2987:, then Alice is entitled to a value of at least 6. To give Alice her due share in a connected piece, we must give her either the three leftmost slices or the three rightmost slices. In both cases George receives a piece with a value of only 1, which is less than his due share of 2. To achieve a WPR division in this case, we must give George his due share in the center of the cake, where his value is relatively large, but then Alice will get two disconnected pieces.
3817:
Note that there exists a connected division in which the ratios between the values of the partners are 3:1 â give Alice the two leftmost slices and 8/11 of the third slice (value 4+16/11=60/11) and give George the remaining 3/11 and the rightmost slice (value 1+9/11=20/11). However, this partition is
2594:
We can do better by letting George cut in the ratio 6:4. If Alice chooses the 4, then the ratio becomes 3:3 and we can use cut-and-choose immediately. If Alice chooses the 6, then the ratio becomes 3:1. Alice cuts in ratio 2:2, George chooses the 2, and we need one more step of cut-and-choose. All in
616:
Case 2: There is a subset of the unmarked pieces whose sum is 8. E.g., if George marks the 3-piece and only one 1-piece. Then, this subset is given to Alice and the remainder is given to George. Alice now has exactly 8 and George has given up a sum of less than 8, so he has at least 5/13.
2590:
Cut-near-halves may need at least four cuts: first, George cuts in the ratio 5:5, and Alice gets 5. Then, Alice cuts in the ratio 3:2; suppose George chooses the 2. Then, George cuts in the ratio 2:1; suppose Alice chooses the 1. Finally, they do cut-and-choose on the
612:
Case 1: There is a subset of the marked pieces whose sum is 5. E.g., if George marks the 3-piece and the three 1-pieces. Then, this subset is given to George and the remainder is given to Alice. George now has at least 5/13 and Alice has exactly 8/13.
3508:
The exact number of required cuts remains an open question. The simplest open case is when there are 3 agents and the weights are 1/7, 2/7, 4/7. It is not known if the number of required cuts is 4 (as in the lower bound) or 5 (as in the upper bound).
3328:
624:
possible cases. I.e, every subset of 5:3:2:1:1:1, EITHER has a subset that sums to 5, OR its complement has a subset that sums to 8. Hence, the above algorithm always finds a WPR allocation with the given ratios. The number of cuts used is only 5.
2772:
Besides the number of required queries, it is also interesting to minimize the number of required cuts, so that the division is not too much fractioned. The
Shishido-Zeng algorithms yield a fair division with at most
2677:
presented an algorithm for dividing a multi-dimensional cake among any number of agents with any entitlements (including irrational entitlements), in a finite number of queries. Their algorithm requires
3530:
proved the existence of proportional cake-cutting with different entitlements even when agents' preferences are described by non-additive preference relations, as long as they satisfy certain axioms.
20:
problem, the partners often have different entitlements. For example, the resource may belong to two shareholders such that Alice holds 8/13 and George holds 5/13. This leads to the criterion of
2452:
2670:
2739:
1190:
585:
3411:
3059:
1677:
1782:
1732:
1433:
1101:
256:
308:
2529:
2288:
3365:
2581:
2039:
3204:
1043:
2985:
2849:
2810:
995:
and the partition is 5:3:2:1:1:1, which is a Ramsey partition. Moreover, this is the shortest Ramsey partition in this case, so it allows us to use a small number of cuts.
897:
824:
798:
754:
1588:
144:
2336:
2149:
2109:
1878:
1527:
1266:
1215:
993:
960:
2887:
2069:
1463:
1814:
1344:
924:
871:
694:
667:
502:
475:
422:
355:
96:
49:
2993:
shows that, if the cake is circular (i.e. the two endpoints are identified) then a connected WPR division for two people is always possible; this follows from the
2210:
2181:
1930:
1904:
1840:
1553:
1292:
1241:
531:
1486:
1312:
1121:
844:
714:
533:
cuts, which may be very large. For example, if Alice is entitled to 8/13 and George is entitled to 5/13, then 13-1=12 cuts are needed in the initial partition.
448:
395:
375:
328:
194:
164:
69:
609:
Now there are two "good" cases - cases in which we can use these pieces to attain a weighted-proportional division respecting the different entitlements:
594:
Suppose a cake has to be divided among Alice and George, Alice is entitled to 8/13 and George is entitled to 5/13. The cake can be divided as follows.
1958:
In both cases, the remaining piece is smaller and the ratio is smaller. Eventually, the ratio becomes 1:1 and the remaining cake can be divided using
3563:
998:
Ramsey partitions always exist. Moreover, there is always a unique shortest Ramsey partition. It can be found using a simple variant of the
3911:
3760:
3972:
2994:
2742:
2753:
When the entitlements are not rational numbers, methods based on cloning cannot be used since the denominator is infinite.
2764:
The algorithm of Cseh and
Fleiner can also be adapted to work with irrational entitlements in a finite number of queries.
2897:=2. A cake made of four consecutive regions has to be divided between Alice and George, whose valuations are as follows:
2341:
2745:; thus it is more efficient than agent-cloning and cut-near-halves. They prove that this runtime complexity is optimal.
2609:
2681:
3473:
539:
1940:
Suppose again that Alice is entitled to 8/13 and George is entitled to 5/13. The cake can be divided as follows.
102:
3370:
1953:
Alice chooses the 6. Then, Alice is entitled to 2 more, and the remaining piece should be divided in ratio 5:2.
1950:
Alice chooses the 7. Then, Alice is entitled to 1 more, and the remaining piece should be divided in ratio 5:1.
1126:
3004:
2295:
At least one of the pieces is worth for Alice at least the value declared by George; give this piece to Alice.
1593:
1737:
1683:
Once a minimal Ramsey partition is found, it can be used to find a WPR division respecting the entitlements.
3595:
McAvaney, Kevin; Robertson, Jack; Webb, William (1992). "Ramsey partitions of integers and fair divisions".
1689:
1355:
1048:
199:
3690:
Shishido, Harunor; Zeng, Dao-Zhi (1999). "Mark-Choose-Cut
Algorithms For Fair And Strongly Fair Division".
261:
3521:
2457:
3323:{\displaystyle {\text{Cuts}}(n+1)\leq \max _{1\leq k\leq n-1}(1+{\text{Cuts}}(k+1)+{\text{Cuts}}(n-k+1))}
2215:
1947:
Alice chooses one of the pieces, which is worth for her at least its declared value. Consider two cases:
3967:
3828:
Segal-Halevi, Erel (2018-03-14). "Cake-Cutting with
Different Entitlements: How Many Cuts are Needed?".
3333:
2538:
1973:
1009:
2957:
2815:
2776:
999:
3189:, and the procedure proceeds recursively. This leads to the following recurrence relation (where
876:
803:
3874:
3855:
3837:
3800:
3766:
3738:
3715:
3669:
3643:
3612:
763:
719:
1561:
108:
3907:
3756:
3707:
3661:
3569:
3559:
1967:
17:
2586:
The cut-near-halves algorithm is not always optimal. For example, suppose the ratio is 7:3.
2301:
2114:
2074:
1845:
1494:
965:
932:
3943:
3899:
3847:
3792:
3748:
3699:
3653:
3604:
2857:
2048:
1442:
3873:
Crew, Logan; Narayanan, Bhargav; Spirkl, Sophie (2019-09-16). "Disproportionate division".
1787:
1317:
902:
849:
672:
645:
480:
453:
400:
333:
74:
27:
3733:
Cseh, Ăgnes; Fleiner, TamĂĄs (2018), "The
Complexity of Cake Cutting with Unequal Shares",
3577:
2189:
2160:
1909:
1883:
1819:
1532:
1271:
1220:
510:
1246:
1195:
2998:
1959:
1471:
1297:
1106:
829:
699:
433:
380:
360:
313:
179:
149:
54:
3461:
as before. If again one of these sets is empty, then we know that all agents value (0,
2761:, that can also handle irrational entitlements, but with an unbounded number of cuts.
3961:
3719:
3673:
636:
3804:
3770:
3616:
2599:
It is an open question how to find the best initial cut for each entitlement ratio.
3859:
3903:
3752:
605:
George marks the pieces that have for him at least the value mentioned by Alice.
3851:
3947:
3928:
3796:
3703:
3105:
Order the agents in increasing order of their mark, breaking ties arbitrarily.
3711:
3665:
377:. Find a proportional cake allocation among them. Finally, give each partner
3783:
Brams, S. J.; Jones, M. A.; Klamler, C. (2007). "Proportional pie-cutting".
3581:
2583:
cuts, so it is always more efficient than the Ramsey-partitions algorithm.
3185:
such that the total weight in each set is exactly 1/2. The cake is cut at
3898:. Theory and Decision Library. Vol. 23. Springer. pp. 259â271.
2212:
is odd then George cuts the cake to two pieces whose valuation-ratio is
3608:
1784:
is the golden ratio. In most cases, this number is better than making
176:
Suppose all the weights are rational numbers, with common denominator
3657:
430:
show a simpler procedure for two partners: Alice cuts the cake into
3879:
3842:
3743:
3648:
3631:
2183:
is even then George cuts the cake to two pieces equal in his eyes;
357:
clones with the same value-measure. The total number of clones is
3573:
477:
most valuable pieces in his eyes, and Alice takes the remaining
1880:
cuts are needed, since the only Ramsey partition of the pair
3894:
Zeng, Dao-Zhi (2000). "Approximate Envy-Free
Procedures".
2954:
Note that the total cake value is 8 for both partners. If
2298:
Suppose the piece taken by Alice is the piece with value
3426:
is empty is analogous). This implies that the weight of
1350:
This lemma leads to the following recursive algorithm.
168:
Several algorithms can be used to find a WPR division.
3149:) at least 1/2, and their total weight is at most 1/2;
3142:) at least 1/2, and their total weight is at most 1/2;
598:
Alice cuts the cake to 6 pieces with valuation-ratios
3896:
Game
Practice: Contributions from Applied Game Theory
3737:, Springer International Publishing, pp. 19â30,
3373:
3336:
3207:
3007:
3001:, it is possible to get a WPR division using at most
2960:
2860:
2818:
2779:
2684:
2612:
2541:
2460:
2344:
2304:
2218:
2192:
2163:
2117:
2077:
2051:
1976:
1912:
1886:
1848:
1822:
1790:
1740:
1692:
1596:
1564:
1535:
1497:
1474:
1445:
1358:
1320:
1300:
1274:
1249:
1223:
1198:
1129:
1109:
1051:
1012:
968:
935:
905:
879:
852:
832:
806:
766:
722:
702:
675:
648:
542:
513:
483:
456:
436:
403:
383:
363:
336:
316:
264:
202:
182:
152:
111:
77:
57:
30:
3632:"The Complexity of Cake Cutting with Unequal Shares"
620:
It is possible to prove that the good cases are the
2447:{\displaystyle c\in \{(a+b-1)/2,(a+b)/2,(a+b+1)/2)}
3405:
3359:
3322:
3119:The first agent that was not added to P is called
3053:
2979:
2881:
2843:
2804:
2733:
2664:
2575:
2523:
2446:
2330:
2282:
2204:
2175:
2143:
2103:
2063:
2033:
1924:
1898:
1872:
1834:
1808:
1776:
1726:
1671:
1582:
1547:
1521:
1480:
1457:
1427:
1338:
1306:
1286:
1260:
1235:
1209:
1184:
1115:
1095:
1037:
987:
954:
918:
891:
865:
838:
818:
792:
748:
708:
688:
661:
579:
525:
496:
469:
442:
416:
389:
369:
349:
322:
302:
250:
188:
158:
138:
90:
63:
43:
3830:Journal of Mathematical Analysis and Applications
3112:. Stop just before the total weight of agents in
2812:cuts, and a strongly-fair division with at most
2665:{\displaystyle n(n-1)\lceil \log _{2}(D)\rceil .}
2154:Ask George to cut the cake to near-halves, i.e.:
1944:George cuts the cake to two pieces in ratios 7:6.
1002:. The algorithm is based on the following lemma:
3818:not WPR since no partner receives his due share.
3232:
2997:. By recursively applying this theorem to find
2734:{\displaystyle 2(n-1)\lceil \log _{2}(D)\rceil }
2555:
1706:
3108:Add the agents in the above order into a set
450:pieces equal in her eyes; George selects the
8:
3488:) for which one of the agents, which is not
2728:
2703:
2656:
2631:
2535:The cut-near-halves algorithm needs at most
2351:
580:{\displaystyle D\lceil \log _{2}(D)\rceil .}
571:
546:
3556:Cake-Cutting Algorithms: Be Fair If You Can
3065:is a power of 2, and a similar number when
1314:is a minimal Ramsey partition for the pair
1268:is a minimal Ramsey partition for the pair
3630:Cseh, Ăgnes; Fleiner, TamĂĄs (2020-06-01).
3434:) at most 1/2. In this case, we find some
2606:agents; the number of required queries is
631:generalize this idea using the concept of
3878:
3841:
3742:
3647:
3430:is at least 1/2, and all agents value (0,
3406:{\displaystyle {\text{Cuts}}(n)\leq 3n-4}
3374:
3372:
3337:
3335:
3291:
3268:
3235:
3208:
3206:
3024:
3006:
2965:
2959:
2859:
2829:
2817:
2790:
2778:
2710:
2683:
2638:
2611:
2546:
2540:
2459:
2433:
2401:
2375:
2343:
2308:
2303:
2272:
2240:
2217:
2191:
2162:
2121:
2116:
2081:
2076:
2050:
1975:
1911:
1885:
1847:
1821:
1789:
1766:
1756:
1739:
1697:
1691:
1595:
1563:
1534:
1496:
1473:
1444:
1357:
1319:
1299:
1273:
1248:
1222:
1197:
1185:{\displaystyle P'=(b,p_{1},\dots ,p_{n})}
1173:
1154:
1128:
1108:
1084:
1065:
1050:
1017:
1011:
973:
967:
940:
934:
910:
904:
878:
857:
851:
831:
805:
784:
771:
765:
740:
727:
721:
701:
680:
674:
653:
647:
553:
541:
512:
488:
482:
461:
455:
435:
408:
402:
382:
362:
341:
335:
315:
288:
269:
263:
240:
234:
213:
207:
201:
181:
151:
128:
116:
110:
82:
76:
56:
35:
29:
3054:{\displaystyle 2n\cdot (\log _{2}n-1)+2}
2899:
1672:{\displaystyle FindMinimalRamsey(a,b-a)}
98:of the resource by their own valuation.
3927:Dall'Aglio, M.; MacCheroni, F. (2009).
3558:. Natick, Massachusetts: A. K. Peters.
3554:Robertson, Jack; Webb, William (1998).
3539:
3520:presented an algorithm for approximate
3453:, and try to partition the agents into
1777:{\displaystyle \phi =(1+{\sqrt {5}})/2}
883:
3549:
3547:
3545:
3543:
2749:Algorithms for irrational entitlements
1727:{\displaystyle \log _{\phi }\min(a,b)}
1428:{\displaystyle FindMinimalRamsey(a,b)}
1096:{\displaystyle P=(p_{1},\dots ,p_{n})}
251:{\displaystyle p_{1}/D,\dots ,p_{n}/D}
303:{\displaystyle p_{1}+\cdots +p_{n}=D}
7:
3785:International Journal of Game Theory
3685:
3683:
2602:The algorithm can be generalized to
2524:{\displaystyle CutNearHalves(b,a-c)}
51:that sum up to 1, and every partner
3197:, not including the clone of agent
2595:all, at most three cuts are needed.
2283:{\displaystyle (a+b-1)/2:(a+b+1)/2}
1966:The general idea is similar to the
696:are positive integers, a partition
71:should receive at least a fraction
3360:{\displaystyle {\text{Cuts}}(2)=2}
3079:-4 using the following protocol:
2576:{\displaystyle \log _{2}\min(a,b)}
2034:{\displaystyle CutNearHalves(a,b)}
536:The number of required queries is
14:
3504:and recurse as in the first case.
24:(WPR): there are several weights
3330:. Adding the initial condition
105:setting, the weights are equal:
3500:. Then, we can cut the cake at
2071:. Suppose Alice is entitled to
1038:{\displaystyle p_{1}<a<b}
826:, either there is a sublist of
507:This simple procedure requires
3692:Group Decision and Negotiation
3636:ACM Transactions on Algorithms
3385:
3379:
3348:
3342:
3317:
3314:
3296:
3285:
3273:
3259:
3225:
3213:
3123:, and the set of agents after
3075:improved this upper bound to 3
3042:
3017:
2980:{\displaystyle w_{A}\geq 0.75}
2876:
2864:
2844:{\displaystyle 4\cdot 3^{n-2}}
2805:{\displaystyle 2\cdot 3^{n-2}}
2757:presented an algorithm called
2725:
2719:
2700:
2688:
2653:
2647:
2628:
2616:
2570:
2558:
2518:
2500:
2441:
2430:
2412:
2398:
2386:
2372:
2354:
2325:
2313:
2269:
2251:
2237:
2219:
2138:
2126:
2098:
2086:
2028:
2016:
1763:
1747:
1721:
1709:
1686:The algorithm needs at least
1666:
1648:
1422:
1410:
1179:
1141:
1090:
1058:
568:
562:
1:
3524:with different entitlements.
3904:10.1007/978-1-4615-4627-6_17
2854:In the worst case, at least
892:{\displaystyle P\setminus L}
819:{\displaystyle L\subseteq P}
629:McAvaney, Robertson and Webb
101:In contrast, in the simpler
3936:Games and Economic Behavior
3753:10.1007/978-3-319-99660-8_3
3367:yields the claimed number
3193:is the number of agents in
3073:Crew, Narayanan and Spirkle
2045:Order the inputs such that
1439:Order the inputs such that
873:, or there is a sublist of
793:{\displaystyle k_{1},k_{2}}
749:{\displaystyle k_{1}+k_{2}}
3989:
3852:10.1016/j.jmaa.2019.123382
3474:intermediate value theorem
2995:StromquistâWoodall theorem
2743:RobertsonâWebb query model
2111:and George is entitled to
3948:10.1016/j.geb.2008.04.006
3797:10.1007/s00182-007-0108-z
3528:Dall'Aglio and MacCheroni
3173:are nonempty, then agent
1583:{\displaystyle b-a\neq 1}
139:{\displaystyle w_{i}=1/n}
103:proportional cake-cutting
3476:, there must be a value
3422:is empty (the case that
3418:The harder case is that
2891:Brams, Jones and Klamler
2889:cuts might be required.
22:weighted proportionality
3973:Fair division protocols
3735:Algorithmic Game Theory
3704:10.1023/a:1008620404353
3156:values both (0,x) and (
3145:All agents in Q value (
2768:Number of required cuts
2331:{\displaystyle c/(a+b)}
2144:{\displaystyle a/(a+b)}
2104:{\displaystyle b/(a+b)}
1873:{\displaystyle b=a+b-1}
1522:{\displaystyle a=b-a=1}
988:{\displaystyle k_{2}=5}
955:{\displaystyle k_{1}=8}
3522:envy-free cake-cutting
3496:) exactly the same as
3407:
3361:
3324:
3055:
2981:
2883:
2882:{\displaystyle 2(n-1)}
2845:
2806:
2735:
2666:
2577:
2525:
2448:
2332:
2284:
2206:
2177:
2145:
2105:
2065:
2064:{\displaystyle a<b}
2035:
1926:
1900:
1874:
1836:
1810:
1778:
1728:
1673:
1584:
1549:
1523:
1482:
1459:
1458:{\displaystyle a<b}
1429:
1340:
1308:
1288:
1262:
1237:
1211:
1186:
1117:
1097:
1039:
989:
956:
929:In the example above,
920:
893:
867:
840:
820:
800:, if for any sub-list
794:
750:
710:
690:
663:
581:
527:
498:
471:
444:
418:
391:
371:
351:
324:
304:
252:
190:
160:
140:
92:
65:
45:
3408:
3362:
3325:
3056:
2982:
2884:
2846:
2807:
2736:
2667:
2578:
2526:
2449:
2333:
2285:
2207:
2178:
2146:
2106:
2066:
2036:
1927:
1901:
1875:
1837:
1811:
1809:{\displaystyle a+b-1}
1779:
1729:
1674:
1585:
1550:
1524:
1483:
1460:
1430:
1341:
1339:{\displaystyle a,b-a}
1309:
1289:
1263:
1238:
1212:
1187:
1118:
1098:
1040:
990:
957:
921:
919:{\displaystyle k_{2}}
894:
868:
866:{\displaystyle k_{1}}
841:
821:
795:
751:
711:
691:
689:{\displaystyle k_{2}}
664:
662:{\displaystyle k_{1}}
582:
528:
499:
497:{\displaystyle p_{A}}
472:
470:{\displaystyle p_{G}}
445:
419:
417:{\displaystyle p_{i}}
392:
372:
352:
350:{\displaystyle p_{i}}
325:
305:
253:
196:. So the weights are
191:
161:
141:
93:
91:{\displaystyle w_{i}}
66:
46:
44:{\displaystyle w_{i}}
3472:. Therefore, by the
3371:
3334:
3205:
3005:
2958:
2893:show an example for
2858:
2816:
2777:
2682:
2610:
2539:
2458:
2342:
2302:
2216:
2190:
2161:
2115:
2075:
2049:
1974:
1910:
1884:
1846:
1820:
1788:
1738:
1690:
1594:
1562:
1533:
1495:
1472:
1443:
1356:
1318:
1298:
1272:
1247:
1221:
1196:
1127:
1107:
1049:
1010:
966:
933:
903:
877:
850:
830:
804:
764:
720:
700:
673:
646:
540:
511:
481:
454:
434:
401:
381:
361:
334:
314:
262:
200:
180:
150:
109:
75:
55:
28:
2205:{\displaystyle a+b}
2176:{\displaystyle a+b}
1925:{\displaystyle b+1}
1906:is a sequence with
1899:{\displaystyle b,1}
1835:{\displaystyle a=1}
1548:{\displaystyle 1,1}
1287:{\displaystyle a,b}
1236:{\displaystyle a+b}
1000:Euclidean algorithm
526:{\displaystyle D-1}
3609:10.1007/bf01204722
3403:
3357:
3320:
3258:
3051:
2977:
2879:
2841:
2802:
2731:
2662:
2573:
2521:
2444:
2328:
2280:
2202:
2173:
2141:
2101:
2061:
2031:
1922:
1896:
1870:
1832:
1806:
1774:
1724:
1669:
1580:
1545:
1519:
1478:
1455:
1425:
1336:
1304:
1284:
1261:{\displaystyle P'}
1258:
1233:
1217:is a partition of
1210:{\displaystyle P'}
1207:
1182:
1113:
1103:is a partition of
1093:
1035:
985:
952:
916:
889:
863:
836:
816:
790:
746:
706:
686:
659:
577:
523:
494:
467:
440:
428:Robertson and Webb
414:
397:the pieces of his
387:
367:
347:
320:
310:. For each player
300:
248:
186:
156:
136:
88:
61:
41:
3642:(3): 29:1â29:21.
3565:978-1-56881-076-8
3377:
3340:
3294:
3271:
3231:
3211:
3177:is split between
3160:) at exactly 1/2.
2952:
2951:
2755:Shishido and Zeng
1968:Even-Paz protocol
1761:
1481:{\displaystyle a}
1307:{\displaystyle P}
1116:{\displaystyle b}
839:{\displaystyle L}
709:{\displaystyle P}
635:(named after the
633:Ramsey partitions
590:Ramsey partitions
443:{\displaystyle D}
390:{\displaystyle i}
370:{\displaystyle D}
323:{\displaystyle i}
189:{\displaystyle D}
159:{\displaystyle i}
64:{\displaystyle i}
18:fair cake-cutting
3980:
3952:
3951:
3933:
3929:"Disputed lands"
3924:
3918:
3917:
3891:
3885:
3884:
3882:
3870:
3864:
3863:
3845:
3825:
3819:
3815:
3809:
3808:
3780:
3774:
3773:
3746:
3730:
3724:
3723:
3687:
3678:
3677:
3651:
3627:
3621:
3620:
3592:
3586:
3585:
3551:
3438:such that agent
3412:
3410:
3409:
3404:
3378:
3375:
3366:
3364:
3363:
3358:
3341:
3338:
3329:
3327:
3326:
3321:
3295:
3292:
3272:
3269:
3257:
3212:
3209:
3060:
3058:
3057:
3052:
3029:
3028:
2986:
2984:
2983:
2978:
2970:
2969:
2900:
2888:
2886:
2885:
2880:
2850:
2848:
2847:
2842:
2840:
2839:
2811:
2809:
2808:
2803:
2801:
2800:
2740:
2738:
2737:
2732:
2715:
2714:
2675:Cseh and Fleiner
2671:
2669:
2668:
2663:
2643:
2642:
2582:
2580:
2579:
2574:
2551:
2550:
2530:
2528:
2527:
2522:
2453:
2451:
2450:
2445:
2437:
2405:
2379:
2337:
2335:
2334:
2329:
2312:
2289:
2287:
2286:
2281:
2276:
2244:
2211:
2209:
2208:
2203:
2182:
2180:
2179:
2174:
2150:
2148:
2147:
2142:
2125:
2110:
2108:
2107:
2102:
2085:
2070:
2068:
2067:
2062:
2040:
2038:
2037:
2032:
1931:
1929:
1928:
1923:
1905:
1903:
1902:
1897:
1879:
1877:
1876:
1871:
1841:
1839:
1838:
1833:
1815:
1813:
1812:
1807:
1783:
1781:
1780:
1775:
1770:
1762:
1757:
1733:
1731:
1730:
1725:
1702:
1701:
1678:
1676:
1675:
1670:
1589:
1587:
1586:
1581:
1554:
1552:
1551:
1546:
1528:
1526:
1525:
1520:
1487:
1485:
1484:
1479:
1464:
1462:
1461:
1456:
1434:
1432:
1431:
1426:
1345:
1343:
1342:
1337:
1313:
1311:
1310:
1305:
1293:
1291:
1290:
1285:
1267:
1265:
1264:
1259:
1257:
1242:
1240:
1239:
1234:
1216:
1214:
1213:
1208:
1206:
1191:
1189:
1188:
1183:
1178:
1177:
1159:
1158:
1137:
1122:
1120:
1119:
1114:
1102:
1100:
1099:
1094:
1089:
1088:
1070:
1069:
1044:
1042:
1041:
1036:
1022:
1021:
994:
992:
991:
986:
978:
977:
961:
959:
958:
953:
945:
944:
925:
923:
922:
917:
915:
914:
898:
896:
895:
890:
872:
870:
869:
864:
862:
861:
845:
843:
842:
837:
825:
823:
822:
817:
799:
797:
796:
791:
789:
788:
776:
775:
758:Ramsey partition
755:
753:
752:
747:
745:
744:
732:
731:
715:
713:
712:
707:
695:
693:
692:
687:
685:
684:
668:
666:
665:
660:
658:
657:
586:
584:
583:
578:
558:
557:
532:
530:
529:
524:
503:
501:
500:
495:
493:
492:
476:
474:
473:
468:
466:
465:
449:
447:
446:
441:
423:
421:
420:
415:
413:
412:
396:
394:
393:
388:
376:
374:
373:
368:
356:
354:
353:
348:
346:
345:
329:
327:
326:
321:
309:
307:
306:
301:
293:
292:
274:
273:
257:
255:
254:
249:
244:
239:
238:
217:
212:
211:
195:
193:
192:
187:
165:
163:
162:
157:
145:
143:
142:
137:
132:
121:
120:
97:
95:
94:
89:
87:
86:
70:
68:
67:
62:
50:
48:
47:
42:
40:
39:
3988:
3987:
3983:
3982:
3981:
3979:
3978:
3977:
3958:
3957:
3956:
3955:
3931:
3926:
3925:
3921:
3914:
3893:
3892:
3888:
3872:
3871:
3867:
3827:
3826:
3822:
3816:
3812:
3782:
3781:
3777:
3763:
3732:
3731:
3727:
3689:
3688:
3681:
3658:10.1145/3380742
3629:
3628:
3624:
3594:
3593:
3589:
3566:
3553:
3552:
3541:
3536:
3515:
3470:
3451:
3369:
3368:
3332:
3331:
3203:
3202:
3116:goes above 1/2.
3096:
3083:Ask each agent
3020:
3003:
3002:
2999:exact divisions
2961:
2956:
2955:
2948:
2943:
2938:
2933:
2923:
2918:
2913:
2908:
2856:
2855:
2825:
2814:
2813:
2786:
2775:
2774:
2770:
2759:mark-cut-choose
2751:
2741:queries in the
2706:
2680:
2679:
2634:
2608:
2607:
2542:
2537:
2536:
2456:
2455:
2340:
2339:
2300:
2299:
2214:
2213:
2188:
2187:
2159:
2158:
2113:
2112:
2073:
2072:
2047:
2046:
1972:
1971:
1938:
1936:Cut-near-halves
1908:
1907:
1882:
1881:
1844:
1843:
1818:
1817:
1786:
1785:
1736:
1735:
1693:
1688:
1687:
1592:
1591:
1560:
1559:
1531:
1530:
1493:
1492:
1470:
1469:
1441:
1440:
1354:
1353:
1316:
1315:
1296:
1295:
1294:if-and-only-if
1270:
1269:
1250:
1245:
1244:
1219:
1218:
1199:
1194:
1193:
1169:
1150:
1130:
1125:
1124:
1105:
1104:
1080:
1061:
1047:
1046:
1013:
1008:
1007:
969:
964:
963:
936:
931:
930:
906:
901:
900:
875:
874:
853:
848:
847:
828:
827:
802:
801:
780:
767:
762:
761:
736:
723:
718:
717:
698:
697:
676:
671:
670:
649:
644:
643:
592:
549:
538:
537:
509:
508:
484:
479:
478:
457:
452:
451:
432:
431:
404:
399:
398:
379:
378:
359:
358:
337:
332:
331:
312:
311:
284:
265:
260:
259:
230:
203:
198:
197:
178:
177:
174:
148:
147:
112:
107:
106:
78:
73:
72:
53:
52:
31:
26:
25:
12:
11:
5:
3986:
3984:
3976:
3975:
3970:
3960:
3959:
3954:
3953:
3919:
3912:
3886:
3865:
3820:
3810:
3775:
3761:
3725:
3698:(2): 125â137.
3679:
3622:
3587:
3564:
3538:
3537:
3535:
3532:
3514:
3511:
3506:
3505:
3468:
3449:
3415:
3414:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3356:
3353:
3350:
3347:
3344:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3290:
3287:
3284:
3281:
3278:
3275:
3267:
3264:
3261:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3234:
3230:
3227:
3224:
3221:
3218:
3215:
3163:
3162:
3161:
3150:
3143:
3134:All agents in
3117:
3106:
3103:
3094:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3027:
3023:
3019:
3016:
3013:
3010:
2976:
2973:
2968:
2964:
2950:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2929:
2928:George's value
2925:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2878:
2875:
2872:
2869:
2866:
2863:
2838:
2835:
2832:
2828:
2824:
2821:
2799:
2796:
2793:
2789:
2785:
2782:
2769:
2766:
2750:
2747:
2730:
2727:
2724:
2721:
2718:
2713:
2709:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2661:
2658:
2655:
2652:
2649:
2646:
2641:
2637:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2597:
2596:
2592:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2549:
2545:
2533:
2532:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2443:
2440:
2436:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2404:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2378:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2327:
2324:
2321:
2318:
2315:
2311:
2307:
2296:
2293:
2292:
2291:
2279:
2275:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2243:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2201:
2198:
2195:
2184:
2172:
2169:
2166:
2152:
2140:
2137:
2134:
2131:
2128:
2124:
2120:
2100:
2097:
2094:
2091:
2088:
2084:
2080:
2060:
2057:
2054:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1964:
1963:
1960:cut and choose
1956:
1955:
1954:
1951:
1945:
1937:
1934:
1921:
1918:
1915:
1895:
1892:
1889:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1831:
1828:
1825:
1805:
1802:
1799:
1796:
1793:
1773:
1769:
1765:
1760:
1755:
1752:
1749:
1746:
1743:
1734:cuts, where
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1700:
1696:
1681:
1680:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1579:
1576:
1573:
1570:
1567:
1556:
1544:
1541:
1538:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1489:
1477:
1466:
1454:
1451:
1448:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1348:
1347:
1335:
1332:
1329:
1326:
1323:
1303:
1283:
1280:
1277:
1256:
1253:
1232:
1229:
1226:
1205:
1202:
1181:
1176:
1172:
1168:
1165:
1162:
1157:
1153:
1149:
1146:
1143:
1140:
1136:
1133:
1112:
1092:
1087:
1083:
1079:
1076:
1073:
1068:
1064:
1060:
1057:
1054:
1034:
1031:
1028:
1025:
1020:
1016:
984:
981:
976:
972:
951:
948:
943:
939:
913:
909:
899:which sums to
888:
885:
882:
860:
856:
846:which sums to
835:
815:
812:
809:
787:
783:
779:
774:
770:
743:
739:
735:
730:
726:
705:
683:
679:
656:
652:
607:
606:
603:
591:
588:
576:
573:
570:
567:
564:
561:
556:
552:
548:
545:
522:
519:
516:
491:
487:
464:
460:
439:
411:
407:
386:
366:
344:
340:
319:
299:
296:
291:
287:
283:
280:
277:
272:
268:
247:
243:
237:
233:
229:
226:
223:
220:
216:
210:
206:
185:
173:
170:
155:
135:
131:
127:
124:
119:
115:
85:
81:
60:
38:
34:
13:
10:
9:
6:
4:
3:
2:
3985:
3974:
3971:
3969:
3966:
3965:
3963:
3949:
3945:
3941:
3937:
3930:
3923:
3920:
3915:
3913:9781461546276
3909:
3905:
3901:
3897:
3890:
3887:
3881:
3876:
3869:
3866:
3861:
3857:
3853:
3849:
3844:
3839:
3835:
3831:
3824:
3821:
3814:
3811:
3806:
3802:
3798:
3794:
3790:
3786:
3779:
3776:
3772:
3768:
3764:
3762:9783319996592
3758:
3754:
3750:
3745:
3740:
3736:
3729:
3726:
3721:
3717:
3713:
3709:
3705:
3701:
3697:
3693:
3686:
3684:
3680:
3675:
3671:
3667:
3663:
3659:
3655:
3650:
3645:
3641:
3637:
3633:
3626:
3623:
3618:
3614:
3610:
3606:
3602:
3598:
3597:Combinatorica
3591:
3588:
3583:
3579:
3575:
3571:
3567:
3561:
3557:
3550:
3548:
3546:
3544:
3540:
3533:
3531:
3529:
3525:
3523:
3519:
3512:
3510:
3503:
3499:
3495:
3491:
3487:
3483:
3479:
3475:
3471:
3464:
3460:
3456:
3452:
3445:
3441:
3437:
3433:
3429:
3425:
3421:
3417:
3416:
3400:
3397:
3394:
3391:
3388:
3382:
3354:
3351:
3345:
3311:
3308:
3305:
3302:
3299:
3288:
3282:
3279:
3276:
3265:
3262:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3228:
3222:
3219:
3216:
3200:
3196:
3192:
3188:
3184:
3180:
3176:
3172:
3168:
3164:
3159:
3155:
3151:
3148:
3144:
3141:
3137:
3133:
3132:
3130:
3126:
3122:
3118:
3115:
3111:
3107:
3104:
3101:
3097:
3090:
3086:
3082:
3081:
3080:
3078:
3074:
3070:
3068:
3064:
3048:
3045:
3039:
3036:
3033:
3030:
3025:
3021:
3014:
3011:
3008:
3000:
2996:
2992:
2988:
2974:
2971:
2966:
2962:
2945:
2940:
2935:
2930:
2927:
2926:
2920:
2915:
2910:
2905:
2903:Alice's value
2902:
2901:
2898:
2896:
2892:
2873:
2870:
2867:
2861:
2852:
2836:
2833:
2830:
2826:
2822:
2819:
2797:
2794:
2791:
2787:
2783:
2780:
2767:
2765:
2762:
2760:
2756:
2748:
2746:
2744:
2722:
2716:
2711:
2707:
2697:
2694:
2691:
2685:
2676:
2672:
2659:
2650:
2644:
2639:
2635:
2625:
2622:
2619:
2613:
2605:
2600:
2593:
2589:
2588:
2587:
2584:
2567:
2564:
2561:
2552:
2547:
2543:
2515:
2512:
2509:
2506:
2503:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2438:
2434:
2427:
2424:
2421:
2418:
2415:
2409:
2406:
2402:
2395:
2392:
2389:
2383:
2380:
2376:
2369:
2366:
2363:
2360:
2357:
2348:
2345:
2322:
2319:
2316:
2309:
2305:
2297:
2294:
2277:
2273:
2266:
2263:
2260:
2257:
2254:
2248:
2245:
2241:
2234:
2231:
2228:
2225:
2222:
2199:
2196:
2193:
2185:
2170:
2167:
2164:
2156:
2155:
2153:
2135:
2132:
2129:
2122:
2118:
2095:
2092:
2089:
2082:
2078:
2058:
2055:
2052:
2044:
2043:
2042:
2025:
2022:
2019:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1969:
1961:
1957:
1952:
1949:
1948:
1946:
1943:
1942:
1941:
1935:
1933:
1919:
1916:
1913:
1893:
1890:
1887:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1829:
1826:
1823:
1816:cuts. But if
1803:
1800:
1797:
1794:
1791:
1771:
1767:
1758:
1753:
1750:
1744:
1741:
1718:
1715:
1712:
1703:
1698:
1694:
1684:
1663:
1660:
1657:
1654:
1651:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1577:
1574:
1571:
1568:
1565:
1557:
1542:
1539:
1536:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1490:
1475:
1467:
1452:
1449:
1446:
1438:
1437:
1436:
1419:
1416:
1413:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1351:
1333:
1330:
1327:
1324:
1321:
1301:
1281:
1278:
1275:
1254:
1251:
1230:
1227:
1224:
1203:
1200:
1174:
1170:
1166:
1163:
1160:
1155:
1151:
1147:
1144:
1138:
1134:
1131:
1110:
1085:
1081:
1077:
1074:
1071:
1066:
1062:
1055:
1052:
1032:
1029:
1026:
1023:
1018:
1014:
1005:
1004:
1003:
1001:
996:
982:
979:
974:
970:
949:
946:
941:
937:
927:
911:
907:
886:
880:
858:
854:
833:
813:
810:
807:
785:
781:
777:
772:
768:
760:for the pair
759:
741:
737:
733:
728:
724:
703:
681:
677:
654:
650:
642:Formally: if
640:
638:
637:Ramsey theory
634:
630:
626:
623:
618:
614:
610:
604:
601:
597:
596:
595:
589:
587:
574:
565:
559:
554:
550:
543:
534:
520:
517:
514:
505:
489:
485:
462:
458:
437:
429:
425:
409:
405:
384:
364:
342:
338:
317:
297:
294:
289:
285:
281:
278:
275:
270:
266:
245:
241:
235:
231:
227:
224:
221:
218:
214:
208:
204:
183:
171:
169:
166:
153:
133:
129:
125:
122:
117:
113:
104:
99:
83:
79:
58:
36:
32:
23:
19:
3968:Cake-cutting
3939:
3935:
3922:
3895:
3889:
3868:
3833:
3829:
3823:
3813:
3791:(3â4): 353.
3788:
3784:
3778:
3734:
3728:
3695:
3691:
3639:
3635:
3625:
3600:
3596:
3590:
3555:
3527:
3526:
3517:
3516:
3507:
3501:
3497:
3493:
3492:, values (0,
3489:
3485:
3481:
3477:
3466:
3462:
3458:
3454:
3447:
3443:
3439:
3435:
3431:
3427:
3423:
3419:
3198:
3194:
3190:
3186:
3182:
3178:
3174:
3170:
3166:
3157:
3153:
3146:
3139:
3135:
3128:
3124:
3120:
3113:
3109:
3099:
3092:
3088:
3084:
3076:
3072:
3071:
3069:is general.
3066:
3062:
2991:Segal-Halevi
2990:
2989:
2953:
2894:
2890:
2853:
2771:
2763:
2758:
2754:
2752:
2674:
2673:
2603:
2601:
2598:
2585:
2534:
2290:in his eyes.
1965:
1939:
1685:
1682:
1529:, then push
1352:
1349:
1243:. Moreover,
997:
928:
757:
756:is called a
641:
632:
628:
627:
621:
619:
615:
611:
608:
599:
593:
535:
506:
427:
426:
175:
167:
100:
21:
15:
3465:) at least
3087:to mark an
1555:and finish.
600:5:3:2:1:1:1
3962:Categories
3880:1909.07141
3843:1803.05470
3836:: 123382.
3744:1709.03152
3649:1709.03152
3603:(2): 193.
3534:References
3446:) exactly
3442:values (0,
3127:is called
3091:such that
3061:cuts when
2591:remainder.
3942:: 57â77.
3720:118080310
3712:0926-2644
3674:218517351
3666:1549-6325
3398:−
3389:≤
3303:−
3252:−
3246:≤
3240:≤
3229:≤
3138:value (0,
3037:−
3031:
3015:⋅
2972:≥
2871:−
2834:−
2823:⋅
2795:−
2784:⋅
2729:⌉
2717:
2704:⌈
2695:−
2657:⌉
2645:
2632:⌈
2623:−
2553:
2513:−
2367:−
2349:∈
2232:−
1865:−
1801:−
1742:ϕ
1704:
1699:ϕ
1661:−
1575:≠
1569:−
1508:−
1331:−
1164:…
1075:…
884:∖
811:⊆
572:⌉
560:
547:⌈
518:−
330:, create
279:⋯
225:…
3805:19624080
3771:19245769
3617:19376212
3582:2730675W
3574:97041258
3513:See also
3165:If both
2338:, where
1255:′
1204:′
1135:′
504:pieces.
424:clones.
146:for all
3860:3901524
3131:. Now:
2454:. Call
1842:, then
1590:, then
1192:, then
258:, with
172:Cloning
16:In the
3910:
3858:
3803:
3769:
3759:
3718:
3710:
3672:
3664:
3615:
3580:
3572:
3562:
3152:Agent
3102:)=1/2.
2851:cuts.
1932:ones.
1123:, and
1045:, and
3932:(PDF)
3875:arXiv
3856:S2CID
3838:arXiv
3801:S2CID
3767:S2CID
3739:arXiv
3716:S2CID
3670:S2CID
3644:arXiv
3613:S2CID
1468:Push
3908:ISBN
3757:ISBN
3708:ISSN
3662:ISSN
3570:LCCN
3560:ISBN
3518:Zeng
3480:in (
3457:and
3376:Cuts
3339:Cuts
3293:Cuts
3270:Cuts
3210:Cuts
3181:and
3169:and
2975:0.75
2056:<
1450:<
1030:<
1024:<
962:and
669:and
622:only
3944:doi
3900:doi
3848:doi
3834:480
3793:doi
3749:doi
3700:doi
3654:doi
3605:doi
3233:max
3201:):
3158:x,1
3147:x,1
3098:(0,
3022:log
2708:log
2636:log
2556:min
2544:log
2186:if
2157:if
1707:min
1695:log
1558:If
1491:If
1006:If
716:of
639:).
551:log
3964::
3940:66
3938:.
3934:.
3906:.
3854:.
3846:.
3832:.
3799:.
3789:36
3787:.
3765:,
3755:,
3747:,
3714:.
3706:.
3694:.
3682:^
3668:.
3660:.
3652:.
3640:16
3638:.
3634:.
3611:.
3601:12
3599:.
3578:OL
3576:.
3568:.
3542:^
2041::
1970::
1435::
926:.
3950:.
3946::
3916:.
3902::
3883:.
3877::
3862:.
3850::
3840::
3807:.
3795::
3751::
3741::
3722:.
3702::
3696:8
3676:.
3656::
3646::
3619:.
3607::
3584:.
3502:z
3498:t
3494:z
3490:t
3486:y
3484:,
3482:x
3478:z
3469:t
3467:w
3463:y
3459:Q
3455:P
3450:t
3448:w
3444:y
3440:t
3436:y
3432:x
3428:t
3424:Q
3420:P
3413:.
3401:4
3395:n
3392:3
3386:)
3383:n
3380:(
3355:2
3352:=
3349:)
3346:2
3343:(
3318:)
3315:)
3312:1
3309:+
3306:k
3300:n
3297:(
3289:+
3286:)
3283:1
3280:+
3277:k
3274:(
3266:+
3263:1
3260:(
3255:1
3249:n
3243:k
3237:1
3226:)
3223:1
3220:+
3217:n
3214:(
3199:t
3195:P
3191:k
3187:x
3183:Q
3179:P
3175:t
3171:Q
3167:P
3154:t
3140:x
3136:P
3129:Q
3125:t
3121:t
3114:P
3110:P
3100:x
3095:i
3093:V
3089:x
3085:i
3077:n
3067:n
3063:n
3049:2
3046:+
3043:)
3040:1
3034:n
3026:2
3018:(
3012:n
3009:2
2967:A
2963:w
2947:1
2942:3
2937:3
2932:1
2922:2
2917:2
2912:2
2907:2
2895:n
2877:)
2874:1
2868:n
2865:(
2862:2
2837:2
2831:n
2827:3
2820:4
2798:2
2792:n
2788:3
2781:2
2726:)
2723:D
2720:(
2712:2
2701:)
2698:1
2692:n
2689:(
2686:2
2660:.
2654:)
2651:D
2648:(
2640:2
2629:)
2626:1
2620:n
2617:(
2614:n
2604:n
2571:)
2568:b
2565:,
2562:a
2559:(
2548:2
2531:.
2519:)
2516:c
2510:a
2507:,
2504:b
2501:(
2498:s
2495:e
2492:v
2489:l
2486:a
2483:H
2480:r
2477:a
2474:e
2471:N
2468:t
2465:u
2462:C
2442:)
2439:2
2435:/
2431:)
2428:1
2425:+
2422:b
2419:+
2416:a
2413:(
2410:,
2407:2
2403:/
2399:)
2396:b
2393:+
2390:a
2387:(
2384:,
2381:2
2377:/
2373:)
2370:1
2364:b
2361:+
2358:a
2355:(
2352:{
2346:c
2326:)
2323:b
2320:+
2317:a
2314:(
2310:/
2306:c
2278:2
2274:/
2270:)
2267:1
2264:+
2261:b
2258:+
2255:a
2252:(
2249::
2246:2
2242:/
2238:)
2235:1
2229:b
2226:+
2223:a
2220:(
2200:b
2197:+
2194:a
2171:b
2168:+
2165:a
2151:.
2139:)
2136:b
2133:+
2130:a
2127:(
2123:/
2119:a
2099:)
2096:b
2093:+
2090:a
2087:(
2083:/
2079:b
2059:b
2053:a
2029:)
2026:b
2023:,
2020:a
2017:(
2014:s
2011:e
2008:v
2005:l
2002:a
1999:H
1996:r
1993:a
1990:e
1987:N
1984:t
1981:u
1978:C
1962:.
1920:1
1917:+
1914:b
1894:1
1891:,
1888:b
1868:1
1862:b
1859:+
1856:a
1853:=
1850:b
1830:1
1827:=
1824:a
1804:1
1798:b
1795:+
1792:a
1772:2
1768:/
1764:)
1759:5
1754:+
1751:1
1748:(
1745:=
1722:)
1719:b
1716:,
1713:a
1710:(
1679:.
1667:)
1664:a
1658:b
1655:,
1652:a
1649:(
1646:y
1643:e
1640:s
1637:m
1634:a
1631:R
1628:l
1625:a
1622:m
1619:i
1616:n
1613:i
1610:M
1607:d
1604:n
1601:i
1598:F
1578:1
1572:a
1566:b
1543:1
1540:,
1537:1
1517:1
1514:=
1511:a
1505:b
1502:=
1499:a
1488:.
1476:a
1465:.
1453:b
1447:a
1423:)
1420:b
1417:,
1414:a
1411:(
1408:y
1405:e
1402:s
1399:m
1396:a
1393:R
1390:l
1387:a
1384:m
1381:i
1378:n
1375:i
1372:M
1369:d
1366:n
1363:i
1360:F
1346:.
1334:a
1328:b
1325:,
1322:a
1302:P
1282:b
1279:,
1276:a
1252:P
1231:b
1228:+
1225:a
1201:P
1180:)
1175:n
1171:p
1167:,
1161:,
1156:1
1152:p
1148:,
1145:b
1142:(
1139:=
1132:P
1111:b
1091:)
1086:n
1082:p
1078:,
1072:,
1067:1
1063:p
1059:(
1056:=
1053:P
1033:b
1027:a
1019:1
1015:p
983:5
980:=
975:2
971:k
950:8
947:=
942:1
938:k
912:2
908:k
887:L
881:P
859:1
855:k
834:L
814:P
808:L
786:2
782:k
778:,
773:1
769:k
742:2
738:k
734:+
729:1
725:k
704:P
682:2
678:k
655:1
651:k
602:.
575:.
569:)
566:D
563:(
555:2
544:D
521:1
515:D
490:A
486:p
463:G
459:p
438:D
410:i
406:p
385:i
365:D
343:i
339:p
318:i
298:D
295:=
290:n
286:p
282:+
276:+
271:1
267:p
246:D
242:/
236:n
232:p
228:,
222:,
219:D
215:/
209:1
205:p
184:D
154:i
134:n
130:/
126:1
123:=
118:i
114:w
84:i
80:w
59:i
37:i
33:w
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