1236:
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lists actions to be performed by the players in terms of the visible data and their valuations. A valid procedure is one that guarantees a fair division for every player who acts rationally according to their valuation. Where an action depends on a player's valuation the procedure is describing the
1283:
a rational player will follow. A player may act as if a piece had a different value but must be consistent. For instance if a procedure says the first player cuts the cake in two equal parts then the second player chooses a piece, then the first player cannot claim that the second player got more.
1327:
No finite protocol (even if unbounded) can guarantee an envy-free division of a cake among three or more players, if each player is to receive a single connected piece. However, this result applies only to the model presented in that work and not for cases where, for example, a mediator has full
718:
is a long narrow cake (modeled as the interval ), then, Alice may assign each subset a value proportional to its length, which means that she wants as much cake as possible, regardless of the icings. Bob may assign value only to subsets of , for example, because this part of the cake contains
1503:, a Russian immigrant asks an Israeli math teacher, how a circular cake can be divided fairly among 7 people? His answer is to make 3 straight cuts through its middle, making 8 equal pieces. Since there are only 7 people, one piece should be discarded, in the spirit of communism.
352:
may be an infinite set representing a divisible resource, for example: money, or a cake. Mathematically, a divisible resource is often modeled as a subset of a real space, for example, the section may represent a long narrow cake, that has to be cut into parallel pieces. The
723:
Based on these subjective value functions, there are a number of widely used criteria for a fair division. Some of these conflict with each other but often they can be combined. The criteria described here are only for when each player is entitled to the same amount:
694:
may assign the value of 1 to the set {car, apartment}, and the value 0 to all other sets except X; this means that he wants to get only the car and the apartment together; the car alone or the apartment alone, or each of them together with the piano, is worthless to
1032:
division means every person feels exactly the same happiness, i.e. the proportion of the cake a player receives by their own valuation is the same for every player. This is a difficult aim as players need not be truthful if asked their valuation:
1258:, i.e. it should be a dominant strategy for the participants to report their true valuations. This requirement is usually very hard to satisfy, especially in combination with fairness and Pareto-efficiency. As a result, it is often weakened to
77:. It demonstrates that two agents with different tastes can divide a cake such that each of them believes that he got the best piece. The research in fair division can be seen as an extension of this procedure to various more complex settings.
80:
There are many different kinds of fair division problems, depending on the nature of goods to divide, the criteria for fairness, the nature of the players and their preferences, and other criteria for evaluating the quality of the division.
1246:
In the real world people sometimes have a very accurate idea of how the other players value the goods and they may care very much about it. The case where they have complete knowledge of each other's valuations can be modeled by
328:
1417:
division for any number of players called 'last-diminisher' was devised in 1944. This was attributed to Banach and
Knaster by Steinhaus when he made the problem public for the first time at a meeting of the
232:
1475:
have both published books with sections about the problem. Martin
Gardner introduced the chore division form of the problem. Ian Stewart has popularized the fair division problem with his articles in
1251:. Partial knowledge is very hard to model. A major part of the practical side of fair division is the devising and study of procedures that work well despite such partial knowledge or small mistakes.
1200:. If different participants have different entitlements (e.g., in a partnership where each partner invested a different amount), then the fairness criteria should be adapted accordingly. See
1014:
839:
927:
485:
reflect the development of complex ideas regarding fairness. However, they are the result of legal debates by rabbis rather than divisions according to the valuations of the claimants.
1188:
1105:
1356:, the cake-cutting problem had been one of the most important open problems in 20th century mathematics, when the most important variant of the problem was finally solved with the
496:
is not possible, as different people may assign different values to each item. Empirical experiments on how people define the concept of fairness have given inconclusive results.
1201:
618:
942:
division guarantees that no-one will want somebody else's share more than their own, i.e. every person gets a share that he values at least as much as all other shares:
660:
1313:
procedures. A discrete procedure would for instance only involve one person at a time cutting or marking a cake. Continuous procedures involve things like one player
556:
34:
to them so that each person receives their due share. That problem arises in various real-world settings such as division of inheritance, partnership dissolutions,
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684:
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in
Washington, D.C., on 17 September 1947. At that meeting he also proposed the problem of finding the smallest number of cuts necessary for such divisions.
1518:
1745:"Le partage des dix-sept chameaux et autres arithmétiques attributes à l'immam 'Alî: Mouvance et circulation de récits de la tradition musulmane chiite"
2004:
1465:. In his book he says a special three-person version of fair division was devised by G. Krochmainy in BerdechĂłw in 1944 and another by Mrs L Kott.
1224:. A division where one player gets everything is optimal by this definition so on its own this does not guarantee even a fair share. See also
2161:
2030:
1872:
1806:
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283:
62:. The central tenet of fair division is that such a division should be performed by the players themselves, without the need for external
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2412:
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2123:
1455:
season 3 episode "One Hour", Charlie talks about the cake-cutting problem as applied to the amount of money a kidnapper was demanding.
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2515:
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1985:
1936:
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1216:, i.e., no other allocation would make someone better off without making someone else worse off. The term efficiency comes from the
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and the other saying "stop". Another type of continuous procedure involves a person assigning a value to every part of the cake.
2786:
2327:
473:. This kind of situation happens quite often with mathematical theories named after real life problems. The decisions in the
2656:
1025:
division guarantees that no subset of agents envies another subset of the same size; it is much stronger than envy-freeness.
2837:
2255:
2230:
1442:
involves the fair division of 17 camels (or elephants, or horses) into the proportions 1/2, 1/3, and 1/9. It is a popular
1298:
It is assumed the aim of each player is to maximize the minimum amount they might get, or in other words, to achieve the
3187:
2613:
2367:
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1537:
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may assign a value of 1/3 to each item, which means that each item is important to her just the same as any other item.
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43:
31:
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736:. For instance, if three people divide up a cake each gets at least a third by their own valuation, i.e. each of the
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The theory of fair division dates back only to the end of the second world war. It was devised by a group of
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1314:
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457:
1446:, often claimed to have an ancient origin, but its first documented publication was in 18th-century Iran.
581:
469:
Most of what is normally called a fair division is not considered so by the theory because of the use of
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2134:
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1262:, which only requires players to report their true valuations if they behave according to a specified
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39:
578:. Often the functions are assumed to be normalized, so that every person values the empty set as 0 (
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information of the players' valuation functions and proposes a division based on this information.
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Based on these distinctions, several general types of fair division problems have been studied:
1733:
Sol
Garfunkel. More Equal than Others: Weighted Voting. For All Practical Purposes. COMAP. 1988
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1981:
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1802:
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1371:
1255:
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74:
460:– dividing lotteries over divisions – is especially common when allocating indivisible goods.
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1963:
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1410:
1263:
1221:
416:
2073:
Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, JĂ©rĂ´me; Procaccia, Ariel D. (2016).
1946:
534:
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for all i) if the items are desirable, and -1 if the items are undesirable. Examples are:
372:
Finally, it is common to make some assumptions about whether the items to be divided are:
368:
heterogeneous – such as a cake that may have different ingredients, different icings, etc.
2960:
454:– dividing waters flowing in an international river among the countries along its stream.
2049:
3035:
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1715:. International Workshop on Internet and Network Economics. Springer. pp. 26–37.
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2011:
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1116:(aka consensus division) is one where all players agree on the value of each share:
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1744:
1361:
1235:
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Recently, the model of fair division has been extended from individual agents to
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2945:
2741:
2726:
2535:
2505:
2440:
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by
Christian Klamler – in Handbook of Group Decision and Negotiation pp 183–202.
2092:
1720:
1383:
1248:
470:
63:
47:
23:
2126:
from the
Discrete Mathematics Project at the University of Colorado at Boulder.
1978:
Mathematics and
Democracy: Designing Better Voting and Fair-Division Procedures
2796:
2450:
1375:
1967:
1928:
1818:
492:, there cannot be an objective measure of the value of each item. Therefore,
2701:
2621:
2445:
2107:
by
Claudia Lindner and Jörg Rothe – in Economics and Computation pp 395–491.
1275:
1217:
939:
354:
70:
51:
27:
1890:
415:. A special case is when the cake is a circle; then the problem is called
3136:
2636:
1923:. Introduction by Alan D. Taylor. Cambridge: Cambridge University Press.
1280:
854:(Such a division exists only if the players have different valuations.):
482:
2139:
2113:
by Jérôme Lang and Jörg Rothe – in
Economics and Computation pp 493–550.
1801:
How to cut a cake and other mathematical conundrums. Ian
Stewart. 2006.
2857:
2847:
2525:
2057:
1645:
1451:
1299:
66:, as only the players themselves really know how they value the goods.
1212:
In addition to fairness, it is sometimes desired that the division be
1394:
1379:
474:
2036:
Hill, T.P. (2000). "Mathematical devices for getting a fair share".
499:
Therefore, most current research on fairness focuses on concepts of
323:{\displaystyle C=\{{\text{piano}},{\text{car}},{\text{apartment}}\}}
2626:
1234:
330:, such that each item should be given entirely to a single person.
1685:
1593:"Game Theoretic Analysis of a bankruptcy Problem from the Talmud"
1254:
An additional requirement is that the fair division procedure be
1882:
2143:
1668:
1196:
All the above criteria assume that the participants have equal
1954:
Barbanel, J. (2010). "A Geometric Approach to Fair Division".
1669:"Envy-free cake divisions cannot be found by finite protocols"
1461:
wrote about a number of variants of fair division in his book
686:
is the set of indivisible items {piano, car, apartment}, then
365:
homogeneous – such as money, where only the amount matters, or
227:{\displaystyle C=X_{1}\sqcup X_{2}\sqcup \cdots \sqcup X_{n}}
1632:
Yaari, M. E.; Bar-Hillel, M. (1984). "On dividing justly".
1386:
involving more than two people are also quite common, the
1709:"The Efficiency of Fair Division with Connected Pieces"
1374:'s origins are undocumented. The related activities of
850:
is one where each player receives strictly more than 1/
280:
may be a finite set of indivisible items, for example:
1846:
Fair division: from cake-cutting to dispute resolution
89:
Formally, a fair division problem is defined by a set
1202:
proportional cake-cutting with different entitlements
1124:
1041:
950:
862:
774:
747:
704:
672:
626:
584:
564:
537:
509:
338:
286:
266:
243:
175:
155:
135:
115:
95:
732:
means that every person gets at least his due share
558:, which assigns a numerical value to each subset of
444:(e.g., rooms in an apartment), and simultaneously a
3145:
3104:
2886:
2830:
2612:
2514:
2421:
2279:
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1182:
1099:
1008:
921:
833:
753:
710:
678:
654:
612:
570:
550:
515:
344:
322:
272:
249:
226:
161:
141:
121:
101:
42:, airport traffic management, and exploitation of
523:people is assumed to have a personal, subjective
440:(aka the housemates problem) – dividing a set of
1769:Mathematical Snapshots. H.Steinhaus. 1950, 1969
433:Combinations and special cases are also common:
1425:For the history of envy-free cake-cutting, see
2105:Cake-Cutting: Fair Division of Divisible Goods
620:for all i), and the entire set of items as 1 (
403:fair division of a single homogeneous resource
2155:
2001:Vincent P. Crawford (1987). "fair division,"
1980:. Princeton, NJ: Princeton University Press.
1591:Aumann, Robert J.; Maschler, Michael (1985).
1294:Select a valid procedure and follow its rules
1009:{\displaystyle V_{i}(X_{i})\geq V_{i}(X_{j})}
397:Fair resource allocation – dividing a set of
8:
1320:For a list of fair division procedures, see
361:Additionally, the set to be divided may be:
317:
293:
2016:The New Palgrave: A Dictionary of Economics
1865:Cake-Cutting Algorithms: Be Fair If You Can
1496:strip is based on the cake-cutting problem.
1291:Agree on their criteria for a fair division
834:{\displaystyle V_{i}(X_{i})\geq V_{i}(C)/n}
719:cherries and Bob only cares about cherries.
2162:
2148:
2140:
1844:Brams, Steven J.; Taylor, Alan D. (1996).
1519:List of unsolved problems in fair division
922:{\displaystyle V_{i}(X_{i})>V_{i}(C)/n}
1684:
1183:{\displaystyle V_{i}(X_{i})=V_{j}(X_{i})}
1171:
1158:
1142:
1129:
1123:
1100:{\displaystyle V_{i}(X_{i})=V_{j}(X_{j})}
1088:
1075:
1059:
1046:
1040:
997:
984:
968:
955:
949:
911:
896:
880:
867:
861:
823:
808:
792:
779:
773:
746:
703:
671:
631:
625:
589:
583:
563:
542:
536:
508:
337:
312:
304:
296:
285:
265:
242:
218:
199:
186:
174:
154:
134:
114:
109:(often called "the cake") and a group of
94:
2076:Handbook of Computational Social Choice
2005:New Palgrave: A Dictionary of Economics
1920:The geometry of efficient fair division
1902:. Cambridge, Massachusetts: MIT Press.
1867:. Natick, Massachusetts: A. K. Peters.
1863:Robertson, Jack; Webb, William (1998).
1673:The Electronic Journal of Combinatorics
1583:
1340:(pre-determined groups) of agents. See
129:players. A division is a partition of
1785:aha! Insight. Martin. Gardner, 1978.
1707:Aumann, Yonatan; Dombb, Yair (2010).
7:
2135:Fair Division: Method of Sealed Bids
2023:The Evolution of the Social Contract
1900:Fair Division and Collective Welfare
734:according to his own value function
613:{\displaystyle V_{i}(\emptyset )=0}
376:goods – such as a car or a cake, or
46:. It is an active research area in
2211:First-player and second-player win
2111:Fair division of indivisible goods
1752:Revue d'histoire des mathématiques
598:
14:
30:among several people who have an
2318:Coalition-proof Nash equilibrium
2130:Fair Division: Method of Markers
1952:Short summary is available at:
1322:Category:Fair division protocols
1956:The College Mathematics Journal
1305:Procedures can be divided into
442:indivisible heterogeneous goods
2328:Evolutionarily stable strategy
2079:. Cambridge University Press.
1848:. Cambridge University Press.
1837:Equity: in theory and practice
1760:; see in particular pp. 13–14.
1713:Internet and Network Economics
1177:
1164:
1148:
1135:
1094:
1081:
1065:
1052:
1003:
990:
974:
961:
908:
902:
886:
873:
820:
814:
798:
785:
762:which he values as at least 1/
643:
637:
601:
595:
1:
2256:Simultaneous action selection
1839:. Princeton University Press.
1390:is a notable recent example.
428:divisible, heterogeneous bad.
413:divisible, heterogeneous good
392:indivisible and heterogeneous
69:The archetypal fair division
3188:List of games in game theory
2368:Quantal response equilibrium
2358:Perfect Bayesian equilibrium
2293:Bayes correlated equilibrium
2025:Cambridge University Press.
1917:Barbanel, Julius B. (2005).
1612:10.1016/0022-0531(85)90102-4
1538:mathematics of apportionment
1440:17-animal inheritance puzzle
1415:proportional (fair division)
1413:in Lvov (then in Poland). A
448:(the rent on the apartment).
379:bads – such as house chores.
44:Earth observation satellites
16:Problem of sharing resources
2657:Optional prisoner's dilemma
2388:Self-confirming equilibrium
1721:10.1007/978-3-642-17572-5_3
1667:Stromquist, Walter (2008).
848:super-proportional division
357:may represent an apple pie.
3250:
3122:Principal variation search
2838:Aumann's agreement theorem
2501:Strategy-stealing argument
2413:Trembling hand equilibrium
2343:Markov perfect equilibrium
2338:Mertens-stable equilibrium
1600:Journal of Economic Theory
1409:, who used to meet in the
1342:fair division among groups
655:{\displaystyle V_{i}(C)=1}
490:subjective theory of value
85:Things that can be divided
3158:Combinatorial game theory
2817:Princess and monster game
2373:Quasi-perfect equilibrium
2298:Bayesian Nash equilibrium
1835:Young, Peyton H. (1995).
1634:Social Choice and Welfare
1514:Fair division experiments
740:people gets a subset of
446:homogeneous divisible bad
401:goods. A special case is
399:divisible and homogeneous
257:can be of various types:
234:, one subset per player.
3173:Evolutionary game theory
2906:Antoine Augustin Cournot
2792:Guess 2/3 of the average
2589:Strictly determined game
2383:Satisfaction equilibrium
2201:Escalation of commitment
2018:, v. 2, pp. 275–76.
2008:, v. 2, pp. 274–75.
1976:Steven J. Brams (2008).
1968:10.4169/074683410x510263
1929:10.1017/CBO9780511546679
1287:What the players do is:
3178:Glossary of game theory
2777:Stackelberg competition
2403:Strong Nash equilibrium
1743:Ageron, Pierre (2013).
1529:Strategic fair division
1260:incentive compatibility
1208:Additional requirements
465:Definitions of fairness
3203:Tragedy of the commons
3183:List of game theorists
3163:Confrontation analysis
2873:Sprague–Grundy theorem
2393:Sequential equilibrium
2313:Correlated equilibrium
1463:Mathematical Snapshots
1427:envy-free cake-cutting
1358:Brams-Taylor procedure
1243:
1239:Berlin divided by the
1226:efficient cake-cutting
1184:
1101:
1010:
923:
835:
755:
712:
680:
656:
614:
572:
552:
517:
458:Fair random assignment
346:
324:
274:
251:
228:
163:
143:
123:
103:
2976:Jean-François Mertens
2021:Bryan Skyrms (1996).
1898:Herve Moulin (2004).
1499:In the Israeli movie
1238:
1185:
1102:
1011:
924:
836:
756:
730:proportional division
713:
681:
657:
615:
573:
553:
551:{\displaystyle V_{i}}
518:
347:
325:
275:
252:
229:
164:
144:
124:
104:
26:of dividing a set of
3105:Search optimizations
2981:Jennifer Tour Chayes
2868:Revelation principle
2863:Purification theorem
2802:Nash bargaining game
2767:Bertrand competition
2752:El Farol Bar problem
2717:Electronic mail game
2682:Lewis signaling game
2226:Hierarchy of beliefs
2014:(1987). "fairness,"
1563:Nash bargaining game
1524:Online fair division
1122:
1039:
948:
860:
772:
766:of the total value:
745:
702:
670:
624:
582:
562:
535:
507:
390:– dividing a set of
388:Fair item assignment
336:
284:
264:
241:
173:
153:
133:
113:
93:
56:social choice theory
40:frequency allocation
3153:Bounded rationality
2772:Cournot competition
2722:Rock paper scissors
2697:Battle of the sexes
2687:Volunteer's dilemma
2559:Perfect information
2486:Dominant strategies
2323:Epsilon-equilibrium
2206:Extensive-form game
2093:free online version
2050:2000AmSci..88..325H
1553:Justice (economics)
1548:International trade
1478:Scientific American
1444:mathematical puzzle
1420:Econometric Society
501:subjective fairness
36:divorce settlements
3132:Paranoid algorithm
3112:Alpha–beta pruning
2991:John Maynard Smith
2822:Rendezvous problem
2662:Traveler's dilemma
2652:Gift-exchange game
2647:Prisoner's dilemma
2564:Large Poisson game
2531:Bargaining problem
2436:Backward induction
2408:Subgame perfection
2363:Proper equilibrium
2095:), chapters 11–13.
2058:10.1511/2000.4.325
2038:American Scientist
1819:"Dinosaur Comics!"
1646:10.1007/BF00297056
1543:Equity (economics)
1433:In popular culture
1388:Potsdam Conference
1382:are also ancient.
1244:
1241:Potsdam Conference
1180:
1097:
1006:
919:
831:
751:
708:
676:
652:
610:
568:
548:
513:
494:objective fairness
481:when an estate is
452:Fair river sharing
342:
320:
270:
247:
224:
169:disjoint subsets:
159:
139:
119:
99:
60:dispute resolution
22:is the problem in
3234:Welfare economics
3211:
3210:
3117:Aspiration window
3086:Suzanne Scotchmer
3041:Oskar Morgenstern
2936:Donald B. Gillies
2878:Zermelo's theorem
2807:Induction puzzles
2762:Fair cake-cutting
2737:Public goods game
2667:Coordination game
2541:Intransitive game
2471:Forward induction
2353:Pareto efficiency
2333:Gibbs equilibrium
2303:Berge equilibrium
2251:Simultaneous game
2031:978-0-521-55583-8
1972:
1874:978-1-56881-076-8
1807:978-0-19-920590-5
1791:978-0-7167-1017-2
1573:Price of fairness
1403:Bronisław Knaster
1372:Divide and choose
1230:price of fairness
754:{\displaystyle C}
711:{\displaystyle C}
679:{\displaystyle C}
571:{\displaystyle C}
516:{\displaystyle n}
488:According to the
409:Fair cake-cutting
345:{\displaystyle C}
315:
307:
299:
273:{\displaystyle C}
250:{\displaystyle C}
162:{\displaystyle n}
142:{\displaystyle C}
122:{\displaystyle n}
102:{\displaystyle C}
75:divide and choose
3241:
3198:Topological game
3193:No-win situation
3091:Thomas Schelling
3071:Robert B. Wilson
3031:Merrill M. Flood
3001:John von Neumann
2911:Ariel Rubinstein
2896:Albert W. Tucker
2747:War of attrition
2707:Matching pennies
2348:Nash equilibrium
2271:Mechanism design
2236:Normal-form game
2191:Cooperative game
2164:
2157:
2150:
2141:
2090:
2069:
1991:
1971:
1951:
1950:
1913:
1894:
1859:
1840:
1823:
1822:
1815:
1809:
1799:
1793:
1783:
1777:
1767:
1761:
1759:
1749:
1740:
1734:
1731:
1725:
1724:
1704:
1698:
1697:
1695:
1693:
1688:
1664:
1658:
1657:
1629:
1623:
1622:
1620:
1614:. Archived from
1597:
1588:
1558:Knapsack problem
1397:mathematicians,
1274:A fair division
1264:solution concept
1222:efficient market
1190:for all i and j.
1189:
1187:
1186:
1181:
1176:
1175:
1163:
1162:
1147:
1146:
1134:
1133:
1107:for all i and j.
1106:
1104:
1103:
1098:
1093:
1092:
1080:
1079:
1064:
1063:
1051:
1050:
1016:for all i and j.
1015:
1013:
1012:
1007:
1002:
1001:
989:
988:
973:
972:
960:
959:
928:
926:
925:
920:
915:
901:
900:
885:
884:
872:
871:
840:
838:
837:
832:
827:
813:
812:
797:
796:
784:
783:
760:
758:
757:
752:
717:
715:
714:
709:
685:
683:
682:
677:
661:
659:
658:
653:
636:
635:
619:
617:
616:
611:
594:
593:
577:
575:
574:
569:
557:
555:
554:
549:
547:
546:
525:utility function
522:
520:
519:
514:
417:fair pie-cutting
351:
349:
348:
343:
329:
327:
326:
321:
316:
313:
308:
305:
300:
297:
279:
277:
276:
271:
256:
254:
253:
248:
233:
231:
230:
225:
223:
222:
204:
203:
191:
190:
168:
166:
165:
160:
148:
146:
145:
140:
128:
126:
125:
120:
108:
106:
105:
100:
3249:
3248:
3244:
3243:
3242:
3240:
3239:
3238:
3214:
3213:
3212:
3207:
3141:
3127:max^n algorithm
3100:
3096:William Vickrey
3056:Reinhard Selten
3011:Kenneth Binmore
2926:David K. Levine
2921:Daniel Kahneman
2888:
2882:
2858:Negamax theorem
2848:Minimax theorem
2826:
2787:Diner's dilemma
2642:All-pay auction
2608:
2594:Stochastic game
2546:Mean-field game
2517:
2510:
2481:Markov strategy
2417:
2283:
2275:
2246:Sequential game
2231:Information set
2216:Game complexity
2186:Congestion game
2174:
2168:
2120:
2087:
2072:
2035:
1998:
1996:Survey articles
1988:
1975:
1953:
1939:
1916:
1910:
1897:
1875:
1862:
1856:
1843:
1834:
1831:
1826:
1817:
1816:
1812:
1800:
1796:
1784:
1780:
1768:
1764:
1747:
1742:
1741:
1737:
1732:
1728:
1706:
1705:
1701:
1691:
1689:
1666:
1665:
1661:
1631:
1630:
1626:
1618:
1595:
1590:
1589:
1585:
1581:
1510:
1493:Dinosaur Comics
1435:
1350:
1334:
1272:
1210:
1167:
1154:
1138:
1125:
1120:
1119:
1084:
1071:
1055:
1042:
1037:
1036:
1023:group-envy-free
993:
980:
964:
951:
946:
945:
892:
876:
863:
858:
857:
804:
788:
775:
770:
769:
743:
742:
700:
699:
668:
667:
627:
622:
621:
585:
580:
579:
560:
559:
538:
533:
532:
505:
504:
467:
334:
333:
282:
281:
262:
261:
239:
238:
214:
195:
182:
171:
170:
151:
150:
131:
130:
111:
110:
91:
90:
87:
17:
12:
11:
5:
3247:
3245:
3237:
3236:
3231:
3226:
3216:
3215:
3209:
3208:
3206:
3205:
3200:
3195:
3190:
3185:
3180:
3175:
3170:
3165:
3160:
3155:
3149:
3147:
3143:
3142:
3140:
3139:
3134:
3129:
3124:
3119:
3114:
3108:
3106:
3102:
3101:
3099:
3098:
3093:
3088:
3083:
3078:
3073:
3068:
3063:
3061:Robert Axelrod
3058:
3053:
3048:
3043:
3038:
3036:Olga Bondareva
3033:
3028:
3026:Melvin Dresher
3023:
3018:
3016:Leonid Hurwicz
3013:
3008:
3003:
2998:
2993:
2988:
2983:
2978:
2973:
2968:
2963:
2958:
2953:
2951:Harold W. Kuhn
2948:
2943:
2941:Drew Fudenberg
2938:
2933:
2931:David M. Kreps
2928:
2923:
2918:
2916:Claude Shannon
2913:
2908:
2903:
2898:
2892:
2890:
2884:
2883:
2881:
2880:
2875:
2870:
2865:
2860:
2855:
2853:Nash's theorem
2850:
2845:
2840:
2834:
2832:
2828:
2827:
2825:
2824:
2819:
2814:
2809:
2804:
2799:
2794:
2789:
2784:
2779:
2774:
2769:
2764:
2759:
2754:
2749:
2744:
2739:
2734:
2729:
2724:
2719:
2714:
2712:Ultimatum game
2709:
2704:
2699:
2694:
2692:Dollar auction
2689:
2684:
2679:
2677:Centipede game
2674:
2669:
2664:
2659:
2654:
2649:
2644:
2639:
2634:
2632:Infinite chess
2629:
2624:
2618:
2616:
2610:
2609:
2607:
2606:
2601:
2599:Symmetric game
2596:
2591:
2586:
2584:Signaling game
2581:
2579:Screening game
2576:
2571:
2569:Potential game
2566:
2561:
2556:
2548:
2543:
2538:
2533:
2528:
2522:
2520:
2512:
2511:
2509:
2508:
2503:
2498:
2496:Mixed strategy
2493:
2488:
2483:
2478:
2473:
2468:
2463:
2458:
2453:
2448:
2443:
2438:
2433:
2427:
2425:
2419:
2418:
2416:
2415:
2410:
2405:
2400:
2395:
2390:
2385:
2380:
2378:Risk dominance
2375:
2370:
2365:
2360:
2355:
2350:
2345:
2340:
2335:
2330:
2325:
2320:
2315:
2310:
2305:
2300:
2295:
2289:
2287:
2277:
2276:
2274:
2273:
2268:
2263:
2258:
2253:
2248:
2243:
2238:
2233:
2228:
2223:
2221:Graphical game
2218:
2213:
2208:
2203:
2198:
2193:
2188:
2182:
2180:
2176:
2175:
2169:
2167:
2166:
2159:
2152:
2144:
2138:
2137:
2132:
2127:
2119:
2118:External links
2116:
2115:
2114:
2108:
2102:
2096:
2085:
2070:
2044:(4): 325–331.
2033:
2019:
2009:
1997:
1994:
1993:
1992:
1986:
1973:
1937:
1914:
1908:
1895:
1873:
1860:
1854:
1841:
1830:
1827:
1825:
1824:
1810:
1794:
1778:
1762:
1735:
1726:
1699:
1659:
1624:
1621:on 2006-02-20.
1606:(2): 195–213.
1582:
1580:
1577:
1576:
1575:
1570:
1565:
1560:
1555:
1550:
1545:
1540:
1531:
1526:
1521:
1516:
1509:
1506:
1505:
1504:
1497:
1488:
1469:Martin Gardner
1466:
1459:Hugo Steinhaus
1456:
1447:
1434:
1431:
1399:Hugo Steinhaus
1349:
1346:
1333:
1330:
1315:moving a knife
1296:
1295:
1292:
1271:
1268:
1214:Pareto optimal
1209:
1206:
1194:
1193:
1192:
1191:
1179:
1174:
1170:
1166:
1161:
1157:
1153:
1150:
1145:
1141:
1137:
1132:
1128:
1114:exact division
1110:
1109:
1108:
1096:
1091:
1087:
1083:
1078:
1074:
1070:
1067:
1062:
1058:
1054:
1049:
1045:
1026:
1019:
1018:
1017:
1005:
1000:
996:
992:
987:
983:
979:
976:
971:
967:
963:
958:
954:
936:
935:
934:
918:
914:
910:
907:
904:
899:
895:
891:
888:
883:
879:
875:
870:
866:
844:
843:
842:
830:
826:
822:
819:
816:
811:
807:
803:
800:
795:
791:
787:
782:
778:
750:
721:
720:
707:
696:
675:
651:
648:
645:
642:
639:
634:
630:
609:
606:
603:
600:
597:
592:
588:
567:
545:
541:
529:value function
512:
503:. Each of the
466:
463:
462:
461:
455:
449:
438:Rental harmony
431:
430:
424:chore division
420:
406:
395:
381:
380:
377:
370:
369:
366:
359:
358:
341:
331:
319:
311:
303:
295:
292:
289:
269:
246:
221:
217:
213:
210:
207:
202:
198:
194:
189:
185:
181:
178:
158:
138:
118:
98:
86:
83:
15:
13:
10:
9:
6:
4:
3:
2:
3246:
3235:
3232:
3230:
3227:
3225:
3224:Fair division
3222:
3221:
3219:
3204:
3201:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3150:
3148:
3146:Miscellaneous
3144:
3138:
3135:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3109:
3107:
3103:
3097:
3094:
3092:
3089:
3087:
3084:
3082:
3081:Samuel Bowles
3079:
3077:
3076:Roger Myerson
3074:
3072:
3069:
3067:
3066:Robert Aumann
3064:
3062:
3059:
3057:
3054:
3052:
3049:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3022:
3021:Lloyd Shapley
3019:
3017:
3014:
3012:
3009:
3007:
3006:Kenneth Arrow
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2986:John Harsanyi
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2956:Herbert Simon
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2922:
2919:
2917:
2914:
2912:
2909:
2907:
2904:
2902:
2899:
2897:
2894:
2893:
2891:
2885:
2879:
2876:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2835:
2833:
2829:
2823:
2820:
2818:
2815:
2813:
2810:
2808:
2805:
2803:
2800:
2798:
2795:
2793:
2790:
2788:
2785:
2783:
2780:
2778:
2775:
2773:
2770:
2768:
2765:
2763:
2760:
2758:
2757:Fair division
2755:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2735:
2733:
2732:Dictator game
2730:
2728:
2725:
2723:
2720:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2683:
2680:
2678:
2675:
2673:
2670:
2668:
2665:
2663:
2660:
2658:
2655:
2653:
2650:
2648:
2645:
2643:
2640:
2638:
2635:
2633:
2630:
2628:
2625:
2623:
2620:
2619:
2617:
2615:
2611:
2605:
2604:Zero-sum game
2602:
2600:
2597:
2595:
2592:
2590:
2587:
2585:
2582:
2580:
2577:
2575:
2574:Repeated game
2572:
2570:
2567:
2565:
2562:
2560:
2557:
2555:
2553:
2549:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2523:
2521:
2519:
2513:
2507:
2504:
2502:
2499:
2497:
2494:
2492:
2491:Pure strategy
2489:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2457:
2456:De-escalation
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2428:
2426:
2424:
2420:
2414:
2411:
2409:
2406:
2404:
2401:
2399:
2398:Shapley value
2396:
2394:
2391:
2389:
2386:
2384:
2381:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2339:
2336:
2334:
2331:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2294:
2291:
2290:
2288:
2286:
2282:
2278:
2272:
2269:
2267:
2266:Succinct game
2264:
2262:
2259:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2212:
2209:
2207:
2204:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2183:
2181:
2177:
2173:
2165:
2160:
2158:
2153:
2151:
2146:
2145:
2142:
2136:
2133:
2131:
2128:
2125:
2124:Fair Division
2122:
2121:
2117:
2112:
2109:
2106:
2103:
2100:
2099:Fair Division
2097:
2094:
2088:
2086:9781107060432
2082:
2078:
2077:
2071:
2067:
2063:
2059:
2055:
2051:
2047:
2043:
2039:
2034:
2032:
2028:
2024:
2020:
2017:
2013:
2010:
2007:
2006:
2000:
1999:
1995:
1989:
1987:9780691133218
1983:
1979:
1974:
1969:
1965:
1961:
1957:
1948:
1944:
1940:
1938:0-521-84248-4
1934:
1930:
1926:
1922:
1921:
1915:
1911:
1909:9780262134231
1905:
1901:
1896:
1892:
1888:
1884:
1880:
1876:
1870:
1866:
1861:
1857:
1855:0-521-55644-9
1851:
1847:
1842:
1838:
1833:
1832:
1828:
1820:
1814:
1811:
1808:
1804:
1798:
1795:
1792:
1788:
1782:
1779:
1776:
1775:0-19-503267-5
1772:
1766:
1763:
1757:
1754:(in French).
1753:
1746:
1739:
1736:
1730:
1727:
1722:
1718:
1714:
1710:
1703:
1700:
1687:
1682:
1678:
1674:
1670:
1663:
1660:
1655:
1651:
1647:
1643:
1639:
1635:
1628:
1625:
1617:
1613:
1609:
1605:
1601:
1594:
1587:
1584:
1578:
1574:
1571:
1569:
1568:Pizza theorem
1566:
1564:
1561:
1559:
1556:
1554:
1551:
1549:
1546:
1544:
1541:
1539:
1535:
1534:Apportionment
1532:
1530:
1527:
1525:
1522:
1520:
1517:
1515:
1512:
1511:
1507:
1502:
1498:
1495:
1494:
1489:
1486:
1485:
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38:, electronic
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20:Fair division
3051:Peyton Young
3046:Paul Milgrom
2961:Hervé Moulin
2901:Amos Tversky
2843:Folk theorem
2756:
2554:-player game
2551:
2476:Grim trigger
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1686:10.37236/735
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1616:the original
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1384:Negotiations
1370:
1362:Steven Brams
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3229:Game theory
3168:Coopetition
2971:Jean Tirole
2966:John Conway
2946:Eric Maskin
2742:Blotto game
2727:Pirate game
2536:Global game
2506:Tit for tat
2441:Bid shading
2431:Appeasement
2281:Equilibrium
2261:Solved game
2196:Determinacy
2179:Definitions
2172:game theory
1692:October 26,
1501:Saint Clara
1473:Ian Stewart
1366:Alan Taylor
1249:game theory
479:entitlement
471:arbitration
64:arbitration
48:mathematics
32:entitlement
24:game theory
3218:Categories
2812:Trust game
2797:Kuhn poker
2466:Escalation
2461:Deterrence
2451:Cheap talk
2423:Strategies
2241:Preference
2170:Topics of
2012:Hal Varian
1962:(4): 268.
1829:Text books
1758:(1): 1–41.
1579:References
1376:bargaining
1332:Extensions
1311:continuous
1270:Procedures
841:for all i.
2996:John Nash
2702:Stag hunt
2446:Collusion
2066:221539202
1654:153443060
1368:in 1995.
1276:procedure
1218:economics
1030:equitable
978:≥
940:envy-free
802:≥
599:∅
355:unit disk
314:apartment
212:⊔
209:⋯
206:⊔
193:⊔
71:algorithm
52:economics
28:resources
3137:Lazy SMP
2831:Theorems
2782:Deadlock
2637:Checkers
2518:of games
2285:concepts
1891:2730675W
1883:97041258
1508:See also
1338:families
1307:discrete
1281:strategy
1228:and the
929:for all
483:bankrupt
237:The set
2889:figures
2672:Chicken
2526:Auction
2516:Classes
2046:Bibcode
1947:2132232
1452:Numb3rs
1348:History
1300:maximin
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1380:barter
475:Talmud
394:goods.
2627:Chess
2614:Games
2062:S2CID
1748:(PDF)
1650:S2CID
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1619:(PDF)
1596:(PDF)
688:Alice
422:Fair
298:piano
149:into
2308:Core
2081:ISBN
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2003:The
1982:ISBN
1933:ISBN
1904:ISBN
1879:LCCN
1869:ISBN
1850:ISBN
1803:ISBN
1787:ISBN
1771:ISBN
1694:2022
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1481:and
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1378:and
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2887:Key
2054:doi
1964:doi
1925:doi
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