Section E.3 The Take
The sum of the crossover probabilities determines the take, which is the share of the betting pool taken by “the house”—that is, the entity that takes the bets.
For example, suppose 1000 people take the first wager and bet $100 each on the Patriots. And 1000 people take the second wager and bet $240 on the Seahawks.
Here’s the total expected value of all of those wagers.
total = expected_value(ps, 100_000, 195_000) + expected_value(1-ps, 240_000, 100_000)
plt.plot(ps, total, label='Total')
plt.axhline(0, color='gray', alpha=0.4)
decorate(xlabel='Actual probability Patriots win',
ylabel='Total expected value of all wagers')
The total expected value is negative for all probabilities (or zero if the Patriots have no chance at all)—which means the house wins.
How much the house wins depends on the actual probability. As an example, suppose the actual probability is the midpoint of the probabilities implied by the odds:
$ p = (p1 + (1-p2)) / 2
p
0.31655034895314055
In that case, here’s the expected take, assuming that the implied probability is correct.
take = -expected_value(p, 100_000, 195_000) - expected_value(1-p, 240_000, 100_000)
take
As a percentage of the total betting pool, it’s a little more than 4%.
$ take / (100_000 + 240_000)
0.04189636971438623
Which we could have approximated by computing the overround, which is the amount that the sum of the implied probabilities exceeds 1.
$ (p1 + p2) - 1
0.04486540378863424
