USCL Season Simulation Midweek 8

It is middle of week 8 in the USCL, and half the teams have played their matches this week. Therefore I ran simulations again.

The interesting thing was that when I ran 100,000 simulations for the Eastern Division, both Philadelphia and New York always made it into the playoffs. There's a chance, however unlikely, that one of those teams will not make it into the playoffs. Therefore, I ran it again, but this time with 2,500,000 simulations. In only one of those simulations, Philadelphia did not make it. So, Philly, don't coast now!

Here are the results.

Western Division (based on 100,000 simulations)

Team	Playoffs	Top 2	Number 1
Arizona	35.4%	1.67%	0%
Chicago	100%	99.981%	99.0%
San Francisco	71.1%	5.67%	0%
St. Louis	1.14%	0%	0%
Seattle	0.584%	0%	0%
Miami	20.6%	0.366%	0%
Los Angeles	99.68%	81.4%	1.22%
Dallas	79.3%	14.9%	0.122%

Eastern Division (based on 2,500,000 simulations)

Team	Playoffs	Top 2	Number 1
New England	37.7%	0.763%	0%
Boston	51.0%	0.274%	0%
New York	99.989%	96.0%	41.6%
Baltimore	38.3%	0.654%	0%
Manhattan	66.4%	7.06%	0.252%
New Jersey	0.0568%	0%	0%
Philadelphia	99.99996%	96.6%	58.9%
Carolina	10.5%	0.792%	0%

A scenario has been requested, and I will mention it in the comments.

USCL Season Simulation 2011

Last year, I created a simulation of possible outcomes for the remainder of the United States Chess League season, to project the chances each team has of making the playoffs.

As there are three weeks left, I decided to run another set of simulations this year.

Once again, here are the assumptions.

1. All teams have the same strength. Therefore, they have the same chances in each match against their opponents.
   
2. The game point results of the match follow the historical distribution of USCL matches (which is here). That is, 2.5-1.5 is more common than 3-1, which is more common than 3.5-0.5, etc.
3. If a team was tied for a playoff spot with the same match and game points, I said they both would make the playoffs -- I didn't go beyond the first tiebreaker.

I ran 100,000 simulations for both the Eastern and Western division. Here are the results.

Western Division

Team	Playoffs	Top 2	Number 1
Arizona	39.322%	6.648%	0%
Chicago	100.000%	99.995%	99.596%
San Francisco	65.377%	16.010%	0%
St. Louis	0.905%	0%	0%
Seattle	11.293%	0.889%	0%
Miami	19.010%	1.242%	0%
Los Angeles	90.735%	51.724%	0.381%
Dallas	80.074%	28.023%	0.146%

Eastern Division

Team	Playoffs	Top 2	Number 1
New England	39.508%	1.464%	0%
Boston	27.179%	0.501%	0%
New York	99.982%	93.277%	43.467%
Baltimore	38.095%	1.378%	0%
Manhattan	67.278%	9.757%	0.357%
New Jersey	0.045%	0%	0%
Philadelphia	99.988%	94.578%	57.171%
Carolina	31.810%	0.792%	0%

Any scenarios you want me to run? Things like: If NJ wins the remainder of their matches 4-0, what's their chance to get in the playoffs... things like that....

Let me know in the comments...

USCL Power Rankings Week 7

The power rankings are finally here this week!

No big changes at the top and the bottom. Chicago and its perfect record outpace the rest of the field, with Philadelphia and New York in second and third.

There is a logjam in the middle of the Eastern Division standings, with four teams having 3.0 match points, with their tiebreaking game points ranking them as follows: New England, Baltimore, Carolina, Boston. However, the power rankings say that the most powerful team in of these four is not New England, but Baltimore instead, and indeed, Boston is more powerful than Carolina.

Until next week!

Rank	Team	Score	Change
1 -	Chicago	1.000	0
2 -	Philadelphia	0.860	0
3 -	New York	0.845	0
4 -	Los Angeles	0.794	+1
5 -	Dallas	0.745	+2
6 -	San Francisco	0.743	0
7 -	Manhattan	0.739	-3
8 -	Baltimore	0.707	0
9 -	New England	0.700	+5
10 -	Arizona	0.681	+1
11 -	Boston	0.678	-2
12 -	Miami	0.671	-2
13 -	Carolina	0.664	-1
14 -	Seattle	0.653	-1
15 -	St. Louis	0.594	0
16 -	New Jersey	0.555	0

USCL Power Rankings Week 6

Because interdivisional week just completed, the power rankings are now combined into one table.

Unsurprisingly, Chicago holds the top spot, with a huge gap between them and the second place team. Indeed, teams two, three, and four are all from the Eastern Division, with Philadelphia barely edging out New York for second. Los Angeles is more powerful than San Francisco, even though the Mechanics are ahead of the Vibe in the official standings. In the battle of the basement, the Knockouts fall below the Arch Bishops.

COMBINED POWER RANKINGS

Rank	Team	Score	Change
1 -	Chicago	1.000	0
2 -	Philadelphia	0.837	0
3 -	New York	0.836	0
4 -	Manhattan	0.789	0
5 -	Los Angeles	0.774	0
6 -	San Francisco	0.760	0
7 -	Dallas	0.748	0
8 -	Baltimore	0.706	0
9 -	Boston	0.697	0
10 -	Miami	0.693	0
11 -	Arizona	0.678	0
12 -	Carolina	0.661	0
13 -	Seattle	0.661	0
14 -	New England	0.657	0
15 -	St. Louis	0.579	0
16 -	New Jersey	0.560	0

USCL Power Rankings Week 5

Exciting baseball tonight. My Orioles (always #1 to me) pulled off a great victory against the Red Sox (my #2 team). Sox collapsed, and the Rays have more of a chance to crush the hapless and worthless Yankees.

Oh wait, this is chess.

So, week 5 in the power rankings show the continued dominance of Chicago in the West. The California teams are closely knotted in second and third, Dallas and Miami are separated by a mere 0.004 in fourth and fifth, and Seattle and Miami by an even smaller 0.001 in sixth and seventh.

In the East, the surprising Philadelphia Inventors grab the top spot in the power rankings, with the NYC teams (Knights and Applesauce) following in second and third. But notice that the power differential between first place Philadelphia and cellar-dwelling eighth place New Jersey is very close to the differential between the first (Chicago) and fourth (Dallas) place teams in the West. This suggests that the East teams are closer in strength to each other, while the West teams are more variable.

EASTERN DIVISION

Rank	Team	Score	Change
1 -	Philadelphia	1.000	+4
2 -	New York	0.981	-1
3 -	Manhattan	0.951	-1
4 -	Baltimore	0.847	0
5 -	Carolina	0.831	+1
6 -	Boston	0.826	-3
7 -	New England	0.769	+1
8 -	New Jersey	0.705	-1

WESTERN DIVISION

Rank	Team	Score	Change
1 -	Chicago	1.000	0
2 -	San Francisco	0.780	0
3 -	Los Angeles	0.777	+2
4 -	Dallas	0.753	-1
5 -	Miami	0.671	-1
6 -	Seattle	0.668	0
7 -	Arizona	0.614	0
8 -	St. Louis	0.568	0

Next week is Inter-division Week, and the power rankings will be combined into one list.

USCL Power Rankings Week 4

Power rankings this week show a lot of movement in the Eastern Division, but holding steady at the top of the Western Division.

In the East, the New York Knights take the top spot away from Boston, which dropped to third. The New Jersey Knockouts move out of the cellar, leaving the Nor'easters in that dubious spot, but just barely, with only 0.005 Power Points separating them.

In the West, the Chicago Blaze remain at the top, but with an astounding 0.220 Power Points separating them from the Mechanics, which remain number two. Despite Finegold's excellent individual record, the Arch Bishops remain at the bottom.

Eastern Division

Rank	Team	Score	Change
1 -	New York	1.000	+2
2 -	Manhattan	0.944	0
3 -	Boston	0.931	-2
4 -	Baltimore	0.912	+2
5 -	Philadelphia	0.902	0
6 -	Carolina	0.831	-2
7 -	New Jersey	0.669	+1
8 -	New England	0.664	-1

Western Division

Rank	Team	Score	Change
1 -	Chicago	1.000	0
2 -	San Francisco	0.780	0
3 -	Dallas	0.777	0
4 -	Miami	0.753	+2
5 -	Los Angeles	0.671	-1
6 -	Seattle	0.668	+1
7 -	Arizona	0.614	-2
8 -	St. Louis	0.568	0

Until next week!

USCL Power Rankings Week 3

United States Chess League, Power rankings. Week 3... Not much surprise this week, as the power rankings exactly match the standings.

Power is pretty evenly spaced in the Eastern Division, except at the top, where Manhattan barely trails Boston.

Chicago sits atop the Western Division, followed by San Francisco and Dallas, with a tightly spaced pack from fourth to seventh.

Eastern Division

Rank	Team	Score	Change
1 -	Boston	1.000	+1
2 -	Manhattan	0.973	+2.5
3 -	New York	0.889	-2
4 -	Carolina	0.827	-2
5 -	Philadelphia	0.782	+1
6 -	Baltimore	0.728	-1.5
7 -	New England	0.667	0
8 -	New Jersey	0.524	0

Western Division

Rank	Team	Score	Change
1 -	Chicago	1.000	0
2 -	San Francisco	0.904	+0.5
3 -	Dallas	0.836	+1
4 -	Los Angeles	0.712	-1.5
5 -	Arizona	0.671	+1
6 -	Miami	0.658	+2
7 -	Seattle	0.630	-2
8 -	St. Louis	0.507	-1

Until next week!

USCL Power Rankings Week 2

Welcome to the 2011 USCL season.

Here are the weekly power rankings for the conclusion of Week 2. Note: Week 1 power rankings were not published (the lack of data makes them rather silly).

The methods to calculate power rankings are the same as in previous years. Until there is inter-divisional play, the rankings will remain separate.

Without further ado...

Eastern Division

Rank	Team	Score	Change
1 -	New York	1.000	+1
2 -	Carolina	0.963	0
3 -	Boston	0.863	-1
4-5 -	Manhattan	0.795	+0.5
4-5 -	Baltimore	0.795	+2.5
6 -	Philadelphia	0.727	-1.5
7 -	New England	0.627	0
8 -	New Jersey	0.590	-1

Western Division

Rank	Team	Score	Change
1 -	Chicago	1.000	0
2-3 -	Los Angeles	0.797	+2
2-3 -	San Francisco	0.797	0
4 -	Dallas	0.786	+0.5
5 -	Seattle	0.759	+3
6 -	Arizona	0.649	-3.5
7 -	Saint Louis	0.535	-0.5
8 -	Miami	0.508	-1.5

Earth versus Space Chess Match

There is an Earth versus Space chess match going on. It is an interesting concept, but poorly executed. They even have a Facebook page, but there's no indication on the FB page of what the current position actually is.

How sad.

Recipe: Coconut Pumpkin Pie

I found some handwritten recipes in two books in some stuff in my garage. Both are in old, spiral bound notebooks. One was my mom's (who died in 1992) and the other is my grandmother's (I'm pretty sure it is hers - she died in 1978).

Anyway, I'm certainly not much of a cook, but I thought I would post these recipes here, as once they go into the cloud, they are likely not lost forever. And who can resist a Coconut Pumpkin Pie? This one if from my grandmother's recipe book.

Note that I am not changing or correcting the spelling or punctuation, because I think it is interesting from a historical perspective.... like in the recipe below, she spells "separated" incorrectly.

Coconut Pumpkin Pie

1 1/3 c. cooked or canned pumpkin
2/3 c. brown sugar
1 tsp. cinnamon
1/4 tsp. ginger
1/4 tsp. nutmeg
1/2 tsp. salt
2 eggs seperated
2 c. bottled milk or 1 c. evap. milk and 1 c. water
1/2 c. shredded coconut
1 9 in pastry lined pie plate.

Combine the pumpkin, brown sugar, cinnamon, ginger, nutmeg, salt and slightly beaten egg yolks. Mix well and add the milk. Fold in the egg whites, beaten stiff, and pour into the lined pie plate. Bake in a very hot oven 450°F for 10 min. The reduce to 325° for 15 min. Sprinkle coconut over top of pie and bake 15 min. longer.

USCL Game of the Year Judging Analysis

I performed some statistics on the judging in United States Chess League's 2010 Game of the Year contest.

There were five judges: Hess, Gustafsson, Johannesson, Melekhina, Young. I will refer to them by the first letter of their last name.

Several analyses were completed.

What are the games for which the judges agreed most and disagreed most?

This can be calculated by looking at the standard deviations of the scores on each game.

The most agreed upon games were:

1. #20, Sammour-Hasbun vs. Kaplan (sd = 2.51)
2. #2, Sammour-Hasbun vs. Kacheishvili (sd = 2.61)
3. #4, Rosen-Guo (sd = 2.97)

The most disagreed upon games were:

1. #13 Schroer vs. Kacheishvili (sd = 7.99)
2. #19 Galofre vs. Milat (sd = 7.80)
3. #12 Friedel vs. Akobian (sd = 7.36)

Which judges were most different?

I calculated which of the judges were "most different" than the combined wisdom of all the judges together. The judges that were the most different could be considered outliers.

There are several ways to do this. I will demonstrate two approaches.

FINDING THE OUTLIER JUDGES

First, I compared the score a judge gave to the average of all the judges, but tempering that by the amount of disagreement of all the judges. For instance Judge Y gave 2 points (19th place) to Schroer-Kacheishvili, while the average number of points was 9.2, and the standard deviation (the amount of disagreement) was 7.99. Therefore, For that game, Judge Y would receive the absolute value of (2 - 9.2)/7.99 or 0.80 "difference points". For each of the twenty games, add up the difference points. The more the difference points, the more different the judge was from the other judges.

The total number of difference points were...
Judge Y: 17.49
Judge J: 11.09
Judge M: 19.40
Judge G: 12.80
Judge H: 16.96

Therefore, Judge Y and Judge M were the most different from the other judges.

Then, we could discard the scores of these two judges, and rescore the contest.

See below for how the results would have changed.


COMPUTE THE MIDDLE SCORES FOR EACH GAME

Another way of rescoring the contest is to do it on a "per game" basis, as opposed to throwing judges as a whole. Instead, discard the high and low scores given to each game, and create a new total.

For example, Golfre-Milat received scores of 1, 1, 1, 5, and 19. If we were to use this method, we would throw out one of the 1s and the 19, and the game would received a revised score of 7.

. . .

The table below shows the original place for each game, as well as the place it would have come it if you used the "Three Judges Only" method, or the "No Hi-Lo" method. Ties were not broken for these alternate methods.

GAME	Original	Three Judges	No Hi-Lo
Sammour-Hasbun vs Kaplan	20	19	19
Galofre vs Milat	19	20	20
Gurevich vs Barcenilla	18	18	18
Akobian vs Friedel	17	T13-14	17
Rosenthal vs Thompson	16	T15-17	15
Krasik vs Balasubramanian	15	T13-14	16
Hungaski vs Schroer	14	T15-17	13
Schroer vs Kacheishvili	13	T15-17	14
Friedel vs Akobian	12	T11-12	12
Shulman - Felecan	11	T11-12	11
Rensch - Abrahamyan	10	T4-5	T7-10
Shankland vs Becerra	9	8	T7-10
Stripunsky vs Erenburg	8	10	T7-10
Christiansen vs Kraai	7	T6-7	T7-10
Schroer vs Christiansen	6	T4-5	4
Kacheishvili vs Shankland	5	9	T5-6
Rosen vs Guo	4	T6-7	T5-6
Shulman vs Khachiyan	3	2	2
Sammour-Hasbun vs Kacheishvili	2	3	3
Akobian vs Shulman	1	1	1

Readers are invited to make their own conclusions.

Ratings Contest

A website called Kaggle have regular contests where they give people a data set, and have them to predict something about the data set. They have done it for a variety of topics (e.g., tourism prediction, grant application success) and, of course, they just finished one for chess ratings.

Spurred on by Jeff Sonas (the Chessmetrics guy), there were a lot of entries, and suffice it to say, someone came up with a method that is 6.6% better than standard Elo ratings.

You can read all about it here, and wait for the follow up contest in January.

USCL Power Rankings: Final Week

The final week of the United States Chess League regular season has come and gone, and the playoffs are set. The power rankings only are for the regular season, so the rankings below are the final ones for this year.

The last week really shook up the rankings, with St. Louis plummeting four places and other teams jockeying for a better position. In the end, the newcomers, the New England Nor'easters are far above the remainder of the field. Interestingly the bottom three places of the rankings are filled by Eastern Division Teams.

Before the 2010 list, let's look at where teams stood in the 2009 list.

Power Rankings 2009 (End of Regular Season)

1 -	New Jersey (8.0 - 1st East)	1.000
2 -	Seattle (7.5 - 1st West)	0.953
3 -	Boston (7.5 - 2nd East)	0.929
4 -	Arizona (6.0 - 3rd West)	0.901
5 -	San Francisco (6.5 - 2nd West)	0.894
6 -	Miami (6.0 - 4th West)	0.879
7 -	New York (5.0 - 3rd East)	0.853
8 -	Chicago (4.0 - 5th West)	0.764
9 -	Philadelphia (4.0 - 5th East)	0.735
10 -	Dallas (3.5 - 6th West)	0.730
11 -	Queens (3.5 - 6th East)	0.726
12 -	Baltimore (4.5 - 4th East)	0.724
13 -	Tennessee (2.0 - 7th West)	0.647
14 -	Carolina (2.0 - 7th East)	0.622

How some teams fortunes have changed in a year, especially New Jersey which went from the top in 2009 to near the bottom this year.

Therefore, here is the final power rankings for the 2010 season.

2010 USCL Power Rankings
End of Regular Season

Rank	Team	Score	Change
1 -	New England	1.000	0
2 -	Boston	0.886	+1
3 -	Arizona	0.857	-1
4 -	Chicago	0.793	+2
5 -	New York	0.793	-1
6 -	Baltimore	0.787	+1
7 -	Miami	0.771	+1
8 -	Seattle	0.760	+2
9 -	St. Louis	0.743	-4
10 -	Los Angeles	0.740	+3
11 -	San Francisco	0.708	-2
12 -	Philadelphia	0.697	+2
13 -	Dallas	0.686	-2
14 -	New Jersey	0.677	-2
15 -	Manhattan	0.638	0
16 -	Carolina	0.607	0

The top eight teams in the power rankings are also the top eight teams in the playoffs. Good luck to all of them.

USCL 2010 Power Rankings Week 9

A little late... okay a LOT late, but I was away.

Notable about this is that New England has completely pulled away from the pack.

Rank	Team	Score	Change
1 -	New England	1.000	0
2 -	Arizona	0.886	0
3 -	Boston	0.857	0
4 -	New York	0.793	0
5 -	St. Louis	0.793	+2
6 -	Chicago	0.787	0
7 -	Baltimore	0.771	-2
8 -	Miami	0.760	+1
9 -	San Francisco	0.743	-1
10 -	Seattle	0.740	+1
11 -	Dallas	0.708	-1
12 -	New Jersey	0.697	+2
13 -	Los Angeles	0.686	-1
14 -	Philadelphia	0.677	+1
15 -	Manhattan	0.638	-2
16 -	Carolina	0.607	0

USCL 2010 Power Rankings Week 8

The power rankings show that New England is clearly the class of the league, with a huge lead in strength over second place Arizona. Baltimore and Dallas were the big gainers this week, while San Francisco took a big tumble.

Only a whisker separates Boston and New York, and when these teams likely play in the playoffs in the first round, it should be a close match.

Not even visible to the human eye (only the fourth decimal place) separates Chicago and St. Louis!

Rank	Team	Score	Change
1 -	New England	1.000	0
2 -	Arizona	0.914	0
3 -	Boston	0.846	0
4 -	New York	0.844	0
5 -	Baltimore	0.810	+2
6 -	Chicago	0.799	0
7 -	St. Louis	0.799	+1
8 -	San Francisco	0.794	-3
9 -	Miami	0.769	0
10 -	Dallas	0.728	+3
11 -	Seattle	0.703	-1
12 -	Los Angeles	0.692	-1
13 -	Manhattan	0.679	-1
14 -	New Jersey	0.658	+1
15 -	Philadelphia	0.651	-1
16 -	Carolina	0.632	0

Remainder of Season Simulation: Post-Week 8

Chance to be at least tied for 4th in match/game points, based on 100,000 simulations.

Western Division
100.00% Arizona
_73.15% St. Louis
_72.37% Chicago
_63.15% San Francisco
_51.96% Miami
_26.43% Dallas
_13.62% Seattle
__4.72% Los Angeles

Eastern Division
100.00% New England
_99.65% Boston
_94.25% New York
_92.55% Baltimore
_13.47% Manhattan
__0.78% New Jersey

USCL Simulation after Week 8 Monday

UPDATED! Now includes Western Division!

Chance of making playoffs (100,000 simulated Wednesday Week 8s and Weeks 9 and 10)

Eastern Division

New England	100.00%
Boston	98.85%
New York	97.44%
Baltimore	92.29%
Manhattan	11.27%
New Jersey	0.79%
Philadelphia	0.00%
Carolina	0.14%

Western Division

Arizona	99.99%
Chicago	85.45%
San Francisco	69.28%
St. Louis	35.04%
Seattle	30.48%
Miami	18.73%
Los Angeles	16.68%
Dallas	49.12%

Note that Arizona made the playoffs in 99,991 of the 100,000 simulations!

Remainder of Season Simulation

I have expanded on my prior projects to determine the likelihood that a team will make the playoffs this season. As I said previously, the ability to be comprehensive in all possibilities of match outcomes eats into years of computing time, so simulation is an obvious answer.

I created a simulation that attempts to model all the matches in the remainder of the USCL season.

The goal was the same -- what is the chance that a team will make the playoffs. This time, however, I included by match points and the game points tiebreaker. (If a team was tied for a playoff spot with the same match and game points, I said they both would make the playoffs -- I didn't go beyond the first tiebreaker.)

To do this, I made several other assumptions.

1. All teams have the same strength. Therefore, they have the same chances in each match against their opponents.
   
2. The game point results of the match follow the historical distribution of USCL matches (which is here). That is, 2.5-1.5 is more common than 3-1, which is more common than 3.5-0.5, etc.

I ran 100,000 simulations for both the Eastern and Western division. Here are the results, where the percentages indicate a tie for fourth place or better.

Eastern Division (Chance to make playoffs)

New England	100.00%
Boston	98.43%
New York	97.49%
Baltimore	70.27%
Manhattan	24.79%
New Jersey	6.92%
Philadelphia	2.69%
Carolina	0.61%

Western Division (Chance to make playoffs)

Arizona	97.18%
Chicago	86.54%
San Francisco	78.07%
St. Louis	45.24%
Seattle	35.50%
Miami	27.39%
Los Angeles	23.44%
Dallas	12.92%

I also included the chance that a team will receive draw odds in the first round of the playoffs (tied for second place or better).

Eastern Division (Chance have draw odds first round of playoffs)

New England	98.37%
Boston	55.19%
New York	41.63%
Baltimore	4.90%
Manhattan	0.23%
New Jersey	0.00%
Philadelphia	0.00%
Carolina	0.00%

Western Division (Chance have draw odds first round of playoffs)

Arizona	80.94%
Chicago	50.92%
San Francisco	40.50%
St. Louis	13.13%
Seattle	7.55%
Miami	6.01%
Los Angeles	3.60%
Dallas	1.25%

Any scenarios you want me to run? Things like: If NJ wins the remainder of their matches 4-0, what's their chance to get in the playoffs... things like that....

Let me know in the comments...

USCL Western Division: Playoff Hunt

In my previous post, I posted the chances of at least tying for fourth place in the USCL Eastern Division. You can read about the method I used in that post.

I ran the same thing for the Western Division. Here is the percentage that each team will finish in at least a tie for fourth in match points.

99.14% Arizona
95.70% Chicago
82.20% San Francisco
61.70% St. Louis
35.96% Seattle
35.69% Miami
34.61% Los Angeles
15.31% Dallas

USCL Eastern Division: Playoff Hunt

There are three weeks to go in the USCL regular season. What are the chances that any particular team will be in playoff contention?

Let's assume for the moment that all teams are equally matched, and that all teams have an equal chance to win, draw, or lose a match with every other team. Therefore, each match can have three possible results.

In the Eastern Division of the USCL, there are twelve matches left and therefore there are 3^12 (three to the twelfth power) possible results (in terms of match results). That's 531, 441 possibilities.

If we make the criterion for "possibly in the playoffs" to be tied for fourth or better, then the percentages below indicate what a team's chances to possibly be in the playoffs.

100.00% - New England
99.55% - Boston
98.27% - New York
74.75% - Baltimore
41.28% - Manhattan
10.32% - New Jersey
3.05% - Philadelphia
1.21% - Carolina

Now, if we take game points (the first tiebreaker) into consideration, that increases the number of possibilities to 9^12 power, which is actually 531,441 times bigger than the last time. Since the simulation presented here took about 4 seconds to run on my computer, the code for including game points is about 50% more complex (increasing time to completion by about 50%), it would take over 553 days for the simulation to run!

I think I'll wait for the actual results.

Anyone interested in me doing this for the Western Division too?