Monday, July 28, 2014
Diane says to Jack: "Don't go outside! It's raining cats and dogs."
Jack goes outside, and Diane finds out. She confronts him: "Jack! I told you not to go outside!" "But the weather cleared up," Jack responds, and Diane is satisfied.
Several days later, a conservative think tank files suit against Jack for lawlessly violating the plain meaning of "Don't go outside!"
Posted by Jonah B. Gelbach at 12:23 PM
Tuesday, November 17, 2009
[Updated to correct mis-spellings of Bill Belichick's name.]
By now everyone knows about Bill Belichick's much criticized decision to go for a first down "on fourth-and-2 from the Pats’ own 28-yard line with a six-point lead and 2:08 remaining in the fourth quarter."
Going against the grain, Steve Levitt argued yesterday that Bill Belichick Is Great for making this decision. Steve bases his argument on a well known paper by Berkeley economist David Romer, titled Do Firms Maximize? Evidence from Professional Football.
I think Levitt is off base. There are two issues.
First, Romer's article (self-consciously) says nothing about late-game situations: it deliberately focuses on first-quarter situations precisely to avoid issues related to the end of games. Thus, even if Levitt is right, Romer's article can't be the reason.
Second, I show below that when one thinks carefully about the actual situation and uses plausible values of the relevant probabilities that NE wins in each situation, the right way to think of this problem is to develop a threshold probability, m*, of making the first down. When the actual probability is above this threshold, NE's win probability is maximized by going for the first down, and otherwise the win probability is maximized by punting. The threshold m* depends only on two conditional probabilities: the probability that NE wins when it punts and the probability that NE wins when it goes for the first down but does not make it. What I regard as plausible bounds on these conditional probabilities yield bounds on the threshold m* of 17 and 78 percent. It seems very likely that the true probability of converting on fourth and 2 is between these bounds. In other words, without knowing the conditional probabilities described above, all we can say is that Levitt might be right, but he might also be wrong.
This issue is one that could be resolved with data -- but not the Romer data that Levitt cites. Here are the details.
I. Romer's Paper
Romer's abstract states:
Play-by-play data and dynamic programming are used to estimate the average payoffs to kicking and trying for a first down under different circumstances. Examination of actual decisions shows systematic, clear-cut, and overwhelmingly statistically significant departures from the decisions that would maximize teams’ chances of winning.
Translated, Romer is saying that teams should go for it on fourth down, much more often than they actually do. Romer's paper is very well done, and it presents compelling evidence. A key fact about Romer's paper is that he accounts not only for the immediate consequences of a team's fourth-down decisions, but also for its subsequent effects:
But Romer is the first to note that, by design, his evidence has little to do with situations like the one in the Indy-Pats game the other night. His paper doesn't and, given its design, can't address late-game situations. Romer writes on page 344 (emphases mine):
The choice between kicking and going for it leads to an immediate payoff in terms of points (which may be zero) and to one team having a first down somewhere on the field. That first down leads to additional scoring (which again may be zero) and to another possession and first down. And so on. (Page 342.)
By describing the values of situations in terms of expected point differences, I am implicitly assuming that a team that wants to maximize its chances of winning should be risk-neutral over points scored. Although this is clearly not a good assumption late in a game, I show in Section IV that it is an excellent approximation for the early part. For that reason, I focus on the first quarter.
In the present context, this is essentially the game's final situation: given that Indy has just one time out remaining, if New England makes the first down, the game is effectively over.* So, it doesn't matter whether New England scores if the Pats make the first down. (*It is possible that NE gets the first down on the play of interest but then turns the ball over to Indy on a subsequent fourth down with a very small number of seconds. But this possibility only reduces the value of going for it now, so I will ignore it.)
Now consider what happens if Indy gets the ball back now, however that happens. Should Indy score, there is no chance Indy would go for 2, so Belichick's relevant lower bound is -7. If we assume that Indy will hit the PAT with probability 1, which is approximately right, then we might as well regard the lower bound as "lose/no-lose".
In sum, Romer's objective function is the wrong one for this situation -- as he obviously understands from the second passage quoted above. So Romer's paper really doesn't provide any empirical support for Levitt's claim that Belichick did the right thing. (This is true even if one thinks that Belichick is risk-neutral over game wins in this situation: what matters is the binary variable representing Indy-gets-a-TD, not expected point difference.)
II. The Right Way to Look at This Situation
Look at it this way.
- Assume the Pats will win the game for sure if they go for it and make the first down. Let m be the probability that NE makes the first down if the Pats go for it. Also let q be the probability that the Pats win even when they go for it and don't get the first down (so q is the probability that NE either prevents an Indy TD or gives one up and then scores themselves).
- If New England punts, NE will either win the game (by not letting Indy score a TD or by letting them do so but subsequently scoring themselves), or NE won't. Let p be the probability that NE does win, if New England punts.
Levitt makes a lot of hay about his belief that Belichick's decision maximized win probability (even under Levitt's postulated principal-agent problem). So let's assume that Belichick's goal indeed is to maximize win probability. That means he should go for it in our situation if and only if the probability of winning when he goes for it is greater than the probability of winning when he kicks (I'll ignore the possibility that the win-probabilities are equal). Given our definitions above:
- If New England goes for it, the probability of winning is m + (1-m)q. That is, the Pats win when they make the first down or, having not made it, win anyway.
- If New England punts, the probability of winning is p.
So, Belichik should go for it whenever m + (1-m)q > p. If we solve this inequality for m, we see that it amounts to the condition
m > (p-q) / (1 - q) = m*.
The threshold's denominator is the probability that NE loses, given that NE goes for the first down and doesn't make it, giving Indy excellent field position. The numerator is the difference in win probabilities given that NE either kicks (p) or goes for it and doesn't make the first down (q). Since p is less than 1 and q is greater than 0 and less than p, m* is always between 0 and 1: our threshold is a proper probability.
If I had to guess, I'd think the following:
- NE's win probability when it punts, p, is at least 50 percent here, but not more than 80 percent (yes, Indy's offense is great, but the Pats had intercepted Manning twice, and Indy did have only 28 points after 58 minutes, after all; plus, there's only 2 minutes left, a punt will put Indy somewhere between its own 20 and 30, and Indy needs a TD, not just a field goal).
- NE's win probability when it goes for it and doesn't get the first down, q, is at least 10 percent and not more than 40 percent. My thinking here is that going 30 yards is a LOT easier than going 70 yards, especially with so little time remaining.
Given these bounds, we can bound the threshold value of m as follows:
- NE wins 50 percent of the time when it punts and 40 percent of the time when it goes for it and doesn't get the first down: p = 0.5, q = 0.4. This can be shown to be the most friendly-to-Belichick-and-Levitt case my bounds allow. It implies that Belichick should go for it if and only if m is greater than 1/6: NE has to have at least a 17 percent chance to convert on 4th and 2.
- NE wins 80 percent of the time when it punts and only 10 percent of the time when it goes for it and doesn't get the first down: p = 0.8, q = 0.1. This implies that Belichick should go for it if and only if m is greater than 7/9: NE has to have at least a 78 percent chance to convert on 4th and 2.
Thus my bounds on the threshold m* are 17 percent and 78 percent. Personally I think the truth is probably closer to the high end, so I am skeptical that the decision was a good one. But even if you disagree, it seems very likely that NE's chance of getting the first down lies between 17 and 78 percent. If you accept my bounds on p and q, then you must agree that it is an empirical question whether Belichick made the right call.
I'll be curious to see what values Levitt thinks p and q take (maybe he'll even estimate them!).
Posted by Jonah B. Gelbach at 6:03 AM
Monday, August 31, 2009
Some politicians like to push public-option health insurance plans by describing them as giving "every citizen the same plan that every member of Congress has". This marketing scheme is fine as far as it goes, but apparently it works only so much.
So here's an alternative: let's give every member of Congress the same plan that a randomly drawn member of the public has.
Most important, let's tally up the share of Americans who go without health insurance each year, and let's randomly assign the same share of House and Senate members to uninsurance status the following year: we, their employers, will not provide them with a health plan.
For example, if 15 percent of Americans are uninsured in 2010, then round(0.15*535,1) = 80 randomly drawn House and Senate members will be denied employer-provided coverage in 2011. Most senators will be ok, since they're almost all rich and can afford individual-market plans. But some of those House members might actually suffer for it and get a taste of the not-so-good life.
Caveats: Yes, I know that the uninsured are not a random draw from the population. And yes, I know that there are real debates over how to interpret survey data concerning the share who are uninsured. But I'm going for simplicity here.
So how about it, all you congressional universal-coverage opponents: are you ready to take the same odds facing the randomly drawn U.S. citizen?
Posted by Jonah B. Gelbach at 7:31 PM
Friday, August 21, 2009
Nancy Pelosi says the House cannot pass a health care bill without a public option. Kent Conrad says there aren't 60 votes in the Senate for a bill with a public option.
So what's a party interested in governing the nation to do?
One compromise Congress sometimes uses in such situations is to enact a demonstration plan with a short-term sunset -- something like five years. That's a bad idea here. Any publicly run plan will require developing a network of providers, which takes time and--more important--money to put together. The fixed costs of setting up public plans around the country make a demonstration with a sunset wasteful. Moreover, there's little reason for providers to take seriously the bargaining power of a program they know may go away in short order.
Here's a better alternative: implement a permanent public option in some parts of the country, but not others. This approach makes both economic and political sense.
First, consider the economic front. Most types of health expenditures go to regional providers: doctors and hospitals generally provide brick-and-mortar service. So, most of the bargaining power and economies of scale that proponents argue a public plan will achieve should be feasible without a nationwide plan. If public-option supporters are right about its virtues, they can be acheived on a regional or state level. (The obvious exceptions are pharmaceuticals and medical devices, though few of the gains that a public plan's supporters project seem to have been based on these goods.)
Second, consider politics. Public plan opponents have conjured up various boogey men, with none appearing quite so often as the specter of "socialized medicine". Apparently a publicly funded health insurance plan will cost a lot and deliver low-quality care.
Let's leave aside a discussion of the status quo's performance on these dimensions for brevity. Instead, I'll point out that under my compromise, Congressional skeptics of a public plan could rest assured that no one in their districts or states had access to a voluntary public plan, preserving the status quo for their constituents. So, folks like Jim DeMint and Michele Bachmann wouldn't have to make constitutional arguments or threats of revolution against a public plan -- they could just make sure that neither South Carolina nor Minnesota is included in the public plan's coverage area. If they're right that the public plan will reduce freedom, kill grandma, destroy capitalism, or whatever, then they'll have an easy time pointing to the medical carnage sure to unfold in areas where people can choose a public plan.
Why is my compromise a good idea for supporters of a public plan? If they're right that the plan will cut costs, keep private insurers honest, improve quality, and so on, then it will do all these things in the parts of the country where it takes effect. People in the rest of the country will be able to see these virtues, and they'll clamor for a piece of the public action -- with the end result being a nationwide public plan.
It goes without saying that each side stands to gain electorally if it turns out to be right enough to convince supporters of the other (see Medicare and Social Security, for an example). And if no one can convince anyone else, it won't be the end of the world: some areas will have a public option and others won't. Other goals like universal coverage can be met with other measures.
How should we choose which areas should be served by a public plan? One novel option would be to include those congressional districts whose House representative votes yay and excludes those whose rep votes no (admittedly, it might be tough to write legislation that effects this without knowing ahead of time who will vote for it).
A more conventional approach would be to let the federal government set a standard package of benefits and premium levels and then allow states to decide via their usual legislative processes whether to participate in a federal-state partnership funded entirely or partly via federal tax dollars. This approach has been used for important federal-state programs like cash assistance and Medicaid, and it has both advantages and disadvantages. The devil would be in the details, but such a system could be designed to satisfy the claims of people on both sides of this issue.
So why not do it? Let's enact a permanent public plan that denies the benefits of a public plan to convervatives and imposes the horrors of socialized (financing of) medicine on progressives.
Update: Looks like Reps. James Clyburn, Louise Slaughter, and John Larson are thinking along similar lines....
Posted by Jonah B. Gelbach at 10:47 AM
Friday, January 30, 2009
The White House released a fact sheet explaining its support for fiscal stimulus plan. Here's an excerpt:
The President’s economic plan has three main goals:No surprise there, right? After all, President Obama has been quite clear about his support for a stimulus plan to achieve these goals, all of which are textbook Keynesian stuff.
- Encourage consumer spending that will continue to boost the economic recovery and create jobs.
- Promote investment by individuals and businesses that will lead to economic growth and job creation.
- Deliver critical help to unemployed citizens.
The White House fact sheet in question was released in the name of former President George W. Bush, on January 7, 2003. You can read the whole thing here (lots of plugs for tax cuts, of course).
Here's an entertaining chunk of text from a page A1 story in the LA Times the next day:
Here's an entertaining chunk of text from a page A1 story in the LA Times the next day:
How things change.