3008:
996:
2612:
2721:
2217:
2357:
2060:
672:
737:
With the establishment of modern decision theory in the 1950s, the model became a key ingredient in the formulation of non-probabilistic decision-making models in the face of severe uncertainty. It is widely used in diverse fields such as
3003:{\displaystyle {\begin{aligned}\max _{d\in D}\min _{s\in S(d)}f(d,s)&=\max _{d\in D}\min _{1\leq s\leq k}\{f_{1}(d),\dots ,f_{k}(d)\}\\&=\max _{d\in D,z\in \mathbb {R} }\{z:z\leq f_{s}(d),\forall s=1,2,\dots ,k\}\end{aligned}}}
37:
is a non-probabilistic decision-making model according to which decisions are ranked on the basis of their worst-case outcomes – the optimal decision is one with the least bad worst outcome. It is one of the most important models in
860:
1883:
545:
1062:
There are many cases where it is convenient to 'organize' the
Maximin/Minimax model as a 'table'. The convention is that the rows of the table represent the decisions, and the columns represent the states.
380:
686:
developed this model as an approach to scenarios in which there is only one player (the decision maker). Player 2 showcases a gloomy approach to uncertainty. In Wald's maximin model, player 1 (the
1620:
1402:
2726:
2349:
3166:
3106:
3068:
2687:
1543:
In cases where is it desirable to live close to the facility, the objective could be to minimize the maximum distance from the facility. This yields the following minimax problem:
1071:
Henri is going for a walk. The sun may shine, or it may rain. Should Henri carry an umbrella? Henri does not like carrying an umbrella, but he dislikes getting wet even more. His "
2238:. By playing it safe, the Maximin model tends to generate conservative decisions, whose price can be high. The following example illustrates this important feature of the model.
1325:
It might be desirable to build the facility so that its shortest distance from an existing dwelling is as large as possible. The maximin formulation of the problem is as follows:
928:
2071:
989:
1208:
Over the years a variety of related models have been developed primarily to moderate the pessimistic approach dictated by the worst-case orientation of the model. For example,
2607:{\displaystyle \max _{d\in D}\min _{s\in S(d)}f(d,s)=\max _{d\,',d\,''}\ \min _{a\leq s\leq b}f(d,s)=\max \ \{\min _{a\leq s\leq b}f(d\,',s),\min _{a\leq s\leq b}f(d\,'',s)\}}
2275:
1200:
The worst case, if Henri goes out without umbrella, is definitely worse than the (best) worst case when carrying an umbrella. Therefore, Henri takes his umbrella with him.
1272:
885:
257:
1052:
1762:
1637:
Experience has shown that the formulation of maximin models can be subtle in the sense that problems that 'do not look like' maximin problems can be formulated as such.
1913:
1449:
556:
3196:
2713:
172:
724:
704:
450:
430:
79:
49:
It is also known by a variety of other titles, such as Wald's maximin rule, Wald's maximin principle, Wald's maximin paradigm, and Wald's maximin criterion. Often '
1518:
221:
137:
108:
3220:
1782:
1727:
1707:
1687:
1667:
1538:
1489:
1469:
1320:
1300:
407:
284:
192:
1907:
Constraints can be incorporated explicitly in the maximin models. For instance, the following is a constrained maximin problem stated in the classic format
780:
2622:
There are no general-purpose algorithms for the solution of maximin problems. Some problems are very simple to solve, others are very difficult.
730:
in which the two players choose their strategies without knowing the other player's choice. The game of Wald's maximin model is also a 2-person
1794:
3406:
L. Joe
Moffitt, John K. Stranlund, and Craig D. Osteen (2008). Robust detection protocols for uncertain introductions of invasive species.
458:
3477:
223:. In many applications the second player represents uncertainty. However, there are maximin models that are completely deterministic.
292:
1549:
1331:
3419:
Jonathan
Rosenhead, Martin Elton, Shiv K. Gupta. (1972). Robustness and Optimality as Criteria for Strategic Decisions.
3482:
231:
1302:
be a finite set representing possible locations of an 'undesirable' public facility (e.g. garbage dump), and let
1626:
1896:
1322:
denote a finite set of locations in the neighborhood of the planned facility, representing existing dwellings.
727:
3111:
3076:
3016:
2633:
2212:{\displaystyle v^{*}:=\max _{d\in D,\,z\in \mathbb {R} }\{z:z\leq f(d,s),g(d,s)\leq 0,\forall s\in S(d)\}.}
1274:
need not represent uncertainty. They can represent (deterministic) variations in the value of a parameter.
39:
890:
1216:
933:
726:
player) knows player 1's decision when he selects his decision. This is a major simplification of the
763:
2282:
2223:
759:
755:
43:
3320:
Sniedovich, M. (2007). The art and science of modeling decision-making under severe uncertainty.
2234:
One of the 'weaknesses' of the
Maximin model is that the robustness that it provides comes with a
2055:{\displaystyle v^{*}:=\max _{d\in D}\min _{s\in S(d)}\ \{f(d,s):g(d,s)\leq 0,\forall s\in S(d)\}.}
1233:
868:
667:{\displaystyle v^{\circ }:=\min _{d\in D,\,z\in \mathbb {R} }\{z:z\geq f(d,s),\forall s\in S(d)\}}
240:
3227:
3223:
1007:
3249:
Wald, A. (1939). Contributions to the theory of statistical estimation and testing hypotheses.
1732:
1410:
3231:
3175:
2692:
1892:
142:
709:
689:
435:
415:
64:
1494:
739:
197:
113:
84:
20:
2255:
3205:
1767:
1712:
1692:
1672:
1652:
1523:
1474:
1454:
1305:
1285:
1219:
743:
392:
269:
177:
3471:
1072:
731:
1002:
855:{\displaystyle \min _{x\in \mathbb {R} }\max _{y\in \mathbb {R} }\ \{x^{2}-y^{2}\}}
683:
29:
3262:
Wald, A. (1945). Statistical decision functions which minimize the maximum risk.
412:
The minimax version of the model is obtained by exchanging the positions of the
263:
24:
1075:", viewing this as a Maximin game pitting Henri against Nature, is as follows.
81:
player plays first. In response, the second player selects the worst state in
751:
2630:
Consider the case where the state variable is an "index", for instance let
747:
1788:
The maximin formulation of this problem, in the MP format, is as follows:
995:
1878:{\displaystyle \max _{Y\subseteq X}\ \{|Y|:g(x)\leq 0,\forall x\in Y\}.}
50:
32:
3460:
Algorithms for Worst-case Design and
Applications to Risk Management,
3333:
Sniedovich, M. (2008). Wald's maximin model: a treasure in disguise!
3199:
3169:
3309:
Decision Theory: An
Introduction to the Mathematics of Rationality,
3432:
Sniedovich, M. (2010). A bird's view of info-gap decision theory.
540:{\displaystyle v^{\circ }:=\min _{d\in D}\max _{s\in S(d)}f(d,s).}
774:
One of the most famous examples of a
Maximin/Minimax model is
1888:
Generic problems of this type appear in robustness analysis.
3380:
Bertsimas D, and Sim, M. (2004). The price of robustness.
375:{\displaystyle v(d):=\min _{s\in S(d)}f(d,s)\ ,\ d\in D}
3393:
Savage, L. (1951). The theory of statistical decision.
230:
format of Wald's maximin model. There is an equivalent
3208:
3178:
3114:
3079:
3019:
2724:
2715:. The associated maximin problem is then as follows:
2695:
2636:
2360:
2285:
2258:
2074:
1916:
1797:
1770:
1735:
1715:
1695:
1675:
1655:
1615:{\displaystyle \min _{d\in D}\max _{s\in S}dist(d,s)}
1552:
1526:
1497:
1477:
1457:
1413:
1397:{\displaystyle \max _{d\in D}\min _{s\in S}dist(d,s)}
1334:
1308:
1288:
1236:
1132:
1077:
1010:
936:
893:
871:
783:
712:
692:
559:
461:
438:
418:
395:
295:
272:
243:
200:
180:
145:
116:
87:
67:
1899:
model are simple instances of Wald's maximin model.
61:This model represents a 2-person game in which the
3348:Robust Discrete Optimization and Its Applications,
3214:
3190:
3160:
3100:
3062:
3002:
2707:
2681:
2606:
2343:
2269:
2211:
2054:
1877:
1776:
1756:
1721:
1701:
1681:
1661:
1614:
1532:
1512:
1483:
1463:
1443:
1396:
1314:
1294:
1266:
1046:
983:
922:
879:
854:
718:
698:
666:
539:
444:
424:
401:
374:
278:
251:
215:
186:
166:
131:
102:
73:
3362:Ben-Tal, A, El Gaoui, L, Nemirovski, A. (2009).
3395:Journal of the American Statistical Association,
2897:
2812:
2796:
2746:
2730:
2556:
2507:
2497:
2455:
2424:
2378:
2362:
2089:
1947:
1931:
1799:
1570:
1554:
1352:
1336:
803:
785:
713:
693:
574:
492:
476:
439:
419:
312:
68:
3322:Decision Making in Manufacturing and Services,
1130: column to the payoff table, we obtain
8:
3293:Choices: an Introduction to Decision Theory,
2993:
2926:
2883:
2833:
2676:
2652:
2601:
2503:
2203:
2119:
2046:
1974:
1869:
1817:
849:
823:
661:
604:
266:, the worst payoff associated with decision
3358:
3356:
3295:University of Minnesota Press, Minneapolis.
887:denotes the real line. Formally we can set
3445:Reemstem, R. and R\"{u}ckmann, J. (1998).
3207:
3177:
3119:
3113:
3092:
3088:
3087:
3078:
3024:
3018:
2945:
2920:
2919:
2900:
2868:
2840:
2815:
2799:
2749:
2733:
2725:
2723:
2694:
2635:
2587:
2559:
2538:
2510:
2458:
2444:
2432:
2427:
2381:
2365:
2359:
2284:
2257:
2113:
2112:
2105:
2092:
2079:
2073:
1950:
1934:
1921:
1915:
1828:
1820:
1802:
1796:
1769:
1734:
1714:
1694:
1674:
1654:
1573:
1557:
1551:
1525:
1496:
1476:
1456:
1412:
1355:
1339:
1333:
1307:
1287:
1235:
1009:
975:
962:
935:
916:
915:
892:
873:
872:
870:
843:
830:
814:
813:
806:
796:
795:
788:
782:
711:
691:
598:
597:
590:
577:
564:
558:
495:
479:
466:
460:
437:
417:
394:
315:
294:
271:
245:
244:
242:
199:
179:
144:
115:
86:
66:
3376:
3374:
3372:
3303:
3301:
3287:
3285:
2065:Its equivalent MP format is as follows:
550:The equivalent MP format is as follows:
3242:
1222:is associated with the payoff regrets.
734:, but the players choose sequentially.
16:Non-probabilistic decision-making model
3462:Princeton University Press, Princeton.
3366:Princeton University Press, Princeton.
706:player) plays first and player 2 (the
3161:{\displaystyle f_{s},s=1,2,\dots ,k,}
3101:{\displaystyle d\in \mathbb {R} ^{n}}
3063:{\displaystyle f_{s}(d)\equiv f(d,s)}
2682:{\displaystyle S(d)=\{1,2,\dots ,k\}}
7:
3408:Journal of Environmental Management,
1001:The optimal solution is the (red)
923:{\displaystyle D=S(d)=\mathbb {R} }
2963:
2182:
2025:
1857:
984:{\displaystyle f(d,s)=d^{2}-s^{2}}
640:
452:operations in the classic format:
14:
3458:Rustem, B. and Howe, M. (2002).
2351:. The model is then as follows:
1645:Consider the following problem:
1138: Sun
1083: Sun
994:
53:' is used instead of 'maximin'.
3421:Operational Research Quarterly,
3346:Kouvelis P, and Yu G. (1997).
3277:Statistical Decision Functions,
2246:Suppose there are two options,
2222:Such models are very useful in
1141: Rain
1086: Rain
3226:problem that can be solved by
3057:
3045:
3036:
3030:
2957:
2951:
2880:
2874:
2852:
2846:
2785:
2773:
2765:
2759:
2646:
2640:
2598:
2580:
2549:
2531:
2491:
2479:
2417:
2405:
2397:
2391:
2338:
2326:
2320:
2309:
2300:
2289:
2200:
2194:
2170:
2158:
2149:
2137:
2043:
2037:
2013:
2001:
1992:
1980:
1966:
1960:
1845:
1839:
1829:
1821:
1745:
1739:
1609:
1597:
1507:
1501:
1438:
1426:
1391:
1379:
1246:
1240:
1211:
1041:
1029:
1023:
1011:
952:
940:
909:
903:
728:classic 2-person zero-sum game
658:
652:
634:
622:
531:
519:
511:
505:
351:
339:
331:
325:
305:
299:
210:
204:
161:
149:
126:
120:
97:
91:
1:
2344:{\displaystyle S(x')=S(x'')=}
1709:, find the largest subset of
3198:is specified by a system of
1491:. Note that in this problem
1267:{\displaystyle S(d),d\in D,}
880:{\displaystyle \mathbb {R} }
252:{\displaystyle \mathbb {R} }
1891:It has been shown that the
1669:and a real valued function
1047:{\displaystyle (x,y)=(0,0)}
3499:
3447:Semi-Infinite Programming,
3311:Ellis Horwood, Chichester.
3264:The Annals of Mathematics,
3251:The Annals of Mathematics,
1903:Constrained maximin models
1757:{\displaystyle g(x)\leq 0}
1633:Maximin models in disguise
139:that minimizes the payoff
3478:Mathematical optimization
3222:, then this problem is a
1444:{\displaystyle dist(d,s)}
682:Inspired by game theory,
3434:Journal of Risk Finance,
3335:Journal of Risk Finance,
1451:denotes the distance of
232:mathematical programming
3230:algorithms such as the
2230:The price of robustness
1212:Savage's minimax regret
259:denotes the real line.
226:The above model is the
3216:
3192:
3191:{\displaystyle d\in D}
3162:
3102:
3064:
3004:
2709:
2708:{\displaystyle d\in D}
2683:
2608:
2345:
2271:
2213:
2056:
1879:
1786:
1778:
1758:
1723:
1703:
1683:
1663:
1616:
1534:
1514:
1485:
1465:
1445:
1398:
1316:
1296:
1268:
1048:
991:. The picture is this
985:
924:
881:
856:
720:
700:
668:
541:
446:
426:
403:
376:
280:
253:
217:
188:
168:
167:{\displaystyle f(d,s)}
133:
104:
75:
40:robust decision making
3291:Resnik, M.D. (1987).
3217:
3193:
3163:
3103:
3065:
3005:
2710:
2684:
2609:
2346:
2272:
2214:
2057:
1897:info-gap's robustness
1880:
1779:
1759:
1724:
1704:
1684:
1664:
1647:
1617:
1535:
1515:
1486:
1466:
1446:
1399:
1317:
1297:
1269:
1204:Variations on a theme
1126: column and a
1049:
986:
925:
882:
857:
721:
719:{\displaystyle \min }
701:
699:{\displaystyle \max }
669:
542:
447:
445:{\displaystyle \min }
427:
425:{\displaystyle \max }
404:
377:
281:
254:
218:
189:
169:
134:
105:
76:
74:{\displaystyle \max }
3382:Operations Research,
3364:Robust Optimization.
3206:
3176:
3112:
3108:, all the functions
3077:
3017:
2722:
2693:
2634:
2358:
2283:
2256:
2072:
1914:
1795:
1768:
1733:
1713:
1693:
1673:
1653:
1550:
1524:
1513:{\displaystyle S(d)}
1495:
1475:
1455:
1411:
1332:
1306:
1286:
1234:
1226:Deterministic models
1220:minimax regret model
1008:
934:
891:
869:
781:
710:
690:
557:
459:
436:
416:
393:
293:
270:
241:
216:{\displaystyle S(d)}
198:
178:
143:
132:{\displaystyle S(d)}
114:
110:, namely a state in
103:{\displaystyle S(d)}
85:
65:
3307:French, S. (1986).
2270:{\displaystyle x''}
2224:robust optimization
1893:radius of stability
1649:Given a finite set
1520:does not vary with
1230:The sets of states
760:operations research
756:robust optimization
44:robust optimization
3275:Wald, A. (1950).
3228:linear programming
3224:linear programming
3212:
3188:
3158:
3098:
3060:
3000:
2998:
2925:
2832:
2810:
2769:
2744:
2705:
2679:
2604:
2576:
2527:
2475:
2450:
2401:
2376:
2341:
2267:
2209:
2118:
2052:
1970:
1945:
1875:
1813:
1774:
1754:
1719:
1699:
1679:
1659:
1625:These are generic
1612:
1584:
1568:
1530:
1510:
1481:
1461:
1441:
1394:
1366:
1350:
1312:
1292:
1264:
1147:Best Worst Payoff
1044:
981:
920:
877:
852:
819:
801:
716:
696:
664:
603:
537:
515:
490:
442:
422:
399:
387:the security level
372:
335:
276:
249:
213:
184:
164:
129:
100:
71:
3483:Optimal decisions
3232:simplex algorithm
3215:{\displaystyle d}
2896:
2811:
2795:
2745:
2729:
2555:
2506:
2502:
2454:
2453:
2423:
2377:
2361:
2088:
1973:
1946:
1930:
1816:
1798:
1777:{\displaystyle x}
1764: for every
1722:{\displaystyle X}
1702:{\displaystyle X}
1682:{\displaystyle g}
1662:{\displaystyle X}
1627:facility location
1569:
1553:
1533:{\displaystyle d}
1484:{\displaystyle d}
1464:{\displaystyle s}
1351:
1335:
1315:{\displaystyle S}
1295:{\displaystyle D}
1198:
1197:
1128:Best Worst Payoff
1120:
1119:
822:
802:
784:
573:
491:
475:
402:{\displaystyle d}
362:
356:
311:
279:{\displaystyle d}
187:{\displaystyle s}
3490:
3463:
3456:
3450:
3443:
3437:
3430:
3424:
3417:
3411:
3404:
3398:
3391:
3385:
3378:
3367:
3360:
3351:
3344:
3338:
3331:
3325:
3324:1(1-2), 111-136.
3318:
3312:
3305:
3296:
3289:
3280:
3273:
3267:
3260:
3254:
3247:
3221:
3219:
3218:
3213:
3197:
3195:
3194:
3189:
3167:
3165:
3164:
3159:
3124:
3123:
3107:
3105:
3104:
3099:
3097:
3096:
3091:
3069:
3067:
3066:
3061:
3029:
3028:
3009:
3007:
3006:
3001:
2999:
2950:
2949:
2924:
2923:
2889:
2873:
2872:
2845:
2844:
2831:
2809:
2768:
2743:
2714:
2712:
2711:
2706:
2688:
2686:
2685:
2680:
2613:
2611:
2610:
2605:
2591:
2575:
2542:
2526:
2500:
2474:
2451:
2449:
2448:
2436:
2400:
2375:
2350:
2348:
2347:
2342:
2319:
2299:
2278:
2276:
2274:
2273:
2268:
2266:
2249:
2218:
2216:
2215:
2210:
2117:
2116:
2084:
2083:
2061:
2059:
2058:
2053:
1971:
1969:
1944:
1926:
1925:
1884:
1882:
1881:
1876:
1832:
1824:
1814:
1812:
1784:in this subset.
1783:
1781:
1780:
1775:
1763:
1761:
1760:
1755:
1728:
1726:
1725:
1720:
1708:
1706:
1705:
1700:
1688:
1686:
1685:
1680:
1668:
1666:
1665:
1660:
1621:
1619:
1618:
1613:
1583:
1567:
1539:
1537:
1536:
1531:
1519:
1517:
1516:
1511:
1490:
1488:
1487:
1482:
1470:
1468:
1467:
1462:
1450:
1448:
1447:
1442:
1403:
1401:
1400:
1395:
1365:
1349:
1321:
1319:
1318:
1313:
1301:
1299:
1298:
1293:
1273:
1271:
1270:
1265:
1133:
1078:
1053:
1051:
1050:
1045:
998:
990:
988:
987:
982:
980:
979:
967:
966:
929:
927:
926:
921:
919:
886:
884:
883:
878:
876:
861:
859:
858:
853:
848:
847:
835:
834:
820:
818:
817:
800:
799:
725:
723:
722:
717:
705:
703:
702:
697:
673:
671:
670:
665:
602:
601:
569:
568:
546:
544:
543:
538:
514:
489:
471:
470:
451:
449:
448:
443:
431:
429:
428:
423:
408:
406:
405:
400:
381:
379:
378:
373:
360:
354:
334:
285:
283:
282:
277:
258:
256:
255:
250:
248:
222:
220:
219:
214:
193:
191:
190:
185:
173:
171:
170:
165:
138:
136:
135:
130:
109:
107:
106:
101:
80:
78:
77:
72:
3498:
3497:
3493:
3492:
3491:
3489:
3488:
3487:
3468:
3467:
3466:
3457:
3453:
3449:Kluwer, Boston.
3444:
3440:
3436:11(3), 268-283.
3431:
3427:
3423:23(4), 413-431.
3418:
3414:
3410:89(4), 293–299.
3405:
3401:
3392:
3388:
3379:
3370:
3361:
3354:
3350:Kluwer, Boston.
3345:
3341:
3332:
3328:
3319:
3315:
3306:
3299:
3290:
3283:
3279:John Wiley, NY.
3274:
3270:
3266:46(2), 265-280.
3261:
3257:
3253:10(4), 299-326.
3248:
3244:
3240:
3204:
3203:
3202:constraints on
3174:
3173:
3115:
3110:
3109:
3086:
3075:
3074:
3020:
3015:
3014:
2997:
2996:
2941:
2887:
2886:
2864:
2836:
2788:
2720:
2719:
2691:
2690:
2632:
2631:
2628:
2620:
2586:
2537:
2443:
2431:
2356:
2355:
2312:
2292:
2281:
2280:
2259:
2254:
2253:
2251:
2247:
2244:
2232:
2075:
2070:
2069:
1917:
1912:
1911:
1905:
1793:
1792:
1766:
1765:
1731:
1730:
1711:
1710:
1691:
1690:
1671:
1670:
1651:
1650:
1643:
1635:
1548:
1547:
1522:
1521:
1493:
1492:
1473:
1472:
1453:
1452:
1409:
1408:
1330:
1329:
1304:
1303:
1284:
1283:
1280:
1232:
1231:
1228:
1214:
1206:
1194:
1189:
1184:
1179:
1167:
1162:
1157:
1116:
1111:
1101:
1096:
1069:
1060:
1058:Decision tables
1006:
1005:
971:
958:
932:
931:
889:
888:
867:
866:
839:
826:
779:
778:
772:
740:decision theory
708:
707:
688:
687:
680:
560:
555:
554:
462:
457:
456:
434:
433:
414:
413:
391:
390:
291:
290:
268:
267:
239:
238:
196:
195:
176:
175:
141:
140:
112:
111:
83:
82:
63:
62:
59:
46:in particular.
42:in general and
21:decision theory
17:
12:
11:
5:
3496:
3494:
3486:
3485:
3480:
3470:
3469:
3465:
3464:
3451:
3438:
3425:
3412:
3399:
3386:
3368:
3352:
3339:
3326:
3313:
3297:
3281:
3268:
3255:
3241:
3239:
3236:
3211:
3187:
3184:
3181:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3122:
3118:
3095:
3090:
3085:
3082:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3027:
3023:
3011:
3010:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2948:
2944:
2940:
2937:
2934:
2931:
2928:
2922:
2918:
2915:
2912:
2909:
2906:
2903:
2899:
2895:
2892:
2890:
2888:
2885:
2882:
2879:
2876:
2871:
2867:
2863:
2860:
2857:
2854:
2851:
2848:
2843:
2839:
2835:
2830:
2827:
2824:
2821:
2818:
2814:
2808:
2805:
2802:
2798:
2794:
2791:
2789:
2787:
2784:
2781:
2778:
2775:
2772:
2767:
2764:
2761:
2758:
2755:
2752:
2748:
2742:
2739:
2736:
2732:
2728:
2727:
2704:
2701:
2698:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2627:
2624:
2619:
2616:
2615:
2614:
2603:
2600:
2597:
2594:
2590:
2585:
2582:
2579:
2574:
2571:
2568:
2565:
2562:
2558:
2554:
2551:
2548:
2545:
2541:
2536:
2533:
2530:
2525:
2522:
2519:
2516:
2513:
2509:
2505:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2473:
2470:
2467:
2464:
2461:
2457:
2447:
2442:
2439:
2435:
2430:
2426:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2399:
2396:
2393:
2390:
2387:
2384:
2380:
2374:
2371:
2368:
2364:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2318:
2315:
2311:
2308:
2305:
2302:
2298:
2295:
2291:
2288:
2265:
2262:
2243:
2240:
2231:
2228:
2220:
2219:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2115:
2111:
2108:
2104:
2101:
2098:
2095:
2091:
2087:
2082:
2078:
2063:
2062:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1968:
1965:
1962:
1959:
1956:
1953:
1949:
1943:
1940:
1937:
1933:
1929:
1924:
1920:
1904:
1901:
1886:
1885:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1831:
1827:
1823:
1819:
1811:
1808:
1805:
1801:
1773:
1753:
1750:
1747:
1744:
1741:
1738:
1718:
1698:
1678:
1658:
1642:
1639:
1634:
1631:
1623:
1622:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1582:
1579:
1576:
1572:
1566:
1563:
1560:
1556:
1529:
1509:
1506:
1503:
1500:
1480:
1460:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1405:
1404:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1364:
1361:
1358:
1354:
1348:
1345:
1342:
1338:
1311:
1291:
1279:
1276:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1227:
1224:
1213:
1210:
1205:
1202:
1196:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1175:
1171:
1170:
1168:
1165:
1163:
1160:
1158:
1155:
1153:
1149:
1148:
1145:
1142:
1139:
1136:
1118:
1117:
1114:
1112:
1109:
1107:
1103:
1102:
1099:
1097:
1094:
1092:
1088:
1087:
1084:
1081:
1068:
1065:
1059:
1056:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
978:
974:
970:
965:
961:
957:
954:
951:
948:
945:
942:
939:
918:
914:
911:
908:
905:
902:
899:
896:
875:
863:
862:
851:
846:
842:
838:
833:
829:
825:
816:
812:
809:
805:
798:
794:
791:
787:
771:
768:
744:control theory
715:
695:
679:
676:
675:
674:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
600:
596:
593:
589:
586:
583:
580:
576:
572:
567:
563:
548:
547:
536:
533:
530:
527:
524:
521:
518:
513:
510:
507:
504:
501:
498:
494:
488:
485:
482:
478:
474:
469:
465:
441:
421:
398:
383:
382:
371:
368:
365:
359:
353:
350:
347:
344:
341:
338:
333:
330:
327:
324:
321:
318:
314:
310:
307:
304:
301:
298:
275:
247:
212:
209:
206:
203:
183:
163:
160:
157:
154:
151:
148:
128:
125:
122:
119:
99:
96:
93:
90:
70:
58:
55:
15:
13:
10:
9:
6:
4:
3:
2:
3495:
3484:
3481:
3479:
3476:
3475:
3473:
3461:
3455:
3452:
3448:
3442:
3439:
3435:
3429:
3426:
3422:
3416:
3413:
3409:
3403:
3400:
3396:
3390:
3387:
3384:52(1), 35-53.
3383:
3377:
3375:
3373:
3369:
3365:
3359:
3357:
3353:
3349:
3343:
3340:
3337:9(3), 287-91.
3336:
3330:
3327:
3323:
3317:
3314:
3310:
3304:
3302:
3298:
3294:
3288:
3286:
3282:
3278:
3272:
3269:
3265:
3259:
3256:
3252:
3246:
3243:
3237:
3235:
3233:
3229:
3225:
3209:
3201:
3185:
3182:
3179:
3171:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3120:
3116:
3093:
3083:
3080:
3071:
3054:
3051:
3048:
3042:
3039:
3033:
3025:
3021:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2960:
2954:
2946:
2942:
2938:
2935:
2932:
2929:
2916:
2913:
2910:
2907:
2904:
2901:
2893:
2891:
2877:
2869:
2865:
2861:
2858:
2855:
2849:
2841:
2837:
2828:
2825:
2822:
2819:
2816:
2806:
2803:
2800:
2792:
2790:
2782:
2779:
2776:
2770:
2762:
2756:
2753:
2750:
2740:
2737:
2734:
2718:
2717:
2716:
2702:
2699:
2696:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2649:
2643:
2637:
2625:
2623:
2617:
2595:
2592:
2588:
2583:
2577:
2572:
2569:
2566:
2563:
2560:
2552:
2546:
2543:
2539:
2534:
2528:
2523:
2520:
2517:
2514:
2511:
2494:
2488:
2485:
2482:
2476:
2471:
2468:
2465:
2462:
2459:
2445:
2440:
2437:
2433:
2428:
2420:
2414:
2411:
2408:
2402:
2394:
2388:
2385:
2382:
2372:
2369:
2366:
2354:
2353:
2352:
2335:
2332:
2329:
2323:
2316:
2313:
2306:
2303:
2296:
2293:
2286:
2279:, and where
2263:
2260:
2241:
2239:
2237:
2229:
2227:
2225:
2206:
2197:
2191:
2188:
2185:
2179:
2176:
2173:
2167:
2164:
2161:
2155:
2152:
2146:
2143:
2140:
2134:
2131:
2128:
2125:
2122:
2109:
2106:
2102:
2099:
2096:
2093:
2085:
2080:
2076:
2068:
2067:
2066:
2049:
2040:
2034:
2031:
2028:
2022:
2019:
2016:
2010:
2007:
2004:
1998:
1995:
1989:
1986:
1983:
1977:
1963:
1957:
1954:
1951:
1941:
1938:
1935:
1927:
1922:
1918:
1910:
1909:
1908:
1902:
1900:
1898:
1894:
1889:
1872:
1866:
1863:
1860:
1854:
1851:
1848:
1842:
1836:
1833:
1825:
1809:
1806:
1803:
1791:
1790:
1789:
1785:
1771:
1751:
1748:
1742:
1736:
1716:
1696:
1676:
1656:
1646:
1640:
1638:
1632:
1630:
1628:
1606:
1603:
1600:
1594:
1591:
1588:
1585:
1580:
1577:
1574:
1564:
1561:
1558:
1546:
1545:
1544:
1541:
1527:
1504:
1498:
1478:
1458:
1435:
1432:
1429:
1423:
1420:
1417:
1414:
1388:
1385:
1382:
1376:
1373:
1370:
1367:
1362:
1359:
1356:
1346:
1343:
1340:
1328:
1327:
1326:
1323:
1309:
1289:
1277:
1275:
1261:
1258:
1255:
1252:
1249:
1243:
1237:
1225:
1223:
1221:
1218:
1209:
1203:
1201:
1191:
1186:
1181:
1176:
1173:
1172:
1169:
1164:
1159:
1154:
1151:
1150:
1146:
1144:Worst Payoff
1143:
1140:
1137:
1135:
1134:
1131:
1129:
1125:
1113:
1108:
1105:
1104:
1098:
1093:
1090:
1089:
1085:
1082:
1080:
1079:
1076:
1074:
1073:payoff matrix
1066:
1064:
1057:
1055:
1038:
1035:
1032:
1026:
1020:
1017:
1014:
1004:
999:
997:
992:
976:
972:
968:
963:
959:
955:
949:
946:
943:
937:
912:
906:
900:
897:
894:
844:
840:
836:
831:
827:
810:
807:
792:
789:
777:
776:
775:
769:
767:
765:
761:
757:
753:
749:
745:
741:
735:
733:
732:zero-sum game
729:
685:
677:
655:
649:
646:
643:
637:
631:
628:
625:
619:
616:
613:
610:
607:
594:
591:
587:
584:
581:
578:
570:
565:
561:
553:
552:
551:
534:
528:
525:
522:
516:
508:
502:
499:
496:
486:
483:
480:
472:
467:
463:
455:
454:
453:
410:
396:
388:
369:
366:
363:
357:
348:
345:
342:
336:
328:
322:
319:
316:
308:
302:
296:
289:
288:
287:
273:
265:
260:
235:
234:(MP) format:
233:
229:
224:
207:
201:
181:
158:
155:
152:
146:
123:
117:
94:
88:
56:
54:
52:
47:
45:
41:
36:
34:
31:
26:
22:
3459:
3454:
3446:
3441:
3433:
3428:
3420:
3415:
3407:
3402:
3394:
3389:
3381:
3363:
3347:
3342:
3334:
3329:
3321:
3316:
3308:
3292:
3276:
3271:
3263:
3258:
3250:
3245:
3072:
3012:
2629:
2621:
2245:
2235:
2233:
2221:
2064:
1906:
1890:
1887:
1787:
1648:
1644:
1636:
1624:
1542:
1406:
1324:
1281:
1229:
1215:
1207:
1199:
1152:No umbrella
1127:
1124:Worst Payoff
1123:
1122:Appending a
1121:
1091:No umbrella
1070:
1061:
1003:saddle point
1000:
993:
864:
773:
736:
684:Abraham Wald
681:
549:
411:
389:of decision
386:
384:
261:
236:
227:
225:
60:
48:
28:
18:
264:game theory
25:game theory
3472:Categories
3397:46, 55–67.
3238:References
2618:Algorithms
1895:model and
1729:such that
1629:problems.
764:philosophy
752:statistics
385:is called
57:Definition
3183:∈
3147:…
3084:∈
3040:≡
2985:…
2964:∀
2939:≤
2917:∈
2905:∈
2859:…
2826:≤
2820:≤
2804:∈
2754:∈
2738:∈
2700:∈
2689:for all
2668:…
2570:≤
2564:≤
2521:≤
2515:≤
2469:≤
2463:≤
2386:∈
2370:∈
2189:∈
2183:∀
2174:≤
2132:≤
2110:∈
2097:∈
2081:∗
2032:∈
2026:∀
2017:≤
1955:∈
1939:∈
1923:∗
1864:∈
1858:∀
1849:≤
1807:⊆
1749:≤
1578:∈
1562:∈
1360:∈
1344:∈
1256:∈
1174:Umbrella
1106:Umbrella
969:−
837:−
811:∈
793:∈
748:economics
647:∈
641:∀
617:≥
595:∈
582:∈
566:∘
500:∈
484:∈
468:∘
367:∈
320:∈
286:, namely
2589:″
2540:′
2446:″
2434:′
2317:″
2297:′
2264:″
1217:Savage's
1193:−5
1188:−5
1183:−5
1166:−9
1161:−9
1115:−5
1100:−9
2626:Example
2277:
2252:
2242:Example
1641:Example
1278:Example
1067:Example
770:Example
766:, etc.
678:History
228:classic
51:minimax
33:maximin
3200:linear
3172:, and
3170:linear
3013:where
2501:
2452:
1972:
1815:
1407:where
865:where
821:
361:
355:
262:As in
237:where
30:Wald's
2236:price
1471:from
174:over
35:model
3168:are
2250:and
1282:Let
930:and
432:and
23:and
3073:If
2898:max
2813:min
2797:max
2747:min
2731:max
2557:min
2508:min
2498:max
2456:min
2425:max
2379:min
2363:max
2090:max
1948:min
1932:max
1800:max
1689:on
1571:max
1555:min
1353:min
1337:max
804:max
786:min
714:min
694:max
575:min
493:max
477:min
440:min
420:max
313:min
194:in
69:max
19:In
3474::
3371:^
3355:^
3300:^
3284:^
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1928::=
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762:,
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309::=
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95:d
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