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0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 307 "" 0 "" {TEXT 366 0 "" }}{PARA 307 "" 0 "" {TEXT 264 0 "" }{TEXT 256 9 "LIEN TRAN" }}{PARA 307 "" 0 "" {TEXT 259 39 "Math 107B - Spring 2002 - Final Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 308 "" 0 "" {TEXT 257 12 "CAKE-CUTTING" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 307 "" 0 "" {TEXT 258 9 "OBJECTIVE" }} {PARA 307 "" 0 "" {TEXT 260 422 "The ability to fairly divide a cake a mong n-people is a great concern of individuals with siblings. Appare ntly, the problem of getting a fair share of cakes is more important t han it sounds. This topic has moved beyond the kitchen and was introd uced into the world of mathematics by Hugo Steinhaus on September 17, \+ 1947. In this final, I will explore the methods in which a cake can b e divided equally among n-people. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 309 "" 0 "" {TEXT 262 7 "SOURCES" }}{PARA 310 "" 0 "" {TEXT 263 20 "1) www.mathworld.com" }}{PARA 310 "" 0 "" {TEXT 261 3 "2) " } {TEXT 266 42 "Cake-Cutting Algoriths: Be Fair If You Can" }{TEXT 265 1 " " }}{PARA 310 "" 0 "" {TEXT 306 39 " by Jack Robertson and Will iam Webb " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 311 "" 0 "" {TEXT 303 14 "WITH HELP FROM" }}{PARA 311 "" 0 "" {TEXT 304 41 "Professor JO HN WAVRIK and STEFAN ERICKSON" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 311 "" 0 "" {TEXT 367 4 "NOTE" }}{PARA 311 "" 0 "" {TEXT 368 70 "This final contains two bit-map images that have been created by me. \+ " }}{PARA 311 "" 0 "" {TEXT 397 56 "So, please use a computer that su pports bit-map image. " }}{PARA 311 "" 0 "" {TEXT 369 102 "The comput ers in the lab do not suppport bit-map images and overwrite the file l eaving out the images." }{TEXT -1 0 "" }}}{EXCHG {PARA 311 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 311 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 313 "" 0 "" {TEXT 371 101 "In this section, I am \+ exploring several possible algorithms that one can use to equally divi de cakes." }}}{SECT 1 {PARA 313 "" 0 "" {TEXT 329 46 "POSSIBLE ALGORIT HMS FOR FAIR DIVISION OF CAKES" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 311 "" 0 "" {TEXT 267 33 "CUT & CHOOSE algorithm (2 people)" } {TEXT 294 0 "" }}{PARA 310 "" 0 "" {TEXT 271 28 "Step1 - person1 cut t he cake" }}{PARA 310 "" 0 "" {TEXT 272 30 "Step2 - person2 chooses fir st " }}{PARA 310 "" 0 "" {TEXT 381 53 " and the remaining piece goes to person1" }}{PARA 320 "" 0 "" {TEXT 273 10 "OUTCOMES (" }{TEXT 389 31 "This algorithm uses (n-1) cuts)" }{TEXT -1 0 "" }} {PARA 310 "" 0 "" {TEXT 295 330 "Person2 will always benefit despite t he fact that the cake is or isn't divided equally. If person1 slips a nd cut one piece of the cake bigger than the other, person2 will benef it from choosing the bigger piece since person2 gets to choose first. \+ Therefore, it is in the benefit of person1 to cut the cake as equally as he can. " }}{PARA 310 "" 0 "" {TEXT 376 0 "" }{TEXT 274 0 "" }} {PARA 310 "" 0 "" {TEXT 268 35 "DOESN'T CUT IT algorithm (3 people)" } }{PARA 310 "" 0 "" {TEXT 275 52 "Step1 - person1 cuts a cake into two \+ pieces so that " }}{PARA 310 "" 0 "" {TEXT 377 63 " piece A=1/3 of the cake and pieceB=2/3 of the cake" }}{PARA 310 "" 0 "" {TEXT 276 103 "Step2 - person2 cuts pieceB into two equal pieces so th at\n pieceB1=1/3=pieceB2 of the cake" }{TEXT 380 0 "" }} {PARA 310 "" 0 "" {TEXT 277 73 "Step3 - person3 chooses one of three p ieces (pieceA, pieceB1, or pieceB2)" }}{PARA 310 "" 0 "" {TEXT 372 39 " then person1 chooses next" }}{PARA 310 "" 0 "" {TEXT 286 48 " and the last piece goes to person2" }}{PARA 310 "" 0 "" {TEXT 293 9 "OUTCOMES " }{TEXT 391 1 "(" }{TEXT 390 31 "This a lgorithm uses (n-1) cuts)" }}{PARA 311 "" 0 "" {TEXT 298 882 "Person3 \+ has the greatest advantage followed by person1 and person2, respective ly. If person1 does not cut the cake into two pieces that are 1/3 and 2/3 of the cake, person3 will benefit from choosing the piece that is >1/3 and leaving the piece that is < 2/3 to be divided among person1 \+ and peson2. In this case, person1 and person2 will not be able to get their fair share of the cake. In the second case, if person1 has cut the cake correctly but person2 does not divide the piece that is =2/3 of the cake into two equal pieces that are 1/3 each, then person3 wil l choose the piece that is >1/3, person1 will choose the piece that =1 /3, and person2 is left with a piece that is <1/3. Therefore, person2 will always get the disadvantages unless both person1 and person2 cut s the cake evenly into three pieces that are =1/3 of the cake so that \+ all three persons will be happy. " }{TEXT -1 0 "" }{TEXT 378 1 " " } {TEXT 330 17 " " }}{PARA 310 "" 0 "" {TEXT 278 0 "" }} {PARA 310 "" 0 "" {TEXT 331 0 "" }{TEXT 269 33 "MOVING KNIFE algorithm (n people)" }}{PARA 310 "" 0 "" {TEXT 279 60 "Step1 - person1 place t he knife on the left end of the cake " }}{PARA 310 "" 0 "" {TEXT 287 59 " and continuously moves the knife to the right " }} {PARA 310 "" 0 "" {TEXT 288 67 " - any other player can say \+ \"STOP\" when he believes that " }}{PARA 310 "" 0 "" {TEXT 289 69 " \+ the portion to the left of the knife is 1/n of the cake " }} {PARA 310 "" 0 "" {TEXT 290 67 " and that piece is given t o the player who said \"STOP\" " }}{PARA 310 "" 0 "" {TEXT 291 34 " \+ and the player is out" }}{PARA 310 "" 0 "" {TEXT 280 50 " \+ - if more than one players say \"STOP,\" " }}{PARA 310 "" 0 " " {TEXT 292 60 " then the piece can go to any one of the p layers" }}{PARA 310 "" 0 "" {TEXT 281 15 "Step2 to (n-1) " }}{PARA 310 "" 0 "" {TEXT 305 69 " - Step1 is repeated for the remai ning portion of the cake " }}{PARA 310 "" 0 "" {TEXT 296 38 " \+ by the remaining players" }}{PARA 310 "" 0 "" {TEXT 282 67 "Stepn - Since there is only one player left, he gets the last piece" }} {PARA 320 "" 0 "" {TEXT 283 0 "" }{TEXT 375 8 "OUTCOMES" }{TEXT -1 2 " (" }{TEXT 392 31 "This algorithm uses (n-1) cuts)" }}{PARA 310 "" 0 " " {TEXT 297 929 "This is not a good algorithm for fair division of cak es because there are three disadvantages in using this algorithm to d ivide cakes. One, the players may take advantage of the method and sa y \"STOP\" when the left portion is more than 1/n of the cake. Second , players who said \"STOP\" at the same time may argue over who gets t he piece. Third, it is unfair for the last player because he is assig ned the last piece of the cake despite its size. The last player will be the one to get the smallest portion because, when it comes to the \+ last two players, the cutter has to keep the knife moving until the ot her player say \"STOP.\" However, the other player does not have to s ay stop until he wants to and this gives him the authority to say stop when the remaining portion to the right of the cake is really small. \+ Therefore, this algorithm does not have any \"check and balance\" sys tem to maintain equality among the players. " }}{PARA 310 "" 0 "" {TEXT 379 0 "" }{TEXT 284 0 "" }}{PARA 310 "" 0 "" {TEXT 270 29 "TRIMM ING algorithm (n people)" }}{PARA 310 "" 0 "" {TEXT 285 45 "Step1 - pe rson1 cuts a piece, 1/n of the cake" }}{PARA 310 "" 0 "" {TEXT 332 45 "Step2 - this piece is passed successively to " }}{PARA 310 "" 0 "" {TEXT 333 46 " person2, person3,...,person(n-1)" }}{PARA 310 "" 0 "" {TEXT 334 66 " - anyone who thinks the piece is \+ bigger than 1/n trims " }}{PARA 310 "" 0 "" {TEXT 335 57 " \+ so that it is 1/n according to the trimmer." }}{PARA 310 "" 0 "" {TEXT 336 42 "Step3 - person(n) takes the trimmed piece " }}{PARA 310 "" 0 "" {TEXT 337 57 " if he thinks it is at least 1/n of the cake" }}{PARA 310 "" 0 "" {TEXT 338 65 " - otherwise th e trimmed piece goes to the last trimmer" }}{PARA 310 "" 0 "" {TEXT 339 64 "Step4 - repeast Step1 to Step3 on the remaining portion of cak e " }}{PARA 310 "" 0 "" {TEXT 340 38 " with n replaced by (n-1)" }}{PARA 310 "" 0 "" {TEXT 341 47 " - repeat until on ly one player left " }}{PARA 320 "" 0 "" {TEXT 299 8 "OUTCOMES" } {TEXT -1 2 " (" }{TEXT 393 36 "This algorithm uses indefinite cuts)" } }{PARA 310 "" 0 "" {TEXT 300 351 "This is not a sufficient way of fair ly dividing cakes into n-pieces. By trimming the cake, the players wi ll end up with small/little pieces or even crumbs as their share of th e cake. Also, this method can take a really long time to finish becau se the players may not agree to each of the portions until these porti ons become little pieces of crumbs." }{TEXT -1 0 "" }{TEXT 343 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 314 "" 0 " " {TEXT 373 379 "Now that we know how to divide cakes equally among n- people. Lets explore the ways in which we could divide cakes equally \+ among two families with unequal number of members using the Ramsey Par titions. The purpose of the Ramsey Partition is to divide a cake into k1/n and k2/n where n=k1+k2. In other words, the cake has to be divi ded into a ratio of k1:k2 for familyA:familyB." }}}{SECT 1 {PARA 314 " " 0 "" {TEXT 344 17 "RAMSEY PARTITIONS" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 307 59 "***************************************************** ******" }}{PARA 315 "" 0 "" {TEXT 345 27 "RAMSEY PARTITION algorithm\n " }{TEXT 308 476 "Step1 - With the ratio k1:k2 given, \n c hoose any Ramsey partition for the ratio k1:k2\nStep2 - Ask either pla yer to cut the cake \n in the ratios given in the Ramsey p artition\nStep3 - Ask the non-cutter to indicate which pieces \n \+ are acceptable\nStep4 - Assign the non-cutter pieces chosen fro m \n the acceptable ones whose values sum to \n \+ the non-cutter's portion and the cutter \n takes the rem aining pieces" }}{PARA 315 "" 0 "" {TEXT 374 60 "********************* **************************************\n" }{TEXT 310 43 "Below is an E XAMPLE of a RAMSEY PARTITION:\n" }{TEXT 311 289 "Let familyA=k1=8 and \+ familyB=k2=5 thus, the entire cake is 8+5=13. For the table below, th e letters (A or B) to the right of a number indicates that this piece \+ has been chosen by family A or B. Also, the number inside the \{ \} i ndicates pieces that are selected from the previous steps. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 315 "" 0 "" {TEXT 382 83 "Starting wit h a cake of 13/13 , the Ramsey Partition below requires 5-cuts/steps. \+ " }}{PARA 315 "" 0 "" {TEXT 383 565 "(1) FamilyB makes the first cut to divide the cake into two pieces: 8/13 and 5/13. Then familyA gets to pick. \nIf familyA thinks that one of the pieces is worth at leas t 8/13 of the cake, then familyA will pick that piece and the 5/13 goe s to familyB. The division process stops here because, now, familyA h as \{8/13\} and familyB has \{5/13\}.\nHowever, if familyA does not th ink that any of the pieces is at least 8/13 of the cake, then familyA \+ will take the piece that is at least 5/13. Now, familyA has \{5/13\} \+ and the piece that is 8/13 is divided in step(2). " }}{PARA 315 "" 0 "" {TEXT 384 575 "(2) FamilyA cuts the remaining piece into two piece s: 5/13 and 3/13. Then familyB gets to pick. \nIf familyB thinks tha t one of the pieces is worth at least 5/13 of the cake, then familyB w ill pick that piece and the 3/13 goes to familyA. The division proces s stops here because, now, familyA has \{5/13, 3/13\} and familyB has \+ \{5/13\}.\nHowever, if familyB does not think that any of the pieces i s at least 5/13 of the cake, then familyB will take the piece that is \+ at least 3/13. Now, familyA has \{5/13\}, familyB has \{3/13\} and th e piece that is 5/13 is divided in step(3)." }}{PARA 315 "" 0 "" {TEXT 309 586 "(3) FamilyB cuts the remaining piece into two pieces: \+ 3/13 and 2/13. Then familyA gets to pick. \nIf familyA thinks that o ne of the pieces is worth at least 3/13 of the cake, then familyA will pick that piece and the 2/13 goes to familyB. The division process s tops here because, now, familyA has \{5/13, 3/13\} and familyB has \{3 /13, 2/13\}.\nHowever, if familyA does not think that any of the piece s is at least 3/13 of the cake, then familA will take the piece that i s at least 2/13. Now, familyA has \{5/13, 2/13\}, familyB has \{3/13 \} and the piece that is 3/13 is divided in step(4)." }}{PARA 315 "" 0 "" {TEXT 386 599 "(4) FamilyA cuts the remaining piece into two pie ces: 2/13 and 1/13. Then familyB gets to pick. \nIf familyB thinks t hat one of the pieces is worth at least 2/13 of the cake, then familyB will pick that piece and the 1/13 goes to familyB. The division proc ess stops here because, now, familyA has \{5/13, 2/13, 1/13\} and fami lyB has \{3/13, 2/13\}.\nHowever, if familyB does not think that any o f the pieces is at least 2/13 of the cake, then familyB will take the \+ piece that is at least 1/13. Now, familyA has \{5/13, 2/13\}, familyB has \{3/13, 1/13\} and the piece that is 2/13 is divided in step(5). " }{TEXT -1 0 "" }}{PARA 315 "" 0 "" {TEXT 387 287 "(5) FamilyB cuts \+ the remaining piece into two pieces: 1/13 and 1/13. Then familyA gets to pick. \nIf familyA thinks that one of the pieces is worth at leas t 1/13 of the cake and the remaining piece goes to familyB. 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