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:
1567:
1384:
919:
717:
554:
1203:
1022:
634:
54:
824:
591:
Govindan, Srihari; Reny, Philip J.; Robson, Arthur J. (2003). "A Short Proof of
Harsanyi's Purification Theorem".
1293:
1163:
834:
1002:
444:{\displaystyle \Pr(a_{i}\leq a^{*})={\frac {{\frac {1}{2+3/A}}+A}{2A}}={\frac {A}{4A^{2}+6A}}+{\frac {1}{2}}.}
1344:
762:
737:
1694:
1120:
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".
1628:
1379:
1349:
1007:
849:
844:
1664:
1587:
1323:
879:
804:
661:
161:
568:
1679:
1412:
1298:
1095:
889:
707:
76:
1482:
1684:
1283:
1253:
909:
697:
1709:
1689:
1669:
1618:
1288:
1193:
1052:
997:
929:
899:
819:
747:
563:
157:
50:
1168:
1153:
727:
86:
1730:
1502:
1487:
1374:
1273:
1258:
1223:
1188:
787:
732:
654:
69:
8:
1659:
1278:
1228:
1065:
992:
972:
829:
712:
545:
1318:
1638:
1497:
1328:
1308:
1158:
1037:
942:
869:
814:
537:
515:
65:
604:
1735:
1623:
1592:
1547:
1442:
1313:
1268:
1243:
1173:
1047:
977:
967:
859:
809:
757:
630:
519:
1704:
1699:
1633:
1597:
1577:
1537:
1507:
1462:
1417:
1402:
1359:
1213:
854:
791:
777:
742:
600:
573:
507:
1602:
1562:
1517:
1432:
1427:
1148:
1100:
987:
752:
722:
692:
38:
1467:
1542:
1532:
1522:
1457:
1447:
1437:
1422:
1218:
1198:
1183:
1178:
1138:
1105:
1090:
1085:
1075:
884:
618:
549:
135:
124:
35:
28:
1724:
1582:
1572:
1527:
1512:
1492:
1263:
1238:
1110:
1080:
1070:
1057:
962:
904:
839:
772:
533:
495:
139:
46:
31:
16:
Mixed strategy equilibria explained as the limit of pure strategy equilibria
1557:
1552:
1407:
982:
1674:
1477:
1472:
1452:
1248:
1233:
1042:
1012:
947:
937:
767:
702:
678:
622:
577:
474:
games, of a pathological nature, for which this condition fails to hold.
20:
646:
1303:
957:
541:
511:
300:, we can calculate the probability of each player playing Cooperate as
1208:
1128:
952:
1643:
1143:
273:. Seeking a symmetric equilibrium where both players cooperate if
75:
The result is also an important aspect of modern-day inquiries in
1364:
1354:
1032:
61:
532:
1133:
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:
1722:
310:
617:
662:
252:. He should therefore himself Cooperate when
669:
655:
676:
567:
237:; his expected utility from Defecting is
494:
1723:
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:
1747:
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
1695:List of games in game theory
875:Quantal response equilibrium
865:Perfect Bayesian equilibrium
800:Bayes correlated equilibrium
7:
1164:Optional prisoner's dilemma
895:Self-confirming equilibrium
593:Games and Economic Behavior
10:
1752:
1629:Principal variation search
1345:Aumann's agreement theorem
1008:Strategy-stealing argument
920:Trembling hand equilibrium
850:Markov perfect equilibrium
845:Mertens-stable equilibrium
296:). Now we have worked out
82:
1665:Combinatorial game theory
1652:
1611:
1393:
1337:
1324:Princess and monster game
1119:
1021:
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:
1680:Evolutionary game theory
1413:Antoine Augustin Cournot
1299:Guess 2/3 of the average
1096:Strictly determined game
890:Satisfaction equilibrium
708:Escalation of commitment
77:evolutionary game theory
1685:Glossary of game theory
1284:Stackelberg competition
910:Strong Nash equilibrium
164:. The probability that
1710:Tragedy of the commons
1690:List of game theorists
1670:Confrontation analysis
1380:Sprague–Grundy theorem
900:Sequential equilibrium
820:Correlated equilibrium
445:
158:incomplete information
64:distribution over the
51:incomplete information
1483:Jean-François Mertens
446:
1612:Search optimizations
1488:Jennifer Tour Chayes
1375:Revelation principle
1370:Purification theorem
1309:Nash bargaining game
1274:Bertrand competition
1259:El Farol Bar problem
1224:Electronic mail game
1189: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
1660:Bounded rationality
1279:Cournot competition
1229:Rock paper scissors
1204:Battle of the sexes
1194:Volunteer's dilemma
1066:Perfect information
993:Dominant strategies
830:Epsilon-equilibrium
713:Extensive-form game
27:was contributed by
1639:Paranoid algorithm
1619:Alpha–beta pruning
1498:John Maynard Smith
1329:Rendezvous problem
1169:Traveler's dilemma
1159:Gift-exchange game
1154:Prisoner's dilemma
1071:Large Poisson game
1038:Bargaining problem
943:Backward induction
915:Subgame perfection
870:Proper equilibrium
512:10.1007/BF01737554
441:
1718:
1717:
1624:Aspiration window
1593:Suzanne Scotchmer
1548:Oskar Morgenstern
1443:Donald B. Gillies
1385:Zermelo's theorem
1314:Induction puzzles
1269:Fair cake-cutting
1244:Public goods game
1174:Coordination game
1048: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:
1743:
1705:Topological game
1700:No-win situation
1598:Thomas Schelling
1578:Robert B. Wilson
1538:Merrill M. Flood
1508:John von Neumann
1418:Ariel Rubinstein
1403:Albert W. Tucker
1254:War of attrition
1214:Matching pennies
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:
1751:
1750:
1746:
1745:
1744:
1742:
1741:
1740:
1721:
1720:
1719:
1714:
1648:
1634:max^n algorithm
1607:
1603:William Vickrey
1563:Reinhard Selten
1518:Kenneth Binmore
1433:David K. Levine
1428:Daniel Kahneman
1395:
1389:
1365:Negamax theorem
1355:Minimax theorem
1333:
1294:Diner's dilemma
1149:All-pay auction
1115:
1101:Stochastic game
1053:Mean-field game
1024:
1017:
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:
1749:
1739:
1738:
1733:
1716:
1715:
1713:
1712:
1707:
1702:
1697:
1692:
1687:
1682:
1677:
1672:
1667:
1662:
1656:
1654:
1650:
1649:
1647:
1646:
1641:
1636:
1631:
1626:
1621:
1615:
1613:
1609:
1608:
1606:
1605:
1600:
1595:
1590:
1585:
1580:
1575:
1570:
1568:Robert Axelrod
1565:
1560:
1555:
1550:
1545:
1543:Olga Bondareva
1540:
1535:
1533:Melvin Dresher
1530:
1525:
1523:Leonid Hurwicz
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1485:
1480:
1475:
1470:
1465:
1460:
1458:Harold W. Kuhn
1455:
1450:
1448:Drew Fudenberg
1445:
1440:
1438:David M. Kreps
1435:
1430:
1425:
1423:Claude Shannon
1420:
1415:
1410:
1405:
1399:
1397:
1391:
1390:
1388:
1387:
1382:
1377:
1372:
1367:
1362:
1360:Nash's theorem
1357:
1352:
1347:
1341:
1339:
1335:
1334:
1332:
1331:
1326:
1321:
1316:
1311:
1306:
1301:
1296:
1291:
1286:
1281:
1276:
1271:
1266:
1261:
1256:
1251:
1246:
1241:
1236:
1231:
1226:
1221:
1219:Ultimatum game
1216:
1211:
1206:
1201:
1199:Dollar auction
1196:
1191:
1186:
1184:Centipede game
1181:
1176:
1171:
1166:
1161:
1156:
1151:
1146:
1141:
1139:Infinite chess
1136:
1131:
1125:
1123:
1117:
1116:
1114:
1113:
1108:
1106:Symmetric game
1103:
1098:
1093:
1091:Signaling game
1088:
1086:Screening game
1083:
1078:
1076:Potential game
1073:
1068:
1063:
1055:
1050:
1045:
1040:
1035:
1029:
1027:
1019:
1018:
1016:
1015:
1010:
1005:
1003:Mixed strategy
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:
1748:
1737:
1734:
1732:
1729:
1728:
1726:
1711:
1708:
1706:
1703:
1701:
1698:
1696:
1693:
1691:
1688:
1686:
1683:
1681:
1678:
1676:
1673:
1671:
1668:
1666:
1663:
1661:
1658:
1657:
1655:
1653:Miscellaneous
1651:
1645:
1642:
1640:
1637:
1635:
1632:
1630:
1627:
1625:
1622:
1620:
1617:
1616:
1614:
1610:
1604:
1601:
1599:
1596:
1594:
1591:
1589:
1588:Samuel Bowles
1586:
1584:
1583:Roger Myerson
1581:
1579:
1576:
1574:
1573:Robert Aumann
1571:
1569:
1566:
1564:
1561:
1559:
1556:
1554:
1551:
1549:
1546:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1528:Lloyd Shapley
1526:
1524:
1521:
1519:
1516:
1514:
1513:Kenneth Arrow
1511:
1509:
1506:
1504:
1501:
1499:
1496:
1494:
1493:John Harsanyi
1491:
1489:
1486:
1484:
1481:
1479:
1476:
1474:
1471:
1469:
1466:
1464:
1463:Herbert Simon
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1419:
1416:
1414:
1411:
1409:
1406:
1404:
1401:
1400:
1398:
1392:
1386:
1383:
1381:
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1351:
1348:
1346:
1343:
1342:
1340:
1336:
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:
1267:
1265:
1264:Fair division
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1245:
1242:
1240:
1239:Dictator game
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:
1130:
1127:
1126:
1124:
1122:
1118:
1112:
1111:Zero-sum game
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1081:Repeated game
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1062:
1060:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1030:
1028:
1026:
1020:
1014:
1011:
1009:
1006:
1004:
1001:
999:
998:Pure strategy
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:
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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:
299:
294:
290:
284:
280:
271:
267:
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259:
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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:
1558:Peyton Young
1553:Paul Milgrom
1468:Hervé Moulin
1408:Amos Tversky
1369:
1350:Folk theorem
1061:-player game
1058:
983:Grim trigger
626:
623:Tirole, Jean
613:
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586:
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528:
503:
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74:
59:
43:
24:
18:
1731:Game theory
1675:Coopetition
1478:Jean Tirole
1473:John Conway
1453:Eric Maskin
1249:Blotto game
1234:Pirate game
1043:Global game
1013: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
1725:Categories
1319:Trust game
1304: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
1503:John Nash
1209:Stag hunt
953:Collusion
564:CiteSeerX
550:Weiss, B.
520:154484458
335:∗
327:≤
223:+ 2(1 − (
125:Hawk–Dove
66:continuum
55:idealized
1736:Theorems
1644:Lazy SMP
1338:Theorems
1289:Deadlock
1144:Checkers
1025:of games
792:concepts
625:(1991).
506:: 1–23.
260:≤ 2 - 3(
1396:figures
1179:Chicken
1033:Auction
1023:Classes
83:Example
62:ex-ante
633:
566:
518:
72:game.
57:game.
23:, the
1134:Chess
1121:Games
516:S2CID
815:Core
631:ISBN
211:+ 3(
176:is (
127:game
1394:Key
601:doi
574:doi
508:doi
454:As
268:)/2
247:)/2
231:)/2
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1727::
1129:Go
621:;
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595:.
572:.
558:.
548:;
544:;
540:;
536:;
514:.
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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
1059:n
670:e
663:t
656:v
639:.
607:.
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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
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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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