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.
Cheap Talk 
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March 18, 2013 at 12:21 am
kerokan
Your intuition is all about Barcelona. What about Milan’s incentives?
I think one must make some strange assumptions about Milan to get this intuition. For instance, Barcelona’s effort to score is higher when B is ahead 1-0 (relative to 0-0), but at the same time Milan’s effort to defend is higher as well. How come those two effects not cancel each other out?
March 18, 2013 at 12:47 am
th.alys
“How come those two effects not cancel each other out?”
1-0 causes anxiety to Milan while it boosts the morale of Barcelona perhaps? Defending well at this level against such a good offensive team is concentration intensive I guess.
For the record, and for those who might not know, Barcelona won 4-0.
March 18, 2013 at 8:15 am
Chris
What am I missing here? Forget about the first leg of this tie for the moment. Isn’t it just possible that 2-0 is the modal outcome for this matchup? Why should we expect the pmf for Barcelona’s goals to be monotonic? Conditional on Milan scoring 0, the implied pmf for Barcelona’s total is single-peaked (at 2 goals), which seems reasonable.
March 18, 2013 at 9:22 am
jeff
Quite right. I guess we were thinking these were conditional probabilities.
March 18, 2013 at 9:33 am
luca
Now, if only someone had written a paper on how equilibrium play in a soccer game depends on the current score…
March 18, 2013 at 1:54 pm
Josh K
Instead of effort, can we think about this instead of substitution between offense and defense? At 1-0, Milan can no longer play for the tie, and must open up its attack. 1 goal is not a comfortable lead, so Barca continues to play balanced. At 2-0, Barca has a comfortable lead and starts playing concervative, making it less likely Barca will mark a third goal.
June 10, 2013 at 7:31 am
Sofia
Not sure you’ll find too many moderately precid hotels in Barcelona unfortunately! Best bet is to shop around on the internet as prices for most hotels seem to vary enormously.The central touristy areas of barcelona are the Barrio Gotico and Born. Both are normally full of people day and night, although Born is more upmarket. You’re probably better off in the Eixample area. It’s an area of broad avenues and large turn-of the century mansion buildings and is still pretty central. Safer too. Day trips?Sitges is a typical daytrip and is certainly worth visiting. Its a pretty seaside town with a nice promenade and interesting older areas.Tarragona is further afield. Has a Roman amphitheatre and some roman/medieval fortifications too. Weather in march? It’s normally sunny most of the time in Barcelona but we get a week of rain sometime in spring. You should be ok though.hope this helps
June 13, 2013 at 10:07 am
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September 30, 2013 at 5:41 am
Keyaan
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October 17, 2013 at 7:53 am
Choudhary
I had a look at your blog and your photos. They are so good. You feel as if you were there. And that is dfcuifilt. Next time you come to Spain do it further south for a real taste of COLOUR, FIESTA and SIESTA, he, he. I live in Andaluceda, close to Granada- Me1laga…I shall keep your blog to follow. How did you find my blog? I am interested to know, just in case it is worth translate into English.Un abrazoSacramento
July 19, 2015 at 3:45 am
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