In many settings employees do not only influence their average performance but also the probability of different performance outcomes.
An investment manager, for instance, can increase the average performance of her portfolio by putting more effort into identifying investment opportunities. Given the set of investment opportunities that she has identified, however, she can also decide whether to invest in high-risk assets that pay a large payoff with a small probability or in low-risk assets that pay a small payoff with a large probability.
Similarly, a salesman can increase his average sales by putting more effort into identifying more potential accounts. Given the set of potential accounts, however, he can also decide whether to focus on a large account that he is likely win with a small probability or a small account that he is likely to win with a large probability.
Firms often motivate their employees to increase their average performance by paying them a bonus if their performance is above a particular threshold. The purpose of this document is to illustrate how such a bonus affects the employees' decisions on what type of investments, customers, and so on to focus on.
You are an investment manager who is in charge of an investment portfolio. The average return of your portfolio is given by $100k (and determined by how much effort you put into identifying investment opportunities, which we are going to take as given). There are five performance outcomes: losing $100k in asset value, not changing asset value, and increasing asset value by $100k, $200k, or $300k. Even though the average performance of your portfolio is fixed, you control the probability with which the different performance outcomes are realized (subject to average performance being $100k and the probabilities of all five outcomes adding up to 100%).
You get paid a bonus of $50k if the asset value is at least $100k. Use the sliders below to identify the probability distribution that maximizes your expected bonus (i.e. the probability that you get the bonus times the bonus).
Next, change the parameters such that you get paid the bonus only if asset value is at least $200k. What is the probability distribution that maximizes your expected bonus?
Finally, keep the bonus threshold at $200k but now reduce the payoff associated with the worst performance outcome (you can pick by how much to reduce it). Notice that in the background the program increases the payoff associated with the best performance outcome to keep the average the same). What is the probability distribution that maximizes your expected bonus?
As you are working on your answers, notice that:
