Using R for a salary negotiation–an extension of decision tree models

my.offer <- function(x1=60,py.x1=.2,x2=70,py.x2=.8,ev.no=50,high=100,p.payoff=1) {
  # return the offer to maximize expected payoff
  # this assumes a game with one decision and one consequence
  # you give an offer, and it is taken or refused. If taken, you receive a salary of
  # (a function of) the offer. If refused, you stay at the old job and receive a
  # salary of ev.no (presumably a current salary, but set to 0 if you are
  # unemployed).
  # the probability of rejection is modeled with a logistic function defined by
  # two points (x1,py.x1) and (x2,py.x2)
  # for example, if you expected a 20% rej. prob. with an offer of 140k, then
  # x1,py.x1 = 140,.2. Similarly with x2,py.x2
  # the expected payoff is modeled as offer*P(Yes|offer) + ev.no*P(No|offer),
  # perhaps with  modifications to account for benefits, negotiation, etc. This
  # is defined in payoff function below.
  # finally, high is defined as anything above what you would be expecting to offer
  # and is used to create the plot limits and set the bounds in the optimization
  # routine.   # model the probability of no given salary offer
  # here we have a logistic function defined by (x1,py.x1) and (x2,py.x2)
  # note that qlogis is the inverse logit function
  # also, matrices in R are defined in column-major form, not row-major form like
  # FORTRAN, so we have to use 1,1,x1,x2 rather than 1,x1,1,x2
  theta <- solve(matrix(c(1,1,x1,x2),nc=2),matrix(qlogis(c(py.x1,py.x2)),nc=1))   # for plot of probability function
  xseq <- seq(ev.no,high,length=100)
  yseq1 <- 1/(1+exp(-theta[1]-theta[2]*xseq))   # model the expected payoff of an offer
  # model negotiations, benefits, and other things here
  # (a simple way to model benefits though is just to change ev.no)
  payoff <- function(x) {
    tmp <- exp(-theta[1]-theta[2]*x)
    return( (ev.no + ifelse(ev.no>x*p.payoff,ev.no,x*p.payoff)*tmp)/(1+tmp) )
  }     yseq <- payoff(xseq)   # plots
  par(mfrow=c(1,2))
  plot(xseq,yseq1,type='l',xlab='Offer',ylab='P(No|X)')
  plot(xseq,yseq,type='l',xlab='Offer',ylab='Expected salary')   # no sense in even discussing the matter if offer < ev.no
  return(optimize(payoff,interval=c(ev.no,high),maximum=TRUE))
}

And here are the graphs and result:

> my.offer()
$maximum
[1] 61.96761

$objective
[1] 58.36087

So this model suggests that the optimum offer is close to $62k, with an expected payoff of around $58k. As a side effect, a couple of graphs are produced: giving the probability of rejection as a function of the asking price, and the expected salary (payoff) as a function of asking price.

So a few comments are in order:

• The value in this model is in varying the inputs and seeing how that affects the optimum asking price.
  
• The function I provided is slightly more complicated than what I presented in this post. You can model things like negotiation (i.e. you may end up at a little less than your asking price if you are not turned down right away), differences in benefits, and so forth. Once you have a simple and reliable baseline model with which to work, you can easily modify it to account for other factors.
  
• Like all models, this is an oversimplification of the salary negotiation process, but a useful oversimplification. There are cases where you want to be more aggressive in your asking, and this model can point those out.
  
• I commented the code profusely, but the side effects are probably not the best programming practice. However, this really is a toy model, so feel free to rip off the code.
  
• This model of course extends to other areas where you have a continuous range of choices with payoffs and/or penalties.
  
• If you are able to gather data on the probability of rejection based on offer, so much the better. You can then, instead of fitting an exact probability model, perform a logistic regression and use that as the basis of the expected payoff calculation.