303:
The strong Nash concept is criticized as too "strong" in that the environment allows for unlimited private communication. As a result of these requirements, Strong Nash rarely exists in games interesting enough to deserve study. Nevertheless, it is possible for there to be multiple strong Nash
83:
is a combination of actions of the different players, in which no coalition of players can cooperatively deviate in a way that strictly benefits all of its members, given that the actions of the other players remain fixed. This is in contrast to simple Nash equilibrium, which considers only
250:
There is a unique Nash equilibrium at (0,0), with payoff vector (0,0). However, it is not SNE as the coalition {1,2} can deviate to (1,1), with payoff vector (1,1). Indeed, coalition consistency is violated at
236:
So, with w1=0.6,w2=0.4 the point (1/3,3/4) is a consistent social-welfare-best-response for all coalitions simultaneously. Therefore, an SNE exists, at the same point (1/3,3/4).
262:, the social-welfare-best-response is either on the line (1,0)--(1,1) or on the line (0,1)--(1,1); but any such point is not a best-response for the player playing 1.
239:
Here is an example in which the coalition consistency fails, and indeed there is no SNE.There are two players, with strategy space . Their utility functions are:
193:
is not an SNE, the condition requires that one can move to a different strategy-profile which is a social-welfare-best-response for all coalitions simultaneously.
210:
which are continuous and convex. It remains to check coalition consistency. For every strategy-tuple x, we check the weighted-best-response of each coalition:
265:
Nessah and Tian also present a necessary and sufficient condition for SNE existence, along with an algorithm that finds an SNE if and only if it exists.
1571:
327:
is a CPNE. Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size
509:
199:
For example, consider a game with two players, with strategy spaces and , which are clearly compact and convex. The utility functions are:
1408:
1225:
760:
558:
319:(CPNE) in which the equilibria are immune to multilateral deviations that are self-enforcing. Every correlated strategy supported by
1044:
863:
665:
316:
1134:
675:
46:
1004:
232:, we can find out that the maximum point is y1=w2/(2*w1) and y2=w1/(2*w2). By taking w1=0.6,w2=0.4 we get y1=1/3 and y2=3/4.
92:, in which there are typically many more players than possible outcomes, and so plain Nash equilibria are far too abundant.
1185:
603:
578:
1535:
961:
715:
705:
640:
221:
For the coalition {2}, we similarly see that for every x1, the maximum payoff is attained at the smallest point, y2=3/4.
755:
735:
218:(-y1 + x2 + 1); it is clear that the maximum is attained at the smallest point of the strategy space, which is y1=1/3.
1220:
312:
that exists, but this is only unique (apart from inconsequential changes) when there is a majority
Condorcet winner.
1469:
1190:
848:
690:
685:
1505:
1428:
1164:
720:
645:
502:
1520:
1253:
1139:
936:
730:
548:
1323:
1525:
1124:
1094:
538:
320:
1459:
1550:
1530:
1510:
1129:
1034:
893:
843:
838:
770:
740:
660:
588:
568:
339:
1009:
994:
415:
1343:
1328:
1215:
1210:
1114:
1099:
1064:
1029:
628:
573:
495:
284:. This can be seen by considering a deviation of the grand coalition - the coalition of all players.
63:
1500:
1119:
1069:
906:
833:
813:
670:
553:
1479:
1338:
1169:
1149:
999:
878:
783:
710:
655:
1464:
1433:
1388:
1283:
1154:
1109:
1084:
1014:
888:
818:
808:
700:
650:
598:
437:
281:
1545:
1540:
1474:
1438:
1418:
1378:
1348:
1303:
1258:
1243:
1200:
1054:
695:
632:
618:
583:
472:
464:
427:
389:
332:
309:
27:
380:
B. D. Bernheim; B. Peleg; M. D. Whinston (1987), "Coalition-Proof
Equilibria I. Concepts",
1443:
1403:
1358:
1273:
1268:
989:
941:
828:
593:
563:
533:
324:
305:
229:
1308:
1383:
1373:
1363:
1298:
1288:
1278:
1263:
1059:
1039:
1024:
1019:
979:
946:
931:
926:
916:
725:
89:
1565:
1423:
1413:
1368:
1353:
1333:
1159:
1104:
1079:
951:
921:
911:
898:
803:
745:
680:
613:
393:
85:
273:
Every SNE is a Nash equilibrium. This can be seen by considering a deviation of the
100:
Nessah and Tian prove that an SNE exists if the following conditions are satisfied:
1398:
1393:
1248:
823:
1515:
1318:
1313:
1293:
1089:
1074:
883:
853:
788:
778:
608:
543:
519:
338:
Confusingly, the concept of a strong Nash equilibrium is unrelated to that of a
116:
105:
73:
31:
432:
342:. That is, a Nash equilibrium can be both strong and weak, either, or neither.
1144:
798:
109:
441:
1049:
969:
793:
468:
1484:
984:
487:
1205:
1195:
873:
477:
974:
88:
in 1959. SNE is particularly useful in areas such as the study of
315:
A relatively weaker yet refined Nash stability concept is called
224:
For the coalition {1,2}, with weights w1,w2, we need to find max
84:
deviations by individual players. The concept was introduced by
491:
455:
D. Moreno; J. Wooders (1996), "Coalition-Proof
Equilibrium",
364:-person games in "Contributions to the Theory of Games IV"
214:
For the coalition {1}, we need to find, for every x2, max
255:=(0,0): for the coalition {1,2}, for any weight-vector
308:, there is always a strong Nash equilibrium for any
1493:
1452:
1234:
1178:
960:
862:
769:
627:
526:
58:
53:
42:
37:
21:
420:Journal of Mathematical Analysis and Applications
49:(if the strong Nash equilibrium is not also weak)
375:
373:
228:(w1*(-y1 + y2 + 1)+w2*(y1 - y2 + 1)). Using the
503:
126:property: there exists a weight-vector-tuple
8:
416:"On the existence of strong Nash equilibria"
414:Nessah, Rabia; Tian, Guoqiang (2014-06-15).
510:
496:
488:
476:
431:
366:, Princeton Univ. Press, Princeton, N.J..
360:Acceptable points in general cooperative
350:
141:, such that for each strategy-profile
115:The payoff function of each player is
18:
104:The strategy space of each player is
7:
409:
407:
405:
403:
559:First-player and second-player win
145:, there exists a strategy-profile
14:
1572:Game theory equilibrium concepts
666:Coalition-proof Nash equilibrium
317:coalition-proof Nash equilibrium
160:) social welfare to members of
676:Evolutionarily stable strategy
47:Evolutionarily stable strategy
1:
604:Simultaneous action selection
304:equilibria. For instance, in
1536:List of games in game theory
716:Quantal response equilibrium
706:Perfect Bayesian equilibrium
641:Bayes correlated equilibrium
394:10.1016/0022-0531(87)90099-8
185:can be taken to be equal to
156:maximizes the weighted (by w
130:, assigning a weight-vector
1005:Optional prisoner's dilemma
736:Self-confirming equilibrium
457:Games and Economic Behavior
137:to each possible coalition
1588:
1470:Principal variation search
1186:Aumann's agreement theorem
849:Strategy-stealing argument
761:Trembling hand equilibrium
691:Markov perfect equilibrium
686:Mertens-stable equilibrium
433:10.1016/j.jmaa.2014.01.030
382:Journal of Economic Theory
1506:Combinatorial game theory
1165:Princess and monster game
721:Quasi-perfect equilibrium
646:Bayesian Nash equilibrium
331:. CPNE is related to the
321:iterated strict dominance
26:
1521:Evolutionary game theory
1254:Antoine Augustin Cournot
1140:Guess 2/3 of the average
937:Strictly determined game
731:Satisfaction equilibrium
549:Escalation of commitment
1526:Glossary of game theory
1125:Stackelberg competition
751:Strong Nash equilibrium
181:is itself an SNE, then
78:strong Nash equilibrium
22:Strong Nash equilibrium
1551:Tragedy of the commons
1531:List of game theorists
1511:Confrontation analysis
1221:Sprague–Grundy theorem
741:Sequential equilibrium
661:Correlated equilibrium
469:10.1006/game.1996.0095
277:singleton coalitions.
66:of more than 2 players
16:Concept in game theory
1324:Jean-François Mertens
340:weak Nash equilibrium
203:u1(x) = - x1 + x2 + 1
124:coalition consistency
64:non-cooperative games
1453:Search optimizations
1329:Jennifer Tour Chayes
1216:Revelation principle
1211:Purification theorem
1150:Nash bargaining game
1115:Bertrand competition
1100:El Farol Bar problem
1065:Electronic mail game
1030:Lewis signaling game
574:Hierarchy of beliefs
287:Every SNE is in the
280:Every SNE is weakly
1501:Bounded rationality
1120:Cournot competition
1070:Rock paper scissors
1045:Battle of the sexes
1035:Volunteer's dilemma
907:Perfect information
834:Dominant strategies
671:Epsilon-equilibrium
554:Extensive-form game
243:u1(x) = -x1 + 2*x2;
206:u2(x) = x1 - x2 + 1
1480:Paranoid algorithm
1460:Alpha–beta pruning
1339:John Maynard Smith
1170:Rendezvous problem
1010:Traveler's dilemma
1000:Gift-exchange game
995:Prisoner's dilemma
912:Large Poisson game
879:Bargaining problem
784:Backward induction
756:Subgame perfection
711:Proper equilibrium
358:R. Aumann (1959),
333:theory of the core
246:u2(x) = 2*x1 - x2.
1559:
1558:
1465:Aspiration window
1434:Suzanne Scotchmer
1389:Oskar Morgenstern
1284:Donald B. Gillies
1226:Zermelo's theorem
1155:Induction puzzles
1110:Fair cake-cutting
1085:Public goods game
1015:Coordination game
889:Intransitive game
819:Forward induction
701:Pareto efficiency
681:Gibbs equilibrium
651:Berge equilibrium
599:Simultaneous game
70:
69:
1579:
1546:Topological game
1541:No-win situation
1439:Thomas Schelling
1419:Robert B. Wilson
1379:Merrill M. Flood
1349:John von Neumann
1259:Ariel Rubinstein
1244:Albert W. Tucker
1095:War of attrition
1055:Matching pennies
696:Nash equilibrium
619:Mechanism design
584:Normal-form game
539:Cooperative game
512:
505:
498:
489:
483:
482:
480:
452:
446:
445:
435:
411:
398:
397:
377:
368:
367:
355:
310:Condorcet winner
282:Pareto-efficient
28:Solution concept
19:
1587:
1586:
1582:
1581:
1580:
1578:
1577:
1576:
1562:
1561:
1560:
1555:
1489:
1475:max^n algorithm
1448:
1444:William Vickrey
1404:Reinhard Selten
1359:Kenneth Binmore
1274:David K. Levine
1269:Daniel Kahneman
1236:
1230:
1206:Negamax theorem
1196:Minimax theorem
1174:
1135:Diner's dilemma
990:All-pay auction
956:
942:Stochastic game
894:Mean-field game
865:
858:
829:Markov strategy
765:
631:
623:
594:Sequential game
579:Information set
564:Game complexity
534:Congestion game
522:
516:
486:
454:
453:
449:
413:
412:
401:
379:
378:
371:
357:
356:
352:
348:
325:Pareto frontier
306:Approval voting
301:
289:weak alpha-core
271:
260:
230:derivative test
227:
217:
173:
159:
154:
135:
119:and continuous;
98:
17:
12:
11:
5:
1585:
1583:
1575:
1574:
1564:
1563:
1557:
1556:
1554:
1553:
1548:
1543:
1538:
1533:
1528:
1523:
1518:
1513:
1508:
1503:
1497:
1495:
1491:
1490:
1488:
1487:
1482:
1477:
1472:
1467:
1462:
1456:
1454:
1450:
1449:
1447:
1446:
1441:
1436:
1431:
1426:
1421:
1416:
1411:
1409:Robert Axelrod
1406:
1401:
1396:
1391:
1386:
1384:Olga Bondareva
1381:
1376:
1374:Melvin Dresher
1371:
1366:
1364:Leonid Hurwicz
1361:
1356:
1351:
1346:
1341:
1336:
1331:
1326:
1321:
1316:
1311:
1306:
1301:
1299:Harold W. Kuhn
1296:
1291:
1289:Drew Fudenberg
1286:
1281:
1279:David M. Kreps
1276:
1271:
1266:
1264:Claude Shannon
1261:
1256:
1251:
1246:
1240:
1238:
1232:
1231:
1229:
1228:
1223:
1218:
1213:
1208:
1203:
1201:Nash's theorem
1198:
1193:
1188:
1182:
1180:
1176:
1175:
1173:
1172:
1167:
1162:
1157:
1152:
1147:
1142:
1137:
1132:
1127:
1122:
1117:
1112:
1107:
1102:
1097:
1092:
1087:
1082:
1077:
1072:
1067:
1062:
1060:Ultimatum game
1057:
1052:
1047:
1042:
1040:Dollar auction
1037:
1032:
1027:
1025:Centipede game
1022:
1017:
1012:
1007:
1002:
997:
992:
987:
982:
980:Infinite chess
977:
972:
966:
964:
958:
957:
955:
954:
949:
947:Symmetric game
944:
939:
934:
932:Signaling game
929:
927:Screening game
924:
919:
917:Potential game
914:
909:
904:
896:
891:
886:
881:
876:
870:
868:
860:
859:
857:
856:
851:
846:
844:Mixed strategy
841:
836:
831:
826:
821:
816:
811:
806:
801:
796:
791:
786:
781:
775:
773:
767:
766:
764:
763:
758:
753:
748:
743:
738:
733:
728:
726:Risk dominance
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
637:
635:
625:
624:
622:
621:
616:
611:
606:
601:
596:
591:
586:
581:
576:
571:
569:Graphical game
566:
561:
556:
551:
546:
541:
536:
530:
528:
524:
523:
517:
515:
514:
507:
500:
492:
485:
484:
447:
426:(2): 871–885.
399:
369:
349:
347:
344:
300:
297:
293:weak-beta core
270:
267:
258:
248:
247:
244:
234:
233:
225:
222:
219:
215:
208:
207:
204:
197:
196:
195:
194:
168:
157:
152:
133:
120:
113:
97:
94:
90:voting systems
68:
67:
60:
56:
55:
51:
50:
44:
40:
39:
35:
34:
24:
23:
15:
13:
10:
9:
6:
4:
3:
2:
1584:
1573:
1570:
1569:
1567:
1552:
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1499:
1498:
1496:
1494:Miscellaneous
1492:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1461:
1458:
1457:
1455:
1451:
1445:
1442:
1440:
1437:
1435:
1432:
1430:
1429:Samuel Bowles
1427:
1425:
1424:Roger Myerson
1422:
1420:
1417:
1415:
1414:Robert Aumann
1412:
1410:
1407:
1405:
1402:
1400:
1397:
1395:
1392:
1390:
1387:
1385:
1382:
1380:
1377:
1375:
1372:
1370:
1369:Lloyd Shapley
1367:
1365:
1362:
1360:
1357:
1355:
1354:Kenneth Arrow
1352:
1350:
1347:
1345:
1342:
1340:
1337:
1335:
1334:John Harsanyi
1332:
1330:
1327:
1325:
1322:
1320:
1317:
1315:
1312:
1310:
1307:
1305:
1304:Herbert Simon
1302:
1300:
1297:
1295:
1292:
1290:
1287:
1285:
1282:
1280:
1277:
1275:
1272:
1270:
1267:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1245:
1242:
1241:
1239:
1233:
1227:
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1207:
1204:
1202:
1199:
1197:
1194:
1192:
1189:
1187:
1184:
1183:
1181:
1177:
1171:
1168:
1166:
1163:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1106:
1105:Fair division
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1080:Dictator game
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
1001:
998:
996:
993:
991:
988:
986:
983:
981:
978:
976:
973:
971:
968:
967:
965:
963:
959:
953:
952:Zero-sum game
950:
948:
945:
943:
940:
938:
935:
933:
930:
928:
925:
923:
922:Repeated game
920:
918:
915:
913:
910:
908:
905:
903:
901:
897:
895:
892:
890:
887:
885:
882:
880:
877:
875:
872:
871:
869:
867:
861:
855:
852:
850:
847:
845:
842:
840:
839:Pure strategy
837:
835:
832:
830:
827:
825:
822:
820:
817:
815:
812:
810:
807:
805:
804:De-escalation
802:
800:
797:
795:
792:
790:
787:
785:
782:
780:
777:
776:
774:
772:
768:
762:
759:
757:
754:
752:
749:
747:
746:Shapley value
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
638:
636:
634:
630:
626:
620:
617:
615:
614:Succinct game
612:
610:
607:
605:
602:
600:
597:
595:
592:
590:
587:
585:
582:
580:
577:
575:
572:
570:
567:
565:
562:
560:
557:
555:
552:
550:
547:
545:
542:
540:
537:
535:
532:
531:
529:
525:
521:
513:
508:
506:
501:
499:
494:
493:
490:
479:
474:
470:
466:
462:
458:
451:
448:
443:
439:
434:
429:
425:
421:
417:
410:
408:
406:
404:
400:
395:
391:
387:
383:
376:
374:
370:
365:
361:
354:
351:
345:
343:
341:
336:
334:
330:
326:
322:
318:
313:
311:
307:
298:
296:
294:
290:
285:
283:
278:
276:
268:
266:
263:
261:
254:
245:
242:
241:
240:
237:
231:
223:
220:
213:
212:
211:
205:
202:
201:
200:
192:
188:
184:
180:
177:Note that if
176:
175:
172:
167:
163:
155:
148:
144:
140:
136:
129:
125:
121:
118:
114:
111:
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25:
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1399:Peyton Young
1394:Paul Milgrom
1309:Hervé Moulin
1249:Amos Tversky
1191:Folk theorem
902:-player game
899:
824:Grim trigger
750:
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54:Significance
38:Relationship
1516:Coopetition
1319:Jean Tirole
1314:John Conway
1294:Eric Maskin
1090:Blotto game
1075:Pirate game
884:Global game
854:Tit for tat
789:Bid shading
779:Appeasement
629:Equilibrium
609:Solved game
544:Determinacy
527:Definitions
520:game theory
323:and on the
291:and in the
74:game theory
32:game theory
1160:Trust game
1145:Kuhn poker
814:Escalation
809:Deterrence
799:Cheap talk
771:Strategies
589:Preference
518:Topics of
478:10016/4408
463:: 80–112,
346:References
269:Properties
1344:John Nash
1050:Stag hunt
794:Collusion
442:0022-247X
299:Criticism
149:in which
96:Existence
43:Subset of
1566:Category
1485:Lazy SMP
1179:Theorems
1130:Deadlock
985:Checkers
866:of games
633:concepts
388:: 1–12,
164:, given
59:Used for
1237:figures
1020:Chicken
874:Auction
864:Classes
117:concave
106:compact
440:
110:convex
975:Chess
962:Games
226:y1,y2
189:. If
81:(SNE)
656:Core
438:ISSN
122:The
108:and
76:, a
62:All
1235:Key
473:hdl
465:doi
428:doi
424:414
390:doi
72:In
30:in
1568::
970:Go
471:,
461:17
459:,
436:.
422:.
418:.
402:^
386:42
384:,
372:^
335:.
295:.
216:y1
174:.
900:n
511:e
504:t
497:v
481:.
475::
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392::
362:n
329:k
275:n
259:S
257:w
253:x
191:x
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183:z
179:x
171:S
169:-
166:x
162:S
158:S
153:S
151:z
147:z
143:x
139:S
134:S
132:w
128:w
112:;
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