Modeling the 2015 Tournament of Champions

I would love to see Elliot Yates in the Tournament of Champions, that quasi-sesquiannual showdown of recent top Jeopardy! players.

But with the next TOC at least a year away, will he be able to hold on to his position on the leaderboard? That’s the question I set out to answer – or to model, anyway.

A couple of notes (September 1, 2015):
1. my assumption was that 4-timer Catherine Hardee had been in the pool for the 2014 TOC, but it appears she is in the current running.
2. the next TOC will air in November 2015, far earlier than I had anticipated.

Note: If you want to jump right to my model’s prediction, click here.

Who qualifies for the TOC?

For the first decade-plus of the Trebek era, the TOC was held each fall, featuring the 15 biggest winners of the previous season. (As special events and reunion tournaments began to creep into the schedule, the frequency of TOCs declined.)

Potential TOC qualifiers are ranked first by number of wins, then by total winnings. The winners of certain tournaments, such as the College Championship and the Teachers Tournament, also receive bids. (The Teen Tournament winner used to get one, too; alas.)

Here are the stats for the last players in for all TOCs since mine, in 2004.

Held	Episodes	Auto bids 1	Last player in	Wins	Winnings
2006 / 05	228 2	2	Aaron Thompson	3	$57,899
2007 / 11	270	1	Nick Swezey	3	$76,202
2009 / 03	276	0	Deborah Fitzgerald 3	4	$55,901
2010 / 05	209	3	Christine Valada 3	4	$68,703
2011 / 11	269	2	Buddy Wright	4	$88,804
2013 / 02	235	3	Dan McShane	4	$62,001
2014 / 11	313	3	Mark Japinga 4	4	$112,600

¹ at present, automatic bids are awarded to the College Champion and the Teachers Tournament champ
² episode count starts the day Ken Jennings lost, November 30, 2004
³ had Joey Beachum not deferred to the 2010 TOC, the 2009 cut-off would have been 4 wins / $68,802, and the 2010 cut-off would have been 3 wins / $111,299 (Kevin Joyce)
⁴ had Jerry Slowik been invited, all regular-play champs would have had at least 5 wins

How long do we have to wait?

My hunch is the next TOC will be held May 2-13, 2016. That would give 18 months between TOCs, which has been the norm, more or less, over the past ten years.

Some back-of-the-envelope math:

1. There have been 12 weeks of regular play since the last TOC.
2. Barring any unannounced special events, there are 17 weeks left in Season 31.
3. In Season 32, assuming 5 weeks for special events in November and February, there will be 28 weeks before the TOC.

That’s a total of 285 games – slightly above average.

This is just our count from home, however. The producers can end a qualification period for a particular TOC whenever they want; in fact, my understanding is the last air date to qualify for November’s TOC was October 10, not November 7, the Friday before the TOC.

Here are all of the players who have won 3 games or more since October 13. (I’ve left in Catherine Hardee and Bill Cossen, who played earlier in Season 31, just in case my source is incorrect.)

Rank	Last game	Name	Wins	Winnings
—	13 Feb 2015	Jennifer Giles	TT	$100,000
1	15 Jan 2015	John Schultz	5	$104,500
2	02 Jan 2015	Vaughn Winchell	5	$103,103
	29 Sep 2014	Catherine Hardee	4	$95,201
3	08 Jan 2015	Elliot Yates	4	$65,000
4	22 Oct 2014	Alexander Persaud	3	$61,000
5	03 Nov 2014	Ryan Alley	3	$56,800
	19 Sep 2014	Bill Cossen	3	$50,602
6	15 Oct 2014	Shane Curtis	3	$42,001
7	25 Feb 2015	Dava-Leigh Brush	3	$31,501

We’re off to a slow start, though. Through 80 games, we have just two 5xers and one 4xer.

Through the first 80 games of the last TOC qualifying period, we’d seen:

1. Joshua Brakhage (5x)
2. Drew Horwood (8x)
3. Rebecca Rider (5x)
4. Mike Lewis (4x, eventual TOC alternate)
5. Andrew Moore (6x)

… and eventual 8xer Ben Ingram had won his first two games.

A really, really, really slow start.

This one goes to twelve

I’m going to further assume that there will be just 12 spots available, with one berth going to Teachers Tournament champ Jennifer Giles, another to the College Champ later this year, and a third to the winner of another tournament (probably Teachers) in the first half of Season 32.

Here are the stats for the twelfth regular qualifier in each of those years.

Held	Episodes	Auto bids	12th player in	Wins	Winnings
2006 / 05	228	2	Michael Falk	3	$59,403
2007 / 11	270	1	Susan Mitchell	4	$67,100
2009 / 03	276	0	Donna Vogel	4	$85,299
2010 / 05	209	3	Christine Valada	4	$68,703
2011 / 11	269	2	Brian Meacham	4	$90,500
2013 / 02	235	3	Dan McShane	4	$62,001
2014 / 11	313	3	Mark Japinga	4	$112,600

Judging from these stats, it doesn’t look great for Elliot. He would have gotten #12 or better just twice.

But this is a new slate, and I’ve got some data from the J! Archive to play with.

Choosing an approach

I considered a few ways to model the next year of Jeopardy! play. For example, Nick Condon, who won 3 games in 2011 , used a Poisson distribution to predict whether he’d make the cut. (He didn’t.)

I went with a different method: figure out how often a X -time champ wins her next game, then use a random-number generator 225 times to see whether our hypothetical champions remain at the first lectern, or if they’re replaced.

Counting champions

Since November 30, 2004, there have been 1,860 games in regular play. Here is the breakdown of how many players have won their X th game, either by win or by tie (R.I.P.).

Win #	Outright Win	Tie	Average	StdDev
1	1019	20	$19,306	$8,277
2	413	10	$19,829	$8,564
3	188	3	$21,802	$7,382
4	90	3	$20,412	$9,139
5	57	1	$21,419	$8,286
6	25		$23,642	$8,178
7	13		$22,646	$8,733
8	9		$23,678	$14,741
9	5		$20,180	$5,972
10+	23		$20,821	$6,256
Grand Total	1842	18	$19,920	$8,355

From that we can determine how frequently champions with certain experience have won again. (I weighted ties as 1/2 or 1/3 to account for their present non-existence.)

# wins	0	1	2	3	4	5	6	7	8	9
p(win)	27.8%	40.6%	45.3%	48.3%	62.8%	43.5%	52.0%	69.2%	55.6%	60.0%

Note that challengers win slightly more than 1/4 of the time; once you’ve got a W under your belt, you can get rolling. Four-time champs also receive a strange boost.

Here’s the model

I then used those numbers to model 225 games of Jeopardy!. I ran the model 100 times, and got these results:

wins	through today	expectations over next 225 games	total	this total or better
1	24	76.11	100.11	172.1
2	16	27.54	43.54	72.99
3	4	11.56	15.56	29.45
4	1	4.05	5.05	13.89
5	2	4.14	6.14	8.84
6	0	1.33	1.33	2.7
7	0	0.54	0.54	1.37
8	0	0.22	0.22	0.83
9	0	0.2	0.2	0.61
10+	0	0.41	0.41	0.41

The model expects 14 players to have 4 or more wins by the end of the qualification period.

I’d take $65,000 any day of the week, but that’s a low total, historically, for 4xers, who average $83,760 over their runs (standard deviation: $21,079). In simulations with two different approaches, Elliot’s total ended up around the 20th percentile.

Conclusion and predictions

Only time will tell if Elliot will make it. He was a popular and likable player, though, so maybe they’ll cut the qualification period short if it looks like he’s going to get dropped.

For what it’s worth, here’s my model’s prediction for the 2016 TOC field:

Rank	Last game	Name	Wins	Winnings
—	13 Feb 2015	Jennifer Giles	TT	$100,000
—		2015 College	CC	$100,000
—		2016 Teachers?	TT	$100,000
1		Simulated	9	$193,685
2		Simulated	7	$159,989
3		Simulated	6	$136,875
4		Simulated	5	$126,213
5		Simulated	5	$106,882
6	15 Jan 2015	John Schultz	5	$104,500
7	02 Jan 2015	Vaughn Winchell	5	$103,103
8		Simulated	5	$94,474
9		Simulated	5	$75,266
10		Simulated	4	$108,995
11		Simulated	4	$88,816
12		Simulated	4	$73,892
13	08 Jan 2015	Elliot Yates	4	$65,000
14		Simulated	4	$51,403