What models are used to get the Next Best Action to convert a physician from a non writer to a write?

Expanding on my comment, the simplest predictive model would be to just compute the conversion rates Pr(conversion | action) for each action {A,B,C,D, ... }.

A solution in R:

library(dplyr)
library(tidyr)

set.seed(100) # for reproducibility
action <- sample(letters[1:5], 100, replace = TRUE) # dummy up sales actions (independent variable)
converted <- as.logical(rbinom(100, 1, prob = .5))  # dummy up physician conversion (dependent variable)

(df <- data.frame(action = action, converted = converted))

# Compute conversion rates for each action
conv_df <- df %>% 
  group_by(action, converted) %>% # unique combinations of action by converted
  summarize(freq = n()) %>% # compute freq of unique combinations
  spread(converted, freq) %>% # spread rows to cols
  mutate(rate = `TRUE`/sum(`TRUE`,`FALSE`)) # compute conversion rate

conv_df[order(-conv_df$rate),] #reverse sort

##Source: local data frame [5 x 4]
##Groups: action [5]
##
##  action FALSE  TRUE      rate
##  (fctr) (int) (int)     (dbl)
##1      d     8    13 0.6190476
##2      e     7    11 0.6111111
##3      a     5     6 0.5454545
##4      c    13    10 0.4347826
##5      b    17    10 0.3703704

The solution above assumes the application is to be some sort of guide for sales reps... if you are wanting to model the distribution of actions in sales contexts, you would need to compute the prevalence of each action as well.

As Brandon suggested I thing the right formalism is to predict the probability increase of the desired target given the prescription was written. I think using causal models is a good go to solution as you can frame the prescriptions as treatments and the models will attempt to reduce the bias within the data. Models such as econML or causalML