Daily Box Score 9/17: Pricing Risk & Giving Incentives

Life is characterized by uncertainty. Maybe you think it's a bummer; maybe it's what gets you out of bed in the morning. But either way, uncertainty is part of daily life. It's involved in personal relationships, business, and yes, even sport.

It's important to keep in mind that uncertainty is a different concept than risk. The difficulty with uncertainty is that it is substantially harder to figure out how to deal with it, whether by preparing emotionally or attaching a price to it (as when you buy insurance). What does this mean for baseball teams?

Table of Contents

A Demonstrative Hypothetical
Arbitrage
Incentives
Discussion Question of the Day

A Demonstrative Hypothetical

For those of you who don't browse the right hand side of the screen very often, you are missing out on the FanPosts section of the site. There, anybody can post their thoughts, questions, and ideas. Anyway, I was reading FanPosts the other day, when I came across this one from (the ghost of) pizzacutter. He proposes a hypothetical:

Imagine for a moment the perfect starting pitcher.  And when I say perfect, I mean he's practically perfect in every way.  He is guaranteed to go out every fifth game and to throw nine perfect innings, striking out all 27 batters on 81 pitches.  Guaranteed. [...]

Pedro Halladay "Three Finger" Gibson-Young will be a free agent this winter.  Imagine that you have a $100M payroll to work with.  How much would you offer him on a per annum basis?  Remember that you do have 24 other roster spots to fill and every million that you offer to Mr. Gibson-Young is another million that you can't spend on the rest of the team.

It's a fun hypothetical, and I'll let you head over there to see my answer and the answers of others. But it's important to point out the ways in which this hypothetical deviates from reality.

First, the production is guaranteed. There is complete certainty of outcome. That is to say, there is no risk. Every time out, he puts up a 9 0 0 0 0 27. (Really, I just wanted to see what that looks like.)

For most pitchers, we have a relatively good idea of what their true talent is, but it comes with a distribution of possible outcomes. We expect C.C. Sabathia to put up a 3.50 ERA (or so), but we can't say for sure that it won't be 4.00 or 3.00. That doesn't mean we were necessarily wrong, just that we didn't know the outcome from the start. But we knew what the most likely outcome was.

But in Mr. Cutter's hypothetical, there is also no uncertainty. Uncertainty is like risk, but without the knowledge of the probabilities. We might still be able to say what the most likely outcome is, but we aren't at all sure what the weighted probabilities are. I think injuries in baseball (at least given current knowledge) are a good example of something that is uncertain. 

As Jeff Sackmann quoted investing guru Michael Mauboussin:

So games of chance like roulette or blackjack are risky, while the outcome of a war is uncertain. Knight said that objective probability is the basis for risk, while subjective probability underlies uncertainty.

But real life is characterized by both risk and uncertainty. Normally, we can apply prices to risk very easily (though you really should be careful to read the warning label). But uncertainty can be trickier.

Here's another example of risk, courtesy of Phil Birnbaum:

The standard deviation of batting average over 500 AB is about 20 points: so even with .286 being correct, there's still a 46% chance that A will hit closer to .290 than .286 next year. There's actually about a 1 in 3 chance that Tejada's average will be below .266 or above .306. For practical purposes, it's impossible to evaluate the two predictions on this one single sample. Even if Bob is omniscient, knowing everything possible about Tejada's talent, health, and diet, it's going to take a lot of evidence to prove that he's a better estimator than the mob, so long as the results of individual at-bats are random.

This is right. Risk characterizes predictions more than anything else, and the only way to reduce it is to increase the time horizon.

But it's also true that if your sample is big enough, you may end up with some funky results that are nevertheless expected. Like, say, the fact that a fund manager beat the S&P 500 for 15 consecutive years could be entirely luck.

Arbitrage

I said that risk can be priced easily. That doesn't mean you can't make mistakes of reasoning, as demonstrated by the classic Martingale fallacy (which actually has a negative expected return). 

But even when you do everything right, and price risk properly, there is still a chance you'll go bust. Take, for example the case of arbitrageurs, the market warriors who enforce the Law of One Price with supercomputers and four-screen displays.

As Felix Salmon describes:

[I]t’s not surprise to learn that [arbitrage] comes with "a high incidence of large negative returns": any arbitrage strategy is ultimately a game of picking up nickels in front of a steamroller. Unless you have unlimited liquidity and never need to worry about margin calls, the market is likely to move against you just until you give up, at which time it will snap back to where you would have made a huge profit.

He points out that the possibility of going bust is why arbitrage is usually reserved for the big boys. And even then, sometimes they lose.

As I was talking with Jonah Keri, who by the way is writing a book about the Rays due next year, he mentioned the idea that baseball teams engage in something like arbitrage. They buy certain intermediate goods (players) to produce some more direct good (wins). They don't really care about how the players score or prevent wins, just that they do. It's not how, as they say, it's how many. 

So we expect the Law of One Price to hold in baseball as well. And, actually, this is pretty close to what we observe. For 2008, the price of a win on the free agent market was $4.5 million. Clearly, as material conditions change (attendance declines, for example), that price will change. But buying assets that are undervalued on the free agent market and selling them once they hit par for the market is a little bit like arbitrage.

But teams that do that, especially if they buy assets that are risky, are playing that same game of "picking up nickels in front of a steamroller." Sometimes, you end up with a team that is 73-73 when they were expected to win 94 games. Didn't your mother ever tell you to stay away from steamrollers?

Incentives

Risk is a fact of the world. Sometimes you're the windshield, but sometimes you're the bug. Wouldn't it be nice to think that we could do something about it? What if we were uncertain if we could do anything about it? How should we act?

J.C. Bradbury asks that question with regards to Adam LaRoche's first-half/second-half splits:

But, what if Adam is a second-half player, and a team wants him to play more like second-half Adam in the first half? How might a team structure a contract to give LaRoche the incentive to do the things he needs to do (e.g., get in shape, practice, take his medication regularly, etc.) to generate higher production. I have a simple solution: offer a big All-Star bonus. Players with strong first halves have an advantage at making the All-Star team over second-half players. Many players have All-Star bonuses in their contracts in small amounts, a few thousands dollars or so. If a full-year of second-half LaRoche is worth an additional $2 million, offer him a $2 million bonus for making the team. If he can fix the problem, it will likely be fixed. If not, you get the same LaRoche as always without having to pay the bonus.

I think this is exactly the right approach, because it hedges the possibility that LaRoche really could just expend a little more effort and become a very good hitter. My only concern would be with the fact that players seem to discount performance bonuses pretty heavily, meaning you might have to increase the value of the bonus.

Similarly, the Rays could do a similar thing with their players, to account for the fact that they (as a result of their low payroll) have to buy lots of risky assets. If they make the playoffs, the added revenue would allow them to pay out. If they don't, they wouldn't have to. I can't think of a reason why more small market teams don't include large playoff bonuses as a matter of course.

Discussion Question of the Day

What reasons can we come up with for why more small market teams don't offer playoff bonuses, considering their revenues are buoyed so much when they do in fact make the playoffs?