Prop Bet Portfolios: an alternative to box pools
I wanted to spice up our Super Bowl party with some gambling-related entertainment beyond the usual Super Bowl Squares (traditional box pool). I came up with a new pool idea that I’m calling Super Bowl Prop Bet Portfolios (open to better suggestions).
Each person gets randomly assigned a Prop Bet Portfolio sheet (I’ll be handing them out to guests as they arrive at the party). Each sheet contains a 4 x 4 grid of 16 prop bets. The 16 bets on each sheet are chosen from 88 possible prop bets. Each bet has a payout that is linked to the odds that were available for that bet on January 31. Even-money bets (+100) are worth 2.0 units if won. For bets with negative odds (better than 50% probability -- vig included), the payout is (100 / abs(odds)) + 1. For bets with positive odds (lower than 50% probability), the payout is (odds/100) + 1. As an example, the odds for Saquon to score at least one TD are -190, so the payout is only 1.5 units (I’m rounding payouts to one decimal place). The odds of a FG or XP attempt hitting an upright is +550, so the payout is 6.5 units. This payout structure ensures that each bet has an expected value of 1 unit and the 16-bet portfolio has an expected value of 16 units.
There are 144 quadrillion unique 16-bet portfolios from 88 prop bets, each one having an expected value of 16 units. But the portfolios can have wildly different variances, depending on the number of long shot bets and the correlation of the bets in the portfolio. If you structure this as a contest where the person with the most points wins a grand prize, the winning portfolio would certainly be one of the portfolios with higher variance.
To try to address this and equilibrate the variance of the portfolios, I constrained the generation of random portfolios in two ways. First, I constrained the sum of the variances (using the binomial distribution) of individual portfolio bets to all be approximately equal. But this doesn't deal with correlation between the bets. Having positively correlated bets in a portfolio (e.g. KC wins, Mahomes MVP) increases portfolio variance. Having negatively correlated bets in a portfolio (e.g. Mahomes MVP and Eagles win) reduces portfolio variance. Actually estimating these correlations seemed impossible, so to address this problem I categorized the 88 bets into two categories: relatively uncorrelated bets and highly correlated bets. Highly correlated bets were put into 1 of 12 subgroups (e.g. one sub group is everything related to the winner of the game). Then I constrained the portfolio generation to have a maximum of one bet from each of the correlated subgroups. This isn't perfect, but these portfolios are a lot more uniformly variant than completely random ones.
I generated 100 random portfolios in a PDF, which will be more than enough for our party. You can download it here: https://drive.google.com/file/d/1PILJHsOI9hRBDxP9J-j3etvzkidkKfh5/view?usp=sharing
And I’ll also be serving a 7-pound smoked brisket from the Pecan Lodge at our Super Bowl party!
