“The Performance Pay Nobel”, Alex Tabarrok2016-10-10 (mechanism design, decision theory; backlinks; similar)⁠:

The 2016 Nobel Prize in economics goes to Oliver Hart and Bengt Holmström for contract theory, the design of incentives.

…Suppose that you are a principal monitoring an agent who produces output. The output depends on the agent’s effort but also on noise…rewarding output alone gets you the worst of all worlds, you have to pay a lot and you don’t get much effort. But perhaps in addition to output, y, you have a signal of effort, call it s. Both y and s signal effort with noise but together they provide more information. First lesson: use s! In fact, the informativeness principle (Holmström1979) says you should use any and all information that might signal the agent’s effort in developing your contract.

But how should you combine the information from y and s? Suppose you write a contract where the agent is paid a wage, w = β₀ + β_yy + β_ss where β₀ is the base wage, β_y is the beta on y, how much weight to put on output and β_s is the weight on the s signal—think of β_y as the performance bonus and β_s as a subjective evaluation bonus. Then it turns out (under some assumptions etc. Canice Prendergast has a good review paper) you should weight β_y and β_s according to the following formula:

$B_{y} = \frac{σ_{s}^{2}}{σ_{s}^{2} + σ_{y}^{2} + r c σ_{s}^{2} σ_{y}^{2}}$

$B_{s} = \frac{σ_{y}^{2}}{σ_{s}^{2} + σ_{y}^{2} + r c σ_{s}^{2} σ_{y}^{2}}$

that looks imposing but it’s really not. σ_s² is the variance of the s signal, σ_y² is the variance of the y signal. Now for the moment assume r is zero so the formula boils down to:

$B_{y} = \frac{σ_{s}^{2}}{σ_{s}^{2} + σ_{y}^{2}}$

$B_{s} = \frac{σ_{y}^{2}}{σ_{s}^{2} + σ_{y}^{2}}$

Ah, now that looks sensible because it’s an optimal information theorem. It says that you should put a high weight on y when the s signal is relatively noisy (notice that β_y goes to 1 as σ_s² increases) and a high weight on s when the y signal is relatively noisy. Notice also that the 2 βs sum to 1 which means that in this world you put all the risk on the agent.

Ok, now let’s return to the first version and fill in the details. What’s r? r is a measure of risk aversion for the agent. If r is 0 then the agent is risk neutral and we are in the second world where you put all the risk on the agent. If the agent is risk averse, however, then r > 0 and so what happens? If r > 0 then you don’t want to put all the risk on the agent because then the agent will demand too much so you take on some risk yourself and tamp down β_y and β_s (notice that the bigger is r the smaller are both β_y and β_s) and instead increase the base wage which acts as a kind of insurance against risk. So the first version combines an optimal information aggregation theorem with the economics of managing the risk-performance-pay tradeoff.

Let’s also discuss some further work which is closely related to Holmström’s approach, tournament theory (Lazear & Rosen 1981). When should you use absolute pay and when should you use relative pay? For example, sometimes we reward salespeople based on their sales and sometimes we reward based on which agent had the most sales, ie. a tournament. Which is better?

The great thing about relative pay is that it removes one type of noise [variance reduction by baseline estimation]. Suppose, for example, that sales depend on effort but also on the state of the economy…But relative pay isn’t always better. If the sales agents come in different ability levels, for example, then relative pay means that neither the high ability nor the low ability agents will work hard. The high ability agents know that they don’t need to exert high effort to win and the low ability agents know that they won’t win even if they do exert high effort. Thus, if there is a lot of risk coming from agent ability then you don’t want to use tournaments. Or to put it differently, tournaments work best when agent ability is similar, which is why in sports tournaments we often have divisions (over 50, under 30) or rounds.

…Holmström’s work has lot of implications for structuring executive pay. In particular, executive pay often violates the informativeness principle. In rewarding the CEO of Ford for example, an obvious piece of information that should used in addition to the price of Ford stock is the price of GM, Toyota and Chrysler stock. If the stock of most of the automaker’s is up then you should reward the CEO of Ford less because most of the gain in Ford is probably due to the economy wide factor rather than to the efforts Ford’s CEO. For the same reasons, if GM, Toyota, and Chrysler are down but Ford is down less, then you might give the Ford CEO a large bonus even though Ford’s stock price is down.

Oddly, however, performance pay for executives rarely works like a tournament. As a result, CEOs are often paid based on noise.

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