478:
can be purified using this method because if there is ever any non-negative probability that the opponent will play a strategy for which the weakly dominated strategy is not a best response, then one will never wish to play the weakly dominated strategy. Hence, the limit fails to hold because it involves a discontinuity.
473:
The main result of the theorem is that all the mixed strategy equilibria of a given game can be purified using the same sequence of perturbed games. However, in addition to independence of the perturbations, it relies on the set of payoffs for this sequence of games being of full measure. There are
477:
The main problem with these games falls into one of two categories: (1) various mixed strategies of the game are purified by different sequences of perturbed games and (2) some mixed strategies of the game involve weakly dominated strategies. No mixed strategy involving a weakly dominated strategy
53:, in which the payoffs of each player are known to themselves but not their opponents. The idea is that the predicted mixed strategy of the original game emerges from the ever-improving approximations of a game which is not observed by the theorist who designed the original,
44:
The purification theorem shows how such mixed strategy equilibria can emerge even if each players plays a pure strategy, so long as players have incomplete information about the payoffs of their opponents. Such strategies arise as the limit of a series of
449:
469:
Harsanyi's proof involves the strong assumption that the perturbations for each player are independent of the other players. However, further refinements to make the theorem more general have been attempted.
306:
461:
Thus, we can think of the mixed strategy equilibrium as the outcome of pure strategies followed by players who have a small amount of private information about their payoffs.
68:
of payoffs that a player can have. As that continuum shrinks to zero, the players' strategies converge to the predicted Nash equilibria of the original, unperturbed,
41:: each player is wholly indifferent between each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent.
60:
The apparently mixed nature of the strategy is actually just the result of each player playing a pure strategy with threshold values that depend on the
142:
equilibria (Defect, Cooperate) and (Cooperate, Defect). It also has a mixed equilibrium in which each player plays
Cooperate with probability 2/3.
156:
from playing
Cooperate, which is uniformly distributed on . Players only know their own value of this cost. So this is a game of
79:
where the perturbed values are interpreted as distributions over types of players randomly paired in a population to play games.
668:
1572:
1389:
919:
717:
554:
1208:
1027:
634:
54:
824:
591:
Govindan, Srihari; Reny, Philip J.; Robson, Arthur J. (2003). "A Short Proof of
Harsanyi's Purification Theorem".
1298:
1168:
834:
1007:
444:{\displaystyle \Pr(a_{i}\leq a^{*})={\frac {{\frac {1}{2+3/A}}+A}{2A}}={\frac {A}{4A^{2}+6A}}+{\frac {1}{2}}.}
1349:
762:
737:
1699:
1125:
874:
864:
799:
458:→ 0, this approaches 2/3 – the same probability as in the mixed strategy in the complete information game.
914:
894:
498:(1973). "Games with randomly disturbed payoffs: a new rationale for mixed-strategy equilibrium points".
1633:
1384:
1354:
1012:
849:
844:
1669:
1592:
1328:
879:
804:
661:
161:
568:
1684:
1417:
1303:
1100:
889:
707:
76:
1487:
1689:
1288:
1258:
909:
697:
1714:
1694:
1674:
1623:
1293:
1198:
1057:
1002:
929:
899:
819:
747:
563:
157:
50:
1173:
1158:
727:
86:
1735:
1507:
1492:
1379:
1278:
1263:
1228:
1193:
787:
732:
654:
69:
8:
1664:
1283:
1233:
1070:
997:
972:
829:
712:
545:
1323:
1643:
1502:
1333:
1313:
1163:
1042:
942:
869:
814:
537:
515:
65:
604:
1740:
1628:
1597:
1552:
1447:
1318:
1273:
1248:
1178:
1052:
977:
967:
859:
809:
757:
630:
519:
1709:
1704:
1638:
1602:
1582:
1542:
1512:
1467:
1422:
1407:
1364:
1218:
992:
854:
791:
777:
742:
600:
573:
507:
1607:
1567:
1522:
1437:
1432:
1153:
1105:
987:
752:
722:
692:
38:
1472:
1547:
1537:
1527:
1462:
1452:
1442:
1427:
1223:
1203:
1188:
1183:
1143:
1110:
1095:
1090:
1080:
884:
618:
549:
135:
124:
35:
28:
1729:
1587:
1577:
1532:
1517:
1497:
1268:
1243:
1115:
1085:
1075:
1062:
962:
904:
839:
772:
533:
495:
139:
46:
31:
16:
Mixed strategy equilibria explained as the limit of pure strategy equilibria
1562:
1557:
1412:
982:
1679:
1482:
1477:
1457:
1253:
1238:
1047:
1017:
947:
937:
767:
702:
678:
622:
577:
474:
games, of a pathological nature, for which this condition fails to hold.
20:
646:
1308:
957:
541:
511:
300:, we can calculate the probability of each player playing Cooperate as
1213:
1133:
952:
1648:
1148:
273:. Seeking a symmetric equilibrium where both players cooperate if
75:
The result is also an important aspect of modern-day inquiries in
1369:
1359:
1037:
61:
532:
1138:
552:(1983). "Approximate Purificaton of Mixed Strategies".
202:, then player 1's expected utility from Cooperating is
309:
34:
in 1973. The theorem justifies a puzzling aspect of
590:
443:
1727:
310:
617:
662:
252:. He should therefore himself Cooperate when
669:
655:
676:
567:
237:; his expected utility from Defecting is
494:
1728:
650:
500:International Journal of Game Theory
464:
49:equilibria for a disturbed game of
13:
718:First-player and second-player win
555:Mathematics of Operations Research
14:
1752:
825:Coalition-proof Nash equilibrium
629:. MIT Press. pp. 233–234.
835:Evolutionarily stable strategy
611:
584:
526:
488:
339:
313:
189:. If player 2 Cooperates when
1:
763:Simultaneous action selection
605:10.1016/S0899-8256(03)00149-0
481:
138:shown here. The game has two
1700:List of games in game theory
875:Quantal response equilibrium
865:Perfect Bayesian equilibrium
800:Bayes correlated equilibrium
7:
1169:Optional prisoner's dilemma
895:Self-confirming equilibrium
593:Games and Economic Behavior
10:
1757:
1634:Principal variation search
1350:Aumann's agreement theorem
1013:Strategy-stealing argument
920:Trembling hand equilibrium
850:Markov perfect equilibrium
845:Mertens-stable equilibrium
296:). Now we have worked out
82:
1670:Combinatorial game theory
1657:
1616:
1398:
1342:
1329:Princess and monster game
1124:
1026:
928:
880:Quasi-perfect equilibrium
805:Bayesian Nash equilibrium
786:
685:
162:Bayesian Nash equilibrium
160:which we can solve using
145:Suppose that each player
121:
1685:Evolutionary game theory
1418:Antoine Augustin Cournot
1304:Guess 2/3 of the average
1101:Strictly determined game
890:Satisfaction equilibrium
708:Escalation of commitment
77:evolutionary game theory
1690:Glossary of game theory
1289:Stackelberg competition
910:Strong Nash equilibrium
164:. The probability that
1715:Tragedy of the commons
1695:List of game theorists
1675:Confrontation analysis
1385:Sprague–Grundy theorem
900:Sequential equilibrium
820:Correlated equilibrium
445:
158:incomplete information
64:distribution over the
51:incomplete information
1488:Jean-François Mertens
446:
1617:Search optimizations
1493:Jennifer Tour Chayes
1380:Revelation principle
1375:Purification theorem
1314:Nash bargaining game
1279:Bertrand competition
1264:El Farol Bar problem
1229:Electronic mail game
1194:Lewis signaling game
733:Hierarchy of beliefs
578:10.1287/moor.8.3.327
307:
286:, we solve this for
149:bears an extra cost
70:complete information
25:purification theorem
1665:Bounded rationality
1284:Cournot competition
1234:Rock paper scissors
1209:Battle of the sexes
1199:Volunteer's dilemma
1071:Perfect information
998:Dominant strategies
830:Epsilon-equilibrium
713:Extensive-form game
27:was contributed by
1644:Paranoid algorithm
1624:Alpha–beta pruning
1503:John Maynard Smith
1334:Rendezvous problem
1174:Traveler's dilemma
1164:Gift-exchange game
1159:Prisoner's dilemma
1076:Large Poisson game
1043:Bargaining problem
943:Backward induction
915:Subgame perfection
870:Proper equilibrium
512:10.1007/BF01737554
441:
1723:
1722:
1629:Aspiration window
1598:Suzanne Scotchmer
1553:Oskar Morgenstern
1448:Donald B. Gillies
1390:Zermelo's theorem
1319:Induction puzzles
1274:Fair cake-cutting
1249:Public goods game
1179:Coordination game
1053:Intransitive game
978:Forward induction
860:Pareto efficiency
840:Gibbs equilibrium
810:Berge equilibrium
758:Simultaneous game
496:Harsanyi, John C.
465:Technical details
436:
423:
389:
372:
132:
131:
1748:
1710:Topological game
1705:No-win situation
1603:Thomas Schelling
1583:Robert B. Wilson
1543:Merrill M. Flood
1513:John von Neumann
1423:Ariel Rubinstein
1408:Albert W. Tucker
1259:War of attrition
1219:Matching pennies
993:Pairing strategy
855:Nash equilibrium
778:Mechanism design
743:Normal-form game
698:Cooperative game
671:
664:
657:
648:
647:
641:
640:
615:
609:
608:
588:
582:
581:
571:
546:Rosenthal, R. W.
530:
524:
523:
492:
450:
448:
447:
442:
437:
429:
424:
422:
412:
411:
395:
390:
388:
380:
373:
371:
367:
349:
346:
338:
337:
325:
324:
295:
285:
272:
251:
236:
201:
188:
87:
1756:
1755:
1751:
1750:
1749:
1747:
1746:
1745:
1726:
1725:
1724:
1719:
1653:
1639:max^n algorithm
1612:
1608:William Vickrey
1568:Reinhard Selten
1523:Kenneth Binmore
1438:David K. Levine
1433:Daniel Kahneman
1400:
1394:
1370:Negamax theorem
1360:Minimax theorem
1338:
1299:Diner's dilemma
1154:All-pay auction
1120:
1106:Stochastic game
1058:Mean-field game
1029:
1022:
988:Markov strategy
924:
790:
782:
753:Sequential game
738:Information set
723:Game complexity
693:Congestion game
681:
675:
645:
644:
637:
619:Fudenberg, Drew
616:
612:
589:
585:
569:10.1.1.422.3903
531:
527:
493:
489:
484:
467:
428:
407:
403:
399:
394:
381:
363:
353:
348:
347:
345:
333:
329:
320:
316:
308:
305:
304:
287:
279:
274:
258:
253:
238:
209:
203:
196:
190:
172:
169:
154:
85:
39:Nash equilibria
17:
12:
11:
5:
1754:
1744:
1743:
1738:
1721:
1720:
1718:
1717:
1712:
1707:
1702:
1697:
1692:
1687:
1682:
1677:
1672:
1667:
1661:
1659:
1655:
1654:
1652:
1651:
1646:
1641:
1636:
1631:
1626:
1620:
1618:
1614:
1613:
1611:
1610:
1605:
1600:
1595:
1590:
1585:
1580:
1575:
1573:Robert Axelrod
1570:
1565:
1560:
1555:
1550:
1548:Olga Bondareva
1545:
1540:
1538:Melvin Dresher
1535:
1530:
1528:Leonid Hurwicz
1525:
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1485:
1480:
1475:
1470:
1465:
1463:Harold W. Kuhn
1460:
1455:
1453:Drew Fudenberg
1450:
1445:
1443:David M. Kreps
1440:
1435:
1430:
1428:Claude Shannon
1425:
1420:
1415:
1410:
1404:
1402:
1396:
1395:
1393:
1392:
1387:
1382:
1377:
1372:
1367:
1365:Nash's theorem
1362:
1357:
1352:
1346:
1344:
1340:
1339:
1337:
1336:
1331:
1326:
1321:
1316:
1311:
1306:
1301:
1296:
1291:
1286:
1281:
1276:
1271:
1266:
1261:
1256:
1251:
1246:
1241:
1236:
1231:
1226:
1224:Ultimatum game
1221:
1216:
1211:
1206:
1204:Dollar auction
1201:
1196:
1191:
1189:Centipede game
1186:
1181:
1176:
1171:
1166:
1161:
1156:
1151:
1146:
1144:Infinite chess
1141:
1136:
1130:
1128:
1122:
1121:
1119:
1118:
1113:
1111:Symmetric game
1108:
1103:
1098:
1096:Signaling game
1093:
1091:Screening game
1088:
1083:
1081:Potential game
1078:
1073:
1068:
1060:
1055:
1050:
1045:
1040:
1034:
1032:
1024:
1023:
1021:
1020:
1015:
1010:
1008:Mixed strategy
1005:
1000:
995:
990:
985:
980:
975:
970:
965:
960:
955:
950:
945:
940:
934:
932:
926:
925:
923:
922:
917:
912:
907:
902:
897:
892:
887:
885:Risk dominance
882:
877:
872:
867:
862:
857:
852:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
796:
794:
784:
783:
781:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
728:Graphical game
725:
720:
715:
710:
705:
700:
695:
689:
687:
683:
682:
674:
673:
666:
659:
651:
643:
642:
635:
610:
599:(2): 369–374.
583:
562:(3): 327–341.
538:Katznelson, Y.
525:
486:
485:
483:
480:
466:
463:
452:
451:
440:
435:
432:
427:
421:
418:
415:
410:
406:
402:
398:
393:
387:
384:
379:
376:
370:
366:
362:
359:
356:
352:
344:
341:
336:
332:
328:
323:
319:
315:
312:
277:
256:
207:
194:
167:
152:
136:Hawk–Dove game
130:
129:
119:
118:
115:
112:
108:
107:
104:
101:
97:
96:
93:
90:
84:
81:
36:mixed strategy
29:Nobel laureate
15:
9:
6:
4:
3:
2:
1753:
1742:
1739:
1737:
1734:
1733:
1731:
1716:
1713:
1711:
1708:
1706:
1703:
1701:
1698:
1696:
1693:
1691:
1688:
1686:
1683:
1681:
1678:
1676:
1673:
1671:
1668:
1666:
1663:
1662:
1660:
1658:Miscellaneous
1656:
1650:
1647:
1645:
1642:
1640:
1637:
1635:
1632:
1630:
1627:
1625:
1622:
1621:
1619:
1615:
1609:
1606:
1604:
1601:
1599:
1596:
1594:
1593:Samuel Bowles
1591:
1589:
1588:Roger Myerson
1586:
1584:
1581:
1579:
1578:Robert Aumann
1576:
1574:
1571:
1569:
1566:
1564:
1561:
1559:
1556:
1554:
1551:
1549:
1546:
1544:
1541:
1539:
1536:
1534:
1533:Lloyd Shapley
1531:
1529:
1526:
1524:
1521:
1519:
1518:Kenneth Arrow
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1499:
1498:John Harsanyi
1496:
1494:
1491:
1489:
1486:
1484:
1481:
1479:
1476:
1474:
1471:
1469:
1468:Herbert Simon
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1419:
1416:
1414:
1411:
1409:
1406:
1405:
1403:
1397:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1351:
1348:
1347:
1345:
1341:
1335:
1332:
1330:
1327:
1325:
1322:
1320:
1317:
1315:
1312:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1292:
1290:
1287:
1285:
1282:
1280:
1277:
1275:
1272:
1270:
1269:Fair division
1267:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1245:
1244:Dictator game
1242:
1240:
1237:
1235:
1232:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1175:
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1140:
1137:
1135:
1132:
1131:
1129:
1127:
1123:
1117:
1116:Zero-sum game
1114:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1092:
1089:
1087:
1086:Repeated game
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1065:
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1035:
1033:
1031:
1025:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1003:Pure strategy
1001:
999:
996:
994:
991:
989:
986:
984:
981:
979:
976:
974:
971:
969:
966:
964:
963:De-escalation
961:
959:
956:
954:
951:
949:
946:
944:
941:
939:
936:
935:
933:
931:
927:
921:
918:
916:
913:
911:
908:
906:
905:Shapley value
903:
901:
898:
896:
893:
891:
888:
886:
883:
881:
878:
876:
873:
871:
868:
866:
863:
861:
858:
856:
853:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
797:
795:
793:
789:
785:
779:
776:
774:
773:Succinct game
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
690:
688:
684:
680:
672:
667:
665:
660:
658:
653:
652:
649:
638:
636:9780262061414
632:
628:
624:
620:
614:
606:
602:
598:
594:
587:
579:
575:
570:
565:
561:
557:
556:
551:
547:
543:
539:
535:
534:Aumann, R. J.
529:
521:
517:
513:
509:
505:
501:
497:
491:
487:
479:
475:
471:
462:
459:
457:
438:
433:
430:
425:
419:
416:
413:
408:
404:
400:
396:
391:
385:
382:
377:
374:
368:
364:
360:
357:
354:
350:
342:
334:
330:
326:
321:
317:
303:
302:
301:
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140:pure strategy
137:
134:Consider the
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47:pure strategy
42:
40:
37:
33:
32:John Harsanyi
30:
26:
22:
1563:Peyton Young
1558:Paul Milgrom
1473:Hervé Moulin
1413:Amos Tversky
1374:
1355:Folk theorem
1066:-player game
1063:
983:Grim trigger
626:
623:Tirole, Jean
613:
596:
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586:
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528:
503:
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74:
59:
43:
24:
18:
1736:Game theory
1680:Coopetition
1483:Jean Tirole
1478:John Conway
1458:Eric Maskin
1254:Blotto game
1239:Pirate game
1048:Global game
1018:Tit for tat
948:Bid shading
938:Appeasement
788:Equilibrium
768:Solved game
703:Determinacy
686:Definitions
679:game theory
627:Game Theory
291:= 1/(2 + 3/
21:game theory
1730:Categories
1324:Trust game
1309:Kuhn poker
973:Escalation
968:Deterrence
958:Cheap talk
930:Strategies
748:Preference
677:Topics of
542:Radner, R.
482:References
123:Fig. 1: a
117:0, 0
114:4, 2
106:2, 4
103:3, 3
1508:John Nash
1214:Stag hunt
953:Collusion
564:CiteSeerX
550:Weiss, B.
520:154484458
335:∗
327:≤
223:+ 2(1 − (
125:Hawk–Dove
66:continuum
55:idealized
1741:Theorems
1649:Lazy SMP
1343:Theorems
1294:Deadlock
1149:Checkers
1030:of games
792:concepts
625:(1991).
506:: 1–23.
260:≤ 2 - 3(
1401:figures
1184:Chicken
1038:Auction
1028:Classes
83:Example
62:ex-ante
633:
566:
518:
72:game.
57:game.
23:, the
1139:Chess
1126:Games
516:S2CID
815:Core
631:ISBN
211:+ 3(
176:is (
127:game
1399:Key
601:doi
574:doi
508:doi
454:As
268:)/2
247:)/2
231:)/2
219:)/2
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1732::
1134:Go
621:;
597:45
595:.
572:.
558:.
548:;
544:;
540:;
536:;
514:.
502:.
311:Pr
298:a*
289:a*
283:a*
281:≤
262:a*
243:+
241:a*
239:4(
227:+
225:a*
215:+
213:a*
199:a*
197:≤
180:+
178:a*
174:a*
171:≤
111:D
100:C
95:D
92:C
1064:n
670:e
663:t
656:v
639:.
607:.
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580:.
576::
560:8
522:.
510::
504:2
456:A
439:.
434:2
431:1
426:+
420:A
417:6
414:+
409:2
405:A
401:4
397:A
392:=
386:A
383:2
378:A
375:+
369:A
365:/
361:3
358:+
355:2
351:1
343:=
340:)
331:a
322:i
318:a
314:(
293:A
278:i
276:a
270:A
266:A
264:+
257:1
255:a
249:A
245:A
235:)
233:A
229:A
221:A
217:A
208:1
206:a
204:−
195:2
192:a
186:A
182:A
168:i
166:a
153:i
151:a
147:i
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