From alexk@princeton.edu Wed Mar 13 19:35:39 2002 Date: Wed, 13 Mar 2002 15:18:47 -0500 From: Alex Kontorovich To: John F Nash Subject: success(2)! Dear Prof. Nash, I corrected the "less_than" function, and found another bug in his code that makes it impossible for the nucleolus to be anything but equal division. Having fixed that, I'm attaching the output file, called TestResults.txt. I first ran this program for precision set to 10 decimal places, which found a nucleolus very similar to yours, but the excess of the solution was still worse (in the 11th decimal place) than your solution nk2. Then I ran it for precision set to 15 decimal places, and seem to have found a nucleolus, lets call it nk0, which is slightly better than nk2 (in the 15th decimal place -- I have printed 16 places in this file). The data in the attached file is presented as follows: nk0 solution for nucleolus excess values for nk0, in lexicographical order nk1 solution for nucleolus excess values for nk1, in lexicographical order nk2 solution for nucleolus excess values for nk2, in lexicographical order results from the better function, saying the ordering of nucleoli is 0>2>1 (where ">" means is better). and finally, the vector, v of coalition values, from your file work.study.n14.n29. Notice how close (but not same) the nk0 solution is to the nk2 solution. I think the best thing to do now is try those examples which you were saying had trouble converging, and see if this program manages to do them. I have a midterm tomorrow, and am giving a talk for Prof. Sinai's seminar on Fri, so maybe we can meet over the weekend or the following Monday? In case you wanted to attend my talk (I know you are very busy, and don't expect to see you there, but feel I should invite you anyway), it will be at 4:30 on Fri in his office, 708 Fine. I look forward to hearing from you. Take care. -Alex -- _______________________ Alex V. Kontorovich '02 13 Prospect Street Tower Club Princeton, NJ 08540 (609)986-9905 http://www.princeton.edu/~alexk [ Part 2: "Attached Text" ] The Prenucleolus0: x1 = 0.2773689273689298 x2 = 0.2440355940355903 x3 = 0.2587782587782601 x4 = 0.2198172198172200 Excesses0: 0.0000000000000000 ( ) -0.0000000000000002 ( 1 2 3 4 ) -0.0769600769600757 ( 1 2 ) -0.0769600769600771 ( 4 ) -0.0769600769600783 ( 3 ) -0.0773689273689236 ( 2 ) -0.0773689273689298 ( 1 ) -0.1002164502164467 ( 2 4 ) -0.1221861471861498 ( 1 4 ) -0.1278138528138504 ( 2 3 ) -0.1361471861471899 ( 1 3 ) -0.1452621452621468 ( 3 4 ) -0.1801827801827802 ( 1 2 3 ) -0.2078884078884068 ( 1 2 4 ) -0.2680856180856159 ( 2 3 4 ) -0.2744829244829285 ( 1 3 4 ) The Prenucleolus1: x1 = 0.2970719095719096 x2 = 0.2764610389610390 x3 = 0.2410113035113035 x4 = 0.1854557479557480 Excesses1: 0.0000000000000000 ( ) 0.0000000000000000 ( 1 2 3 4 ) -0.0425986050986051 ( 4 ) -0.0591931216931217 ( 3 ) -0.0931337181337181 ( 3 4 ) -0.0970719095719096 ( 1 ) -0.0982804232804233 ( 2 4 ) -0.1075276575276576 ( 1 4 ) -0.1097943722943723 ( 2 ) -0.1290885040885041 ( 1 2 ) -0.1380832130832131 ( 1 3 ) -0.1424723424723424 ( 2 3 ) -0.2145442520442521 ( 1 2 3 ) -0.2256553631553632 ( 1 2 4 ) -0.2420574795574796 ( 1 3 4 ) -0.2483826358826358 ( 2 3 4 ) The Prenucleolus2: x1 = 0.2769600769600770 x2 = 0.2444444444444444 x3 = 0.2587782587782588 x4 = 0.2198172198172198 Excesses2: 0.0000000000000000 ( ) 0.0000000000000000 ( 1 2 3 4 ) -0.0769600769600770 ( 3 ) -0.0769600769600770 ( 1 ) -0.0769600769600770 ( 4 ) -0.0769600769600770 ( 1 2 ) -0.0777777777777778 ( 2 ) -0.1006253006253006 ( 2 4 ) -0.1217772967772968 ( 1 4 ) -0.1282227032227032 ( 2 3 ) -0.1357383357383357 ( 1 3 ) -0.1452621452621453 ( 3 4 ) -0.1801827801827802 ( 1 2 3 ) -0.2078884078884079 ( 1 2 4 ) -0.2684944684944685 ( 2 3 4 ) -0.2740740740740741 ( 1 3 4 ) 0 bet 1? = 1, 1 bet 0? = 0 0 bet 2? = 1, 2 bet 0? = 0 1 bet 2? = 0, 2 bet 1? = 1 Vector v: 0.0000000 ( ) 0.1428571 ( 4 ) 0.1818182 ( 3 ) 0.3333333 ( 3 4 ) 0.1666667 ( 2 ) 0.3636364 ( 2 4 ) 0.3750000 ( 2 3 ) 0.4545455 ( 2 3 4 ) 0.2000000 ( 1 ) 0.3750000 ( 1 4 ) 0.4000000 ( 1 3 ) 0.4814815 ( 1 3 4 ) 0.4444444 ( 1 2 ) 0.5333333 ( 1 2 4 ) 0.6000000 ( 1 2 3 ) 1.0000000 ( 1 2 3 4 )