108:-1 types. Based on experimental data, most of the players only use one model to predict the behavior of all the other players. Although the hierarchy of types could be indefinite, the benefits of higher levels would decrease substantially while incurring a much greater cost. Because of the limit of players' expectation level and players' priors, it is possible to reach an equilibrium in games without communication.
65:". There is nothing that makes Grand Central Terminal a location with a higher payoff because people could just as easily meet at another public location, such as a bar or a library, but its tradition as a meeting place raises its salience and therefore makes it a natural "focal point". Later, Schelling's informal experiments have been replicated under controlled conditions with monetary incentives by Judith Mehta.
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to go straight, swerve to the left or swerve to the right. Both players want to avoid crashing, but neither knows what the other will do. In this case, the decision to swerve right can serve as a focal point which leads to the winning right-right outcome. It seems a natural focal point in places using
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Focal points can also have real-life applications. For example, imagine two bicycles headed towards each other and in danger of crashing. Avoiding collision becomes a coordination game where each player's winning choice depends on the other player's choice. Each player, in this case, has the choice
330:
Adding repetition to the game introduces a focal point at the Nash equilibrium solution of 0. This was shown by
Camerer as, “ the game is played multiple times with the same group, the average moves close to 0.” Introducing the iterative aspect to the game forces all players onto higher levels of
47:
often concert their intentions or expectations with others if each knows that the other is trying to do the same" in a cooperative situation (p. 57), so their action would converge on a focal point which has some kind of prominence compared with the environment. However, the conspicuousness of the
218:. The red square is the "right" square to select only if a player can be sure that the other player has selected it, but by hypothesis neither can. However, it is the most salient and notable square, so—lacking any other one—most people will choose it, and this will in fact (often) work.
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game shows the level-n theory in practice. In this game, players are tasked with guessing an integer from 0 to 100 inclusive which they believe is closest to 2/3 of the average of all players’ guesses. A Nash equilibrium can be found by thinking through each level:
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Bacharach argued that people could find a focal point because they act as members of a team instead of individuals in a cooperative game. With the identity changed, the player follows the prescription of an imaginary group leader to maximize the group interest.
56:
The existence of the focal point is first demonstrated by
Schelling with a series of questions. Here is one example: to determine the time and place to meet a stranger in New York City, but without being able to communicate in person beforehand. In this
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The cognitive hierarchy (CH) theory is a derivation of level-n theory. A level-n player from the CH model would assume that their strategy is the most sophisticated and that the levels 0, 1, 2, ..., n-1 on which their opponents play follow a normalized
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Split money game: Two players share $ 100. They first write down their individual claims on a sheet of paper. If their claims add to $ 100 or less, both of them will get exactly what they claimed, but if the sum is higher than $ 100 they get
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game. A player would be able to determine the value which they should play based on the assumed distribution of lower-level players described by the
Poisson distribution. Another example of a game involving CH theory is the
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A level-0 player will choose actions regardless of the actions of other players. A level-1 player believes that all other players are level-0 types. A level-n player estimates that all other players are level-0, 1, 2, ...,
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These games suggest that focal points have some saliency. These characteristics make them preferable choices to people. Furthermore, people would assume each other has also noticed the saliency and make the same decision.
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one, they will each receive a prize. Three of the squares are blue and one is red. Assuming they each know nothing about the other player, but that they each do want to win the prize, then they will, reasonably,
250:, which involves two cars racing toward each other on a collision course and in which the driver who first decides to swerve is seen as a coward, while no driver swerving results in a fatal collision for both.
61:, any place and time in the city could be an equilibrium solution. Schelling asked a group of students this question, and found the most common answer was "noon at (the information booth at)
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Stahl and Wilson argue that a focal point is formed because players would try to predict how other players act. They model the level of "rational expectation" players by their ability to
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Although the concept of a focal point has been widely accepted in game theory, it is still unclear how a focal point forms. The researchers have proposed theories from two aspects.
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In a simple example, two people unable to communicate with each other are each shown a panel of four squares and asked to select one; if and only if they both select the
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Letter order game: Give an order to letters A, B, and C. If the three players give the same order, they win an award, otherwise they get nothing.
451:
Mehta, Judith; Starmer, Chris; Sugden, Robert (1994). "The Nature of
Salience: An Experimental Investigation of Pure Coordination Games".
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As N grows, 2/3 of the average will trend towards zero. At this point, the only Nash equilibrium is for all players to guess 0
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Head-tail game: Name "heads" or "tails". If the two players name the same, they win an award, otherwise, they get nothing.
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For the three players, A, B, and C, in letter order game. 9 out of 12 A, 10 out of 12 B, and 14 out of 16 C wrote "ABC".
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For the players to claim part of the $ 100. 36 out of 40 chose $ 50. 2 of the remainder chose $ 49 and $ 49.99.
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611:"Focal Points (or Schelling Points): How We Naturally Organize in Games of Coordination – Mind Your Decisions"
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Bacharach, Michael (1 June 1999). "Interactive team reasoning: A contribution to the theory of co-operation".
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Level N: Assuming all other players reason similarly, 2/3 of the maximum average will never be higher than
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For the two players, A and B, in head-tail game. 16 out of 22 A and 15 out of 22 B chose "heads".
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Here is a subset of the questions raised by
Schelling to prove the existence of a focal point.
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Level 2: The average can be in , which is 2/3 of the maximum average of level 1
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Level 1: The average can be in , which is 2/3 of the maximum average of level 0
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Camerer, Colin F.; Ho, Teck-Hua; Chong, Juin-Kuan (1 August 2004).
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thinking which allows them all to play guesses trending towards 0.
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The existence of focal points can help explain the use of
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form priors (models) about the behavior of other players;
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Pastin, Ivan; Pastine, Tuvana; Humberstone, Tom (2017).
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427:(First ed.). Cambridge: Harvard University Press.
35:. The concept was introduced by the American economist
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TED community experiment on focal / Schelling points
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397:. United Kingdom: Icon Books Inc. pp. 54–61.
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486:Stahl, Dale O.; Wilson, Paul W. (1 July 1995).
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317:{\displaystyle 100\cdot ({\tfrac {2}{3}})^{N}}
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99:choose the best responses given these priors.
167:The results of the informal experiments are
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535:"A Cognitive Hierarchy Model of Games"
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43:(1960). Schelling states that "eople
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651:Schelling Points for Alien Contact
210:The red square is not in a sense a
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729:First-player and second-player win
542:The Quarterly Journal of Economics
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836:Coalition-proof Nash equilibrium
346:Coordination failure (economics)
267:Level 0: The average can be in
254:“Guess 2/3 of the average” game
846:Evolutionarily stable strategy
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112:The cognitive hierarchy theory
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774:Simultaneous action selection
423:Schelling, Thomas C. (1960).
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1711:List of games in game theory
886:Quantal response equilibrium
876:Perfect Bayesian equilibrium
811:Bayes correlated equilibrium
653:https://youtu.be/3lwlNWMl86M
453:The American Economic Review
395:Game Theory: A Graphic Guide
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1180:Optional prisoner's dilemma
906:Self-confirming equilibrium
495:Games and Economic Behavior
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856:Mertens-stable equilibrium
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194:Coordination game example
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1315:Guess 2/3 of the average
1112:Strictly determined game
901:Satisfaction equilibrium
719:Escalation of commitment
554:10.1162/0033553041502225
425:The strategy of conflict
260:Guess 2/3 of the average
246:is also apparent in the
128:Keynesian beauty contest
123:Guess 2/3 of the average
72:, including traditional
41:The Strategy of Conflict
1701:Glossary of game theory
1300:Stackelberg competition
921:Strong Nash equilibrium
641:Common Entries Contests
207:choose the red square.
1726:Tragedy of the commons
1706:List of game theorists
1686:Confrontation analysis
1396:Sprague–Grundy theorem
911:Sequential equilibrium
831:Correlated equilibrium
589:10.1006/reec.1999.0188
507:10.1006/game.1995.1031
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244:anti-coordination game
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230:Collision game example
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63:Grand Central Terminal
16:Concept in game theory
1499:Jean-François Mertens
637:Rare Entries Contests
615:mindyourdecisions.com
577:Research in Economics
366:Equilibrium selection
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148:Schelling's questions
1628:Search optimizations
1504:Jennifer Tour Chayes
1391:Revelation principle
1386:Purification theorem
1325:Nash bargaining game
1290:Bertrand competition
1275:El Farol Bar problem
1240:Electronic mail game
1205:Lewis signaling game
744:Hierarchy of beliefs
361:Surprisingly popular
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186:In coordination game
119:Poisson distribution
33:coordination failure
1676:Bounded rationality
1295:Cournot competition
1245:Rock paper scissors
1220:Battle of the sexes
1210:Volunteer's dilemma
1082:Perfect information
1009:Dominant strategies
841:Epsilon-equilibrium
724:Extensive-form game
1655:Paranoid algorithm
1635:Alpha–beta pruning
1514:John Maynard Smith
1345:Rendezvous problem
1185:Traveler's dilemma
1175:Gift-exchange game
1170:Prisoner's dilemma
1087:Large Poisson game
1054:Bargaining problem
954:Backward induction
926:Subgame perfection
881:Proper equilibrium
371:Rendezvous problem
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134:The team reasoning
88:The level-n theory
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1609:Suzanne Scotchmer
1564:Oskar Morgenstern
1459:Donald B. Gillies
1401:Zermelo's theorem
1330:Induction puzzles
1285:Fair cake-cutting
1260:Public goods game
1190:Coordination game
1064:Intransitive game
989:Forward induction
871:Pareto efficiency
851:Gibbs equilibrium
821:Berge equilibrium
769:Simultaneous game
639:(an example) and
434:978-0-674-84031-7
404:978-1-78578-082-0
356:Simultaneous game
351:Coordination game
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59:coordination game
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1721:Topological game
1716:No-win situation
1614:Thomas Schelling
1594:Robert B. Wilson
1554:Merrill M. Flood
1524:John von Neumann
1434:Ariel Rubinstein
1419:Albert W. Tucker
1270:War of attrition
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1004:Pairing strategy
866:Nash equilibrium
789:Mechanism design
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709:Cooperative game
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1303:
1301:
1298:
1296:
1293:
1291:
1288:
1286:
1283:
1281:
1280:Fair division
1278:
1276:
1273:
1271:
1268:
1266:
1263:
1261:
1258:
1256:
1255:Dictator game
1253:
1251:
1248:
1246:
1243:
1241:
1238:
1236:
1233:
1231:
1228:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1206:
1203:
1201:
1198:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1176:
1173:
1171:
1168:
1166:
1163:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1143:
1142:
1140:
1138:
1134:
1128:
1127:Zero-sum game
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1097:Repeated game
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1076:
1072:
1070:
1067:
1065:
1062:
1060:
1057:
1055:
1052:
1050:
1047:
1046:
1044:
1042:
1036:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1014:Pure strategy
1012:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
985:
982:
980:
977:
975:
974:De-escalation
972:
970:
967:
965:
962:
960:
957:
955:
952:
950:
947:
946:
944:
942:
938:
932:
929:
927:
924:
922:
919:
917:
916:Shapley value
914:
912:
909:
907:
904:
902:
899:
897:
894:
892:
889:
887:
884:
882:
879:
877:
874:
872:
869:
867:
864:
862:
859:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
808:
806:
804:
800:
796:
790:
787:
785:
784:Succinct game
782:
780:
777:
775:
772:
770:
767:
765:
762:
760:
757:
755:
752:
750:
747:
745:
742:
740:
737:
735:
732:
730:
727:
725:
722:
720:
717:
715:
712:
710:
707:
705:
702:
701:
699:
695:
691:
683:
678:
676:
671:
669:
664:
663:
660:
654:
650:
648:
645:
642:
638:
635:
634:
616:
612:
606:
598:
594:
590:
586:
582:
578:
571:
563:
559:
555:
551:
547:
543:
536:
529:
527:
525:
516:
512:
508:
504:
500:
496:
489:
482:
474:
470:
466:
462:
458:
454:
447:
445:
436:
430:
426:
419:
417:
415:
406:
400:
396:
389:
387:
382:
372:
369:
367:
364:
362:
359:
357:
354:
352:
349:
347:
344:
342:
339:
338:
332:
328:
309:
298:
295:
286:
283:
275:
272:
269:
266:
265:
264:
261:
251:
249:
245:
242:This idea of
240:
238:
228:
219:
217:
213:
208:
206:
201:
192:
183:
176:
173:
170:
169:
168:
161:
158:
155:
154:
153:
140:
131:
129:
124:
120:
109:
107:
98:
95:
94:
93:
85:
77:
75:
71:
66:
64:
60:
49:
46:
42:
38:
34:
30:
26:
22:
1574:Peyton Young
1569:Paul Milgrom
1484:Hervé Moulin
1424:Amos Tversky
1366:Folk theorem
1077:-player game
1074:
994:Grim trigger
618:. Retrieved
614:
605:
580:
576:
570:
545:
541:
498:
494:
481:
456:
452:
424:
394:
329:
326:
257:
243:
241:
233:
211:
209:
204:
199:
197:
180:
166:
151:
137:
115:
105:
102:
91:
83:
74:gender roles
70:social norms
67:
55:
44:
40:
39:in his book
28:
24:
18:
1747:Game theory
1691:Coopetition
1494:Jean Tirole
1489:John Conway
1469:Eric Maskin
1265:Blotto game
1250:Pirate game
1059:Global game
1029:Tit for tat
959:Bid shading
949:Appeasement
799:Equilibrium
779:Solved game
714:Determinacy
697:Definitions
690:game theory
341:Game theory
25:focal point
21:game theory
1335:Trust game
1320:Kuhn poker
984:Escalation
979:Deterrence
969:Cheap talk
941:Strategies
759:Preference
688:Topics of
620:2017-12-12
378:References
1519:John Nash
1225:Stag hunt
964:Collusion
597:1090-9443
562:0033-5533
515:0899-8256
465:0002-8282
287:⋅
52:Existence
1741:Category
1660:Lazy SMP
1354:Theorems
1305:Deadlock
1160:Checkers
1041:of games
803:concepts
335:See also
163:nothing.
143:Examples
80:Theories
1412:figures
1195:Chicken
1049:Auction
1039:Classes
473:2118074
595:
560:
513:
471:
463:
431:
401:
212:better
1150:Chess
1137:Games
538:(PDF)
491:(PDF)
469:JSTOR
826:Core
593:ISSN
558:ISSN
511:ISSN
461:ISSN
429:ISBN
399:ISBN
258:The
205:both
200:same
27:(or
23:, a
1410:Key
585:doi
550:doi
546:119
503:doi
284:100
45:can
19:In
1743::
1145:Go
613:.
591:.
581:53
579:.
556:.
544:.
540:.
523:^
509:.
499:10
497:.
493:.
467:.
457:84
455:.
443:^
413:^
385:^
239:.
130:.
1075:n
681:e
674:t
667:v
623:.
599:.
587::
564:.
552::
517:.
505::
475:.
437:.
407:.
310:N
306:)
299:3
296:2
290:(
106:n
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