Dynamic effects of price promotions: field evidence, consumer search, and supply-side implications

Abstract

This paper investigates the dynamic effects of price promotions in a retail setting through the use of a large-scale field experiment varying the promotion depths of 170 products across 17 categories in 10 supermarkets of a major retailer in Chile. In the intervention phase of the experiment, treated customers were exposed to deep discounts (approximately 30%), whereas control customers were exposed to shallow discounts (approximately 10%). In the subsequent measurement phase, the promotion schedule held discount levels constant across groups. We find that treated customers were 22.4% more likely to buy promoted items than their control counterparts, despite facing the same promotional deals. Strikingly, the magnitude of the dynamic effects of price promotions (when promotional depths are equal across conditions) is 61% of the promotional effects induced by offering shallow vs. deep discounts during the intervention phase. The result is robust to other concurrent dynamic forces, including consumer stockpiling behavior and state dependence. We use the experimental variation and historical promotional activities to inform a demand-side model in which consumers search for deals, and a supply-side model in which firms compete for those consumers. We find that small manufacturers can benefit from heightened promotion sensitivity by using promotions to induce future consideration. However, when unit margins are high, heightened promotion sensitivity leads to fierce competition, making all firms worse off.

Appendices

Appendix A: Criteria for category selection

We selected categories with the goal of providing the maximum amount of useful variation. First, we wanted to limit the influence of stockpiling behavior on the response to the promotion stimulus. If consumers respond to promotions by anticipating purchases, then severe post promotion dips could affect our estimates. On this basis, we excluded a few categories for which households’ inventory costs were deemed to be very low (e.g., soups) and others for which consumers could keep the product in inventory for a period of time, well beyond the post-promotion period (e.g., coffee). A second related consideration for including a category was the length of the typical interpurchase time observed in the category. In particular, we excluded those categories for which typical interpurchase times exceeded 5 weeks on average. Third, we only included categories that had already been promoted on a regular basis. Since our focus is on the effects of changes in promotion depth, we wanted to keep the frequency with which products were placed on promotion as constant as possible. This requirement led us to exclude categories such as “baked goods” which were rarely, if ever, placed on promotion. Fourth, we included categories that were purchased across different demographic segments (i.e., heterogeneous in terms of socioeconomic groups and ages). By imposing this requirement, we wanted to ensure that the same categories would be relevant across all stores included in the experimental design. Fifth, we chose categories in which consumers were unlikely to use the presence of promotion as an input in their assessment of a product’s quality. For instance, the presence of promotions in specific categories (e.g., fresh produce) can be interpreted as a negative quality signal, e.g., the product is about to expire or does not sell well, and the promotion is seen as an attempt to sell it rapidly. Sixth, we chose categories with different degrees of brand loyalty, e.g., soft drinks are well-known for having a few star brands with very loyal consumers, whereas milk exhibits more generic products, likely to be considered close substitutes by more consumers. Other considerations that played a role in our choice of categories were avoiding categories in which stockouts were known to occur more frequently and avoiding categories with a small number of brands.

Appendix B: Robustness/alternative mechanisms

2.1 B.1 Continuous treatment variable

One of the consequences of the experimental protocol not having been implemented by store managers exactly as designed is that different stores effectively received different treatment intensities (see Table 3). Therefore, it is useful to verify whether the same results hold when the empirical treatment, rather than the theoretical one, is introduced into the econometric specification. We consider two approaches. First, we calculate the average promotional product price for each store during the first half, across the experimental goods. Using this statistic as the treatment variable takes into account that consumers who shop at different stores may have been effectively exposed to different treatment intensities.

Table 16 summarizes the results (even-numbered columns) and compares them with the main regression results based on the treatment indicator variable (odd-numbered columns). We find that an increase in the discount depth (i.e., lower prices through promotions) during the intervention phase leads to a higher purchase likelihood during the second half of the experiment. Consumers are 10.4% more likely to buy a promoted item during the second half of the experiment for each dollar received through discounts during the first half. The same qualitative results are found in terms of the composition of consumers’ basket, in terms of the percentage of items and of expenditure.

Table 16 Effect of Treatment on Customer Behavior - Intervention Prices

In order to provide further robustness of our results, we also perform the same analysis at the category level. In this case, the regressor of interest is the average promotional price across experimental goods of each category. In this case, an observation is an individual/category combination. Because most individuals do not buy from most categories, this leads to many zeros on the left-hand side.

The results are presented in Table 17. All results remain statistically significant, and consistent with the original directions, although a slight loss in power may have been induced by the number of zeros introduced by this approach. Overall, our results are robust to using the empirical treatment intensities.

Table 17 Effect of Treatment on Customer Behavior - Average Prices by Category

2.2 B.2 Improving efficiency through matching

We complement our analysis with a matching procedure. Much like the use of control regressors, a matching method reduces the variance of the unobserved error term by taking advantage of the correlation between observable and unobservable characteristics. This approach shifts the emphasis from the cluster to the individual level by matching, within pairwise randomized stores, those individuals who have balanced covariates before the experimental period (Rubin 1973, 1979; Imbens and Rubin 2015).

In our case, the matching technique allows us to construct a large subsample of statistical twins (one twin buying at a control store and the other at a treated one) to ensure identical pre-treatment purchasing behavior between treatment and control at the individual level.³¹ Notice that in our setting, the exogeneity of the treatment is guaranteed by the experimental design and matching is only needed to identify similar customers based on historical data.³²

2.2.1 B.2.1 Matching framework

To construct our sample of statistical twins, we introduce a recent matching technique developed by Zubizarreta (2012). This matching technique takes advantage of new developments in optimization to match individuals in multiple aspects, which until recently was an unfeasible task due to the large dimensionality of the problem. Matching individuals on several dimensions encompasses other matching techniques, such as propensity score, by creating a superior and easily interpretable matching sample.

Formally, let $\mathcal {T}=\{t_{1},\ldots ,t_{T}\}$ be the set of treated units, and $\mathcal {C}=\{c_{1},\ldots ,c_{C}\}$, the set of potential controls. Without loss of generality, suppose $T\leq C$. Each treated unit $t\in \mathcal {T}$ has a P dimensional vector of observed covariates x_t = {x_t,1,...,x_t,P}, and each control $c\in \mathcal {C}$ has a similar vector $\mathbf {x}_{c}=\{x_{c,1},...,x_{c,P}\}$. Let the assignment indicator $a_{t,c}$ be equal to 1 if treated unit t is assigned to control c, and 0 otherwise; and denote the entire assignment matrix by a.³³ The optimal assignment problem is given by:

$$\begin{array}{@{}rcl@{}} \min_{\mathbf{a}}\sum\limits_{t\in\mathcal{T}}\sum\limits_{c\in\mathcal{C}}\delta_{t,c}a_{t,c} \end{array} $$

(14)

$$\begin{array}{@{}rcl@{}} \text{subject to}\sum\limits_{c\in\mathcal{C}}a_{t,c}= 1 & , & t\in\mathcal{T} \end{array} $$

(15)

$$\begin{array}{@{}rcl@{}} \sum\limits_{t\in\mathcal{T}}a_{t,c}\leq1 & , & c\in\mathcal{C} \end{array} $$

(16)

$$\begin{array}{@{}rcl@{}} a_{t,c}\in\{0,1\} & , & t\in\mathcal{T},c\in\mathcal{C} \end{array} $$

(17)

$$\begin{array}{@{}rcl@{}} \left|\sum\limits_{t\in\mathcal{T}}\sum\limits_{c\in\mathcal{C}}\frac{x_{c,j} a_{t,c}}{T}-\overline{x}_{\mathcal{T},j}\right|\leq\varepsilon_{j} , j\in\{1,..,P\} \end{array} $$

(18)

where $\delta _{t,c}\in \left [0,\infty \right )$ is a distance function between treated and control units (e.g. Euclidean distance), ${\sum }_{t\in \mathcal {T}}{\sum }_{c\in \mathcal {C}}\frac {x_{c,j}a_{t,c}}{T}$ denotes the average covariate j of assigned controls and $\overline {x}_{\mathcal {T},j}$ denotes the average covariate j across all treated individuals.

The goal of the matching program is to minimize the total sum of distances between treated units and matched controls as stated in expression (14). The first three constraints describe the integer nature of the assignment problem: Each treatment unit is paired with one control unit (Eq. 15) and not all control units should be used (Eq. 16).³⁴ The set of constraints given by expression (18) introduces an upper bound on the difference allowed between treatment and control individuals for each covariate, according to $\varepsilon _{j}>0$, the pre-determined tolerance level for covariate $j\in \{1,..,P\}$. This last set of constraints is a distinctive feature of the mixed-integer programming (MIP) matching approach proposed by Zubizarreta (2012).

2.2.2 B.2.2 Matching supermarket customers

We use the individual-level data available through the retailer’s loyalty card to match pairs of control and treated consumers based on the pre-experimental records according to the procedure described above. To construct statistical twins, we consider demographic and behavior-based covariates: Age, gender, the weekly average of total expenditure, the weekly average of total spending in the experimental categories, and the frequency of trips to the store. We then applied the criteria that customers are required to buy an item in at least one of the 31 main categories during the first half of the experiment and not visit multiple stores during the experimental period. The historical dataset used for this task covers a 46 week period, ending approximately one year before the experiment.

Consumers’ historical behavior toward promoted goods is a natural variable to include in this analysis.³⁵ In our case, the supermarket chain does not hold records that allow us to identify promotions during the pre-experimental period directly. We construct the promotional variable for the pre-experimental period using the algorithm proposed by Kehoe and Midrigan (2015). This algorithm identifies regular prices in a posted price time series as the modal price within a rolling window of five weeks centered on the current week.³⁶ We define a promotional event for a given price trajectory as a period when the posted price for a given UPC-store lies below the regular price (for the same UPC-store) derived from the Kehoe and Midrigan (2015) price filter.³⁷ We use the depth measurements to create two new variables: the share of promoted items bought by each consumer during the historical dataset, and the number of promoted items bought per consumer-visit during the same period.

Table 18 presents the sample sizes of the universe of customers before matching and those who were matched by the MIP matching procedure. Columns (1) and (2) present the number of customers who faced the experimental promotional activity in each store pair and for whom we have historical data. Overall, as shown in columns (3), (4) and (5), the MIP matching generated 13,482 one-to-one customer pairs (one control and one treated customer), distributed across 10 stores of our retail chain. Table 19 presents the resulting covariates of the final matched sample.³⁸ The last column reports the p-value for the null hypothesis of identical means. We obtain close averages in total expenditure, expenditure in promoted categories, and age between treatment and control matched individuals. Given the large sample size of individuals, the tests reject nearly same means in gender and number of trips, although the table shows that the actual values are quite similar. As for the purchase behavior metrics related to promoted goods (last two variables in in Table 19) added later per the review team’s suggestion, we do not reject that those variables have the same means across conditions. Moreover, the absolute differences for these variables are slim, if anything, favoring the control group towards being more active in purchasing promoted goods (thus making our results potentially conservative).

Table 18 Universe of Potential Pairs

Table 19 Pre-treatment Covariates of Control and Treated Matched Individuals

In order to further test the effects of the pre-experimental promotion-related variables, we include the historical share of promoted products bought as well as the historical mean number of promoted products purchased per visit in the main analysis as control variables. We report the results in Table 20. We find virtually no differences to the treatment effects (compare to Table 8). We interpret these findings, also in connection with those reported in Table 19, as the matching procedure having performed well in matching consumers overall, through the five dimensions mentioned above, such that the remaining dimensions had little effect on the treatment estimates.

Table 20 Effect of Treatment on Customer Behavior with Promotion-Related Controls - Matched Sample

2.3 B.3 State dependence

To understand how state dependence could explain our results, consider a pair of similar customers who only differ on the experimental condition they were exposed to in the intervention phase. Assume that the treated customer bought a promoted product because of the deep discount, whereas the control customer decided not to purchase it, given its lower promotional discount (10%). It is possible that, during the second half of the experiment, both customers visited the store on the week that the same product was on promotion once again, at a shallow level. In this case, the treated customer may be more likely to buy the product in the second half, not because of heightened promotion sensitivity, but rather because of state dependence.³⁹

We repeat the main analysis, but now only consider, for each consumer, purchases of goods that were not bought during the first half of the experiment. While this procedure is expected to mechanically decrease the treatment effect due to ignoring relevant data, it has the merit of parsing out the effect of state dependence. Table 21 summarizes the results for the full sample: the treatment effect in column (1) remains statistically insignificant, and all treatment effect estimates decrease slightly. However, statistic $E\left [\left .\widehat {y}_{i}\right |X_{i},T_{i}= 1\right ]\div E\left [\left .\widehat {y}_{i}\right |X_{i},T_{i}= 0\right ]$ produces an estimate of a 21.2% relative increase of purchases of promoted products, which is similar to the one of 22.4% found before, when state dependence was not controlled for.⁴⁰

Table 21 Effect of Treatment on Customer Behavior for New Purchases

Given the slight changes in significance and minimal changes to treatment effect estimates, we believe the new results are due to an overly-stringent test rather than state-dependence being responsible for the results. In order to investigate this issue further, we consider the same analysis on the matched sample, discussed in the previous section, and present the results in Table 22. All results of interest (columns (1)–(3)) remain statistically significant in this case, with the treatment estimates falling only slightly. Taken together, the results imply that state dependence may play a role in our measurement, but is unlikely to be responsible for the finding of heightened promotion sensitivity.

Table 22 Effect of Treatment on Customer Behavior for New Purchases - Matched Sample

2.4 B.4 Stockpiling behavior

Our results are likely to be made conservative by consumer stockpiling behaviors. The reason is that treated consumers are likely to hold higher inventories than control ones, due to the exposure to deep promotions during the intervention phase. As a result, stockpiling is expected to dampen treated consumer purchases during the second half of the experiment, including those of promoted products.

2.5 B.5 Placebo test

In this section, we introduce a placebo test designed to assess whether our experimental intervention is likely to be effectively responsible for the differences in consumer behavior across treated and control pools. We focus the analysis on customers who did not visit the supermarket, or alternatively, did not use their loyalty cards during the first half of the experiment. Since these customers are less likely to have been exposed to the differential treatment conditions, we expect to find lower magnitudes and very few statistical significance of treatment effects (i.e., less than 5% of measured effects should be significant).

The results of the analysis are presented in Table 23 and are quite different from those in Table 6. First, only two behaviors are found to be marginally significant across different dependent variables and methods of calculating standard errors. Second, all point estimates fall below the original ones, often by an order of magnitude. If taken ‘as is’, the point estimate in column (1) would set a lower bound on the real treatment effect of 0.024, or a 10.7% increase $\left (0.024\div E\left [\left .\widehat {y}_{i}^{full sample}\right |X_{i},T_{i}= 0\right ]\right )$ in relative terms.

Table 23 Effect of Treatment on Placebo Customers’ Behavior

These results are positive in the sense that consumers who are less likely to have been exposed to the intervention exhibit lower treatment effects, generally without statistical significance. The results are reassuring in terms of assessing experimental validity, but not without limitations. In particular, the choice of not purchasing from a major category during the first half of the experiment is unlikely to be exogenous. Because of this, the lack of a statistically significant treatment effect can source from selection, i.e., this test may sample from consumers who respond less to price promotions in the first place, and who exhibit lower promotion sensitivity effects as well. However, a countervailing force is the fact that it is impossible to rule out that some of these customers were exposed to our intervention but purchased without using their loyalty card during the first half of the experiment. This force would contaminate the results in the opposite direction, making the treatment effects in Table 23 conservative (in other words, the analysis did not focus on truly Placebo consumers).

Appendix C: Search model

3.1 C.1 Proposition: Logistic uncertainty

The logistic p.d.f. and c.d.f. with unit scale parameter are given by $f\left (x\right )=\frac {e^{-\left (x-\mu \right )}}{\left (1+e^{-\left (x-\mu \right )}\right )^{2}}$ and $F\left (x\right )=\frac {1}{1+e^{-\left (x-\mu \right )}}$ respectively. We seek the solution to equation

$$z=-c+{\int}_{z}^{\infty}xdF_{j}\left( x\right)+F_{j}\left( z\right)z $$

with respect to z. Plugging in the expressions above yields

$$ z=-c+{\int}_{z}^{\infty}\frac{xe^{-\left( x-\mu\right)}}{\left( 1+e^{-\left( x-\mu\right)}\right)^{2}}dx+\frac{z}{1+e^{-\left( z-\mu\right)}}. $$

(19)

Integration by parts yields

$$ {\int}_{z}^{\infty}\frac{xe^{-\left( x-\mu\right)}}{\left( 1+e^{-\left( x-\mu\right)}\right)^{2}}dx=\log\left( e^{z}+e^{\mu}\right)+\frac{z}{1+e^{z-\mu}}-z $$

(20)

and the reservation value Eq. 19 becomes

$$\begin{array}{@{}rcl@{}} & & z=-c+\log\left( e^{z}+e^{\mu}\right)+\underset{= 0}{\underbrace{\frac{z}{1+e^{z-\mu}}-z+\frac{z}{1+e^{-\left( z-\mu\right)}}}}\\ & \Leftrightarrow & z=-c+\log\left( e^{z}+e^{\mu}\right)\\ & \Rightarrow & z^{*}=\log\left( \frac{e^{\mu}}{e^{c}-1}\right)=\mu-\log\left( e^{c}-1\right) \end{array} $$

and the solution is unique for

3.2 C.2 Theorem: Contraction mapping

Let u be a random variable with continuous p.d.f. and c.d.f. $f\left (\cdot \right )$, $F\left (\cdot \right )$ respectively. The indifference condition is given by

$$\begin{array}{@{}rcl@{}} z^{*} & = & -c+Pr\left( u\geq z^{*}\right)E\left[u\left|u\geq z^{*}\right.\right]+Pr\left( u<z^{*}\right)z^{*}\\ & = & -c+{\int}_{z^{*}}^{\infty}uf\left( u\right)du+z^{*}F\left( z^{*}\right) \end{array} $$

Define ${\Gamma }\left (z\right )=-c+{\int }_{z}^{\infty }uf\left (u\right )du+zF\left (z\right )$. Under standard continuity assumptions, ${\Gamma }\left (z\right )$ is a contraction mapping if ${\Gamma }^{\prime }\left (z\right )\in \left [0,1\right )$. In our case,

$$\begin{array}{@{}rcl@{}} {\Gamma}^{\prime}\left( z\right) & = & \frac{d}{dz}\left( -c+{\int}_{z}^{\infty}uf\left( u\right)du+zF\left( z\right)\right)\\ & = & 0-zf\left( z\right)+zf\left( z\right)+F\left( z\right)\\ & = & F\left( z\right) \end{array} $$

which is bounded between zero and one. The use of the Leibniz integral rule implies integration must be interchangeable with differentiation. The contraction mapping applies for a large class of differentiable distributions, as long as ${\int }_{z_{n}}^{\infty }uf\left (u\right )du$ is finite , which is also implied by the original Weitzman (1979) model. So, the theorem applies to most distributions used in empirical work.

Using the proposition above, it is easy to show that in the case of mixture of logistics given by Eq. 8, the reservation value can be found through contraction

$${\Gamma}\left( z\right)=-c+\omega_{ijt}^{\kappa}\log\left( e^{z}+e^{v_{ijt}+\gamma^{D}}\right)+\left( 1-\omega_{ijt}^{\kappa}\right)\log\left( e^{z}+e^{v_{ijt}+\gamma^{S}}\right) $$

where $\omega _{ijt}^{\kappa }$ is consumer i’s belief associated with finding a deep discount for a given promotional history $\kappa \in \left \{ S,D\right \}$.

3.3 C.3 Likelihood and estimation

We now characterize the likelihood of an alternative being chosen, which involves adding over search sequences. First, we rank the inside alternatives by their reservation values such that $z_{1}>z_{2}>...>z_{n}$, where n is equal to the number of inside alternatives in the choice set. We depict the potential search paths consistent with a choice of alternative j in the diagram of Fig. 7, where searching an additional option corresponds to a lateral movement, and a downward one depicts the purchase of alternative j.⁴¹ For a consumer to be willing to search option j with reservation value $z_{j}$, she must have inspected options with higher reservation values before and have found that it was worthwhile searching option j nonetheless. The reason is that options are ordered by their reservation values, and so if a consumer did not search option $j-1$ then she prefers not to search option j either. The sequence of events leading the consumer to arrive to node j is given by

$$\begin{array}{@{}rcl@{}} & z_{1}>u_{0}\wedge z_{2}>\max\left\{ u_{0},u_{1}\right\} \wedge z_{3}>\max\left\{ u_{0},u_{1},u_{2}\right\} \wedge...\wedge z_{j}>\max\left\{ u_{0}..u_{j-1}\right\} \\ & =z_{j}>\max\left\{ u_{0}..u_{j-1}\right\} \end{array} $$

(21)

The identity above can be shown by induction. If a consumer searched option 2 for example, then $z_{2}>\max \left \{ u_{0},u_{1}\right \}$. This implies the consumer also searched option 1 because

$$z_{2}>\max\left\{ u_{0},u_{1}\right\} \Rightarrow z_{1}>u_{0} $$

since $z_{2}<z_{1}$. For the consumer to prefer option j to the options searched before, we require $u_{j}>\max \left \{ u_{0}..u_{j-1}\right \}$, and so a consumer searches alternative j and considers it the best option up to that stage if and only if

$$ Reach Node_{j}: \min\left\{ z_{j},u_{j}\right\} >\max\left\{ u_{0}..u_{j-1}\right\} $$

(22)

Conditional on searching option j and preferring it up to that stage, many subsequent search paths can lead to a final choice of j. For example, the consumer may choose alternative j without searching any further, or do so after searching option $j + 1$, options $j + 1$ and $j + 2$, etc. Let $Buy_{j\left |k\right .}$ be each of such subsequent paths, where j is the chosen product and $k\geq j$ is the last product searched by the consumer. Then, the probability of choosing option j, which informs our likelihood function, is equal to

$$\begin{array}{@{}rcl@{}} Pr\left( Choose_{j}\right) & = & Pr\left\{ Reach Node_{j}\wedge\left( \left.Buy_{j}\right|Reach Node_{j}\right)\right\} \\ & = & Pr\left( \min\left\{ z_{j},u_{j}\right\} >\max\left\{ u_{0}..u_{j-1}\right\} \wedge\left( \bigvee_{k=j}^{n}Buy_{j\left|k\right.}\right)\right) \end{array} $$

(23)

We now characterize each of the paths, where movements referred to as ‘down’ and ‘right’ are related to the ones in the Fig. 7:

$$\begin{array}{@{}rcl@{}} Buy_{j\left|j\right.} & = & \underset{Path Down_{j}}{\underbrace{u_{j}>z_{j + 1}}}\\ Buy_{j\left|j + 1\right.} & = & \underset{Path Right_{j}}{\underbrace{\left( \sim Path Down_{j}\right)\wedge u_{j}>u_{j + 1}}}\wedge\underset{Path Down_{j + 1}}{\underbrace{u_{j}>z_{j + 2}}}\\ Buy_{j\left|j + 2\right.} & = & Path Right_{j}\wedge\underset{Path Right_{j + 1}}{\underbrace{\left( \sim Path Down_{j + 1}\right)\wedge u_{j}>u_{j + 2}}}\wedge\underset{Path Down_{j + 2}}{\underbrace{u_{j}>z_{j + 3}}}\\ & \vdots\\ Buy_{j\left|k\right.} & = & \left\{\begin{array}{llll} \left( \bigwedge_{l=j}^{k-1}Path Right_{l}\right)\wedge Path Down_{k}, & j\leq k<n\\ \left( \bigwedge_{l=j}^{k-1}Path Right_{l}\right), & j\leq k=n \end{array}\right. \end{array} $$

We have characterized the likelihood function. It remains to maximize it with respect to parameters, conditional on the data. Because utilities are probabilistic, we use simulation to generate u’s and construct the likelihood. Moreover, the need to investigate multiple search paths led us to employ 10,000 draws per choice-alternative.

In order to account for heterogeneity in search sequences, we add a noise parameter $\eta \sim N\left (0,1\right )$ to the reservation values. For example, in some circumstances, consumers may not include some products in their consideration sets, which is equivalent to those products featuring very low reservation values. This assumption also provides the demand function with smoothness for purposes of the counterfactual analysis.

An additional difficulty with ‘accept/reject choice simulation’ is that small changes in parameters do not affect simulated outcomes, even for large sets of draws.⁴² Moreover, the log-likelihood function exhibits saddle points that make finding the global maximum challenging.

We implement a patterned grid search across a wide range of parameter values, and ensure that the bounds set for the parameters were never achieved during the estimation procedure. Calculation of the standard errors required additional smoothing. For this purpose, following McFadden (1989), we smoothed out the likelihood function by use of a kernel function, which in our case is analogous to adding a low-variance extreme-value noise to each u and z component.⁴³ For illustration purposes, suppose we observe option $n-1$ being chosen. The probability of this choice is

$$\begin{array}{@{}rcl@{}} Pr\left( Choose_{n-1}\right) & = & Pr\left\{ \min\left\{ z_{n-1},u_{n-1}\right\} >\max\left\{ u_{0}..u_{n-2}\right\} \wedge\left( \bigvee_{k=n-1}^{n}Buy_{n-1\left|k\right.}\right)\right\} \\ & = & Pr\left\{ \min\left\{ z_{n-1},u_{n-1}\right\} >\max\left\{ u_{0}..u_{n-2}\right\}\right.\\ &&\left. \wedge\left( u_{n-1}>z_{n}\vee\left( u_{n-1}<z_{n}\wedge u_{n-1}>u_{n}\right)\right)\right\} \end{array} $$

Given a parameter guess, we generate R sets of simulations of u’s. For each set r, we calculate

$$\begin{array}{@{}rcl@{}} p^{r}\left( Choose_{n-1}\right)=K\left( \min\left\{ z_{n-1},u_{n-1}^{r}\right\} -\max\left\{ {u_{0}^{r}}..u_{n-2}^{r}\right\} \right)\\.\left[K\left( u_{n-1}^{r}-z_{n}\right)+K\left( z_{n}-u_{n-1}^{r}\right).K\left( u_{n-1}^{r}-{u_{n}^{r}}\right)\right] \end{array} $$

where

$$K\left( x\right)=\frac{1}{1+\exp\left( -\frac{x}{\sigma}\right)} $$

is the logistic kernel with smoothing parameter $\sigma = 0.001$. We used the smoothing parameter to calculate standard errors. During estimation, we used $K\left (x\right )= 1\left (x>0\right )$ instead, because the grid search algorithm does not require smoothing out the objective function.

Finally, we average across simulation results to calculate the choice probability, i.e.

$$Pr\left( Choose_{n-1}\right)\simeq\frac{1}{R}\sum\limits_{r = 1}^{R}p^{r}\left( Choose_{n-1}\right). $$

McFadden (1989) characterizes the estimator above as well as its consistency in detail.