Comments on: Choosing up sides http://bit-player.org/2007/choosing-up-sides An amateur's outlook on computation and mathematics. Fri, 29 Aug 2008 07:44:15 +0000 http://wordpress.org/?v=1.5.2 Comment on Choosing up sides by: mark http://bit-player.org/2007/choosing-up-sides#comment-1692 Mon, 02 Jun 2008 13:38:59 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1692 Hi there- Isn't this the Morse-Thue sequence? ABBA-BAAB-BAAB-ABBA... Check out "Fair Division" works by Brams and Taylor. I think they prove some cool things about Morse-Thue and its relation to the playground picking. Hi there-

Isn’t this the Morse-Thue sequence?

ABBA-BAAB-BAAB-ABBA…

Check out “Fair Division” works by Brams and Taylor. I think they prove some cool things about Morse-Thue and its relation to the playground picking.

]]> Comment on Choosing up sides by: Seb http://bit-player.org/2007/choosing-up-sides#comment-1449 Fri, 20 Apr 2007 17:40:23 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1449 Barry, I made some new experiments and I think you are right. Team B seems to be favored by the algorithm ABBABA. Thank you. You should continue trusting your "strength" analysis. Barry, I made some new experiments and I think you are right. Team B seems to be favored by the algorithm ABBABA. Thank you. You should continue trusting your “strength” analysis.

]]> Comment on Choosing up sides by: Barry Cipra http://bit-player.org/2007/choosing-up-sides#comment-1448 Tue, 17 Apr 2007 18:45:18 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1448 Seb's algorithm is equivalent to ABBABA, which works best when n is a multiple of 6, say n=6m. My "strength" analysis says this algorithm produces a bias of m in favor of team B, which suggests the discrepancy should also favor team B (but not as much as team A is favored by the ABAB algorithm, which has bias 3m). So if Seb's program shows his algorithm still favors team A, then either there's something amiss in his program, or my "strength" analysis is worthless. The latter possibility seems the more likely.... Seb’s algorithm is equivalent to ABBABA, which works best when n is a multiple of 6, say n=6m. My “strength” analysis says this algorithm produces a bias of m in favor of team B, which suggests the discrepancy should also favor team B (but not as much as team A is favored by the ABAB algorithm, which has bias 3m). So if Seb’s program shows his algorithm still favors team A, then either there’s something amiss in his program, or my “strength” analysis is worthless. The latter possibility seems the more likely….

]]> Comment on Choosing up sides by: Seb http://bit-player.org/2007/choosing-up-sides#comment-1443 Sat, 07 Apr 2007 12:13:03 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1443 The following algorithm (call it ABCABC-BABA) uses a third bully in the following way: Do ABCABC until all the players have been distributed in three groups. Then, take the players in C and distribute them using the BABA algorithm. According to my program this one is better than the ABAB but not as good as the ABBA. May be some kind of 'buffer' like this can improve the result but the team with the first choice has still a big advantage. The following algorithm (call it ABCABC-BABA) uses a third bully in the following way:
Do ABCABC until all the players have been distributed in three groups. Then, take the players in C and distribute them using the BABA algorithm. According to my program this one is better than the ABAB but not as good as the ABBA. May be some kind of ‘buffer’ like this can improve the result but the team with the first choice has still a big advantage.

]]> Comment on Choosing up sides by: Barry Cipra http://bit-player.org/2007/choosing-up-sides#comment-1438 Wed, 28 Feb 2007 18:35:38 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1438 Brian, I missed your use of "two bullies." My "bully's algorithm" has just one bully, who picks all the best players, leaving the scrawny kids for the other team -- that's what 11110000 (or AAAABBBB, in your notation) does with 8 kids. My point, if I have one, is that you should be able to get some sense of how big the expected discrepancy will be by measuring what I call the "strength" of each teams and taking the difference (which I call the "bias"). My "bully's algorithm" is quadratically biased (it grows with the square of the number of players), the classic ABAB algorithm is linearly biased, and your ABBA and my ABA'B' algorithms are essentially unbiased (i.e., either 0 or 1). Brian, I missed your use of “two bullies.” My “bully’s algorithm” has just one bully, who picks all the best players, leaving the scrawny kids for the other team — that’s what 11110000 (or AAAABBBB, in your notation) does with 8 kids.

My point, if I have one, is that you should be able to get some sense of how big the expected discrepancy will be by measuring what I call the “strength” of each teams and taking the difference (which I call the “bias”). My “bully’s algorithm” is quadratically biased (it grows with the square of the number of players), the classic ABAB algorithm is linearly biased, and your ABBA and my ABA’B’ algorithms are essentially unbiased (i.e., either 0 or 1).

]]> Comment on Choosing up sides by: brian http://bit-player.org/2007/choosing-up-sides#comment-1437 Tue, 27 Feb 2007 02:58:48 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1437 Hi guys. I'm getting confused by my own blog. What is the "bully's algorithm"? I had used the term "two bullies algorithm" (or words to that effect) to mean essentially the same thing as the ABAB algorithm. Barry, you have something else in mind? Or maybe I've just dropped a stitch. I've been a bit out of it, in more than one way -- now including being out of town. Because I may not be able to attend to this forum from moment to moment over the next two weeks, I've set comments to post automatically, without my intervention. So please carry on without me. (But the spam filter remains in place.) --Brian Hi guys. I’m getting confused by my own blog. What is the “bully’s algorithm”? I had used the term “two bullies algorithm” (or words to that effect) to mean essentially the same thing as the ABAB algorithm. Barry, you have something else in mind? Or maybe I’ve just dropped a stitch. I’ve been a bit out of it, in more than one way — now including being out of town. Because I may not be able to attend to this forum from moment to moment over the next two weeks, I’ve set comments to post automatically, without my intervention. So please carry on without me. (But the spam filter remains in place.)

–Brian

]]> Comment on Choosing up sides by: Barry Cipra http://bit-player.org/2007/choosing-up-sides#comment-1436 Mon, 26 Feb 2007 20:17:13 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1436 An equivalent way to run my algorithm is to alternate picks with A going first and B second for roughly half the rounds, then switching to B first, A second for the rest (thus giving B two consecutive picks midway through). E.g, for n=8, the picks go ABABBABA, while for n=10 they go ABABABBABA. In general, any "playground algorithm" amounts to assigning to each (even) n, an n-bit binary number (letting A=1, B=0) with equal numbers of 1's and 0's. There is, for example, the bully's algorithm, which gives numbers like 11110000. The expected discrepancy for the bully's algorithm is obviously going to be quite large! Let's define the "value" of a player to be the number of players on the other team that he or she is better than -- i.e., the "value" of each 1 is the number of 0's to its right, and likewise the "value of each 0 is the number of 1's to its right. The "strength" of a team is the sum of its players' values. The "bias" of an algorithm is the difference between the two teams' strengths. It is a function of n. For the bully's algorithm, the bias is a whopping k^2, where k=n/2. For the classic ABAB algorithm, it's simply k. And for Brian's ABBA and my ABA'B' approach, it's 0 when k is even and 1 when k is odd. (It's a nice exercise to show that you cannot get 0 bias when k is odd, no matter what you do.) My guess is that, at least asymptotically, the expected discrepancy for any playground algorithm is purely a function of its bias divided by k. I'm not sure what I mean by "purely." I'll leave that for someone else to figure out.... An equivalent way to run my algorithm is to alternate picks with A going first and B second for roughly half the rounds, then switching to B first, A second for the rest (thus giving B two consecutive picks midway through). E.g, for n=8, the picks go ABABBABA, while for n=10 they go ABABABBABA.

In general, any “playground algorithm” amounts to assigning to each (even) n, an n-bit binary number (letting A=1, B=0) with equal numbers of 1’s and 0’s. There is, for example, the bully’s algorithm, which gives numbers like 11110000. The expected discrepancy for the bully’s algorithm is obviously going to be quite large!

Let’s define the “value” of a player to be the number of players on the other team that he or she is better than — i.e., the “value” of each 1 is the number of 0’s to its right, and likewise the “value of each 0 is the number of 1’s to its right. The “strength” of a team is the sum of its players’ values. The “bias” of an algorithm is the difference between the two teams’ strengths. It is a function of n. For the bully’s algorithm, the bias is a whopping k^2, where k=n/2. For the classic ABAB algorithm, it’s simply k. And for Brian’s ABBA and my ABA’B’ approach, it’s 0 when k is even and 1 when k is odd. (It’s a nice exercise to show that you cannot get 0 bias when k is odd, no matter what you do.)

My guess is that, at least asymptotically, the expected discrepancy for any playground algorithm is purely a function of its bias divided by k. I’m not sure what I mean by “purely.” I’ll leave that for someone else to figure out….

]]> Comment on Choosing up sides by: John http://bit-player.org/2007/choosing-up-sides#comment-1435 Mon, 26 Feb 2007 14:42:57 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1435 I'm not sure Barry's algorithm would be as acceptable in a playground situation. Is it more insulting to be picked last or to be nominated as the worst player while there are still many more to be selected :) I’m not sure Barry’s algorithm would be as acceptable in a playground situation. Is it more insulting to be picked last or to be nominated as the worst player while there are still many more to be selected :)

]]> Comment on Choosing up sides by: Barry Cipra http://bit-player.org/2007/choosing-up-sides#comment-1434 Thu, 22 Feb 2007 17:47:17 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1434 I wonder how the following algorithm, which I'll call the ABA'B' method, does: Captain A picks first, followed by Captain B. They obviously pick the top two players. Then Captain B makes A's next choice -- picking the *worst* player available -- followed by A doing the same for B's team. Then repeat. In effect this pairs players from the outside in (assuming the number of players n is a multiple of 4 -- otherwise the innermost pair go to opposite teams, with the better of those two going to team A). My guess is, it'll be virtually indistinguishable, expected-discrepancy-wise, from the ABBA algorithm. I wonder how the following algorithm, which I’ll call the ABA’B’ method, does: Captain A picks first, followed by Captain B. They obviously pick the top two players. Then Captain B makes A’s next choice — picking the *worst* player available — followed by A doing the same for B’s team. Then repeat. In effect this pairs players from the outside in (assuming the number of players n is a multiple of 4 — otherwise the innermost pair go to opposite teams, with the better of those two going to team A).

My guess is, it’ll be virtually indistinguishable, expected-discrepancy-wise, from the ABBA algorithm.

]]> Comment on Choosing up sides by: brian http://bit-player.org/2007/choosing-up-sides#comment-1433 Thu, 22 Feb 2007 13:43:54 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1433 Suresh: The abilities are known to both captains, and furthermore the captains agree in all their assessments. I assume that the captains always choose the highest-ranked player from those remaining. Suresh: The abilities are known to both captains, and furthermore the captains agree in all their assessments. I assume that the captains always choose the highest-ranked player from those remaining.

]]> Comment on Choosing up sides by: Suresh http://bit-player.org/2007/choosing-up-sides#comment-1432 Wed, 21 Feb 2007 22:46:56 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1432 I'm not clear on the rules of the game here. are the player "qualities" hidden from the choosers, or does A always choose the top ranked player, B chooses the next two, and so on? If not, do A and B pick randomly from all players at each step ? I’m not clear on the rules of the game here. are the player “qualities” hidden from the choosers, or does A always choose the top ranked player, B chooses the next two, and so on? If not, do A and B pick randomly from all players at each step ?

]]> Comment on Choosing up sides by: clem http://bit-player.org/2007/choosing-up-sides#comment-1430 Tue, 20 Feb 2007 22:08:35 +0000 http://bit-player.org/2007/choosing-up-sides#comment-1430 The ABBA method is basically a two player version of the "snake draft" customarily used to pick teams in fantasy football leagues. Team drafting order is selected either randomly or based on the previous season's final standings. After the first round of picks, the last picker picks first in the second round, thus picking twice in a row, and the sequence is reversed. So in a four team league it would go ABCDDCBAABCD.... The ABBA method is basically a two player version of the “snake draft” customarily used to pick teams in fantasy football leagues. Team drafting order is selected either randomly or based on the previous season’s final standings. After the first round of picks, the last picker picks first in the second round, thus picking twice in a row, and the sequence is reversed. So in a four team league it would go ABCDDCBAABCD….

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