# Pulp Fiction (Our Esteemed Leader’s cut)

The previous post had a cinematic theme. That got me remembering an offsite a while back where Matthew.Benham@smartbapps.co.uk gave a talk that I think he called ‘Do the Right Thing’, which is the title of a 1989 Spike Lee film. Midway through his talk Matthew gave a premiere screening of his own version of a scene from Pulp Fiction. Unfortunately, I’ve been unable to get hold of a copy of Matthew’s cut, so we’ll just have to make do with the inferior original….

The theme of Matthew’s talk was the importance of always acting in relation to best knowledge, even if it contradicts previous actions taken when different information was available. So, given the knowledge and information you had at the start of a game, you might have bet on team A. But if the game evolves in such a way that a bet on team B becomes positive value, you should do that. Always do the right thing. And the point of the scene from Pulp Fiction? Don’t let pride get in the way of that principle.

These issues will make a great topic for this blog sometime. But this post is about something else…

Dependence is a big issue in Statistics, and we’re likely to return to it in different ways in future posts. Loosely speaking, two events are said to be independent if knowing the outcome of one, doesn’t affect the probabilities of the outcomes of the other. For example, it’s usually reasonable to treat the outcomes of two different football matches taking place on the same day as independent. If we know one match finished 3-0, that information is unlikely to affect any judgements we might have about the possible outcomes of a later match. Events that are not independent are said to be dependent: in this case, knowing the outcome of one will affect the outcome of the other.  In tennis matches, for example, the outcome of one set tends to affect the chances of who will win a subsequent set, so set winners are dependent events.

With this in mind, let’s follow-up the discussion in the previous 2 posts (here and here) about accumulator bets. By multiplying prices from separate bets together, bookmakers are assuming that the events are independent. But if there were dependence between the events, it’s possible that an accumulator offers a value bet, even if the individual bets are of negative value. This might be part of the reason why Mark Kermode has been successful in several accumulator bets over the years (or would have been if he’d taken his predictions to the bookmaker and actually placed an accumulator bet).

Let me illustrate this with some entirely made-up numbers. Let’s suppose ‘Pulp Fiction (Our Esteemed Leader’s cut)’, is up for a best movie award, and its upstart director, Matthew Benham, has also been nominated for best director. The numbers for single bets on PF and MB are given in the following table. We’ll suppose the bookmakers are accurate in their evaluation of the probabilities, and that they guarantee themselves an expected profit by offering prices that are below the fair prices (see the earlier post).

True Probability Fair Price Bookmaker Price
Best Movie: PF 0.4 2.5 2
Best Director: MB 0.25 4 3.5

Because the available prices are lower than the fair prices and the probabilities are correct, both individual bets have negative value (-0.2 and -0.125 respectively for a unit stake). The overall price for a PF/MB accumulator bet is 7, which assuming independence is an even poorer value bet, since the expected winnings from a unit stake are

$0.4 \times 0.25 \times 7 -1 = -0.3$

However, suppose voters for the awards tend to have similar preferences across categories, so that if they like a particular movie, there’s an increased chance they’ll also like the director of that movie. In that case, although the table above might be correct, the probability of MB winning the director award if PF (MB cut) is the movie winner is likely to be greater than 0.25. For argument’s sake, let’s suppose it’s 0.5. Then, the expected winnings from a unit stake accumulator bet become

$0.4 \times 0.5 \times 7 -1 = 0.4$

That’s to say, although the individual bets are still both negative value, the accumulator bet is extremely good value. This situation arises because of the implicit assumption of independence in the calculation of accumulator prices. When the assumption is wrong, the true expected winnings will be different from those implied by the bookmaker prices, potentially generating a positive value bet.

Obviously with most accumulator bets – like multiple football results – independence is more realistic, and this discussion is unhelpful. But for speciality bets like the Oscars, or perhaps some political bets where late swings in votes are likely to affect more that one region, there may be considerable value in accumulator bets if available.

If anyone has a copy of Our Esteemed Leader’s cut of the Pulp Fiction scene on a pen-drive somewhere, and would kindly pass it to me, I will happily update this post to include it.

## 3 thoughts on “Pulp Fiction (Our Esteemed Leader’s cut)”

1. Ian Rutherford says:

An example of dependency in sports betting that made headlines in the press was Warwickshire in the 1994 cricket season. There were four individual competitions in English Cricket back then and some astute punters had secured approximately 64,000-1 about Warwickshire winning all of the them in an accumulator.

Halfway through the season Warwickshire had won one and were well placed in the other three. Bookmakers then announced that these accumulators broke their contingency rules and they would not be paying out at anywhere near 64,000-1 if Warwickshire completed the Grand Slam.

Fair, to an extent, there was no offer to increase the original price given on bets placed on any of the other teams.

1. Ian Rutherford says:

Expanding on my fair to an extent comment – if my memory serves me correctly doubles and trebles placed on Warwickshire were also subject to the contingency rule, and the bookies stance on this came across as like it or lump it in the sports betting media, who were up in arms about this “robbery” against the honest punter. I am not as naive as I was back then to believe every word written in a newspaper!!

Would it not have been fairer and better PR to say all doubles and above placed on the 1994 cricket season where no team has been duplicated in a bet will receive a 5% bonus on winnings as a result of our mistake in originally giving the wrong price for dependency bets?

1. Stuart Coles says:

I wonder if the bookmakers would have been so strict about returning stakes due to their `contingency rules having been broken’ if Warwickshire had lost the first of those competitions. I might be naive, but it seems to me that bookmakers should honour bets they are happy to take, in the absence of match-fixing etc.