## Friday, November 8, 2013

### Could a playoff seed ever be determined by a coin flip?

In order to produce my playoff seed projections for the NFL, I had to do some rather tedious coding of the tiebreaker rules. For example, here are the divisional tiebreaker rules when two teams have the same win-loss-tie record:
2. Best won-lost-tied percentage in games played within the division.
3. Best won-lost-tied percentage in common games.
4. Best won-lost-tied percentage in games played within the conference.
5. Strength of victory.
6. Strength of schedule.
7. Best combined ranking among conference teams in points scored and points allowed.
8. Best combined ranking among all teams in points scored and points allowed.
9. Best net points in common games.
10. Best net points in all games.
11. Best net touchdowns in all games.
12. Coin toss
Most casual fans are aware of the implications head-to-head, divisional records, and conference records have for playoff seeding. But things get pretty esoteric as you move further down the list. I had to look up what "Strength of Victory" meant (it's the combined winning percentage of the opponents you beat), and frankly had to make a bit of a guess as to what "combined ranking" meant for #7 and #8. At the very bottom of the list is "Coin Toss", which I even went through the trouble of simulating with a random number generator.

But could the "Coin Toss" scenario ever come in to play? (doubtful) Would ESPN televise it? (probably) Should they have replaced "Coin Toss" with "Dance Off"? (definitely)

Thanks to my borderline-OCD determination to code each tiebreaker scenario, I can attempt to answer the first question. My playoff projections are based on a 10,000 run Monte Carlo simulation, so it was easy enough to output how frequently each tiebreaker was actually used. The table below summarizes the frequency (per simulated season) of each tiebreaker rule.

tiebreakerfrequency
conference record3.1433
common games1.3530
division record1.2119
strength of victory0.7248
strength of schedule0.0375
points rank (conf)0.0030
points rank0.0001
net points in common games0.0000
net points0.0000
net touchdowns0.0000
coin toss0.0000

Not too surprisingly, head to head was the most commonly applied tiebreaker. In 10,000 simulated seasons, the "Combined ranking among all teams in points scored and allowed" tiebreaker was only needed once. And the remaining three (net points in common game, net points, and net touchdowns) were never used, making it highly unlikely that a playoff seed would ever come down to a coin toss; unless, of course, the NFL manages to stick around for the next 100,000 years. Not that I'd put it past them.

1. Funny, my friend and I were having a beer and looking at the playoff standings and rules, which led us to wonder just how unlikely a playoff decision being decided by a coin toss is?

We started from where you left of with a .0001 probability of getting down to the points rank. We then made a model that ran simulations of points and TDs scored in a season assuming both TDs and FGs per game follow a Poisson distribution with means of 2.4 and 1.7 respectively. For each TD there was a 0.0584 chance of going for 2 with a .47 chance of success. Extra points had a success probability of 0.9849. Then we ran a shit load of simulations to estimate the probability of two teams having the same # of TDs and points. We also assumed the opposing team did not differ in the expected number of TDs or points. We came up with there being a 4.4% chance two teams have the same net TDs and 0.58% chance of having the same net. The probability that net TDs and net points were the same for two teams was 0.7589e-4.

The tricky part was actually net points in common games. We drank too many beers and just assumed that games had a .7 chance of being counted as a game against a common opponent. The probability that two teams had the same net points, same net TDs, and same common game net score was (1.48e-5). This is sensitive to the number of games against common opponents, with values closer to 0.5 making it even more unlikely for all tie conditions being met.

If we make the assumption that the probability of getting to step 8 on the tiebreaker list and tying on the last 3 steps are independent (it’s a stretch, but hey we had some beers as I mentioned) then we have 0.001 * 1.48e-5 = 1.48e-8. So we might expect to see a coin flip decide a playoff spot once every 55 million seasons or so (maybe only 5 million if some of our assumptions of independence are off).

Science! Beer!

2. Looks legit. You may have inadvertently stumbled upon the Ballmer Peak.

3. I'm late to the party but I've been wondering this for a long time. What I was wondering, though, is HOW the incredibly unlikely coin toss would be handled. Do they literally flip a coin? I think the two teams should gather in a public area and it should be documented for posterity

4. also I'd like to see how they could ever do a "coin toss" to resolve a three-way tie

5. Three Way Coin Tosses can be handled pretty simply. All three people (teams) flip a coin, then whomever is the odd coin wins. For example: Two "heads" and a "tail" means the "tail" wins (or two "tails" and a "head" means the "head" wins. If all three come up the same, then just flip again. This way, no one has to win the flip more than one time.

6. There are two categories of "Best combined ranking in points scored and points allowed;" 1.) among conference teams & 2.) among all teams. HOU and IND flirted with the AFC points rank this year, 2015.

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