation algorithm

Setup: K types of events, sequencially simulate points within the same day. No decay within a day.

• 1. Draw parameters $\alpha ^{i},\beta ^{i},\kappa ^{i},\gamma ^{i},\phi ^{i},\psi ^{i},\delta ^{i},\pi ^{i},\lambda _{p}^{i},\zeta ^{i},\omega ^{i}$ with replacement from the posterior samples.

• 2. Simulate a point process in [0, T] given $\alpha ^{i},\beta ^{i},\kappa ^{i},\gamma ^{i},\phi ^{i},\psi ^{i},\delta ^{i},\pi ^{i},\lambda _{p}^{i},\zeta ^{i},\omega ^{i}$,

  • 1. Divide [0,T] into daily intervals. At beginning of each day (t), compute $\lambda _{t,k}^{i}=exp\{\delta ^{i}+\Pi _{t}^{i}X_{t}^{i}\!+\!\sum _{j=1}^{K}\sum _{t<t'}\alpha _{jk}^{i}g(t-t';\beta _{jk}^{i},\kappa _{jk}^{i})+\sum _{j=1}^{K}\sum _{t<t'}\gamma _{jk}^{i}Z_{jt}^{i}g(t-t';\phi _{jk}^{i},\psi _{jk}^{i})\}$ for $k=1,2,\cdots ,K$.

  • 2. Simulate $h\sim exp(\lambda _{t}^{i})$ where $\lambda _{t}^{i}=\sum _{k}\lambda _{t,k}^{i},k=1,2,\cdots ,K$.

  • 3. If $t+h<t'$ where $t<t'<t+1$ is the time of a behavior-based push in the same day, simulate $k\sim multinomial(p_{1},\cdots ,\,p_{K})$, where $p_{k}=\frac{\lambda _{t,k}^{i}}{\lambda _{t}^{i}},k=1,2,\cdots ,K$. Keep the event k at $t+h$. Reset t to $t+h$. Repeat from (a).

  • 4. If $t+h>t'$, reset t to $t'$. Repeat from (a).

  • 5. If $t+h>t+1$, reset t to $t+1$, Repeat from (a).

  • 6. Continue until $t>T$.

• 7. The simulation output is event times $\{t_{1},\cdots ,t_{N}\}$ of event types $\{k_{1,\cdots ,}k_{N}\}$.

Appendix E. Supplementary results

1.1 Means and variances of heterogeneous parameters in the model

Table 14 Effects of Variables Influencing App Pulls Outside Malls

Table 15 Effects of Variables Influencing Mall Visits

Table 16 Effects of Variables Influencing App Pulls inside Malls

1.2 Model estimates with half-day decaying effects

Table 17 Effects of Variables Influencing App Pulls Outside Malls

Table 18 Effects of Variables Influencing Mall Visits

Table 19 Effects of Variables Influencing App Pulls inside Malls