39:
of a nation for an unknown number of years into the future. In such cases, it is often convenient to model the future outcomes as an infinite stream. Then, it may be required to compare two infinite streams and decide which one of them is better (for example, in order to decide on a policy). The
2337:
717:
892:
1584:
534:
2166:
1850:
1285:
1198:
183:
1677:
1461:
570:
1059:
993:
1643:
1427:
758:
2373:
1743:
125:
98:
1621:
447:
421:
327:
281:
2161:
1988:
1915:
1383:
926:
753:
565:
395:
369:
205:
2105:
2048:
1872:
1487:
474:
349:
2400:
2132:
2075:
2019:
1946:
1774:
1716:
235:
1968:
1895:
1357:
1330:
1310:
1107:
1087:
301:
255:
65:
1497:
35:
Often, the decisions of a policy-maker may have influences that extend to the far future. Economic decisions made today may influence the
2344:
As explained above, some sequences may be incomparable by the overtaking criterion. This is why the overtaking criterion is defined as a
2332:{\displaystyle \exists T_{0}:\forall T>T_{0}:(x_{1},\ldots ,x_{T},0,0,0,\ldots )\prec (y_{1},\ldots ,y_{T},0,0,0,\ldots )}
479:
1779:
2078:
2473:
67:
is the set of possible outcomes. E.g., it may be the set of positive real numbers, representing the possible annual
1204:
1117:
2422:
2464:
Carlson, D. A.; Haurie, A. B.; Leizarowitz, A. (1991). "Definition of
Optimality on an Unbounded Time Interval".
130:
1648:
712:{\displaystyle \exists N_{0}:\forall N>N_{0}:\sum _{t=1}^{N}u_{t}(x_{t})<\sum _{t=1}^{N}u_{t}(y_{t})}
24:
is used to compare infinite streams of outcomes. Mathematically, it is used to properly define a notion of
1432:
999:
2589:
936:
887:{\displaystyle 0<\lim \inf _{N\to \infty }\sum _{t=1}^{N}u_{t}(x_{t})-\sum _{t=1}^{N}u_{t}(y_{t})}
2607:
1626:
1388:
2351:
1721:
103:
76:
2421:
theory, as an alternative to the limit-of-means criterion and the discounted-sum criterion. See
1594:
426:
400:
306:
260:
2492:
Brock, William A. (1970). "An
Axiomatic Basis for the Ramsey–Weizsäcker Overtaking Criterion".
2341:
Every partial order that satisfies these axioms, also satisfies the first cardinal definition.
476:
is called the "overtaking criterion" if there is an infinite sequence of real-valued functions
68:
2140:
1973:
1900:
1362:
905:
732:
544:
374:
354:
190:
2084:
2027:
1857:
1466:
459:
334:
2378:
2110:
2053:
1997:
1924:
1752:
1694:
8:
1991:
214:
2509:
1953:
1880:
1342:
1315:
1295:
1092:
1072:
286:
240:
50:
2545:
2532:
Rubinstein, Ariel (1979). "Equilibrium in supergames with the overtaking criterion".
2469:
1066:
This shows that a difference in a single time period may affect the entire sequence.
2571:
2541:
2501:
2439:
1336:
2444:
2434:
2411:
36:
29:
1335:
This also shows that the overtaking criterion cannot be represented by a single
25:
2418:
1918:
2601:
1680:
208:
1874:
is called the "overtaking criterion" if it satisfies the following axioms:
2575:
2513:
100:
is the set of infinite sequences of possible outcomes. Each element in
1579:{\displaystyle (a,a,\ldots )\prec (a+1,a,\ldots )\prec (b,b,\ldots )}
17:
2505:
2562:
Rubinstein, A. (1980). "Strong perfect equilibrium in supergames".
2588:
See papers by: Gale, Koopmans, McKenzie, von
Weizsacker, and
1312:
are larger, then smaller, then equal to the partial sums of
1645:. In contrast, every set of disjoint nonempty segments in
40:
overtaking criterion is one option to do this comparison.
2463:
2163: iff
1591:
Hence, there is a set of disjoint nonempty segments in
755: iff
567: iff
2079:
Debreu theorems#Additivity of ordinal utility function
2381:
2354:
2169:
2143:
2113:
2087:
2056:
2030:
2000:
1976:
1956:
1927:
1903:
1883:
1860:
1782:
1755:
1724:
1697:
1651:
1629:
1597:
1500:
1469:
1435:
1391:
1365:
1345:
1318:
1298:
1207:
1120:
1095:
1075:
1002:
939:
908:
761:
735:
573:
547:
529:{\displaystyle u_{1},u_{2},\ldots :X\to \mathbb {R} }
482:
462:
429:
403:
377:
357:
337:
309:
289:
263:
243:
217:
193:
133:
106:
79:
53:
1332:, so none of these sequences "overtakes" the other.
2134:are essential (have an effect on the preferences).
1845:{\displaystyle (x_{1},\ldots ,x_{T},0,0,0,\ldots )}
2394:
2367:
2331:
2155:
2126:
2099:
2069:
2042:
2013:
1982:
1962:
1940:
1909:
1889:
1866:
1844:
1768:
1737:
1710:
1671:
1637:
1615:
1578:
1481:
1455:
1421:
1377:
1351:
1324:
1304:
1279:
1192:
1101:
1081:
1053:
987:
920:
886:
747:
711:
559:
528:
468:
441:
415:
389:
363:
343:
321:
295:
275:
249:
229:
199:
177:
119:
92:
59:
2599:
1339:function. I.e, there is no real-valued function
772:
768:
2557:
2555:
2527:
2525:
2523:
1280:{\displaystyle y=(3,3,3,3,3,3,3,3,3,\ldots )}
1193:{\displaystyle x=(4,1,4,4,1,4,4,1,4,\ldots )}
2552:
2520:
2487:
2485:
1623:with a cardinality like the cardinality of
2561:
2531:
1656:
1631:
1449:
522:
2482:
178:{\displaystyle x=(x_{1},x_{2},\ldots )}
2600:
1749:elements are nonzero. Each element of
452:
2491:
2423:Folk theorem (game theory)#Overtaking
1686:
1672:{\displaystyle (\mathbb {R} ,\prec )}
2564:International Journal of Game Theory
2410:The overtaking criterion is used in
1429:. One way to see this is: for every
2468:. Berlin: Springer. pp. 9–17.
2107:, at least three of the factors in
2081:for a definition). Also, for every
2077:is preferentially-independent (see
1456:{\displaystyle a,b\in \mathbb {R} }
13:
2375:, and a complete ordering only on
2360:
2186:
2170:
1730:
782:
590:
574:
112:
85:
14:
2619:
329:) or that they are incomparable.
2466:Infinite Horizon Optimal Control
1054:{\displaystyle y=(-1,2,0,0,...)}
2405:
988:{\displaystyle x=(0,0,0,0,...)}
211:. Given two infinite sequences
32:on an unbounded time interval.
2582:
2457:
2326:
2270:
2264:
2208:
1839:
1783:
1666:
1652:
1610:
1598:
1573:
1555:
1549:
1525:
1519:
1501:
1416:
1410:
1401:
1395:
1274:
1214:
1187:
1127:
1048:
1009:
982:
946:
881:
868:
831:
818:
779:
706:
693:
656:
643:
518:
172:
140:
1:
2450:
1069:2. In the following example,
902:1. In the following example,
724:An alternative condition is:
2546:10.1016/0022-0531(79)90002-4
1638:{\displaystyle \mathbb {R} }
1422:{\displaystyle U(x)<U(y)}
7:
2428:
2368:{\displaystyle X^{\infty }}
1994:in the obvious topology on
1738:{\displaystyle X^{\infty }}
120:{\displaystyle X^{\infty }}
93:{\displaystyle X^{\infty }}
43:
10:
2624:
2534:Journal of Economic Theory
1616:{\displaystyle (X,\prec )}
442:{\displaystyle y\preceq x}
416:{\displaystyle x\preceq y}
322:{\displaystyle y\succeq x}
276:{\displaystyle x\succeq y}
351:is the strict variant of
2156:{\displaystyle x\prec y}
1983:{\displaystyle \preceq }
1910:{\displaystyle \preceq }
1745:in which only the first
1378:{\displaystyle x\prec y}
921:{\displaystyle x\prec y}
748:{\displaystyle x\succ y}
560:{\displaystyle x\prec y}
390:{\displaystyle x\prec y}
364:{\displaystyle \preceq }
200:{\displaystyle \preceq }
2100:{\displaystyle T\geq 3}
2396:
2369:
2333:
2157:
2128:
2101:
2071:
2044:
2043:{\displaystyle T>1}
2015:
1984:
1964:
1942:
1911:
1891:
1868:
1867:{\displaystyle \prec }
1846:
1770:
1739:
1712:
1673:
1639:
1617:
1580:
1483:
1482:{\displaystyle a<b}
1457:
1423:
1379:
1353:
1326:
1306:
1281:
1194:
1103:
1083:
1055:
989:
922:
888:
857:
807:
749:
713:
682:
632:
561:
530:
470:
469:{\displaystyle \prec }
443:
417:
391:
365:
345:
344:{\displaystyle \prec }
323:
297:
277:
251:
237:, it is possible that
231:
201:
179:
121:
94:
69:gross domestic product
61:
2397:
2395:{\displaystyle X_{T}}
2370:
2334:
2158:
2129:
2127:{\displaystyle X_{T}}
2102:
2072:
2070:{\displaystyle X_{T}}
2045:
2016:
2014:{\displaystyle X_{T}}
1985:
1965:
1943:
1941:{\displaystyle X_{T}}
1912:
1892:
1869:
1847:
1771:
1769:{\displaystyle X_{T}}
1740:
1713:
1711:{\displaystyle X_{T}}
1674:
1640:
1618:
1581:
1484:
1458:
1424:
1380:
1354:
1327:
1307:
1282:
1195:
1104:
1084:
1056:
990:
923:
889:
837:
787:
750:
714:
662:
612:
562:
531:
471:
444:
418:
392:
366:
346:
324:
298:
278:
252:
232:
202:
180:
122:
95:
62:
2379:
2352:
2167:
2141:
2111:
2085:
2054:
2028:
1998:
1974:
1954:
1925:
1901:
1881:
1858:
1780:
1753:
1722:
1695:
1649:
1627:
1595:
1498:
1467:
1433:
1389:
1363:
1343:
1316:
1296:
1292:The partial sums of
1205:
1118:
1093:
1073:
1000:
937:
906:
759:
733:
571:
545:
480:
460:
427:
401:
375:
355:
335:
307:
287:
261:
241:
215:
191:
131:
104:
77:
51:
22:overtaking criterion
2417:It is also used in
1992:continuous relation
453:Cardinal definition
230:{\displaystyle x,y}
71:. It is normalized
2576:10.1007/BF01784792
2392:
2365:
2329:
2153:
2124:
2097:
2067:
2040:
2011:
1980:
1960:
1938:
1907:
1887:
1864:
1842:
1766:
1735:
1708:
1687:Ordinal definition
1669:
1635:
1613:
1576:
1479:
1453:
1419:
1375:
1349:
1322:
1302:
1277:
1190:
1109:are incomparable:
1099:
1079:
1051:
985:
918:
884:
786:
745:
709:
557:
526:
466:
439:
413:
387:
361:
341:
319:
303:is weakly better (
293:
273:
257:is weakly better (
247:
227:
197:
175:
117:
90:
57:
1963:{\displaystyle T}
1890:{\displaystyle T}
1718:as the subset of
1352:{\displaystyle U}
1325:{\displaystyle y}
1305:{\displaystyle x}
1102:{\displaystyle y}
1082:{\displaystyle x}
771:
296:{\displaystyle y}
250:{\displaystyle x}
60:{\displaystyle X}
28:for a problem of
2615:
2592:
2586:
2580:
2579:
2559:
2550:
2549:
2529:
2518:
2517:
2489:
2480:
2479:
2461:
2440:Cardinal utility
2401:
2399:
2398:
2393:
2391:
2390:
2374:
2372:
2371:
2366:
2364:
2363:
2338:
2336:
2335:
2330:
2301:
2300:
2282:
2281:
2239:
2238:
2220:
2219:
2204:
2203:
2182:
2181:
2162:
2160:
2159:
2154:
2133:
2131:
2130:
2125:
2123:
2122:
2106:
2104:
2103:
2098:
2076:
2074:
2073:
2068:
2066:
2065:
2049:
2047:
2046:
2041:
2020:
2018:
2017:
2012:
2010:
2009:
1989:
1987:
1986:
1981:
1969:
1967:
1966:
1961:
1947:
1945:
1944:
1939:
1937:
1936:
1916:
1914:
1913:
1908:
1896:
1894:
1893:
1888:
1873:
1871:
1870:
1865:
1851:
1849:
1848:
1843:
1814:
1813:
1795:
1794:
1775:
1773:
1772:
1767:
1765:
1764:
1744:
1742:
1741:
1736:
1734:
1733:
1717:
1715:
1714:
1709:
1707:
1706:
1678:
1676:
1675:
1670:
1659:
1644:
1642:
1641:
1636:
1634:
1622:
1620:
1619:
1614:
1585:
1583:
1582:
1577:
1488:
1486:
1485:
1480:
1462:
1460:
1459:
1454:
1452:
1428:
1426:
1425:
1420:
1384:
1382:
1381:
1376:
1358:
1356:
1355:
1350:
1337:cardinal utility
1331:
1329:
1328:
1323:
1311:
1309:
1308:
1303:
1286:
1284:
1283:
1278:
1199:
1197:
1196:
1191:
1108:
1106:
1105:
1100:
1088:
1086:
1085:
1080:
1060:
1058:
1057:
1052:
994:
992:
991:
986:
927:
925:
924:
919:
893:
891:
890:
885:
880:
879:
867:
866:
856:
851:
830:
829:
817:
816:
806:
801:
785:
754:
752:
751:
746:
718:
716:
715:
710:
705:
704:
692:
691:
681:
676:
655:
654:
642:
641:
631:
626:
608:
607:
586:
585:
566:
564:
563:
558:
535:
533:
532:
527:
525:
505:
504:
492:
491:
475:
473:
472:
467:
448:
446:
445:
440:
422:
420:
419:
414:
396:
394:
393:
388:
370:
368:
367:
362:
350:
348:
347:
342:
328:
326:
325:
320:
302:
300:
299:
294:
282:
280:
279:
274:
256:
254:
253:
248:
236:
234:
233:
228:
206:
204:
203:
198:
184:
182:
181:
176:
165:
164:
152:
151:
127:is of the form:
126:
124:
123:
118:
116:
115:
99:
97:
96:
91:
89:
88:
66:
64:
63:
58:
2623:
2622:
2618:
2617:
2616:
2614:
2613:
2612:
2608:Economic growth
2598:
2597:
2596:
2595:
2587:
2583:
2560:
2553:
2530:
2521:
2506:10.2307/1909701
2490:
2483:
2476:
2462:
2458:
2453:
2445:Ordinal utility
2435:Debreu theorems
2431:
2412:economic growth
2408:
2386:
2382:
2380:
2377:
2376:
2359:
2355:
2353:
2350:
2349:
2296:
2292:
2277:
2273:
2234:
2230:
2215:
2211:
2199:
2195:
2177:
2173:
2168:
2165:
2164:
2142:
2139:
2138:
2118:
2114:
2112:
2109:
2108:
2086:
2083:
2082:
2061:
2057:
2055:
2052:
2051:
2029:
2026:
2025:
2005:
2001:
1999:
1996:
1995:
1975:
1972:
1971:
1955:
1952:
1951:
1932:
1928:
1926:
1923:
1922:
1902:
1899:
1898:
1882:
1879:
1878:
1859:
1856:
1855:
1809:
1805:
1790:
1786:
1781:
1778:
1777:
1776:is of the form
1760:
1756:
1754:
1751:
1750:
1729:
1725:
1723:
1720:
1719:
1702:
1698:
1696:
1693:
1692:
1689:
1655:
1650:
1647:
1646:
1630:
1628:
1625:
1624:
1596:
1593:
1592:
1499:
1496:
1495:
1468:
1465:
1464:
1448:
1434:
1431:
1430:
1390:
1387:
1386:
1364:
1361:
1360:
1344:
1341:
1340:
1317:
1314:
1313:
1297:
1294:
1293:
1206:
1203:
1202:
1119:
1116:
1115:
1094:
1091:
1090:
1074:
1071:
1070:
1001:
998:
997:
938:
935:
934:
907:
904:
903:
875:
871:
862:
858:
852:
841:
825:
821:
812:
808:
802:
791:
775:
760:
757:
756:
734:
731:
730:
700:
696:
687:
683:
677:
666:
650:
646:
637:
633:
627:
616:
603:
599:
581:
577:
572:
569:
568:
546:
543:
542:
521:
500:
496:
487:
483:
481:
478:
477:
461:
458:
457:
455:
428:
425:
424:
402:
399:
398:
376:
373:
372:
356:
353:
352:
336:
333:
332:
308:
305:
304:
288:
285:
284:
262:
259:
258:
242:
239:
238:
216:
213:
212:
192:
189:
188:
160:
156:
147:
143:
132:
129:
128:
111:
107:
105:
102:
101:
84:
80:
78:
75:
74:
52:
49:
48:
46:
37:economic growth
30:optimal control
12:
11:
5:
2621:
2611:
2610:
2594:
2593:
2581:
2551:
2519:
2500:(6): 927–929.
2481:
2474:
2455:
2454:
2452:
2449:
2448:
2447:
2442:
2437:
2430:
2427:
2419:repeated games
2407:
2404:
2389:
2385:
2362:
2358:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2299:
2295:
2291:
2288:
2285:
2280:
2276:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2237:
2233:
2229:
2226:
2223:
2218:
2214:
2210:
2207:
2202:
2198:
2194:
2191:
2188:
2185:
2180:
2176:
2172:
2152:
2149:
2146:
2121:
2117:
2096:
2093:
2090:
2064:
2060:
2039:
2036:
2033:
2008:
2004:
1979:
1959:
1935:
1931:
1919:complete order
1906:
1886:
1863:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1812:
1808:
1804:
1801:
1798:
1793:
1789:
1785:
1763:
1759:
1732:
1728:
1705:
1701:
1688:
1685:
1668:
1665:
1662:
1658:
1654:
1633:
1612:
1609:
1606:
1603:
1600:
1589:
1588:
1587:
1586:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1478:
1475:
1472:
1451:
1447:
1444:
1441:
1438:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1374:
1371:
1368:
1348:
1321:
1301:
1290:
1289:
1288:
1287:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1200:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1098:
1078:
1064:
1063:
1062:
1061:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
995:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
917:
914:
911:
897:
896:
895:
894:
883:
878:
874:
870:
865:
861:
855:
850:
847:
844:
840:
836:
833:
828:
824:
820:
815:
811:
805:
800:
797:
794:
790:
784:
781:
778:
774:
770:
767:
764:
744:
741:
738:
722:
721:
720:
719:
708:
703:
699:
695:
690:
686:
680:
675:
672:
669:
665:
661:
658:
653:
649:
645:
640:
636:
630:
625:
622:
619:
615:
611:
606:
602:
598:
595:
592:
589:
584:
580:
576:
556:
553:
550:
524:
520:
517:
514:
511:
508:
503:
499:
495:
490:
486:
465:
454:
451:
438:
435:
432:
412:
409:
406:
386:
383:
380:
360:
340:
318:
315:
312:
292:
272:
269:
266:
246:
226:
223:
220:
196:
174:
171:
168:
163:
159:
155:
150:
146:
142:
139:
136:
114:
110:
87:
83:
56:
45:
42:
9:
6:
4:
3:
2:
2620:
2609:
2606:
2605:
2603:
2591:
2585:
2577:
2573:
2569:
2565:
2558:
2556:
2547:
2543:
2539:
2535:
2528:
2526:
2524:
2515:
2511:
2507:
2503:
2499:
2495:
2488:
2486:
2477:
2475:3-540-54249-3
2471:
2467:
2460:
2456:
2446:
2443:
2441:
2438:
2436:
2433:
2432:
2426:
2424:
2420:
2415:
2413:
2403:
2387:
2383:
2356:
2347:
2342:
2339:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2297:
2293:
2289:
2286:
2283:
2278:
2274:
2267:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2235:
2231:
2227:
2224:
2221:
2216:
2212:
2205:
2200:
2196:
2192:
2189:
2183:
2178:
2174:
2150:
2147:
2144:
2135:
2119:
2115:
2094:
2091:
2088:
2080:
2062:
2058:
2037:
2034:
2031:
2022:
2006:
2002:
1993:
1977:
1957:
1950:2. For every
1948:
1933:
1929:
1920:
1904:
1884:
1877:1. For every
1875:
1861:
1853:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1810:
1806:
1802:
1799:
1796:
1791:
1787:
1761:
1757:
1748:
1726:
1703:
1699:
1684:
1682:
1681:countable set
1663:
1660:
1607:
1604:
1601:
1570:
1567:
1564:
1561:
1558:
1552:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1522:
1516:
1513:
1510:
1507:
1504:
1494:
1493:
1492:
1491:
1490:
1476:
1473:
1470:
1445:
1442:
1439:
1436:
1413:
1407:
1404:
1398:
1392:
1372:
1369:
1366:
1346:
1338:
1333:
1319:
1299:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1211:
1208:
1201:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1124:
1121:
1114:
1113:
1112:
1111:
1110:
1096:
1076:
1067:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1006:
1003:
996:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
943:
940:
933:
932:
931:
930:
929:
915:
912:
909:
900:
876:
872:
863:
859:
853:
848:
845:
842:
838:
834:
826:
822:
813:
809:
803:
798:
795:
792:
788:
776:
765:
762:
742:
739:
736:
729:
728:
727:
726:
725:
701:
697:
688:
684:
678:
673:
670:
667:
663:
659:
651:
647:
638:
634:
628:
623:
620:
617:
613:
609:
604:
600:
596:
593:
587:
582:
578:
554:
551:
548:
541:
540:
539:
538:
537:
515:
512:
509:
506:
501:
497:
493:
488:
484:
463:
450:
436:
433:
430:
410:
407:
404:
384:
381:
378:
358:
338:
330:
316:
313:
310:
290:
270:
267:
264:
244:
224:
221:
218:
210:
209:partial order
194:
186:
169:
166:
161:
157:
153:
148:
144:
137:
134:
108:
81:
72:
70:
54:
41:
38:
33:
31:
27:
23:
19:
2584:
2567:
2563:
2537:
2533:
2497:
2494:Econometrica
2493:
2465:
2459:
2416:
2409:
2406:Applications
2348:ordering on
2345:
2343:
2340:
2136:
2024:3. For each
2023:
1949:
1876:
1854:
1746:
1690:
1590:
1334:
1291:
1068:
1065:
901:
898:
723:
456:
331:
187:
73:
47:
34:
21:
15:
536:such that:
2451:References
1679:must be a
1359:such that
899:Examples:
283:) or that
26:optimality
2361:∞
2324:…
2287:…
2268:≺
2262:…
2225:…
2187:∀
2171:∃
2148:≺
2092:≥
1978:⪯
1905:⪯
1862:≺
1837:…
1800:…
1731:∞
1664:≺
1608:≺
1571:…
1553:≺
1547:…
1523:≺
1517:…
1446:∈
1370:≺
1272:…
1185:…
1013:−
913:≺
839:∑
835:−
789:∑
783:∞
780:→
740:≻
664:∑
614:∑
591:∀
575:∃
552:≺
519:→
510:…
464:≺
434:⪯
408:⪯
382:≺
359:⪯
339:≺
314:⪰
268:⪰
195:⪯
170:…
113:∞
86:∞
18:economics
2602:Category
2570:: 1–12.
2429:See also
2414:theory.
423:and not
371:, i.e.,
44:Notation
2540:: 1–9.
2514:1909701
2346:partial
1691:Define
2512:
2472:
20:, the
2590:Brock
2510:JSTOR
1990:is a
1917:is a
207:is a
2470:ISBN
2193:>
2035:>
1474:<
1463:and
1405:<
1385:iff
1089:and
766:<
660:<
597:>
2572:doi
2542:doi
2502:doi
2137:4.
1921:on
773:inf
769:lim
397:if
16:In
2604::
2566:.
2554:^
2538:21
2536:.
2522:^
2508:.
2498:38
2496:.
2484:^
2425:.
2402:.
2050:,
2021:.
1970:,
1897:,
1852:.
1683:.
1489::
928::
449:.
185:.
2578:.
2574::
2568:9
2548:.
2544::
2516:.
2504::
2478:.
2388:T
2384:X
2357:X
2327:)
2321:,
2318:0
2315:,
2312:0
2309:,
2306:0
2303:,
2298:T
2294:y
2290:,
2284:,
2279:1
2275:y
2271:(
2265:)
2259:,
2256:0
2253:,
2250:0
2247:,
2244:0
2241:,
2236:T
2232:x
2228:,
2222:,
2217:1
2213:x
2209:(
2206::
2201:0
2197:T
2190:T
2184::
2179:0
2175:T
2151:y
2145:x
2120:T
2116:X
2095:3
2089:T
2063:T
2059:X
2038:1
2032:T
2007:T
2003:X
1958:T
1934:T
1930:X
1885:T
1840:)
1834:,
1831:0
1828:,
1825:0
1822:,
1819:0
1816:,
1811:T
1807:x
1803:,
1797:,
1792:1
1788:x
1784:(
1762:T
1758:X
1747:T
1727:X
1704:T
1700:X
1667:)
1661:,
1657:R
1653:(
1632:R
1611:)
1605:,
1602:X
1599:(
1574:)
1568:,
1565:b
1562:,
1559:b
1556:(
1550:)
1544:,
1541:a
1538:,
1535:1
1532:+
1529:a
1526:(
1520:)
1514:,
1511:a
1508:,
1505:a
1502:(
1477:b
1471:a
1450:R
1443:b
1440:,
1437:a
1417:)
1414:y
1411:(
1408:U
1402:)
1399:x
1396:(
1393:U
1373:y
1367:x
1347:U
1320:y
1300:x
1275:)
1269:,
1266:3
1263:,
1260:3
1257:,
1254:3
1251:,
1248:3
1245:,
1242:3
1239:,
1236:3
1233:,
1230:3
1227:,
1224:3
1221:,
1218:3
1215:(
1212:=
1209:y
1188:)
1182:,
1179:4
1176:,
1173:1
1170:,
1167:4
1164:,
1161:4
1158:,
1155:1
1152:,
1149:4
1146:,
1143:4
1140:,
1137:1
1134:,
1131:4
1128:(
1125:=
1122:x
1097:y
1077:x
1049:)
1046:.
1043:.
1040:.
1037:,
1034:0
1031:,
1028:0
1025:,
1022:2
1019:,
1016:1
1010:(
1007:=
1004:y
983:)
980:.
977:.
974:.
971:,
968:0
965:,
962:0
959:,
956:0
953:,
950:0
947:(
944:=
941:x
916:y
910:x
882:)
877:t
873:y
869:(
864:t
860:u
854:N
849:1
846:=
843:t
832:)
827:t
823:x
819:(
814:t
810:u
804:N
799:1
796:=
793:t
777:N
763:0
743:y
737:x
707:)
702:t
698:y
694:(
689:t
685:u
679:N
674:1
671:=
668:t
657:)
652:t
648:x
644:(
639:t
635:u
629:N
624:1
621:=
618:t
610::
605:0
601:N
594:N
588::
583:0
579:N
555:y
549:x
523:R
516:X
513::
507:,
502:2
498:u
494:,
489:1
485:u
437:x
431:y
411:y
405:x
385:y
379:x
317:x
311:y
291:y
271:y
265:x
245:x
225:y
222:,
219:x
173:)
167:,
162:2
158:x
154:,
149:1
145:x
141:(
138:=
135:x
109:X
82:X
55:X
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