Applying Incentives

Executive leadership wants to apply additional incentives on sales reps to keep prices high². They task you with setting-up a tiered compensation scheme whereby deals at the top-end of the distribution get a higher compensation rate compared to deals at the bottom end of the distribution.

Applying such an additional incentive on sales teams has the potential advantage of pushing some proportion of deals to a higher price³. There are also Trade-offs associated with such an initiative (indicated in the Appendix), these will be ignored for the purposes of this exercise.

Say you decide to set cut-points to split the distribution into quartiles such that sales reps get larger bonuses if their deals fall into higher quartiles.

data_distribution %>% 
  mutate(cum_dens = cumsum(dens_scaled),
         quartile = (cum_dens) %/% 0.2500001 + 1,
         quartile = as.factor(quartile)) %>% 
  rename(initial_quartile = quartile) %>% 
  ggplot(aes(x = price, y = dens_scaled, fill = initial_quartile))+
  geom_col()+
  scale_fill_discrete(type = c("deeppink", "orange", "yellow2", "royalblue"))+
  ylim(c(0, 0.15))+
  labs(y = "density")

Applying incentives is likely to lead to a different distribution for future deals.

• Consider what the relevant factors and assumptions are in influencing the existing distribution.
• Take a moment to hypothesize what the new distribution will look like after incentives are applied.

After applying incentives the resulting distribution is likely to depend on:

1. The starting distribution⁴ of deals.
2. What the incentives are and how they influence the initial distribution.
3. How this influence degrades the farther away the starting position of a deal is from the next tier up in incentives.

You could paramaterize this problem and model the expected distribution. Making some Simple Assumptions (described in the Appendix), the chart below shows a (potential) resulting distribution after applying the incentives.

data_distribution <- data_distribution %>% 
  mutate(cum_dens = cumsum(dens_scaled),
         incentive = ((cum_dens) %/% 0.2500001) * 5,
         incentive_delta = ifelse(incentive == 15, 0, 5)) # hard coded

# very inefficient approach computationally
get_row_bump <- function(price_current){
  current_incentive <- data_distribution %>% 
    filter(price == {{price_current}}) %>% 
    pull(incentive)

  output <- data_distribution %>% 
    filter(price > {{price_current}}, incentive > current_incentive) %>% 
    head(1) %>% 
    pull(price)
  
  if(length(output) == 0) output <- NA
  
  output
}

proportion_raised <- function(density, incentive_delta, distance){
  decay <- 0.75^(distance - 1)
  incentive <- incentive_delta * 0.15
  # incentive <- ifelse(incentive > 1, 1, incentive)
  
  density * decay * incentive
}

data_distribution %>% 
  mutate(quartile = (cum_dens) %/% 0.2500001 + 1,
         quartile = as.factor(quartile)) %>% 
  mutate(price_next = map_int(price, get_row_bump),
         price_dist = price_next - price) %>% 
  mutate(density_convert = proportion_raised(dens_scaled, incentive_delta, price_dist)) %>% 
  select(-dens) %>% 
  # add converted density to nearest point
  group_by(incentive) %>% 
  mutate(dens_convert_total = sum(density_convert),
         price_switch = price == min(price)) %>% 
  ungroup() %>% 
  mutate(dens_convert_total = lag(dens_convert_total),
         dens_convert_total = ifelse(price_switch, dens_convert_total, 0)) %>%
  # na's to 0's
  mutate(across(c(density_convert, dens_convert_total), ~ifelse(is.na(.x), 0, .x))) %>% 
  # adjust percentages
  mutate(dens_adj = dens_scaled - density_convert + dens_convert_total) %>%
  rename(initial_quartile = quartile) %>% 
  # graph
  ggplot(aes(x = price))+
  geom_col(aes(y = dens_adj, fill = initial_quartile))+
  scale_fill_discrete(type = c("deeppink", "orange", "yellow2", "royalblue"))+
  ylim(c(0, 0.15))+
  labs(y = "density")

Gray bars in the chart below indicate where (on the original distribution) movement to a higher tier will occur.

data_distribution %>% 
  mutate(quartile = (cum_dens) %/% 0.2500001 + 1,
         quartile = as.factor(quartile)) %>% 
  mutate(price_next = map_int(price, get_row_bump),
         price_dist = price_next - price) %>% 
  mutate(density_convert = proportion_raised(dens_scaled, incentive_delta, price_dist)) %>% 
  select(-dens) %>% 
  rename(initial_quartile = quartile) %>% 
  ggplot(aes(x = price))+
  geom_col(aes(y = dens_scaled, fill = initial_quartile))+
  geom_col(aes(y = density_convert), fill = "gray")+
  scale_fill_discrete(type = c("deeppink", "orange", "yellow2", "royalblue"))+
  ylim(c(0, 0.15))+
  labs(y = "density")

data_distribution %>% 
  mutate(quartile = (cum_dens) %/% 0.2500001 + 1,
         quartile = as.factor(quartile)) %>% 
  mutate(price_next = map_int(price, get_row_bump),
         price_dist = price_next - price) %>% 
  mutate(density_convert = proportion_raised(dens_scaled, incentive_delta, price_dist)) %>% 
  select(-dens) %>% 
  mutate(incentive = case_when(incentive == 0 ~ "pink to orange",
                               incentive == 5 ~ "orange to yellow",
                               incentive == 10 ~ "yellow to blue",
                               TRUE ~ "stayed blue") %>% fct_inorder()) %>% 
  group_by(incentive) %>% 
  summarise(density_converted = sum(density_convert) %>% round(3)) %>% 
  knitr::kable()

Simple Assumptions

For this example, we will say the incentives you established are higher compensation rates depending on which quartile the deal falls in. If the deal falls in the lowest quartile they get no increase, in the 2nd quartile they get a 5 percentage point (ppt) increase in pay, the 3rd a 10 ppt increase, the 4th a 15 ppt increase⁹.

For now I’ll pick some overly simple but sensible values for each question:

1. As indicated, we are assuming the ‘natural’ distribution of prices is roughly normal

2. We will assume that for every 1 ppt change in incentive that 15% of the deals immediately to the right of the cut-off will be moved up to the cut-off value.

3. We will assume that this influence degrades by 25% for every dollar you move from the cut-point¹⁰. I am ignoring the possibility of deals jumping more than on level (e.g. deals moving from the 1st quartile to the 3rd quartile)¹¹.

Trade-offs

• A sales rep may have been able to sell more product at a lower price.
  • The additional incentive causes some deals (those selling for lower prices) to be passed on because the incentive to close on the deal for reps has been lowered (this may be intentional in that the impact on price erosion of these deals is worth the decrease in sales…).
• You may have to pay your sales reps more
• Applying such incentives may create additional bureaucratic hurdles in closing deals that increase the friction of closing deals, causing some percentage of deals to be lost
  • It could be that deals don’t have slack in them and are already optimal…
  • Any change in pricing behavior has the risk of upsetting customers or having downstream affects.

Ideally the organization is able to take into account risks and advantages in pricing and set-up incentives that are focused on overall profitability and firm growth (not just in terms of a single factor).