You are currently browsing jeff’s articles.
As far as I know. Anyway I always assumed that the Ely Lecture at the AEA meetings was named after me.
But changing the subject, Adriana Lleras-Muney writes to me:
From Henry Miller
“To be intelligent may be a boon, but to be completely trusting, gullible to the point of idiocy, to surrender without reservation is of of the supreme joys of life”
I think Henry Miller is confusing correlation with causation. Its probably true that in our happiest moments (among those moments we are with other people–I might even dispute that those moments are the happiest unconditionally) we are trusting, gullible and idiotically surrendering. But that’s likely because we are with a certain person and in a certain blissful state that we respond by surrendering. Its the person and the state that brings us the supreme joy and our surrender is just a symptom of that joy. I might go far as to say that the surrender is a complementary good but its enough to think about surrendering to the very next person who knocks on your office door to convince you that the surrender is not itself the source of joy.
Would it be possible to make a statistical model of a jazz solo and use it to create new ones? Take a standard, and let’s focus on the saxaphone, say. Go to the solo and estimate a Markov transition kernel which tells you the probability distribution over the next tone conditional on the previous tone. In particular you want the joint probability distribution over the following note (or just the interval) and the note’s value (eighth, quarter, etc.) Feed it tons and tons of recordings of sax solos for the same tune (that’s why you want a standard.)
Once you have estimated your kernel, simulate it. Will it be music? How much of an improvement do you get if the state variable is the last two notes instead of just one? If your state variable is the last n notes, at what n are improvements no longer noticeable?
In my kids’ tennis class they are getting good enough to have actual rallies. The coach feeds them a ball and has them play out points. Each rally is worth 1 point and they play to 10. To stop them from trying to hit winners on the first shot and in attempt to get them to play longer rallies, the coaches tried out an interesting rule. ”The ball must cross the net four times before the point begins. If your shot goes out before that, its 2 points for the other side.”
One form of mental accounting is where you give yourself separate budgets for things like food, entertainment, gas, etc. It’s suboptimal because these separate budgets make you less flexible in your consumption plans. For example in a month where there are many attractive entertainment offerings, you are unable to reallocate spending away from other goods in favor of entertainment.
But it could be understood as a second-best solution when you have memory limitations. Suppose that when you decide how much to spend on groceries, you often forget or even fail to think of how much you have been spending on gas this month. If so, then its not really possible to be as flexible as you would be in the first-best because there’s no way to reduce your grocery expenditures in tandem with the increased spending on gas.
That means that you should not increase your spending on gas. In other words you should stick to a fixed gas budget.
Now memory is associative, i.e. current experiences stimulate memories of related experiences. This can give some structure to the theory. It makes sense to have a budget for entertainment overall rather than separate budgets for movies and concerts because when you are thinking of one you are likely to recall your spending on the other. So the boundaries of budget categories should be determined by an optimal grouping of expenditures based on how closely associated they are in memory.
(Discussion with Asher Wolinsky and Simone Galperti)
The NRA successfully lobbied to stop gun control legislation. Several Democrats sided with Republicans to defeat it. But the NRA seems to have spent more than necessary to defeat the measures because they failed by more than a one-vote margin. It would have been enough to buy exactly the number of Senators necessary to prevent the bill from progressing through the Senate, no more than that.
But in fact the cost of defeating legislation is decreasing in the number of excess votes purchased. If the NRA has already secured enough votes to win, the next vote cannot be pivotal and so the Senator casting that vote takes less blame for the defeat. Indeed if enough Senators are bought so that the bill goes down by at least two votes, no Senator is pivotal.
Here’s a simple model. Suppose that the political cost of failing to pass gun control is c. If the NRA buys the minimum number of votes needed to halt the legislation it must pay c to each Senator it buys. That’s because each of those Senators could refuse to vote for the NRA and avoid the cost c. But if the NRA buys one extra vote, each Senator incurs the cost c whether or not he goes along with the NRA and his vote has just become cheaper by the amount c.
For the Vapor Mill: What is the voting rule that maximizes the cost of defeating popular legislation?
Amnesty –forgiving all of the current and previous violators but renewing a threat to punish future violators– always seems like a reputation fail. If we are granting amnesty today then doesn’t that signal that we will eventually be granting amnesty again in the future?
But there is at least one environment in which a once-only amnesty is incentive compatible and effective: when crime has bandwagon effects. For example, suppose there’s a stash of candy in the pantry and my kids have taken to raiding it. I catch one red-handed but I can’t punish her because she rightly points out that since everybody’s doing it she assumed we were looking the other way. A culture of candy crime had taken hold.
An amnesty (bring me your private stash and you will be forgiven) moves us from the everyone’s a criminal because everyone’s a criminal equilibrium to the one in which nobody’s a criminal. The latter is potentially stable if its easier to single out and punish a lone offender than one of many.
- Capitalize on money illusion (and take a step toward eventually dispelling it) by having the Tooth Fairy leave 1000 Korean Won rather than 1 US Dollar.
- Security teams employ hackers to find flaws in their software before taking it live. Marketing teams should hire comedians to find potential bastardizations of new brand names/marketing campaigns under consideration.
- I would like to see brain images of pianists while they play. How does hand independence work in the brain? Are both sides continuously active or is the brain switching back and forth monitoring the hands separately?
- Even though I am the game theorist my wife does all the bargaining because she is better at “acting” irrational.
Say you are speaking for an hour to an audience of 100. Its just a fact of human nature that nobody in the audience is going to be paying close attention to what you are saying for more than 1/4 of the time. The other 45 minutes of the time people will be thinking, talking, or just daydreaming. You must accept this as an unavoidable constraint.
Absent any intervention on your part then you will get a randomly selected 15 minutes of attention from each member of the audience. This means that at any one point in time you will have the attention of only 1/4 of your audience or 25 out of the 100 people. The very important things you will have to say will be processed and potentially remembered by 1/4 of your audience, the same fraction that will be paying attention to the least important things you have to say.
So what you should do to prepare is ask yourself what are the three important things you have to say and you want remembered. Each of them should take you five minutes to say. Then imagine you have a sign that will flash above you which tells everyone in the audience whether now is the time to be paying close attention or now is an opportunity to doze off. With that sign you could coordinate their attention so that all of them are listening during the same 15 minutes, those 15 minutes when you will be saying your three important things.
Now you probably won’t be bringing that sign with you. But you can achieve the same effect by using the way that you stand, the way that you talk, and the style of your slides. When you are saying something important you speak slowly and loudly and you walk up and down the room and make eye contact and your slides have just one or two things on them so that they are easy to read and process.
You are telling them with your demeanor that now is the time to listen. Later, when you are saying something less important you lower your voice, go faster, stand still and read off your busy slides. You are doing these things to tell your audience that now is the time to think, talk or doodle and rest up for the next important moment.
If you are like me you subscribe to many RSS feeds and you find them accumulating articles faster than you can read them. What do you do when MetaFilter has 500 unread articles?
Use the glance and scroll method. Look at an article and read it if it looks interesting. But if it doesnt then don’t keep browsing. Give a hard flick on your mousewheel, trackpad, phone screen whatever and scroll rapidly past a large number of articles and glance at the next one you land on. Repeat.
The point is that all articles, regardless of their position in the feed are equally likely to be interesting, other things equal. So if you can only read a fraction of what’s there you can’t do any worse than this. But in fact other things are not equal.
The quality of articles is endogenously serially correlated. Because any blog runs the best material available that day/couple of days/week. If you landed on a lame article then chances are you landed on a lame day. Scroll your way out of that trough and hope you land on a peak.
Non-peer reviewed, inaccessible data, and punditry that can’t tell the difference between P&P and a regular AER article can’t be good for the reputation of the journal, the AEA, or the profession.
Oliver Sacks on the social costs of plagiarism stigma:
Helen Keller was accused of plagiarism when she was only twelve.2 Though deaf and blind from an early age, and indeed languageless before she met Annie Sullivan at the age of six, she became a prolific writer once she learned finger spelling and Braille. As a girl, she had written, among other things, a story called “The Frost King,” which she gave to a friend as a birthday gift. When the story found its way into print in a magazine, readers soon realized that it bore great similarities to “The Frost Fairies,” a children’s short story by Margaret Canby. Admiration for Keller now turned into accusation, and Helen was accused of plagiarism and deliberate falsehood, even though she said that she had no recollection of reading Canby’s story, and thought she had made it up herself. The young Helen was subjected to a ruthless inquisition, which left its mark on her for the rest of her life.
There is a subtle defense of plagiarism in the connection he draws with false memories, and the value of ignoring the source.
Indifference to source allows us to assimilate what we read, what we are told, what others say and think and write and paint, as intensely and richly as if they were primary experiences. It allows us to see and hear with other eyes and ears, to enter into other minds, to assimilate the art and science and religion of the whole culture, to enter into and contribute to the common mind, the general commonwealth of knowledge. This sort of sharing and participation, this communion, would not be possible if all our knowledge, our memories, were tagged and identified, seen as private, exclusively ours. Memory is dialogic and arises not only from direct experience but from the intercourse of many minds.
Kepi kiss: David Olson.
Spouse A (henceforth “she”, the driver) prefers the air inside the vehicle to be a little warmer than the preferred temperature of Spouse B (“he”, the navigator, not because he is a worse driver –quite the contrary– but because he is an even better passenger.) In their regular confrontation with this dilemma they are seemingly blessed with the optional dual-zone climate control in their decked out Volvo SUV.
And indeed there is an equilibrium of the dual climate-zone game in which each spouse enjoys his/her temperature bliss point. This equilibrium is unfortunately highly unstable. Because of the exchange of heat across the thermal gradient the only way each can maintain the constant target temperature is to adjust their controllers so that the air blown out their respective vents deviates slightly from that target further in the direction of the extreme. Hers must be set somewhat warmer and his somewhat cooler.
Now from that starting point, the slightest perturbation upsets the delicate balance and can set off a dangerous chain reaction. Consider for example what happens when, due to random alterations in air flow she begins to feel a bit on the cool side of her comfort zone. Her response is to adjust her controller one peg toward the red. This restores her comfort level but very soon as a result he will begin to feel the discomfort of unexpectedly hot and dry air blowing into his zone and he will react by moving his controller one peg toward the blue.
This is not likely to end well.
Star Michigan guard Trey Burke collected two personal fouls in the early minutes of the National Championship game against Louisville and he was promptly benched and sat out most of the remaining first half. The announcers didn’t bother to say why because its common wisdom that you don’t want your best players fouling out early.
But the common wisdom requires some scrutiny because on its surface it actually looks absurd. You fear your best player fouling out because then his playing time might be limited. So in response you guarantee his playing time will be limited by benching him. Jonathon Weinstein once made this point.
But just because basketball commentators, and probably even basketball coaches, don’t properly understand the rationale for the strategy doesn’t mean the strategy is unsound. In fact it follows from a very basic strategic idea: information is valuable.
Suppose the other team is scoring points at some random rate. If they are lucky they score a lot and if they are less lucky they score fewer. If the other team scores a lot your team should start shooting threes and go for short possessions to catch up. If the other team scores fewer you should go for safer shots and run down the clock. But you only know which of these you should do at the end of the game. If your best players are on the bench at that time you cannot capitalize on this information.
And I have been to many very good Chinese restaurants in China, Taiwan, Singapore etc. This place is called Peter Chang’s China Grill. Here’s Wikipedia about the chef. Here are the badly misguided Yelp reviews. Note that from the look of the restaurant, the location, the service you would never guess what was in store for you. Indeed I was terrified when the folks at UVA told me we were driving off campus to go have Chinese for lunch, even moreso when I saw the place they were taking me to. But the food was a revelation. You probably do need to know what and how to order. For that I suggest getting invited to give a talk at the University of Virginia Economics department.
- Econ grad schools should add as a requirement for the PhD that students take a recent published experiment and attempt to replicate it.
- Not accepting American Express is a local public good. Because once enough of them do it, I stop expecting that it will be accepted and stop offering it on the first try.
- Homosexuality is a puzzle for evolution not because of the question “how does it survive despite not reproducing?” Because for a gene reproduction *is* survival. The puzzle is “how do they reproduce despite not wanting to?”
- Books get made into films all the time, why do poems rarely get made into pop songs?
- Why is it easier to apologize when the offended party isn’t actually upset?
- 40 blurbs from negative Ebert reviews. My favorite so far: “Mad Dog Time is the first movie I have seen that does not improve on the sight of a blank screen viewed for the same length of time. Oh, I’ve seen bad movies before. But they usually made me care about how bad they were. Watching Mad Dog Time is like waiting for the bus in a city where you’re not sure they have a bus line….Mad Dog Time should be cut into free ukulele picks for the poor.”
- Outtakes from Siskel and Ebert. Watch the hilarious game of The Dozens that breaks out midway through.
- Pilot for Three’s Company.
- Just scroll through these pictures and you will figure it out.
- Old Harmony Korine appearances on Letterman.
In our paper, Alex Frankel, Emir Kamenica and I argue that soccer is among the most suspenseful sports according to our theoretical measure. Now, via Matt Dickenson, comes an empirical validation of this finding using German cardiac arrest data:
The red line shows the spike in heart attacks on the dates of 2006 World Cup matches involving the German national team. Note that point 7 is the third place match against Portugal after Germany had been eliminated in their semi-final match against Italy (point 6.)
Steve Tadelis, Distinguished Economist at eBay and his colleagues have done an experiment and seem to have concluded that its a waste of money to pay for sponsored links on Google when Google’s regular search algorithm shows links to eBay for free.
Before you read the rest of this post, go to Google and try searching for “Amazon.” You’ll probably notice that the top two listings are both for Amazon’s website, with the first appearing on a light beige background. If you click on the first — a paid search ad — Amazon will pay Google for attracting your business. If you click on the second, Amazon gets your business but Google gets nothing. Try “Macys,” “Walgreens,” and “Sports Authority” — you’ll see the same thing.
If you search for eBay, though, you’ll find only a single listing — an unpaid one. Odds are, after marketers at Amazon, Walgreens and elsewhere catch wind of a preliminary study released on Friday, their search listings will start to look a lot more like eBay’s. The study — by eBay Research Labs economists Thomas Blake, Chris Nosko, and Steve Tadelis — analyzed eBay sales after shutting down purchases of search ads on Google and elsewhere, while maintaining a control set of regions where search ads continued unchanged. Their findings suggest that many paid ads generate virtually no increase in sales, and even for ones that do, the sales benefits are far eclipsed by the cost of the ads themselves.
This is a dilemma for Google. Because it suggests that the way to capitalize on the popularity of their search algorithm is to discriminate against those who don’t pay for ads. But unless Google reinvents itself as a full-blown paid advertising site, such discrimination is likely to raise legal issues.
Here I am on the podcast Hang Up And Listen with hosts Stefan Fatsis, Mike Pesca and Josh Levin. Its a sports oriented podcast and I am talking about Purple Pricing. They had some very good questions. I come in at about the 25 minute mark.
NEWRY, Maine — A Finnish couple has added to their victories by taking first place in the North American Wife Carrying Championship at Maine’s Sunday River ski resort.
Taisto Miettinen and Kristina Haapanen traveled from Helsinki, Finland – where they won the World Wife Carrying Championship – for Saturday’s contest. The Sun Journal (bit.ly/Q30QWq) reports that the couple finished with a time of 52.58 seconds on a course that includes hurdles, sand traps and a water hole.
The winners receive the woman’s weight in beer and five times her weight in cash.
The model: At date 0 each of N husbands decides how fat his wife should be. At date 1 they run a wife-carrying race, where the husband’s speed is given by some function f(s,w) where s is the strength of the husband, and w is the weight of his wife. The function f is increasing in its first argument and decreasing in the second. The winner gets K times his wife’s weight in cash and beer. Questions
- If the husbands are symmetric what is the equilibrium distribution of wife weights?
- Under what conditions on f does a stronger husband have a fatter wife?
- Derive the comparative statics with respect to K.
Professional soccer leagues tend to be dominated year after year by a small number of top teams. Major League Baseball on the other hand, has seen World Series appearances by the Detroit Tigers, the Texas Rangers, the Philadelphia Phillies, the Tampa Bay Rays, etc. It seems like any team can assemble a champion.
These two sports have very different production functions. A baseball team is basically a collection of individuals. Among team sports its the closest thing to an individual sport. A team’s output is basically the sum of individual outputs with very little complementarity. In baseball there is very little talk of one player making his teammates better. The production function is additive and, offensively, players are perfectly substitutable.
Soccer is at the other extreme where players are highly complementary. Scoring a goal is a total team effort. Even the best striker needs good chances. In soccer, the best players are more productive when there are other good players on the team. The production function is closer to Leontif.
These differences in production explain the differences in market structure. Consider competitive bidding for a top player in the two sports. In baseball that player’s marginal product is the same on any team he would play for so many teams will compete and any team could land him. In soccer that player’s marginal value is highest for the team that is already the best. The competition is going to be very weak and he is likely to sign with the best team.
In baseball competition levels the playing field. In soccer it tilts it even further.
It’s called Coffee Places Where You Can Think, and its right there. A word of caution though. Ariel is a connoisseur of coffee houses but his preferences are guided mostly by the atmosphere of the place and not at all by the quality of the coffee. Indeed his bad taste in coffee rivals only his bad taste in web site designs. So use this guide accordingly. (Here is a series of pictures of Ariel’s tin can of instant coffee traveling to exotic locales across the world.)
But the picture above is from The Mudhouse in Charlottesville, Virginia which is a place I can also highly recommend, having been there and had an exquisite cappuccino just last month with Federico Ciliberto.
We have coordinated on April 1 as the date where everyone gets to indulge their latent desire to say something false and hope that it gets believed. The problem of course is that as a result nothing you say on April 1 is believed. Credibility has a public good aspect to it and the social optimum would conserve enough of it so that at least some of us could feed our hilarious public deception jones.
(You might argue otherwise. By reducing credibility across the board we set the stage for the truly exceptional liars to show their stuff fooling people even though everyone was expecting them to do exactly that.)
It soon becomes tempting to start making up stuff on the day before April 1, and then eventually sooner. Whether, how, and why this unfolds depends on just what are the basic forces at work here, a question about which we can only theorize. We know by revealed preference that people want to say made-up stuff. But its more than that, you want people to believe it, pass it on and then get called out for believing a fake story. And the best kind of April Fools story is the kind that is ex post so obviously fake that the foolee looks especially gullible.
All of these suggest that April 1 is the least ideal day for April Fools. So then we can ask why are April Fools pranks perpetrated on April 1 and not some other day? Is it because April 1 gives you reputational cover for reporting bogus stories? It sounds like a good theory, and if it were true we would not have to worry about March 31 Fools and the eventual Year-Round Fools. But it doesn’t seem to survive closer scrutiny. If my reputation for legitimate reporting is safe for that one day it must be because nobody expects me to do legitimate reporting that day. But then April 1 is the last day I would want to be dropping my April Fools.
Instead I think its something more subtle. The perfect April Fools prank works by first roping in the reader but then slowly revealing clues that remind her that its April Fools and you have been had. April 1 plays a crucial role in this development. The Reveal would not come with the same impact on any July 19. Indeed the best April Fools pranks tell you the date somewhere along the way. Its how you say to the reader “look at you, you forgot about April Fools, and I got you” without actually saying that. And it provides ammunition when he blindly passes the story on and his friends can say “check the date.”
So April Foolers need the following epistemic infrastructure for their pranks: 1) The possibility of surprise, 2) The expectation of surprise. And clearly it cannot be an equilibrium to have both of these if the source of #2 is to be contained in a single date April 1. Once there is too much of #2, there’s precious too little of #1.
And that’s when the pre-emption begins. By publishing your April Fool on March 31, you get more of 1), and still pretty much the same amount of 2). Until of course everybody starts doing that and the unraveling continues. (We are already getting close.)
There is one bright side to all of this. When everybody has come to expect that whatever you say on April 1 is false, you may indeed no longer be able to fool people into believing something you made up. But the flipside is a welcome opportunity for many wishing to come clean at minimum cost. You now have a day when you can tell the truth and have nobody believe you.
Check out these pictures of bodysurfers ducking under waves.
I grew up body surfing and boogie boarding in Southern California. Riding waves is exhilarating and its the #1 reason you are there but there’s one other unforgettable experience that comes with it and that’s ducking under a passing wave.
A crashing wave is an awesomely powerful thing. So its such an incredible feeling of freedom that with just a little agility and perfect timing you can get yourself down below a 1 or 2 foot protective layer of water where all of that massive power can only just roll over you at most gently rocking you forward and back. And then you pop right back up to the surface to get ready for the next one.
With bodysurfing this is everything: ducking under waves until you are in position to catch just the right one. With boogie boarding there’s a little bit of paddling in between but the board is still small and light enough that when necessary you can get under even the most threatening wave.
But for sure the hardest thing about surfing is that this manoeuver is no longer available to you. For one thing you would be putting others in danger if you bail off the board and let the wave throw it around. But anyway the board is so bulky that the wave is going to drag you along with it.
Forget about learning to ride a wave. That’s as easy as anything else when you can get enough repetitions in. The bloody hard thing about surfing is that it can take months to get that many repetitions in because every time you fall you have to paddle back out through those waves. Yeah there are still tricks (turtle roll, duck dive), but even getting good enough at those for them to be useful takes weeks. The first few weeks you are lucky to get into position for 1 or 2 shots per session at actually catching a wave.
Chullo chortle: Kottke.
Dear Northwestern Economics community. I was among the first to submit my bracket and I have already chosen all 16 teams seeded #1 through #4 to be eliminated in the first round of the NCAA tournament. In case you don’t believe me:
Now that i got that out of the way, consider the following complete information strategic-form game. Someone will throw a biased coin which comes up heads with probability 5/8. Two people simultaneously make guesses. A pot of money will be divided equally among those who correctly guessed how the coin would land. (Somebody else gets the money if both guess incorrectly.)
In a symmetric equilibrium of this game the two players will randomize their guesses in such a way that each earns the same expected payoff. But now suppose that player 1 can publicly announce his guess before player 2 moves. Player 1 will choose heads and player 2′s best reply is to choose tails. By making this announcement, player 1 has increased his payoff to a 5/8 chance of winning the pot of money.
This principle applies to just about any variety of bracket-picking game, hence my announcement. In fact in the psychotic version we play in our department, the twisted-brain child of Scott Ogawa, each matchup in the bracket is worth 1000 points to be divided among all who correctly guess the winner, and the overall winner is the one with the most points. Now that all of my colleagues know that the upsets enumerated above have already been taken by me their best responses are to pick the favorites and sure they will be correct with high probability on each, but they will split the 1000 points with everyone else and I will get the full 1000 on the inevitable one or two upsets that will come from that group.
Remember how Mr. Miyagi taught The Karate Kid how to fight? Wax on/Wax off. Paint the fence. Don’t forget to breathe. A coach is the coach because he knows what the student needs to do to advance. A big problem for coaches is that the most precocious students also (naturally) think they know what they need to learn.
If Mr. Miyagi told Daniel that he needed endless repetition of certain specific hand movements to learn karate, Daniel would have rebelled and demanded to learn more and advance more quickly. Mr. Miyagi used ambiguity to evade conflict.
An artist with natural gift for expression needs to learn convention. But she may disagree with the teacher about how much time should be spent learning convention. If the teacher simply gives her exercises to do without explanation her decision to comply will be on the basis of an overall judgment of whether this teacher, on average, knows best. To instead say “You must learn conventions, here are some exercises for that” runs the risk that the student moderates the exercises in line with her own judgment about the importance of convention.
this a screenshot, from a few minutes ago (ed: last week), of bwin.com. the bets here are on goals in regular time of the barcelona-milan to be played in a little while. barcelona lost 2-0 in milan so barcelona needs at least 2 goals to force extra-time/penalty kicks. this is for the champions league.
as you can see from the screenshot barcelona winning 1-0 pays 10, 2-0 pays 7.5, 3-0 pays 8.75, while 4-0 pays 12.
what can we learn from this non-monotonicity? gamblers anticipate that barcelona’s extra incentives to score the 2-0 goal make it a more likely event than the 1-0 result (even though they have to score an extra goal!). once they have scored the 2-0, those extra incentives vanish so we are back to the intuition that a result with more goals is less likely.
How could this effect play out in real time? Here’s a model. It takes effort to increase the probability of scoring a goal. An immediate implication is that if the score is 0-0 with little time left, Barcelona will stop spending effort and the game will end 0-0. Too late in the game and it becomes so unlikely they can score two goals that the effort cost isn’t worth it. But if the score is 1-0 they will continue to spend effort beyond that point. So there is some interval near the end of the game where the conditional probability of scoring a goal is positive if the score is 1-0 but close to zero if the score is 0-0.
I would be interested in seeing some numbers calibrated to generated the betting odds above. We need three parameters. The first two are the probability of scoring a goal in a given minute of game time when Barcelona spends effort, and when it does not. The second is Barcelona’s rate of substitution between effort and win-probability. This could be expressed as follows. Over the course of a minute of play what is the minimum increase in win probability that would give Barcelona sufficient incentive to spend effort. These three parameters will determine when Barcelona stops spending effort in the 1-0 versus 0-0 scenarios and given this will then determine the probabilities of 1-0, 2-0, 3-0 etc. scores.