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Topic: 2006 World Championship (Read 5934 times) |
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99of9
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Re: 2006 World Championship
« Reply #60 on: Nov 30th, 2005, 6:21pm » |
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on Nov 30th, 2005, 5:54pm, Fritzlein wrote:| But in any case, if we switch to the finer granularity of one hour time slots and four or five levels of preference, then the preferences themselves should do better at unambiguously determining the best time. It may become superfluous to make assumptions on behalf of the players about what is best for them. |
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Yes, I think this is the best solution. I think at least 5 different levels of preference is probably normal for most people. For example:
1) These times are ideal for me, no drawbacks.
2) Times like this would make a particular day busy, but there's no actual clash.
3) I can play these times, but I may be slightly tired and may not play at my best.
4) I would need reschedule other committments in order to play at these times.
5) This is in the middle of the night. I am available, but would otherwise be sleeping!
I also agree with how you were proposing to order pairs Fritz. But I do wonder if there's a slightly more mathematical way of treating cases where one person has only made a few slots available at that level, whereas the other has put up lots of options. I'll try to think of something and come back to this (or look up the voting systems that were mentioned).
(Because under your system one person could effectively veto most times by proposing level 1 for limited set of slots, and level 5 for all the rest of his/her 30. It would be quite unfair if that was encouraged by the system.)
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| « Last Edit: Nov 30th, 2005, 6:35pm by 99of9 » |
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99of9
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Re: 2006 World Championship
« Reply #61 on: Nov 30th, 2005, 6:48pm » |
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Now that I think a little longer, perhaps it would be sufficient to make a rule that everyone needs to choose at least 5 slots of type 1, at least 10 of type 2 or higher, at least 15 of type 3 or higher, at least 20 of type 4 or higher, and obviously at least 30 of type 5 or higher.
(double this if the number of slots is doubled)
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Fritzlein
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Re: 2006 World Championship
« Reply #62 on: Nov 30th, 2005, 7:23pm » |
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on Nov 30th, 2005, 6:21pm, 99of9 wrote:| (Because under your system one person could effectively veto most times by proposing level 1 for limited set of slots, and level 5 for all the rest of his/her 30. It would be quite unfair if that was encouraged by the system.) |
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I didn't think about this problem. I'm not sure how bad it is, though. If you put most of your slots at the lowest preference, then you are really gambling. If it happens that none of your few top preferences overlaps with my available times, then my preferences will completely control the choice among the slots that do overlap. This could stick you with a hideous time when you could have had one that is merely bad.
Therefore, I'm not convinced my system does encourage putting all your eggs in one basket. Yes, it would give you a better chance of getting your most preferred slot, but also a better chance chance of getting your least preferred slot. If I understand the sentiment that has been expressed on this topic, people are more worried about avoiding a handful of awful times than they are worried about securing a particularly good time. Someone who has to pick four terrible times just to round out the thirty would be well advised to set only those four at the lowest preference.
We could require everyone to select exactly 1/5 of each of the five categories. I suspect that most of the time this would actually be forcing people to act in their own best interest. The related game theory would be fascinating. On the other hand, there might be rare cases where an extreme assignment of preferences reflects reality, in which case it would be too bad to forbid players from making the relevant choices. I would merely advise players that it is probably wisest to spread out over the categories, and I wouldn't impose any restrictions.
Remember, there will still be times the players overlap only in one slot, and that slot happens to be terrible for both players, or terrible for one and perfect for the other. Because bad situations and unfair situations can't be avoided entirely, we should temper our enthusiasm for complicating the current scheme. We may end up putting in a lot of effort for only a little benefit. 
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| « Last Edit: Nov 30th, 2005, 7:55pm by Fritzlein » |
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99of9
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Re: 2006 World Championship
« Reply #63 on: Nov 30th, 2005, 7:58pm » |
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[EDIT: Fritz's recent post came in as I was typing this one up. I agree with most of his sentiments therein.]
on Nov 30th, 2005, 7:23pm, Fritzlein wrote:| On the other hand, there will still be times the players overlap only in one slot, and that slot happens to be terrible for both players, or terrible for one and perfect for the other. Maybe the fact that bad situations or unfair situations can't be avoided should temper our enthusiasm for complicating the current scheme. We may end up putting in a lot of effort for only a little benefit. |
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Of course you are right, but my feeling is it is worth making at least some of the changes, especially the easy ones. Doubling the number of timeslots and increasing the number of preference gradations seems like it would be easy. Perhaps we just "request" that people provide an even distribution, rather than trying to enforce it in software.
My feeling comes from the fact that as an Australian I am perhaps the least likely to fit in with everyone else's natural timezone preferences. Despite this, and despite being unable to choose a full 30 slots on any of the 3 weeks, I have been reasonably satisfied with the timeslots I got. This week's is the worst, when I have to start a game at 11pm on a Saturday night before flying interstate on Sunday. Obviously that's still not too bad.
The point is, that although we can't mathematically guarantee good overlaps, from a statistical standpoint reasonable overlaps are likely if both players supply 30 choices.
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| « Last Edit: Nov 30th, 2005, 8:03pm by 99of9 » |
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Ryan_Cable
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Re: 2006 World Championship
« Reply #64 on: Dec 1st, 2005, 4:49am » |
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1 Fritzlein 2273
2 99of9 . 2142
3 Ryan_Cable 2105
4 PMertens 2045
5 Adanac . 2039
6 robinson 2026
7 Belbo . 2009
8 omar . 1860
9 BlackKnight 1818
10 megamau 1651
11 jdb . 1584
12 MrBrain 1369
The following tables list: probability of as of the start of round 3, additive change from the start of tournament, multiplicative change from the start of tournament, additive change from last round, multiplicative change from last round.
With me @ 1800, omar +100, and 99of9 +75:
StDev 0:
Fritzlein 0.56991 -0.00097 0.99830 0.01135 1.020320109
99of9 . 0.33813 0.00967 1.02944 0.00905 1.027500912
Ryan_Cable 0.00043 -0.00003 0.93478 -0.00008 0.843137255
PMertens 0.02101 -0.01315 0.61504 -0.01322 0.613789074
Adanac . 0.03317 0.00663 1.24981 0.00322 1.107512521
robinson 0.02614 0.00341 1.15002 -0.00266 0.907638889
Belbo . 0.00857 -0.00365 0.70130 -0.00521 0.62191582
omar . 0.0025 -0.00170 0.59523 -0.00239 0.511247444
BlackKnight 0.00014 0.00004 1.4 . 0.00007 2
megamau 0 . -0.00001 0 . 0 . #DIV/0!
jdb . 0 . 0 . #DIV/0! 0 . #DIV/0!
MrBrain 0 . 0 . #DIV/0! 0 . #DIV/0!
StDev 50:
Fritzlein 0.54837 0.00020 1.00036 0.01639 1.030809429
99of9 . 0.34058 0.01571 1.04835 0.01408 1.043124043
Ryan_Cable 0.00067 -0.00020 0.77011 -0.00010 0.87012987
PMertens 0.02399 -0.01680 0.58813 -0.02031 0.541534989
Adanac . 0.04047 0.00710 1.21276 0.00398 1.109070978
robinson 0.03170 0.00369 1.13173 -0.00200 0.940652819
Belbo . 0.01064 -0.00602 0.63865 -0.00755 0.584936778
omar . 0.00346 -0.00322 0.51796 -0.00425 0.448767834
BlackKnight 0.00011 -0.00012 0.47826 -0.00002 0.846153846
megamau 0.00001 0.00001 #DIV/0! 0.00001 #DIV/0!
jdb . 0 . 0 . #DIV/0! 0 . #DIV/0!
MrBrain 0 . 0 . #DIV/0! 0 . #DIV/0!
StDev 100:
Fritzlein 0.50146 0.00873 1.01771 0.02694 1.05677316
99of9 . 0.33179 0.01770 1.05635 0.01768 1.056286014
Ryan_Cable 0.00141 -0.00071 0.66509 -0.00090 0.61038961
PMertens 0.03304 -0.02319 0.58758 -0.02568 0.5626703
Adanac . 0.05913 0.01178 1.24878 0.00706 1.13558671
robinson 0.04811 0.00527 1.12301 -0.00189 0.9622
Belbo . 0.01707 -0.01129 0.60190 -0.01406 0.548345647
omar . 0.00738 -0.00624 0.54185 -0.00856 0.462986198
BlackKnight 0.00058 -0.00050 0.53703 0 . 1
megamau 0.00003 0 . 1 . 0.00003 #DIV/0!
jdb . 0 . -0.00002 0 . 0 . #DIV/0!
MrBrain 0 . 0 . #DIV/0! 0 . #DIV/0!
I have cut the number of simulation results that I am listing to save myself the effort of formatting them. The 99of9 +50 sims reduced 99of9 ~5% and increased Fritzlein ~4%. The 99of9 +100 sims increased 99of9 ~6% and reduced Fritzlein ~4%.
On the whole, little has changed since the beginning of the tournament, since there have been no titan upsets. Fritzlein has come back to where he was at the beginning of the tournament, and 99of9 has gained (additively) ~1.3%. The undefeated sub-titans (Adanac and robinson) have gained (additively) 1% total. Most of these points have come at the expense of PMertens, with Belbo and omar making smaller losses. The eliminated players never had much chance of winning. One slightly strange result is that robinson lost (multiplicatively) ~7%, since the last round, despite winning; I think this comes from the lack of a titan upset.
The probability density of upsets for round 2:
0 0.31449
1 0.41612
2 0.21079
3 0.051735
4 0.0064502
5 0.00038965
6 0.0000090180
Expectation 1.02072
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MrBrain
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Re: 2006 World Championship
« Reply #65 on: Dec 1st, 2005, 7:31am » |
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on Nov 30th, 2005, 6:21pm, 99of9 wrote:
5) This is in the middle of the night. I am available, but would otherwise be sleeping!
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Other than Tuesday, this was my situation for this week.
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Fritzlein
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Re: 2006 World Championship
« Reply #66 on: Dec 1st, 2005, 8:04am » |
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Ryan,
Thanks for continuing to post the results of your simulations. To me it feels like beating a tough player like Omar must have boosted my chances significantly, because I now have to survive one round fewer to win it all, but I guess the sims have most of my losses coming to 99of9. Further reinforcement for me to root for Adanac on Saturday. 
on Dec 1st, 2005, 4:49am, Ryan_Cable wrote:The probability density of upsets for round 2:
0 0.31449
1 0.41612
2 0.21079
3 0.051735
4 0.0064502
5 0.00038965
6 0.0000090180
Expectation 1.02072 |
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My predictions add up to a cumulative 1.65 upsets. It's good to know that I am on the conservative side of things, as I would wish to be when my standing is fairly good.
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Fritzlein
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Re: 2006 World Championship
« Reply #67 on: Dec 2nd, 2005, 7:10pm » |
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We're two games into the third round, just over half the games have been played, and already the tournament has become a changing of the guard. All three of the medalists from the 2004 championship are out, and two of the three medalists from the 2005 championship are out. There are so many strong Arimaa players around nowadays, "the field" in this tournament is a veritable minefield. I'm quaking in my boots, myself.
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Ryan_Cable
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Re: 2006 World Championship
« Reply #68 on: Dec 2nd, 2005, 10:29pm » |
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on Dec 1st, 2005, 8:04am, Fritzlein wrote:| To me it feels like beating a tough player like Omar must have boosted my chances significantly, because I now have to survive one round fewer to win it all, but I guess the sims have most of my losses coming to 99of9. |
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Not really, you actually had a 0.3123 chance of loosing to one of MrBrain, omar + 100, and robinson. Likewise, 99of9 +75 had a 0.4395 chance of loosing once to acheron, Belbo, and Adanac. Thus, the chance of you two meeting in round 4 was 0.3855 (assuming the possibility of different pairings comes out in the wash). Even at round 3, there is only a 0.5928 chance that round 4 will have a battle of the titans.
However, a titan is expected to have no trouble beating the bottom of the looser bracket for rounds 2 and 3, and very little trouble in round 4. Thus, the titans are both expected to survive to round 5 (>90%), which creates a focusing effect, where round 5 is likely to have one undefeated titan and one wounded titan. It doesn’t really matter how you get there; the wounded titan takes a big hit in his chances, but all of this must go to the undefeated titan. Only if a titan is eliminated or (less unlikely) a non-titan takes the undefeated spot does the field make significant gains, and still not big gains.
Consider, if Adanac beats 99of9, then beats Fritzlein to take the undefeated spot. Then let 99of9 be eliminated, and let Fritzlein survive to face an undefeated Adanac for the title. At this point, Fritzlein would have a 0.6299 chance of winning the tournament!!! If it were 99of9 instead of Fritzlein, he would have a 0.5415 chance of winning! Yet if it were the titan that was undefeated, Adanac would only have a 0.04258 or 0.06976 chance to win. The square of a small fraction is a very small fraction, but the square of a big fraction is still a fairly large fraction.
The main advantage of any field is numerical superiority, which was strong enough to make FIDE call Alexander Khalifman and Rustam Kasimdzhanov world champions. But in a 16 player FDE, the focusing effect means that the most intense portion of the tournament comes after the field is likely to have been cut down to just 3 non-titans.
on Dec 2nd, 2005, 7:10pm, Fritzlein wrote:| There are so many strong Arimaa players around nowadays, "the field" in this tournament is a veritable minefield. I'm quaking in my boots, myself. |
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And the minefield is covered by nearly impenetrable interlocking fires. 
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| « Last Edit: Dec 2nd, 2005, 10:30pm by Ryan_Cable » |
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Fritzlein
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Re: 2006 World Championship
« Reply #69 on: Dec 3rd, 2005, 9:30am » |
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on Dec 2nd, 2005, 10:29pm, Ryan_Cable wrote:
However, a titan is expected to have no trouble beating the bottom of the looser bracket for rounds 2 and 3, and very little trouble in round 4. Thus, the titans are both expected to survive to round 5 (>90%), which creates a focusing effect, where round 5 is likely to have one undefeated titan and one wounded titan. |
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OK, I understand better now what your sims are saying.
Quote:| Consider, if Adanac beats 99of9, then beats Fritzlein to take the undefeated spot. Then let 99of9 be eliminated, and let Fritzlein survive to face an undefeated Adanac for the title. At this point, Fritzlein would have a 0.6299 chance of winning the tournament!!! If it were 99of9 instead of Fritzlein, he would have a 0.5415 chance of winning! Yet if it were the titan that was undefeated, Adanac would only have a 0.04258 or 0.06976 chance to win. The square of a small fraction is a very small fraction, but the square of a big fraction is still a fairly large fraction. |
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Part of my issue is that I never quite believed the size of the fractions the pre-tournament ratings indicated. Apparently you are simulating me at 80% to beat each of Robinson and Adanac, for about a 64% chance of getting the round 5 bye, but I sure won't be betting that much on myself in the prediction contest.
Quote:| And the minefield is covered by nearly impenetrable interlocking fires. |
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Well, we have one "titan" upset now, which means that 99of9 and I can no longer face each other until round seven. While I'm thrilled about that, it is hardly far-fetched that there will be three more such upsets, giving someone else the crown without the "titans" ever having played each other. I'll give myself a slight boost for surviving two rounds, but at best I'm still only 55% to win it all, with say 15% to 99of9 and 30% to the field. Only if I can beat Robinson this round will I boost myself to 65% at the field's expense.
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99of9
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Re: 2006 World Championship
« Reply #70 on: Dec 3rd, 2005, 3:58pm » |
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on Dec 3rd, 2005, 9:30am, Fritzlein wrote:| at best I'm still only 55% to win it all, with say 15% to 99of9 and 30% to the field. Only if I can beat Robinson this round will I boost myself to 65% at the field's expense. |
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Adanac should no longer be called field, since he clearly has a greater % than me now.
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Fritzlein
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Re: 2006 World Championship
« Reply #71 on: Dec 11th, 2005, 4:17pm » |
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Well, so much for the "titans". The "field" is looking pretty strong right now. Indeed, the changing of the guard is complete. The two undefeated players are folks who just learned this year! Well done, Robinson and Adanac. Have fun duking it out for the fifth round bye!
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Janzert
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Re: 2006 World Championship
« Reply #72 on: Dec 11th, 2005, 8:41pm » |
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I logged into the game room several times over the last couple days and never figured out that the games had been successfully rescheduled yet.  Oh well, on to the next round.
Janzert
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99of9
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Re: 2006 World Championship
« Reply #73 on: Dec 11th, 2005, 9:33pm » |
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We missed some crackers Janzert. I was asleep for most of the games. Better observation next time .
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Ryan_Cable
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Re: 2006 World Championship
« Reply #74 on: Dec 12th, 2005, 1:28am » |
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1 Fritzlein 2273
2 99of9 . 2142
3 Ryan_Cable 2105
4 PMertens 2045
5 Adanac . 2039
6 robinson 2026
7 BlackKnight 1818
8 megamau 1651
Well, that was an exciting two weeks. Seeds 5 and 6 facing off for the undefeated spot is certainly unexpected. For the first time, we have 4 players with >10% chances, and we are likely to still have 3 players with >10% chances after round 4. While the upsets have quite sensibly received the most comments, it should also be noted that all seeds 6 and above are still in the tournament, and except for me all of them are very likely to still be there after round 4. The remainder of the tournament is going to be about as exciting as is mathematically possible.
The following tables list: probability of winning as of the start of round 4, additive change from the start of tournament, multiplicative change from the start of tournament, additive change from last round, multiplicative change from last round.
With me @ 1800 and 99of9 +75:
StDev 0:
Fritzlein 0.43808 -0.13280 0.76737 -0.13183 0.76868
99of9 . 0.27875 -0.04971 0.84865 -0.05938 0.82438
Ryan_Cable 0.00032 -0.00014 0.69565 -0.00011 0.74418
PMertens 0.03909 0.00493 1.14432 0.01808 1.86054
Adanac . 0.13046 0.10392 4.91559 0.09729 3.93307
robinson 0.11298 0.09025 4.97052 0.08684 4.32211
BlackKnight 0.00032 0.00022 3.2 . 0.00018 2.28571
megamau 0 . -0.00001 0 . 0 . #DIV/0!
StDev 50:
Fritzlein 0.42400 -0.12417 0.77348 -0.12437 0.77320
99of9 . 0.27234 -0.05253 0.83830 -0.06824 0.79963
Ryan_Cable 0.00069 -0.00018 0.79310 0.00002 1.02985
PMertens 0.04312 0.00233 1.05712 0.01913 1.79741
Adanac . 0.13791 0.10454 4.13275 0.09744 3.40770
robinson 0.12139 0.09338 4.33380 0.08969 3.82933
BlackKnight 0.00054 0.00031 2.34782 0.00043 4.90909
megamau 0.00001 0.00001 #DIV/0! 0 . 1
StDev 100:
Fritzlein 0.39195 -0.10078 0.79546 -0.10951 0.78161
99of9 . 0.26192 -0.05217 0.83390 -0.06987 0.78941
Ryan_Cable 0.00191 -0.00021 0.90094 0.00050 1.35460
PMertens 0.05339 -0.00284 0.94949 0.02035 1.61592
Adanac . 0.15233 0.10498 3.21710 0.09320 2.57618
robinson 0.13689 0.09405 3.19537 0.08878 2.84535
BlackKnight 0.00143 0.00035 1.32407 0.00085 2.46551
megamau 0.00018 0.00015 6 . 0.00015 6
Adanac and robinson have increased their chances ~4 fold. Yet the titans only take a ~20% multiplicative decrease, with the biggest loss coming from Fritzlein, since he was most likely to be the final undefeated. With my standard ratings estimates, the titans retain a 70% chance of winning the tournament.
Raising Adanac to 2114 and robinson to 2131 (their end of round 3 ratings):
StDev 0:
Fritzlein 0.35121
99of9 . 0.21372
Ryan_Cable 0.00024
PMertens 0.02635
Adanac . 0.18289
robinson 0.22547
BlackKnight 0.00012
megamau 0
StDev 50:
Fritzlein 0.34206
99of9 . 0.21006
Ryan_Cable 0.00033
PMertens 0.02954
Adanac . 0.18740
robinson 0.23036
BlackKnight 0.00023
megamau 0.00002
StDev 100:
Fritzlein 0.31712
99of9 . 0.20547
Ryan_Cable 0.00120
PMertens 0.03793
Adanac . 0.20133
robinson 0.23595
BlackKnight 0.00098
megamau 0.00002
Now, robinson is actually >1% above 99of9, and Adanac is only 3% behind 99of9. Fritzlein is still 12% above anyone else. The titans are reduced to a 55% chance of winning.
Personally, I think the odds are about Fritzlein 41%, 99of9 22%, robinson 18%, Adanac 17%, PMertens 2%. 99of9's main advantage in the tournament was his being slightly better than Fritzlein on the board, and it is now unlikely for there to be even one Battle of the Titans game. The only thing he has left going for him (aside from just the skill that shows up in his rating) is the fact that he is PMertens's Achilles' heal. On the other hand, Fritzlein's strength comes from the fact that he is expected to be able to beat any non-titan 75% of the time, so the loss he suffered is somewhat offset by the loss 99of9 suffered. I think Adanac and robinson are probably about half way between their pre and post ratings at ~2085. PMertens will have to make good on his Houdini nickname to win the tournament.
Here is the probability density of upsets for round 4 with me @ 1800 and 99of9 +75:
0 0.36856
1 0.47928
2 0.13997
3 0.011930
4 0.00023389
Expectation 0.79597
With Adanac @ 2114 and robinson @ 2131, counting robinson as the favorite:
0 0.37265
1 0.47672
2 0.13859
3 0.011794
4 0.00023110
Expectation 0.79022
Nearly all of the upset probability comes form the Adanac v. robinson tossup. There is a 0.88399 chance that both titans win, and a 0.71056 chance that there are no loser bracket upsets. Thus it is reasonable to go ahead and run some simulations for round 5 assuming no loser bracket upsets.
robinson undefeated with 99of9 +75, Adanac @ 2114 and robinson @ 2131:
StDev 0:
Fritzlein 0.32348
99of9 . 0.23887
PMertens 0.03140
Adanac . 0.05803
robinson 0.34822
StDev 50:
Fritzlein 0.31855
99of9 . 0.23578
PMertens 0.03561
Adanac . 0.06382
robinson 0.34624
StDev 100:
Fritzlein 0.30697
99of9 . 0.23197
PMertens 0.04739
Adanac . 0.07918
robinson 0.33449
Adanac undefeated with 99of9 +75, Adanac @ 2114 and robinson @ 2131:
StDev 0:
Fritzlein 0.37196
99of9 . 0.20700
PMertens 0.02903
Adanac . 0.31068
robinson 0.08133
StDev 50:
Fritzlein 0.35945
99of9 . 0.20987
PMertens 0.03391
Adanac . 0.30954
robinson 0.08723
StDev 100:
Fritzlein 0.34039
99of9 . 0.21272
PMertens 0.04540
Adanac . 0.30125
robinson 0.10024
There are two likely pairings for round 5: Adanac v. Fritzlein and 99of9 v. PMertens or PMertens v. Fritzlein and 99of9 v. robinson. (There is also a 20% chance I could be in PMertens’s spot.) 99of9 would prefer the former; PMertens and probably Fritzlein would prefer the latter. Collectively the titans are probably better off with robinson taking the undefeated spot, though the simulations say otherwise.
There have been 6 upsets in 22 games. 4 of these were in games with a 70%+ favorite. My standard rating estimates expected 4.14676 and 2.22388 respectively. They gave a 0.21056 chance of >=6 upsets and a 0.167998149 chance of >=4 big upsets. Part of this comes from the fact that I did not bother to adjust the ratings of MrBrain and jdb, despite the fact they were known to be underrated before the tournament began. The rest comes from the fact that Adanac and robinson have outperformed their pre-tournament ratings. It is difficult to say how much of this outperformance is due to luck and how much is due to underrating. Still, I think my rating estimates have performed reasonably well so far.
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