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MedFinVentures.org
It's common to have one offer in-hand while contemplating another. How do you determine if you should take or pass on the first offer when you don't have a commitment from the second one? Let's use probability to help us understand the situation better.
A typical scenario.
Understanding the scenario
In the case above, our healthcare professional contemplates 2 offers: one that s/he has (from company 1) and one that s/he may want more (company 2). S/he has already received an offer from company 1: let's value that offer at $x where x is the annual salary offered by company 1.
Company 2 is offering more money, and has offered but not yet conducted an interview with our healthcare pro. Let's call the offer from company 2 $x*(1+y) where y is the percent increase in salary compared to that offered by company 1.
Expected salaries
In order for our healthcare professional to turn down the offer from company 1, s/he has to know the probability of obtaining an offer from company 2: let's call that p(company_2). The probability of obtaining an offer, along with the salary, tells us what s/he can *expect* from each company.
E(company_1) = salary from company 1 * p(company_1)
E(company_2) = salary from company 2 * p(company_2)
Expected salary from company 1
The salary offered to our healthcare pro is known to him/her, but unknown to us. As a result, we labeled that salary as $x. And since we know that s/he was offered the role, his/her probability of getting an offer from the company is 100% = 1.0
E(salary from company_1) = 100%*x = $x
Expected salary from company 2
Company 2 is offering more than company 1: we represent that percent increase as y.
Thus, the salary for company 2 is $x*(1+y). However, this salary isn't a guarantee: it is only realized if our healthcare professional gets the offer: represented as p(company_2).
So, the expected salary from company 2 is:
E(salary from company_2) = p(company_2)*$x*(1+y).
Indifference point
Now we're at the point of comparing the salaries, in expectation, from companies 1 and 2. Based on the increase in salary from company 2, we can determine the probability needed to get that job in order to make it worth turning down the offer from company 1.
E(salary from company_1) = E(salary from company_2)
Final equation that shows the relationship between the probability needed to get the role from company 2 relative to the increase in salary company 2 offers.
Understanding the final equation
In the final equation, the probability of getting the role from company 2 is influenced by the increase in salary that company 2 offers. The greater the increase in salary (compared to company 1), the lower the probability needed of obtaining the role.
For example, say the salary bump company 2 offers is 10%. With such a nominal increase, our healthcare pro would need to reach a high probability (~ 91%) of getting the role in order to justify turning down the role from company 1.
As the salary bump increases, the probability that s/he needs to reach decreases. If the salary increase is 50% over company 1, our healthcare pro would need to pass a 67% probability of getting the job in order to justify turning down the role in company 1.
Note that the p(company_2) doesn't depend on the actual salary that company 1 offered ($x); it only depends on the percent increase offered by company 2 (y).
It's now up to our healthcare professional to determine how far s/he is from the target probability of getting the job from company 2. The more company 2 is willing to pay, the more risk s/he can take.
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