28:
307:
756:
659:). These represent player I and player II's (mixed) strategies. Thus, player I can assure himself of a payoff of at least 3/7 if he knows player II's strategy, and player II can hold the payoff down to 1/3 if he knows player I's strategy.
146:
610:
537:
1017: + 1/2) (the payoff along the two discontinuities) to either +1 or −1, making the payoff upper or lower semicontinuous, respectively. If this is done, the game then has a value.
864:
806:
976:
929:
689:
684:
1025:
Subsequent work by Heuer discusses a class of games in which the unit square is divided into three regions, the payoff function being constant in each of the regions.
459:
342:
653:
633:
381:
424:
404:
141:
121:
383:
is interpreted as a point on the unit square, the figure shows the payoff to player I. Player I may adopt a mixed strategy, choosing a number according to a
302:{\displaystyle K(x,y)={\begin{cases}-1&{\text{if }}x<y<x+1/2,\\0&{\text{if }}x=y{\text{ or }}y=x+1/2,\\1&{\text{otherwise.}}\end{cases}}}
542:
88:) but this is not necessarily the case if the game has an infinite set of strategies. There follows a simple example of a game with no minimax value.
472:
31:
Game square (that is, the payoff to player I) for a game with no value, due to Sion and Wolfe. The payoff is 0 along the two diagonal lines
1051:
1154:
1149:
993:
The payoff function of Sion and Wolfe's example is not semicontinuous. However, it may be made so by changing the value of
384:
74:
55:
1042:
Sion, Maurice; Wolfe, Phillip (1957), "On a game without a value", in
Dresher, M.; Tucker, A. W.; Wolfe, P. (eds.),
758:. Dasgupta and Maskin assert that the game values are achieved if player I puts probability weight only on the set
1075:
811:
869:
761:
934:
887:
877:
873:
751:{\displaystyle \varepsilon <{\frac {1}{2}}\left({\frac {3}{7}}-{\frac {1}{3}}\right)\simeq 0.0476}
612:
These are the maximal and minimal expectations of the game's value of player I and II respectively.
176:
663:
669:
1092:
17:
1047:
54:
to one of the players when both play a perfect strategy (which is to choose from a particular
1123:
1084:
429:
312:
85:
638:
618:
354:
1066:
43:
880:
payoff function has a value (in this context, an upper (lower) semicontinuous function
461:, player II to minimize the payoff, and each player is aware of the other's objective.
409:
389:
126:
106:
51:
1128:
1111:
1143:
656:
655:
respectively take the supremum and infimum over pdf's on the unit interval (actually
62:
1073:(1986). "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory".
91:
The existence of such zero-sum games is interesting because many of the results of
70:
1046:, Annals of Mathematics Studies 39, Princeton University Press, pp. 299–306,
27:
1070:
983:
92:
36:
80:
Zero-sum games with a finite number of pure strategies are known to have a
979:
345:
40:
1096:
81:
66:
47:
1088:
605:{\displaystyle \inf _{g}\sup _{f}\iint K\,df\,dg={\frac {3}{7}}.}
532:{\displaystyle \sup _{f}\inf _{g}\iint K\,df\,dg={\frac {1}{3}}}
295:
143:
respectively, between 0 and 1. The payoff to player I is
937:
890:
814:
764:
692:
672:
641:
621:
545:
475:
432:
412:
392:
357:
315:
149:
129:
109:
309:
That is, after the choices are made, player II pays
970:
923:
858:
800:
750:
678:
647:
627:
604:
531:
453:
418:
398:
375:
336:
301:
135:
115:
95:become inapplicable if there is no minimax value.
642:
622:
557:
547:
487:
477:
406:, and similarly player II chooses from a pdf
8:
965:
938:
918:
891:
1112:"Three-part partition games on rectangles"
1127:
936:
889:
837:
823:
813:
779:
763:
727:
714:
699:
691:
671:
640:
620:
589:
579:
572:
560:
550:
544:
519:
509:
502:
490:
480:
474:
431:
426:. Player I seeks to maximize the payoff
411:
391:
356:
314:
287:
267:
247:
233:
213:
187:
171:
148:
128:
108:
1044:Contributions to the Theory of Games III
859:{\displaystyle \left\{1/4,1/2,1\right\}}
26:
1034:
801:{\displaystyle \left\{0,1/2,1\right\}}
7:
971:{\displaystyle \{P\mid K(P)>c\}}
924:{\displaystyle \{P\mid K(P)<c\}}
61:This article gives an example of a
872:shows that any zero-sum game with
808:and player II puts weight only on
385:probability density function (pdf)
25:
18:Example of a game without a value
103:Players I and II choose numbers
956:
950:
909:
903:
448:
436:
370:
358:
331:
319:
165:
153:
1:
1129:10.1016/S0304-3975(00)00404-7
39:, in particular the study of
1116:Theoretical Computer Science
679:{\displaystyle \varepsilon }
344:to player I (so the game is
84:value (originally proved by
1171:
1076:Review of Economic Studies
657:Borel probability measures
469:Sion and Wolfe show that
884:is one in which the set
666:for sufficiently small
46:, not every game has a
972:
925:
860:
802:
752:
680:
649:
629:
606:
533:
455:
454:{\displaystyle K(x,y)}
420:
400:
377:
338:
337:{\displaystyle K(x,y)}
303:
137:
117:
32:
1155:Mathematical examples
1150:Non-cooperative games
973:
926:
861:
803:
753:
681:
650:
648:{\displaystyle \inf }
630:
628:{\displaystyle \sup }
607:
534:
456:
421:
401:
378:
376:{\displaystyle (x,y)}
339:
304:
138:
118:
30:
1110:G. A. Heuer (2001).
935:
888:
878:lower semicontinuous
870:Glicksberg's theorem
812:
762:
690:
670:
639:
619:
543:
473:
430:
410:
390:
355:
313:
147:
127:
107:
50:value. This is the
35:In the mathematical
686:, specifically, if
664:epsilon equilibrium
968:
921:
856:
798:
748:
676:
645:
625:
602:
565:
555:
529:
495:
485:
451:
416:
396:
373:
334:
299:
294:
133:
113:
33:
735:
722:
707:
597:
556:
546:
527:
486:
476:
419:{\displaystyle g}
399:{\displaystyle f}
290:
250:
236:
190:
136:{\displaystyle y}
116:{\displaystyle x}
16:(Redirected from
1162:
1134:
1133:
1131:
1107:
1101:
1100:
1063:
1057:
1056:
1039:
977:
975:
974:
969:
930:
928:
927:
922:
865:
863:
862:
857:
855:
851:
841:
827:
807:
805:
804:
799:
797:
793:
783:
757:
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754:
749:
741:
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736:
728:
723:
715:
708:
700:
685:
683:
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677:
654:
652:
651:
646:
634:
632:
631:
626:
611:
609:
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603:
598:
590:
564:
554:
538:
536:
535:
530:
528:
520:
494:
484:
460:
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397:
382:
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343:
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308:
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291:
288:
271:
251:
248:
237:
234:
217:
191:
188:
142:
140:
139:
134:
122:
120:
119:
114:
86:John von Neumann
69:. It is due to
44:continuous games
21:
1170:
1169:
1165:
1164:
1163:
1161:
1160:
1159:
1140:
1139:
1138:
1137:
1109:
1108:
1104:
1089:10.2307/2297588
1065:
1064:
1060:
1054:
1041:
1040:
1036:
1031:
1023:
1021:Generalizations
933:
932:
886:
885:
819:
815:
810:
809:
769:
765:
760:
759:
713:
709:
688:
687:
668:
667:
637:
636:
617:
616:
541:
540:
471:
470:
467:
428:
427:
408:
407:
388:
387:
353:
352:
311:
310:
293:
292:
285:
279:
278:
231:
225:
224:
185:
172:
145:
144:
125:
124:
105:
104:
101:
37:theory of games
23:
22:
15:
12:
11:
5:
1168:
1166:
1158:
1157:
1152:
1142:
1141:
1136:
1135:
1102:
1058:
1052:
1033:
1032:
1030:
1027:
1022:
1019:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
854:
850:
847:
844:
840:
836:
833:
830:
826:
822:
818:
796:
792:
789:
786:
782:
778:
775:
772:
768:
747:
744:
740:
734:
731:
726:
721:
718:
712:
706:
703:
698:
695:
675:
644:
624:
601:
596:
593:
588:
585:
582:
578:
575:
571:
568:
563:
559:
553:
549:
526:
523:
518:
515:
512:
508:
505:
501:
498:
493:
489:
483:
479:
466:
463:
450:
447:
444:
441:
438:
435:
415:
395:
372:
369:
366:
363:
360:
333:
330:
327:
324:
321:
318:
296:
286:
284:
281:
280:
277:
274:
270:
266:
263:
260:
257:
254:
249: or
246:
243:
240:
232:
230:
227:
226:
223:
220:
216:
212:
209:
206:
203:
200:
197:
194:
186:
184:
181:
178:
177:
175:
170:
167:
164:
161:
158:
155:
152:
132:
112:
100:
97:
52:expected value
24:
14:
13:
10:
9:
6:
4:
3:
2:
1167:
1156:
1153:
1151:
1148:
1147:
1145:
1130:
1125:
1121:
1117:
1113:
1106:
1103:
1098:
1094:
1090:
1086:
1082:
1078:
1077:
1072:
1068:
1062:
1059:
1055:
1053:9780691079363
1049:
1045:
1038:
1035:
1028:
1026:
1020:
1018:
1016:
1012:
1008:
1004:
1000:
996:
991:
989:
985:
981:
962:
959:
953:
947:
944:
941:
915:
912:
906:
900:
897:
894:
883:
879:
875:
871:
867:
852:
848:
845:
842:
838:
834:
831:
828:
824:
820:
816:
794:
790:
787:
784:
780:
776:
773:
770:
766:
745:
742:
738:
732:
729:
724:
719:
716:
710:
704:
701:
696:
693:
673:
665:
660:
658:
613:
599:
594:
591:
586:
583:
580:
576:
573:
569:
566:
561:
551:
524:
521:
516:
513:
510:
506:
503:
499:
496:
491:
481:
464:
462:
445:
442:
439:
433:
413:
393:
386:
367:
364:
361:
349:
347:
328:
325:
322:
316:
282:
275:
272:
268:
264:
261:
258:
255:
252:
244:
241:
238:
228:
221:
218:
214:
210:
207:
204:
201:
198:
195:
192:
182:
179:
173:
168:
162:
159:
156:
150:
130:
110:
98:
96:
94:
89:
87:
83:
78:
76:
72:
68:
64:
63:zero-sum game
59:
57:
53:
49:
45:
42:
38:
29:
19:
1119:
1115:
1105:
1080:
1074:
1061:
1043:
1037:
1024:
1014:
1010:
1006:
1002:
998:
994:
992:
987:
881:
868:
662:There is no
661:
614:
468:
351:If the pair
350:
102:
90:
79:
65:that has no
60:
34:
1122:: 639–661.
1083:(1): 1–26.
1067:P. Dasgupta
984:real number
93:game theory
1144:Categories
1029:References
465:Game value
289:otherwise.
1071:E. Maskin
945:∣
898:∣
743:≃
725:−
694:ε
674:ε
567:∬
497:∬
180:−
982:for any
346:zero-sum
235:if
189:if
99:The game
41:zero-sum
1097:2297588
1013:,
1001:,
82:minimax
48:minimax
1095:
1050:
1005:) and
986:
931:(resp
746:0.0476
1093:JSTOR
978:) is
874:upper
75:Wolfe
67:value
1069:and
1048:ISBN
980:open
960:>
913:<
697:<
635:and
615:The
539:but
202:<
196:<
123:and
73:and
71:Sion
1124:doi
1120:259
1085:doi
990:).
876:or
643:inf
623:sup
558:sup
548:inf
488:inf
478:sup
348:).
77:.
58:).
56:PDF
1146::
1118:.
1114:.
1091:.
1081:53
1079:.
866:.
1132:.
1126::
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1087::
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1011:x
1009:(
1007:K
1003:x
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997:(
995:K
988:c
966:}
963:c
957:)
954:P
951:(
948:K
942:P
939:{
919:}
916:c
910:)
907:P
904:(
901:K
895:P
892:{
882:K
853:}
849:1
846:,
843:2
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791:1
788:,
785:2
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777:1
774:,
771:0
767:{
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733:3
730:1
720:7
717:3
711:(
705:2
702:1
600:.
595:7
592:3
587:=
584:g
581:d
577:f
574:d
570:K
562:f
552:g
525:3
522:1
517:=
514:g
511:d
507:f
504:d
500:K
492:g
482:f
449:)
446:y
443:,
440:x
437:(
434:K
414:g
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371:)
368:y
365:,
362:x
359:(
332:)
329:y
326:,
323:x
320:(
317:K
283:1
276:,
273:2
269:/
265:1
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259:x
256:=
253:y
245:y
242:=
239:x
229:0
222:,
219:2
215:/
211:1
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205:x
199:y
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169:=
166:)
163:y
160:,
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151:K
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
