RideTheEdge
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Joined: Apr 30, 2020
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January 10th, 2022 at 2:29:24 PM permalink

I went into the final week with a 1.5 game lead, but came in second to the sharp Johnzimbo (JZ below) after taking advantage of the option to skip making picks the final week. Was I foolish to skip?

A couple weeks ago I created a simple model of my chances of winning with 2 weeks to go. This model considered the choice I'd have of playing one or both of the remaining weeks. Absent any other information, always playing just one week was superior without considering the option to play week 18 if I fell behind or tied after week 17. My decision to skip week 18 was informed by this model, along with the realization that JZ would have to go 5-0 or 6-0 to win (4-0 or 5-1 didn't cut it.) I'll describe that model in another post.

Using similar calculations I revisited the decision to skip week 18 to soothe my regret and convince myself that being lazy and complacent was indeed the rational thing to do.

First thing to note is that confidence picks are probably not relevant. I think I should not make a confidence pick to reduce the variance of my outcomes and lower the chance of being caught. JZ should probably not make a confidence pick because it increases the remote possibility of falling into third place with an 0-6 result, and the 6-0 record does just as well as 5-0 when it comes to winning the contest (assuming I skip). I might be wrong here, as the equity of the perfection bonus might exceed the lost equity of falling to third place (both less than \$2), and I didn't calculate the added possibilities a confidence pick with increased likelihood of success would introduce. In any case, passing up on the confidence picks can't be very far wrong for either of us and it greatly simplifies the analysis.

So, with each of us choosing 5 games of equal value, all that is needed is the probability of each outcome and the resulting winner. A couple simplifying assumptions:
- Each pick has the same probability of winning, though the two of us can have different probabilities. For example I can model my picks at 50% correct and JZ's at 60%.
- No ties.

Each of us can go 5-0, 4-1, 3-2, 2-3, 4-1, or 0-5. The binomial distribution (available as a formula in excel or google sheets) calculates the probability of each of these results happening under the simplifying assumptions above. It just so happens that Bob Dancer wrote an article on the binomial distribution around the same time as I was having these thoughts. See https://www.lasvegasadvisor.com/gambling-with-an-edge/how-do-you-figure-4/

First step is to work out all the probabilities. This chart lists the probability of each outcome. For example, the top left value of 0.0977% is the chance that we both get 5 correct. Each cell in the spreadsheet below is the product of each of us getting the given number of picks correct. Each of our win rates is configurable, these numbers are for 50% picks correct for both of us:
`	5	4	3	2	1	05	0.0977%	0.4883%	0.9766%	0.9766%	0.4883%	0.0977%4	0.4883%	2.4414%	4.8828%	4.8828%	2.4414%	0.4883%3	0.9766%	4.8828%	9.7656%	9.7656%	4.8828%	0.9766%2	0.9766%	4.8828%	9.7656%	9.7656%	4.8828%	0.9766%1	0.4883%	2.4414%	4.8828%	4.8828%	2.4414%	0.4883%0	0.0977%	0.4883%	0.9766%	0.9766%	0.4883%	0.0977%`

With the standings after week 17, JZ needs 2 more correct picks than me to win. So, if he goes 3-2 and I go 1-4 he wins, but if I go 2-3 it's my game.

Running this a few times with various levels of my/JZ's skill and adding up the cells where JZ lacks sufficient correct picks, the probability of winning the contest is:
- 50/50 (aka "the AZDuffman"): 83%
- 53/53 (my estimate of my ability): 83%
- 53/58 (my estimate of me vs. a skilled handicapper): 79%
- 50/60 (pessimistic bias): 73%
- 60/50 (optimistic bias): 90%
- 53/50 (me vs. coin flips): 85%

JZ's chance of a perfect week is just his win probability to the 5th power. Anywhere from 3% to 8% for a 50-60% success rate. Results to the contrary, long run picks over 60%, even with the opportunity to use Thursday's line, seems too high to me. Happy to discuss any of these estimated percentages discuss further.

Even if I'm vastly superior in my picking ability my winning chance if I make picks in week 18 is only 90%. I think a better guess is 79%. Against a very skillful opponent skipping is 92%. Better off skipping.
JohnnyQ
• Posts: 4032
Joined: Nov 3, 2009
January 10th, 2022 at 2:31:06 PM permalink
Hey, 2nd place is nothing to sneeze at, right ? Better "variance" next time if M146 is still so gracious.
There's emptiness behind their eyes There's dust in all their hearts They just want to steal us all and take us all apart
RideTheEdge
• Posts: 180
Joined: Apr 30, 2020
January 10th, 2022 at 2:38:18 PM permalink
Quote: JohnnyQ

Hey, 2nd place is nothing to sneeze at, right ? Better "variance" next time if M146 is still so gracious.

Very true, and the entire event was a lot of fun.

I enjoy the modeling and sharing the work with people who are more likely than the general population to be receptive to the number crunching.

Still stings though...
Mission146
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Joined: May 15, 2012
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January 10th, 2022 at 2:52:09 PM permalink
Quote: RideTheEdge

Quote: JohnnyQ

Hey, 2nd place is nothing to sneeze at, right ? Better "variance" next time if M146 is still so gracious.

Very true, and the entire event was a lot of fun.

I enjoy the modeling and sharing the work with people who are more likely than the general population to be receptive to the number crunching.

Still stings though...

Such is the nature of making the best play possible. If it's the best play, then it's the best play...if the best play always performed optimally, then you wouldn't be gambling.
https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
RideTheEdge
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Joined: Apr 30, 2020
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January 11th, 2022 at 10:18:04 AM permalink
There is more to all this modeling effort than licking a wound.

Prior to week 17, JZ and I started to discuss making a deal. Thanks to JZ for agreeing to share the existence of those discussions. Had we come to terms we would have asked Mission if that was allowed of course. At that point I was ahead by a 1.5 games. So I put some thought into what a fair price would be.

Some considerations before crunching numbers:
- What is the effect of confidence picks? This seemed pretty complicated, so I just punted.
- Should I skip a week?
- Easier if I ignore ties and assume the contest always has a winner. Since we resolved an unequal number of games during the season, a confidence pick or an unresolved game (i.e. tie against the spread or a total hit on the nose) would be necessary to end up tied.

So, I ran numbers twice. First with each of us picking 10 games, and then with me picking 5 and JZ picking 10.

Using the same binomial distribution calculations as above here are the results for each of us picking 10 games using the AZDuffman picking technique:
`	10	9	8	7	6	5	4	3	2	1	010	0.0001%	0.0010%	0.0043%	0.0114%	0.0200%	0.0240%	0.0200%	0.0114%	0.0043%	0.0010%	0.0001%9	0.0010%	0.0095%	0.0429%	0.1144%	0.2003%	0.2403%	0.2003%	0.1144%	0.0429%	0.0095%	0.0010%8	0.0043%	0.0429%	0.1931%	0.5150%	0.9012%	1.0815%	0.9012%	0.5150%	0.1931%	0.0429%	0.0043%7	0.0114%	0.1144%	0.5150%	1.3733%	2.4033%	2.8839%	2.4033%	1.3733%	0.5150%	0.1144%	0.0114%6	0.0200%	0.2003%	0.9012%	2.4033%	4.2057%	5.0468%	4.2057%	2.4033%	0.9012%	0.2003%	0.0200%5	0.0240%	0.2403%	1.0815%	2.8839%	5.0468%	6.0562%	5.0468%	2.8839%	1.0815%	0.2403%	0.0240%4	0.0200%	0.2003%	0.9012%	2.4033%	4.2057%	5.0468%	4.2057%	2.4033%	0.9012%	0.2003%	0.0200%3	0.0114%	0.1144%	0.5150%	1.3733%	2.4033%	2.8839%	2.4033%	1.3733%	0.5150%	0.1144%	0.0114%2	0.0043%	0.0429%	0.1931%	0.5150%	0.9012%	1.0815%	0.9012%	0.5150%	0.1931%	0.0429%	0.0043%1	0.0010%	0.0095%	0.0429%	0.1144%	0.2003%	0.2403%	0.2003%	0.1144%	0.0429%	0.0095%	0.0010%0	0.0001%	0.0010%	0.0043%	0.0114%	0.0200%	0.0240%	0.0200%	0.0114%	0.0043%	0.0010%	0.0001%`

With various skill levels, here is the resulting probability of winning the contest if we each pick 10 games, starting from 1.5 games ahead:
- 50/50: 75%
- 53/53: 75%
- 53/58: 67%
- 50/60: 59%
- 60/50: 87%
- 53/50: 79%

Next, here are the probabilities when I always pick 5 games and JZ always picks 10, starting with our actual records after week 17. There are quirks when we pick unequal numbers of games that affect the contest winning percentages that I might get into in another post.
`	5	4	3	2	1	010	0.00%	0.02%	0.03%	0.03%	0.02%	0.00%9	0.03%	0.15%	0.31%	0.31%	0.15%	0.03%8	0.14%	0.69%	1.37%	1.37%	0.69%	0.14%7	0.37%	1.83%	3.66%	3.66%	1.83%	0.37%6	0.64%	3.20%	6.41%	6.41%	3.20%	0.64%5	0.77%	3.85%	7.69%	7.69%	3.85%	0.77%4	0.64%	3.20%	6.41%	6.41%	3.20%	0.64%3	0.37%	1.83%	3.66%	3.66%	1.83%	0.37%2	0.14%	0.69%	1.37%	1.37%	0.69%	0.14%1	0.03%	0.15%	0.31%	0.31%	0.15%	0.03%0	0.00%	0.02%	0.03%	0.03%	0.02%	0.00%`

Again varying skill levels:
- 50/50: 85%
- 53/53: 83%
- 53/58: 76%
- 50/60: 70%
- 60/50: 91%
- 53/50: 87%

So at first blush, I'm better off always skipping a week rather than never skipping a week, and my probability of success is 70 to 85%.

I was quite surprised that my paltry 1.5 game lead with two weeks left gave me such a large advantage. Maybe it's something like flipping coins with JZ needing to win 2 flips in a row. I could do as badly as 1-4 in week 17 but still win the contest if JZ went 6-6 or 7-5 the last two weeks.

Some qualitative considerations:
- Confidence picks favor JZ since he can increase variance and his confidence picks will have a higher likelihood of being correct. So, shave off a few percent.
- I have a strategic option to make picks week 18 if JZ catches up after week 17. Add a few percent back.
- I assume the above effects rougly cancel out, but one can add a few percent to the estimate of JZ's skill to compensate in case that is wrong.

It's not hard to calculate potential deals from the above numbers, but I'd rather not delve into the details of the negotiation other than to say it was friendly and we decided to let the contest play out.
Mission146
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January 11th, 2022 at 10:28:43 AM permalink
This is all more very impressive and in-depth analysis!!!

To answer the question that wasn't exactly asked, players can make whatever arrangements that they want to vis-a-vis prize spitting amongst themselves, but they would be on their own honor to adhere to the agreement (as I am confident they would) and I would have no hand in enforcing the agreement. For my part, I would pay out the players what they have won in accordance with the rules of the game and everything else is between them.
https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
Johnzimbo
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January 11th, 2022 at 11:04:40 AM permalink
I have been keeping a buddy of mine informed weekly on how I was doing in the contest and going into the last week I told him the guy I was trying to catch would not be making picks in week 18 because it was the right move, and that I would need to be almost perfect in the last week to pass him, so yeah, sitting out the last week was the right move I think.
gordonm888
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January 11th, 2022 at 11:55:17 AM permalink
I started doing an almost identical analysis on the 10 people (other than me) who had a chance of finishing with a record of 0.500 and thus be splitting the \$75 Mr Average award payout with me. I didn't have a decision to make, but the purpose was to estimate my EV and my probability distribution of equity. I had two contestants who needed a 3-3, one who needed a 3-2, one that needed a 1-4, etc. I was planning at looking at assumptions of picking rates of 50% and then 55% and 60%, with certain contestants picking games they thought would fail.

I lost interest and didn't complete the analysis.

I had wondered whether contestants were better off shooting for a 6-0 perfection bonus of \$20 rather than a score of, say, 2-4 that they needed to share the Mr. Average payout. I decided that that was unlikely, because a perfect score for Round 18 would be hard to achieve and thus rarely occur, and because the Mr. Average award was so much larger.

What never occurred to me is that more than one of the contestants who needed a 3-2, 1-4 or 5-0 to win the Mr. Average would instead opt for the confidence pick, thus eliminating any chance at sharing the Mr. Average award (except for the unlikely scenario that one of the picked games was a push.)

And what I am still surprised at is that AZDuffman, needing a 5-0 to share the \$75 Average payout with me, instead designated a confidence pick and, incredibly, went 6-0. And fate is cruel, because two others also went 6-0 and thus AZ only won a 3-way split of \$20 -while literally no one other than me was left standing at 0.500 to share the Mr. Average. AZ has already explained his thinking on that. It's just . . . that I never would have correctly anticipated AZ's decision as part of analyzing my EV.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
RideTheEdge