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.