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In the bottom of the 8th inning of tonight’s game 3 of the NLDS, Anthony Rizzo blooped a single to shallow left and Leonys Martin scored what proved to be the winning run.  The noteworthy thing is that Martin scored from second base on a ball that was so shallow it was playable by the shortstop.

The reason he was able to score from second is that there were two outs and he was free to run without worrying about potentially tagging if the ball was caught. So this raises an interesting question:  was this a situation in which the Cubs were better off with two outs (as opposed to one or zero)?

Hint:  neither answer is cut and dry correct but I do claim there is a right answer.

You are in a negotiation and you are offered X. You really want both X and Y but you have just been given the option to take X now and then continue negotiating for Y.

In this story you are the House of Representatives, X is repeal of the mandate and Y is rollback of Medicaid. You have been offered X in the form of the “skinny repeal”

Will you take X? That is, will the House just pass the skinny bill as Lindsey Graham and John McCain worry they might?

A one player extensive-form game with perfect information.  The player’s name is Zeno. The game has an infinite horizon.  For every non-negative integer N there is a node at which Zeno chooses either to Continue or Quit.  If he Quits at node N his payoff is -N/(N+1). If he continues he moves on to node N+1.

There is also a terminal node at infinity.  This node is reached if and only if Zeno Continues at all finite nodes.  The payoff at infinity is 1.

Here is the story:  Zeno stands at the starting line of a Marathon.  After running the distance he decides whether to quit, etc.  His goal is to finish the Marathon and run into the arms of his proud and adoring fans.  If he doesn’t get to the finish line (the node at infinity) then all he has done is make himself tired with no compensating adoration.  The farther he runs before quitting the more tired he is.

The game has a unique subgame perfect equilibrium:  Zeno completes the Marathon.  But it has another strategy which is unimprovable by a one-stage deviation:  Zeno quits at every opportunity.

This latter strategy has a nice behavioral interpretation.  Zeno lacks confidence in his determination to complete the Marathon.  In particular he wants to complete it but he expects that if he runs another half of the distance he will wind up quitting once he gets there so why bother.  And indeed the reason he knows he will quit after running half the distance is that he knows that when he gets there he will know that after running another half the distance he will still quit so why bother.

This is a great example for teaching the One Stage Deviation Principle, which asserts that strategies that are unimprovable are also SPE.  The OSDP requires the game to be continuous at infinity.  The Marathon game is not continuous at infinity.

To make it continuous at infinity, assume that Zeno’s fans will be almost as proud of him if he runs 26.1999 miles as they would be if he ran the remaining one-millionth of a mile.  If so, then Zeno’s payoff from nearly completing the Marathon is positive and the Quitting strategy becomes improvable.

On Saturday The New York Times published the story that Donald Trump Jr together with Paul Manafort and Jared Kushner met in Trump Tower with a Russian Lawyer.  That was basically the entire content of the story in terms of new information being reported.  The rest of the story was recap of allegations about collusion together with some hint-hint dot connecting.

Junior responded by acknowledging the meeting and saying that it was about Russian adoptions.

The Times followed up on Sunday by reporting that this was in fact a lie, that Junior was promised information about Hillary Clinton that would be valuable for his campaign.

Junior acknowledged this, but minimized the significance of the meeting through a number of statements.

Finally the Times was set to publish accounts of an email thread that apparently showed that Trump was told in advance that this was an effort by the Russian government to provide information that would help his father get elected.

Before they could do that Junior published the actual emails himself.

Someone, either the Times’ source, or the Times journalists themselves had all of that information at the very beginning.  And rather than do the normal journalistic thing of reporting what they knew they released it in little bits.

The first bit was barely meaningful to every reader in the world except for Trump, Manafort Kushner.  It was a message that we have some information that you might not like and it was a challenge to find the right response.

Their dilemma was that first and foremost they don’t want to admit to have colluded with the Russian government.  But, if that was going to be proven anyway they would rather not first lie about it and then have to admit it.

They tried lying.  The Times released another little piece which did two things.  First it proved they lied, and second it proved that they have more information than they had in their original story.

Now Junior and Co. had to guess just how much information.  They guessed wrong.  Indeed the Times had all of the information and in the end Junior was forced to admit what he could have admitted at the beginning and along the way he was caught in two lies.  Indeed in the end he pre-empted the Times revelation by revealing everything himself.

This is a very clever tactic by journalists who are technically doing the journalistically-ethical thing of publishing only the truth,  but being strategic about it by publishing in little pieces to maximize the damage, probably because the Trump administration has made them the enemy.

Note how truly powerful this tactic is once it has been deployed.  The next time the Times has some but not all of the information that would prove guilt they will again release a little bit of it.  Trump will have to guess again how much they are holding back.  Trump will have to trade off the probability of being caught in another lie (if The Times truly has all the dirt) versus credible denial (if they don’t).  If used optimally, this bluffing tactic can get the subject to pre-emptively confess (i.e. to fold to extent the poker analogy) even when the damaging evidence isn’t there.

(One could write a simple dynamic Bayesian persuasion model to calculate the Times’ optimal probability of bluffing in order to maximize the probability that the subject’s best-response is to pre-emptively confess.)

Everyone who has armchair-theorized why movie theaters don’t sell assigned seats in advance is now obligated to explain why this has changed and how that’s consistent with their model.

I will start.  My theory was based on the value of advertising to movie-goers who must arrive early to get preferred seats and then are a captive audience.  This has become significantly less valuable now that said movie-goers can bring their own screens and be captive to some other advertiser.

Indeed today at the movies we arrived on time to our assigned seat, there were no ads and just a few previews.  (Spoiler alert:  the movie with Stringer Bell and the Titanic girl is going to suck, Bladerunner is going to be great and Wonder Woman was awesome for the first and last 10 minutes but apart from that was basically Splash meets Hot Lead and Cold Feet.)

1. One reason time seems to go faster when you are older. “Has it already been a year?” You have so many more things going on and only a few of them can reside in working memory. When some milestone brings a buried one back to the surface it feels like time has passed more quickly than if it were on your mind the whole time.
2. A 50 meter tall swimmer would win every race.  Generally taller swimmers have less ground to cover and so the distance is shorter for them.  Is there any kind of race other than swimming like that?
3. If you have your brain cut into two parts so that one side is unaware of what the other is doing can you tickle yourself?

The anticlimax that is/will be the Comey testimony proves something I have always thought about skeletons in the closet. They should be released early when not everyone is paying attention.

The premise is that outrage is something that needs to be coordinated. Its not enough for everyone to feel outraged. Its not even enough for everyone to know everyone is feeling outraged.  The feeling of outrage has to be common knowledge. Only that way enough people know that when they act on their outrage there will be enough others acting on their outrage for it to be a movement that can have impact.  (Outrage is a scarce resource.)

So skeletons released sometime in the past when nobody was actually looking for skeletons is a way to dampen that coordination. Because then when the time comes and people are actually focused on you and any skeletons you might have, the old already-uncloseted skeletons can’t have the same impact as a brand new one released right now when everyone sees it for the first time.  They are “old news” (and its no wonder that’s a go-to line of defense against resurrected skeletons.)  Strategically everyone infers that since the skeletons didn’t cause outrage at the time they must not be that outrageous to that many people and so they are not a call to outrage now.

So despite the several outrageous things in the Comey statement, they are all old news and that feeling of anti-climax is a symptom of the above logic. If Comey had not previewed his testimony in the preceding weeks but instead dropped it as one brand new bombshell on TV in front of a Super Bowl audience (paging Michael Chwe) the same skeletons would cause significantly more outrage.

An interesting corollary is that leaks are actually Trump’s ally. Leaking the scandal little by little through varied and segregated media channels is a way of getting the skeletons out with minimal impact well in advance of any chance of outrage.

(There’s a new paper by Gratton, Holden and Kolotilin that also looks at bombshells but I think the argument is a little different.)

1. Watch for the key leading indicator of impending Presidential downfall:  the Vice President going silent.  We will reach a point where the VP’s objective switches from defending the President to preserving his own viability as President.  Right around that turning point he will try his best to avoid having to comment.
2. That means that before reaching that point, Democrats will do their very best to get him to comment while they still have a chance.  Knowing this of course makes the VP want to hide even sooner, etc.  The unraveling may have already reached back to the present moment.
3. At some point the VP himself becomes a threat to the President because Republicans in Congress will try to find the smoothest path to a fresh Presidency with the possibility of delivering on their agenda intact.  This will involve coordination with the VP.
4. That means the President himself will try to get the VP to publicly support him in order to ensure that the VP is tarred with the same brush, removing the above branch of the tree and maintaining the threat of a messy impeachment proceeding that congressional Republicans will want to avoid.  Of course the first sign that the President needs the VP is the first signal to the VP that his best respond is to run far away, accelerating the unraveling further.
5. This implies another leading indicator of impending doom.  There will come a time when its the GOP that wants to hasten the proceedings and the Democrats will try to slow it down.
6. Trump of course is the major wild-card (sic) in all of this.  He has no loyalty to anybody and he has a massive megaphone whether he is in office our out.  There is the prospect of Trump busting out tapes of conversations with GOP leaders expressing support.  There is the possibility that he publicly turns on Pence before actually being removed from office.
7. And of course the GOP is by no means out of the water once Trump is gone.  He will be tweeting.  He will have his “tapes”.  He could even run again.

This paper by Finkelstein, Hendren, and Shepard finds that the uninsured have such a low willingness to pay for health insurance that they wouldn’t even cover the costs they impose on insurers:

Yesterday just before the start of a seminar I went to the thermostat to try to warm up the room.  The speaker saw me fiddling with it and nodded with approval.  I said nothing, turned up the heat and returned to my seat.

The thing is, she couldn’t see whether I was raising the temperature or lowering it.  She just assumed that I would be moving the temperature in the same direction as her preference.  And of course with good reason.  We both want a comfortable room.  We each get private signals about whether the room is currently too cold or too warm.  Our private signals are correlated. If her private signal suggests the room is too warm then she knows that more likely than not my private signal says the same thing, and since I am acting on my private signal and nothing more I am most likely adjusting the thermostat exactly how she wants it.

I know all of this and therefore I strictly prefer to say nothing about what I am doing.  By saying nothing I allow her to continue to believe that I am doing what she wants me to do even though I don’t know for sure what it is she wants.  Her signal of course could be different than mine.  If instead I tell her I am raising the temperature then in the best case I am just confirming what she already believed and nothing is gained.  But there is the risk that I inform her that actually I am doing the opposite of what she wants and then we have a conflict.

Her seminar was awesome.

There was 20 seconds left, Vanderbilt had just scored a layup to go ahead by 1 and Northwestern’s Bryant Mcintosh was racing to midcourt to set-up a final chance to regain the lead and win the game.  Vanderbilt’s Matthew Fisher-Davis intentionally fouled him, sending McIntosh to the line and the commentators and all of social media into a state of bewilderment.  Yes, we understand intentionally fouling when you are down 1 with 20 seconds to go, but when you are ahead by 1?

But it was a brilliant move and it failed only because the worst-case scenario (for Vanderbilt) realized:  McIntosh made two clutch free throws and Vanderbilt did not score on the ensuing possession.

(Before we get into the analysis, a simple way to understand the logic of the play is to notice that intentionally fouling late in the game very often is the right strategic move when you are down by a few points and there is no reason that should change precipitously when the point differential goes from slightly negative to slightly positive.The tactic is based on a  tradeoff between giving away (random) points and getting (for sure) possession.  The factors in that tradeoff are continuous as a function of the current scoring margin.)

Let be the probability that a team scores (at least two points) on a possession.  Let be the probability that Bryant McIntosh makes a free throw.  Roughly, the probability that Vanderbilt wins if they do not foul is because Northwestern is going to play for the final shot and win if they make a field goal.

What is the probability that Vanderbilt wins when Fisher-Davis fouls? There are multiple, mutually-exclusive ways they could win.  First, McIntosh might miss both free-throws.  This happens with probability .  The other simple case is McIntosh makes both free-throws, a probability event, in which case Vanderbilt wins by scoring on the following possession, which they do with probability . Thus, the total probability Vanderbilt wins in this second case is .

The third possibility is McIntosh makes one free-throw.  This has probability . (I am pretty sure McIntosh was shooting two, i.e. Northwestern was in the double bonus, but if it was a one-and-one this would make Fisher-Davis’ case even stronger.)  Now there are two sub-cases.  First, Vanderbilt could score on the ensuing and win.  Second, even if they don’t score, it will be tied and the game will be sent into overtime. Let’s say Vanderbilt wins with probability in overtime, a conservative number since Vanderbilt had all the momentum at that stage of the game.

Then the total probability of a Vanderbilt win in this third case is .  Adding up all of these probabilities, Vanderbilt wins using the Fisher-Davis foul with probability

Fisher-Davis made the right move provided the above expression exceeds .  Let’s start by noticing some basic properties.  First, if then fouling is always the right move, no matter what is.  (If Northwestern is going to score for sure, you want to foul and get possession so that you can score for sure and win.)  If then again fouling is the right strategy, regardless of .  (If he’s going to miss his free-throws then send him to the line.)

Next, notice that the probability Vanderbilt wins when Fisher-Davis fouls is monotonically increasing in . Since the probability Vanderbilt wins without fouling is decreasing in , the larger it is the better the Fisher-Davis gambit looks.

Finally, even if , so that McIntosh is surely going to sink two free-throws, Fisher-Davis made the right move as long as .

Ok so what are the actual values of and .  McIntosh is an 85% free-throw shooter so .  Its harder to estimate but here are some guidelines.  First, both teams were scoring (at least two points) on just about every possession down the stretch of that game.  An estimate based on the last 3 minutes of data would put at at least , in any case certainly larger than .

More generally, I googled a bit and found something basketball stat guys call offensive efficiency.  It’s an estimate of the number of points scored per possession.  Northwestern and Vanderbilt have very similar numbers here, about 1.03.  A crude way to translate that into the number we are interested, namely probability of at least 2 points in a possession, is to simply divide that number in half, again giving .  (This would be exactly right if you could only ever score 2 points.  But of course there are three-point possessions and one-point possessions.)  A third way is to notice that Northwestern was shooting a 49% field goal percentage for the game.  This doesn’t equal field goals per possession of course because some possessions lead to turnovers hence no field goal attempt, and on the other side some possessions lead to multiple field goal attempts due to offensive rebounds.

So as far as I know there isn’t one convincing measure of but its pretty reasonable to put it above at that phase of the game.  This would be enough to justify Fisher-Davis even if McIntosh was certain to make both free throws.  (I used Wolfram Alpha to figure out what would be required given the precise value and it is about .45).

Finally, even if is below say around it means that the foul lowered Vanderbilt’s win probability but not by very much at all. Probably less than every single time in the game that someone missed a shot.  Certainly less than a few seconds later when LaChance missed the winning shot on the final possession.  Its interesting how in close games the specific things we focus our attention on when in fact pretty much every single play in the game turned out to be pivotal.

How do you assess whether a probabilistic forecast was successful?  Put aside the question of sequential forecasts updated over time.  That’s a puzzle in itself but on Monday night each forecaster will have its final probability estimate and there remains the question of deciding, on Wednesday morning, which one was “right.”

Give no credibility to pronouncements by, say 538, that they correctly forecasted X out of 50 states.  According to 538’s own model these are not independent events.  Indeed the distinctive feature of 538’s election model is that the statewide errors are highly correlated.  That’s why they are putting Trump’s chances at 35% as of today when a forecast based on independence would put that probability closer to 1% based on the large number of states where Clinton has a significant (marginal) probability of winning.

So for 538 especially (but really for all the forecasters that assume even moderate correlation) Tuesday’s election is one data point.  If I tell you the chance of a coin coming up Armageddon Tails is 35%, you toss it once and it comes up Tails you certainly have not proven me right.

The best we can do is set up a horserace among the many forecasters.  The question is how do you decide which forecaster was “more right” based on Tuesday’s outcome?  Of course if Trump wins then 538 was more right than every other forecaster but we do have more to go on than just the binary outcome.

Each forecaster’s model defines a probability distribution over electoral maps. Indeed they produce their estimates by simulating their models to generate that distribution and then just count the fraction of maps that come out with an Electoral win for Trump.  The outcome on Tuesday will be a map.  And we can ask based on that map who was more right.

What yardstick should be used?  I propose maximum likelihood.  Each forecaster on Monday night should publish their final forecasted distribution of maps.  Then on Wednesday morning we ask which forecaster assigned the highest probability to the realized map.

That’s not the only way to do it of course, but (if you are listening 538, etc) whatever criterion they are going to use to decide whether their model was a success they should announce it in advance.

1. Suppose one forecaster says the probability Trump wins is q and the other says the probability is p>q.  If Trump in fact wins, who was “right?”
2. Suppose one forecaster says the probability is q and the other says the probability is 100%.  If Trump in fact wins, who was right?
3. Suppose one forecaster said q in July and then revised to p in October.  The other said q’ < q in July but then also revised to p in October.  Who was right?
4. Suppose one forecaster continually revised their probabilistic forecast then ultimately settled on p<1.  The other forecaster steadfastly insisted the probability was 1 from beginning to end.  Trump wins.  Who was right?
5. Suppose one forecaster’s probability estimates follow a martingale (as the laws of probability say that a true probability must do) and settles on a forecast of q.  The other forecaster‘s “probability estimates” have a predictable trend and eventually settles on a forecast of q’>q.  Trump wins.  Who was right?
6. Suppose there are infinitely many forecasters so that for every possible sequence of events there is at least one forecaster who predicted it with certainty.  Is that forecaster right?

What I wrote yesterday:

When Fox broadcasts the Super Bowl they advertise for their shows, like American Idol. But those years in which, say, ABC has the Super Bowl you will never see an ad for American Idol during the Super Bowl broadcast.

This is that sort of puzzle whose degree of puzzliness is non-monotonic in how good your economic intuition is.

If you don’t think of it in economic terms at all it doesn’t seem at all like a puzzle. Try it: ask your grandpa if he thinks that its odd that you never see networks advertising their shows on other networks.  Of course they don’t do that.

When you apply a little economics to it that’s when it starts to look like a puzzle. There is a price for advertising. The value of the ad is either higher or lower than the price. If its higher you advertise.  If its another network that price is the cost of advertising. If its your own network that price is still a cost: the opportunity cost is the price you would earn if instead you sold the ad to a third-party.  If it was worth it to advertise American Idol when your own network has the Super Bowl then it should be worth it when some other network has it too.

But a little more economics removes the puzzle.  Networks have market power.  The way to use that market power for profit is to artificially restrict quantity and set price above marginal cost. (The marginal cost of running another 30 second ad is the cost in terms of viewership that would come from shortening, say, the halftime show by 30 seconds.)

When a network chooses whether to run an ad for its own show on its own Super Bowl broadcast it compares the value of the ad to that marginal cost.  When a network chooses whether to run an ad on another network’s Super Bowl broadcast it compares the value to the price.

Indeed even if the total time for ads is given and not under control of the network (i.e. total quantity is fixed) the profit maximizing price for ads will typically only sell a fraction of that ad time.  Then the marginal (opportunity) cost of the additional ads to pad that time is zero and even very low value ads like for American Idol will be shown when Fox has the Super Bowl and not when any other network does.

In fact that last observation and the fact that you never ever see any network advertise its shows on another network tells us that the value of advertising television shows is very low.  Perhaps that in fact tells us that the networks themselves understand (but their paying advertisers don’t) that the value of advertising in general is very low.

When Fox broadcasts the Super Bowl they advertise for their shows, like American Idol. But those years in which, say, ABC has the Super Bowl you will never see an ad for American Idol during the Super Bowl broadcast.

More generally, networks advertise their own shows on their own network but never pay to advertise their shows on other networks.  I never understood this.  But I think I finally figured it out, there’s some very simple economics behind this.

Right now at Primary.guide, you can read the current betting market odds for a “contested convention” and a “brokered convention.”  The definitions are as follows.  A contested convention means that no candidate has 1237 delegates by the end of the last primary.  A brokered convention means that no candidate wins on the first ballot at the convention.

Right now the odds of a brokered convention are 50%.  Note also that the odds of a Trump nomination are 50% as well.  And Trump is the only candidate with any chance of winning a majority on the first ballot (even if he doesn’t get 1237 bound delegates he will be close and no other candidate could combine their bound delegates with unbound delegates to get to a majority.)

Thus, if there is no brokered convention Trump is the nominee.  The probability of no brokered convention is 50%.  Thus the entire 50% probability of a Trump nomination is accounted for by the event that he wins on the first ballot.

In other words there is zero probability, according to betting markets, that Trump wins a brokered convention.

The odds of a contested convention are 80%. That means that betting markets think there is a 30% chance Trump fails to get 1237 bound delegates but still wins on the first round.  I.e. according to betting markets we have the following three mutually exclusive events:

1. Trump gets to 1237 by June 7.  20% odds
2. Trump fails to get 1237 bound delegates but wins on the first ballot.  30%
3. Nobody wins on the first ballot and Trump is not the nominee.  50%