399:. Intuitively, since Player 1 expects Player 2 to grab the penny, he is not concerned about leaving Player 2 uncertain about whether it is heads up or tails up. However, it can be seen that the unique proper equilibrium of this game is the one where Player 1 hides the penny heads up with probability 1/2 and tails up with probability 1/2 (and Player 2 grabs the penny). This unique proper equilibrium can be motivated intuitively as follows: Player 1 fully expects Player 2 to grab the penny. However, Player 1 still prepares for the unlikely event that Player 2 does not grab the penny and instead for some reason decides to make a guess. Player 1 prepares for this event by making sure that Player 2 has no information about whether the penny is heads up or tails up, exactly as in the original
389:
Player 1 (row player) hides a penny and if Player 2 (column player) guesses correctly whether it is heads up or tails up, he gets the penny. In this variant, Player 2 has a third option: Grabbing the penny without guessing. The
394:
of the game are the strategy profiles where Player 2 grabs the penny with probability 1. Any mixed strategy of Player 1 is in (Nash) equilibrium with this pure strategy of Player 2. Any such pair is even
249:
121:
312:
292:
269:
166:
145:
170:
if, whenever a player has two pure strategies s and s' such that the expected payoff of playing s is smaller than the expected payoff of playing s' (that is
427:
of an extensive form game. It was shown by van Damme that a normal form proper equilibrium of an extensive form game is behaviorally equivalent to a
1549:
487:
173:
1386:
1203:
738:
536:
416:
396:
82:
43:
1022:
841:
643:
94:
1112:
1554:
982:
653:
821:
85:
by assuming that more costly trembles are made with significantly smaller probability than less costly ones.
1163:
581:
556:
456:
A relationship between perfect equilibria in extensive form games and proper equilibria in normal form games
1513:
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100:
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1107:
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638:
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The strategy profile of the game is said to be a proper equilibrium if it is a limit point, as
254:
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987:
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1122:
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in two different ways, completely analogous to the two different ways
1027:
947:
771:
1462:
962:
1183:
1173:
851:
952:
419:
is applied to extensive games. This leads to the notions of
406:
244:{\displaystyle u(s,\sigma _{-i})<u(s',\sigma _{-i})}
300:
280:
257:
176:
154:
133:
103:
306:
286:
263:
243:
160:
139:
115:
251:), then the probability assigned to s is at most
1541:
481:
442:Refinements of the Nash equilibrium concept
488:
474:
495:
411:One may apply the properness notion to
1542:
322:The game to the right is a variant of
271:times the probability assigned to s'.
77:. Proper equilibrium further refines
469:
460:International Journal of Game Theory
446:International Journal of Game Theory
407:Proper equilibria of extensive games
13:
537:First-player and second-player win
83:trembling hand perfect equilibrium
44:Trembling hand perfect equilibrium
14:
1566:
425:extensive form proper equilibrium
329:
1550:Game theory equilibrium concepts
644:Coalition-proof Nash equilibrium
294:approaches 0, of a sequence of
654:Evolutionarily stable strategy
421:normal form proper equilibrium
328:
238:
211:
202:
180:
116:{\displaystyle \epsilon >0}
1:
582:Simultaneous action selection
434:
331:Matching Pennies with a twist
88:
1514:List of games in game theory
694:Quantal response equilibrium
684:Perfect Bayesian equilibrium
619:Bayes correlated equilibrium
382:
379:
376:
366:
363:
360:
7:
983:Optional prisoner's dilemma
714:Self-confirming equilibrium
314:-proper strategy profiles.
10:
1571:
1448:Principal variation search
1164:Aumann's agreement theorem
827:Strategy-stealing argument
739:Trembling hand equilibrium
669:Markov perfect equilibrium
664:Mertens-stable equilibrium
317:
1484:Combinatorial game theory
1471:
1430:
1212:
1156:
1143:Princess and monster game
938:
840:
747:
699:Quasi-perfect equilibrium
624:Bayesian Nash equilibrium
605:
504:
429:quasi-perfect equilibrium
417:trembling hand perfection
307:{\displaystyle \epsilon }
287:{\displaystyle \epsilon }
264:{\displaystyle \epsilon }
161:{\displaystyle \epsilon }
54:
49:
39:
34:
23:
18:
1499:Evolutionary game theory
1232:Antoine Augustin Cournot
1118:Guess 2/3 of the average
915:Strictly determined game
709:Satisfaction equilibrium
527:Escalation of commitment
1504:Glossary of game theory
1103:Stackelberg competition
729:Strong Nash equilibrium
140:{\displaystyle \sigma }
1529:Tragedy of the commons
1509:List of game theorists
1489:Confrontation analysis
1199:Sprague–Grundy theorem
719:Sequential equilibrium
639:Correlated equilibrium
397:trembling hand perfect
308:
288:
265:
245:
162:
141:
117:
1555:Non-cooperative games
1302:Jean-François Mertens
309:
289:
266:
246:
163:
142:
118:
97:game and a parameter
1431:Search optimizations
1307:Jennifer Tour Chayes
1194:Revelation principle
1189:Purification theorem
1128:Nash bargaining game
1093:Bertrand competition
1078:El Farol Bar problem
1043:Electronic mail game
1008:Lewis signaling game
552:Hierarchy of beliefs
413:extensive form games
298:
278:
255:
174:
152:
131:
101:
1479:Bounded rationality
1098:Cournot competition
1048:Rock paper scissors
1023:Battle of the sexes
1013:Volunteer's dilemma
885:Perfect information
812:Dominant strategies
649:Epsilon-equilibrium
532:Extensive-form game
448:, 15:133-154, 1978.
333:
69:is a refinement of
1458:Paranoid algorithm
1438:Alpha–beta pruning
1317:John Maynard Smith
1148:Rendezvous problem
988:Traveler's dilemma
978:Gift-exchange game
973:Prisoner's dilemma
890:Large Poisson game
857:Bargaining problem
762:Backward induction
734:Subgame perfection
689:Proper equilibrium
440:Roger B. Myerson.
304:
284:
261:
241:
158:
137:
127:strategy profile
113:
67:Proper equilibrium
19:Proper equilibrium
1537:
1536:
1443:Aspiration window
1412:Suzanne Scotchmer
1367:Oskar Morgenstern
1262:Donald B. Gillies
1204:Zermelo's theorem
1133:Induction puzzles
1088:Fair cake-cutting
1063:Public goods game
993:Coordination game
867:Intransitive game
797:Forward induction
679:Pareto efficiency
659:Gibbs equilibrium
629:Berge equilibrium
577:Simultaneous game
387:
386:
147:is defined to be
64:
63:
1562:
1524:Topological game
1519:No-win situation
1417:Thomas Schelling
1397:Robert B. Wilson
1357:Merrill M. Flood
1327:John von Neumann
1237:Ariel Rubinstein
1222:Albert W. Tucker
1073:War of attrition
1033:Matching pennies
674:Nash equilibrium
597:Mechanism design
562:Normal-form game
517:Cooperative game
490:
483:
476:
467:
466:
401:Matching Pennies
334:
324:Matching Pennies
313:
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310:
305:
293:
291:
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167:
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122:
120:
119:
114:
81:'s notion of a
75:Roger B. Myerson
71:Nash Equilibrium
59:Roger B. Myerson
25:Solution concept
16:
15:
1570:
1569:
1565:
1564:
1563:
1561:
1560:
1559:
1540:
1539:
1538:
1533:
1467:
1453:max^n algorithm
1426:
1422:William Vickrey
1382:Reinhard Selten
1337:Kenneth Binmore
1252:David K. Levine
1247:Daniel Kahneman
1214:
1208:
1184:Negamax theorem
1174:Minimax theorem
1152:
1113:Diner's dilemma
968:All-pay auction
934:
920:Stochastic game
872:Mean-field game
843:
836:
807:Markov strategy
743:
609:
601:
572:Sequential game
557:Information set
542:Game complexity
512:Congestion game
500:
494:
462:13:1--13, 1984.
437:
409:
392:Nash equilibria
320:
299:
296:
295:
279:
276:
275:
256:
253:
252:
229:
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175:
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153:
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132:
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79:Reinhard Selten
12:
11:
5:
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1558:
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1521:
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1428:
1427:
1425:
1424:
1419:
1414:
1409:
1404:
1399:
1394:
1389:
1387:Robert Axelrod
1384:
1379:
1374:
1369:
1364:
1362:Olga Bondareva
1359:
1354:
1352:Melvin Dresher
1349:
1344:
1342:Leonid Hurwicz
1339:
1334:
1329:
1324:
1319:
1314:
1309:
1304:
1299:
1294:
1289:
1284:
1279:
1277:Harold W. Kuhn
1274:
1269:
1267:Drew Fudenberg
1264:
1259:
1257:David M. Kreps
1254:
1249:
1244:
1242:Claude Shannon
1239:
1234:
1229:
1224:
1218:
1216:
1210:
1209:
1207:
1206:
1201:
1196:
1191:
1186:
1181:
1179:Nash's theorem
1176:
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1125:
1120:
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1105:
1100:
1095:
1090:
1085:
1080:
1075:
1070:
1065:
1060:
1055:
1050:
1045:
1040:
1038:Ultimatum game
1035:
1030:
1025:
1020:
1018:Dollar auction
1015:
1010:
1005:
1003:Centipede game
1000:
995:
990:
985:
980:
975:
970:
965:
960:
958:Infinite chess
955:
950:
944:
942:
936:
935:
933:
932:
927:
925:Symmetric game
922:
917:
912:
910:Signaling game
907:
905:Screening game
902:
897:
895:Potential game
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874:
869:
864:
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854:
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838:
837:
835:
834:
829:
824:
822:Mixed strategy
819:
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789:
784:
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769:
764:
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742:
741:
736:
731:
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716:
711:
706:
704:Risk dominance
701:
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631:
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584:
579:
574:
569:
564:
559:
554:
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547:Graphical game
544:
539:
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529:
524:
519:
514:
508:
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502:
501:
493:
492:
485:
478:
470:
464:
463:
452:Eric van Damme
449:
436:
433:
431:of that game.
408:
405:
385:
384:
381:
378:
375:
369:
368:
365:
362:
359:
353:
352:
347:
345:Guess tails up
342:
340:Guess heads up
337:
319:
316:
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260:
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21:
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2:
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1500:
1497:
1495:
1492:
1490:
1487:
1485:
1482:
1480:
1477:
1476:
1474:
1472:Miscellaneous
1470:
1464:
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1459:
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1454:
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1444:
1441:
1439:
1436:
1435:
1433:
1429:
1423:
1420:
1418:
1415:
1413:
1410:
1408:
1407:Samuel Bowles
1405:
1403:
1402:Roger Myerson
1400:
1398:
1395:
1393:
1392:Robert Aumann
1390:
1388:
1385:
1383:
1380:
1378:
1375:
1373:
1370:
1368:
1365:
1363:
1360:
1358:
1355:
1353:
1350:
1348:
1347:Lloyd Shapley
1345:
1343:
1340:
1338:
1335:
1333:
1332:Kenneth Arrow
1330:
1328:
1325:
1323:
1320:
1318:
1315:
1313:
1312:John Harsanyi
1310:
1308:
1305:
1303:
1300:
1298:
1295:
1293:
1290:
1288:
1285:
1283:
1282:Herbert Simon
1280:
1278:
1275:
1273:
1270:
1268:
1265:
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1260:
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1197:
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1177:
1175:
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1155:
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1111:
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1101:
1099:
1096:
1094:
1091:
1089:
1086:
1084:
1083:Fair division
1081:
1079:
1076:
1074:
1071:
1069:
1066:
1064:
1061:
1059:
1058:Dictator game
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
999:
996:
994:
991:
989:
986:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
954:
951:
949:
946:
945:
943:
941:
937:
931:
930:Zero-sum game
928:
926:
923:
921:
918:
916:
913:
911:
908:
906:
903:
901:
900:Repeated game
898:
896:
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873:
870:
868:
865:
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853:
850:
849:
847:
845:
839:
833:
830:
828:
825:
823:
820:
818:
817:Pure strategy
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
782:De-escalation
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
754:
752:
750:
746:
740:
737:
735:
732:
730:
727:
725:
724:Shapley value
722:
720:
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715:
712:
710:
707:
705:
702:
700:
697:
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690:
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632:
630:
627:
625:
622:
620:
617:
616:
614:
612:
608:
604:
598:
595:
593:
592:Succinct game
590:
588:
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583:
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563:
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558:
555:
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520:
518:
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491:
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477:
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471:
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443:
439:
438:
432:
430:
426:
422:
418:
414:
404:
402:
398:
393:
374:
373:Hide tails up
371:
370:
358:
357:Hide heads up
355:
354:
351:
348:
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343:
341:
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197:
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186:
183:
177:
169:
155:
134:
126:
125:totally mixed
110:
107:
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96:
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84:
80:
76:
72:
68:
60:
57:
53:
48:
45:
42:
38:
33:
30:
26:
22:
17:
1377:Peyton Young
1372:Paul Milgrom
1287:Hervé Moulin
1227:Amos Tversky
1169:Folk theorem
880:-player game
877:
802:Grim trigger
688:
459:
445:
424:
420:
410:
388:
372:
356:
349:
344:
339:
330:
321:
273:
148:
92:
66:
65:
50:Significance
35:Relationship
1494:Coopetition
1297:Jean Tirole
1292:John Conway
1272:Eric Maskin
1068:Blotto game
1053:Pirate game
862:Global game
832:Tit for tat
767:Bid shading
757:Appeasement
607:Equilibrium
587:Solved game
522:Determinacy
505:Definitions
498:game theory
95:normal form
55:Proposed by
29:game theory
1544:Categories
1138:Trust game
1123:Kuhn poker
792:Escalation
787:Deterrence
777:Cheap talk
749:Strategies
567:Preference
496:Topics of
435:References
350:Grab penny
89:Definition
1322:John Nash
1028:Stag hunt
772:Collusion
302:ϵ
282:ϵ
259:ϵ
231:−
227:σ
195:−
191:σ
156:ϵ
135:σ
105:ϵ
40:Subset of
1463:Lazy SMP
1157:Theorems
1108:Deadlock
963:Checkers
844:of games
611:concepts
219:′
93:Given a
1215:figures
998:Chicken
852:Auction
842:Classes
318:Example
168:-proper
403:game.
383:-1, 1
380:-1, 1
367:-1, 1
361:-1, 1
953:Chess
940:Games
377:0, 0
364:0, 0
634:Core
423:and
206:<
123:, a
108:>
1213:Key
458:."
454:. "
326:.
73:by
27:in
1546::
948:Go
444:.
878:n
489:e
482:t
475:v
239:)
234:i
223:,
216:s
212:(
209:u
203:)
198:i
187:,
184:s
181:(
178:u
111:0
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