40:
5282:"The axioms should not be too numerous, their system is to be as simple and transparent as possible, and each axiom should have an immediate intuitive meaning by which its appropriateness may be judged directly. In a situation like ours this last requirement is particularly vital, in spite of its vagueness: we want to make an intuitive concept amenable to mathematical treatment and to see as clearly as possible what hypotheses this requires." –
867:
855:
5308:
events, sacrificing much personal wealth or well-being, and all of these actions will factor into the construction/definition of the agent's VNM-utility function. In other words, both what is naturally perceived as "personal gain", and what is naturally perceived as "altruism", are implicitly balanced in the VNM-utility function of a VNM-rational individual. Therefore, the full range of
5307:
Because the theorem assumes nothing about the nature of the possible outcomes of the gambles, they could be morally significant events, for instance involving the life, death, sickness, or health of others. A von
Neumann–Morgenstern rational agent is capable of acting with great concern for such
4983:"Many economists will feel that we are assuming far too much ... Have we not shown too much? ... As far as we can see, our postulates plausible ... We have practically defined numerical utility as being that thing for which the calculus of mathematical expectations is legitimate." –
5441:
are concerned with quantities called the "total utility" and "average utility" of collectives, and characterize morality in terms of favoring the utility or happiness of others with disregard for one's own. These notions can be related to, but are distinct from, VNM-utility:
2149:
4628:
4978:
Von
Neumann and Morgenstern anticipated surprise at the strength of their conclusion. But according to them, the reason their utility function works is that it is constructed precisely to fill the role of something whose expectation is maximized:
3204:. Note that every sure outcome can be seen as a lottery: it is a degenerate lottery in which the outcome is selected with probability 1. Hence, by the Completeness and Transitivity axioms, it is possible to order the outcomes from worst to best:
1963:
4125:
3094:
2393:
2921:
2717:
5527:
cannot be formulated free of contradiction on this level. This may seem to be a paradoxical assertion. But anybody who has seriously tried to axiomatize that elusive concept, will probably concur with it." –
5453:
2) VNM-utility is not canonically additive across multiple individuals (see
Limitations), so "total VNM-utility" and "average VNM-utility" are not immediately meaningful (some sort of normalization assumption is
5116:
3264:
2596:
5214:
4722:
5470:. Since morality affects decisions, a VNM-rational agent's morals will affect the definition of its own utility function (see above). Thus, the morality of a VNM-rational agent can be characterized by
3810:
4340:
3537:
2043:
5629:) canonically true or false. In particular, the aforementioned "total VNM-utility" and "average VNM-utility" of a population are not canonically meaningful without normalization assumptions.
2295:
2782:
1869:
4478:
4484:
4181:
3968:
1762:
1668:
4966:
2477:
3733:
1172:
1830:
3307:
3202:
985:(it is unique up to affine transformations i.e. adding a constant and multiplying by a positive scalar). No claim is made that the agent has a "conscious desire" to maximize
3912:
3624:
3582:
5563:
are only determined up to additive constants and multiplicative positive scalars, the theorem does not provide any canonical way to compare the two. Hence expressions like
1880:
1086:
4826:
1616:
1589:
1562:
1510:
1483:
1371:
5786:
5429:. An agent-focused von Neumann–Morgenstern rational agent therefore cannot favor more equal, or "fair", distributions of utility between its own possible future selves.
4901:
3394:
3352:
2419:
2036:
5427:
5812:
3980:
1334:
1294:
1268:
5833:
4750:
2521:
1707:
4262:
4235:
4208:
3860:
3681:
3654:
3435:
1204:
4779:
2221:
5383:
5360:
5337:
4866:
4846:
4360:
3833:
3455:
3155:
2315:
2192:
2172:
1998:
1456:
1436:
2948:
2320:
2812:
5511:
is simply "expanded out" and considered a lottery itself, the VNM formalism ignores what may be experienced as "nested gambling". This is related to the
5278:
is to provide "modest conditions" (i.e. axioms) describing when the expected utility hypothesis holds, which can be evaluated directly and intuitively:
1209:
The outcomes in a lottery can themselves be lotteries between other outcomes, and the expanded expression is considered an equivalent lottery: 0.5(0.5
2197:
Note that the "only if" direction is necessary for the theorem to work. Without that, we have this counterexample: there are only two outcomes
5025:
For example, a person who only possesses $ 1000 in savings may be reluctant to risk it all for a 20% chance odds to win $ 10,000, even though
2602:
5031:
7250:
7206:
897:
829:
6104:
5289:
As such, claims that the expected utility hypothesis does not characterize rationality must reject one of the VNM axioms. A variety of
6005:
3210:
969:
and the individual will always prefer actions that maximize expected utility. That is, they proved that an agent is (VNM-)rational
5135:
3744:
5956:
5677:
5266:
published a treatise in which he posits that rational behavior can be described as maximizing the expectation of a function
4634:
4267:
2144:{\displaystyle L\preceq N\quad {\text{if and only if}}\quad \,(1-p)\,L+{\underline {pM}}\preceq (1-p)\,N+{\underline {pM}}}
2532:
977:
defined by possible outcomes such that every preference of the agent is characterized by maximizing the expected value of
6254:
6177:
3463:
942:
function. This function is known as the von
Neumann–Morgenstern utility function. The theorem forms the foundation of
5866:
3269:
We assume that at least one of the inequalities is strict (otherwise the utility function is trivial—a constant). So
1792:. It says that any separation in preference can be maintained under a sufficiently small deviation in probabilities:
5474:
of the agent's VNM-utility with the VNM-utility, E-utility, or "happiness" of others, among other means, but not by
3309:. We use these two extreme outcomes—the worst and the best—as the scaling unit of our utility function, and define:
3157:
can be built. Here we outline the construction process for the case in which the number of sure outcomes is finite.
5309:
1113:) = 75% (and exactly one of them will occur). More generally, for a lottery with many possible outcomes
7300:
3105:
can be uniquely determined (up to adding a constant and multiplying by a positive scalar) by preferences between
890:
5845:
1788:
Instead of continuity, an alternative axiom can be assumed that does not involve a precise equality, called the
6688:
5702:
5641:
has been shown to have imperfect predictive accuracy in a set of lab based empirical experiments, such as the
4623:{\displaystyle M'=\left(\sum _{i}p_{i}q_{i}\right)\cdot A_{n}+\left(\sum _{i}p_{i}(1-q_{i})\right)\cdot A_{1}}
2226:
6658:
6648:
6341:
5998:
5450:: it is that according to which one decides, and thus by definition cannot be something which one disregards.
1835:
228:
6869:
490:
6808:
6781:
5467:
5290:
4133:
3920:
1714:
1640:
7235:
7216:
6793:
6638:
6604:
6589:
6568:
6563:
6028:
5638:
5257:
4906:
839:
59:
49:
3689:
1128:
7210:
6786:
6476:
6466:
6144:
6119:
4368:
1802:
883:
833:
363:
353:
3272:
3167:
1958:{\displaystyle \,(1-\varepsilon )L+\varepsilon N\,\prec \,M\,\prec \,\varepsilon L+(1-\varepsilon )N.}
7290:
6356:
6079:
243:
5462:
for "experience utility" has been coined to refer to the types of "hedonistic" utility like that of
5246:
3868:
3587:
3545:
7305:
7240:
7186:
6929:
6884:
6723:
6594:
6471:
5991:
2727:
1973:
Independence assumes that a preference holds independently of the probability of another outcome.
1053:
812:
550:
505:
358:
127:
71:
6944:
5270:, which in particular need not be monetary-valued, thus accounting for risk aversion. This is the
4787:
1594:
1567:
1540:
1488:
1461:
565:
6894:
6728:
6718:
6708:
6698:
6436:
6426:
6386:
6376:
6249:
6170:
6139:
5814:. To make such relations explicit in the axioms, Kreps (1988) chapter 2 denotes indifference by
5125:
the person is VNM-rational, such facts are automatically accounted for in their utility function
4120:{\displaystyle u(M)=u\left(\sum _{i}p_{i}A_{i}\right)=\sum _{i}p_{i}u(A_{i})=\sum _{i}p_{i}q_{i}}
3121:
having only two outcomes. Conversely, any agent acting to maximize the expectation of a function
1347:
943:
515:
323:
313:
273:
263:
170:
93:
5759:
4879:
3357:
3315:
2398:
2003:
7064:
6909:
6769:
6712:
6668:
6631:
6381:
6321:
6296:
6266:
6239:
6134:
6099:
6049:
5438:
5388:
2434:
958:
685:
530:
268:
208:
165:
112:
5791:
1310:
1273:
1247:
7196:
6879:
6854:
6839:
6813:
6752:
6431:
6371:
6351:
6346:
6114:
5817:
2482:
1673:
1024:
819:
500:
475:
460:
318:
258:
238:
233:
7280:
7245:
6949:
6693:
6663:
6616:
6579:
6505:
6456:
6421:
6361:
6326:
6261:
6244:
6129:
6033:
5897:
5245:($ 2) without contradicting VNM-rationality at all. This leads to a quantitative theory of
4240:
4213:
4186:
3838:
3659:
3632:
3402:
1789:
1182:
570:
392:
343:
308:
248:
213:
117:
107:
54:
6194:
5965:
4755:
1970:
Only one of (3) or (3′) need to be assumed, and the other will be implied by the theorem.
8:
7175:
6984:
6803:
6703:
6683:
6643:
6599:
6584:
6540:
6481:
6406:
6396:
6366:
6289:
6064:
2200:
1527:
1401:
1397:
796:
605:
427:
368:
293:
283:
253:
180:
88:
6844:
6276:
5365:
5342:
4730:
465:
7295:
7230:
7201:
7159:
6964:
6673:
6653:
6621:
6535:
6530:
6510:
6461:
6401:
6391:
6336:
6331:
6163:
6109:
5927:
5919:
5322:
4851:
4831:
4345:
3818:
3440:
3140:
2300:
2177:
2157:
1983:
1521:
1441:
1421:
1030:
824:
780:
585:
422:
417:
397:
348:
288:
278:
223:
218:
190:
185:
155:
23:
3089:{\displaystyle Eu(p_{1}A_{1}+\cdots +p_{n}A_{n})=p_{1}u(A_{1})+\cdots +p_{n}u(A_{n}).}
2388:{\displaystyle {\frac {1}{2}}A+{\frac {1}{2}}B\succeq {\frac {1}{2}}B+{\frac {1}{2}}B}
7275:
7094:
7069:
6979:
6859:
6747:
6550:
6486:
6451:
6441:
6311:
5952:
5945:
5931:
5862:
5672:
4237:. But, by our assumption, the decision maker is indifferent between the sure outcome
2916:{\displaystyle L\prec M\qquad \mathrm {if\,and\,only\,if} \qquad E(u(L))<E(u(M)),}
954:
715:
690:
600:
480:
437:
373:
338:
328:
160:
122:
78:
7019:
640:
7285:
7144:
7089:
7074:
7059:
7044:
6974:
6954:
6934:
6889:
6496:
6446:
6416:
6411:
6301:
6227:
6094:
6059:
5911:
5719:
5711:
5668:
5512:
5302:
5263:
962:
950:
939:
765:
710:
695:
680:
665:
595:
575:
555:
510:
383:
333:
303:
298:
7024:
645:
7255:
7149:
7114:
7079:
7014:
6939:
6924:
6818:
6774:
6611:
6545:
6520:
6515:
6491:
6222:
6207:
6054:
6014:
2792:
For any VNM-rational agent (i.e. satisfying axioms 1–4), there exists a function
927:
911:
859:
770:
735:
700:
635:
560:
545:
432:
407:
402:
378:
150:
145:
7169:
7154:
7119:
7104:
7084:
7054:
6874:
6525:
6234:
6202:
6089:
5741:
5642:
5463:
5018:
1012:
1008:
966:
935:
790:
775:
740:
725:
705:
675:
495:
412:
102:
98:
6904:
1033:
outcomes, a lottery is a scenario where each outcome will happen with a given
525:
7269:
7134:
7124:
7099:
7039:
7034:
7029:
7009:
6999:
6969:
6959:
6864:
6764:
6737:
6501:
5846:
http://www.econport.org/content/handbook/decisions-uncertainty/basic/von.html
5008:
2424:
The independence axiom implies the axiom on reduction of compound lotteries:
1405:
755:
745:
720:
660:
655:
650:
630:
620:
590:
580:
485:
388:
5715:
5229:"), the three possible situations the individual could face. In particular,
39:
7164:
7109:
7004:
6994:
6989:
6914:
6759:
6284:
6212:
5902:
5225:
4752:
is, in effect, a lottery in which the best outcome is won with probability
871:
785:
730:
625:
615:
610:
535:
83:
16:
Any individual whose preferences satisfy four axioms has a utility function
5977:
An
Introduction to Decision Theory (Cambridge Introductions to Philosophy)
5293:
theories have arisen, most of which drop or relax the independence axiom.
5274:. As stated, the hypothesis may appear to be a bold claim. The aim of the
7139:
7129:
6919:
6798:
6742:
6217:
6124:
1034:
931:
760:
750:
540:
175:
7049:
6849:
6626:
5961:
5923:
5362:, a von Neumann–Morgenstern rational agent must be indifferent between
670:
470:
5724:
2712:{\displaystyle {\text{then}}\quad pL+(1-p)N\sim pqL'+p(1-q)N'+(1-p)N.}
1411:
Completeness assumes that an individual has well defined preferences:
6899:
6829:
6186:
5756:
Implicit in denoting indifference by equality are assertions like if
5111:{\displaystyle 20\%(\$ 10\,000)+80\%(\$ 0)=\$ 2000>100\%(\$ 1000)}
2174:
cancel out and don't affect our decision, because the probability of
1396:. These axioms, apart from continuity, are often justified using the
520:
451:
31:
5915:
4342:. So, by the Reduction axiom, he is indifferent between the lottery
6678:
5251:
1769:
where the notation on the left side refers to a situation in which
1007:. It is related, but not necessarily equivalent to, the utility of
5983:
3437:, define a lottery that selects the best outcome with probability
1530:
assumes that preferences are consistent across any three options:
1037:, all probabilities summing to one. For example, for two outcomes
965:, where such an individual's preferences can be represented on an
6074:
6069:
4130:
To see why this utility function makes sense, consider a lottery
1622:
Continuity assumes that there is a "tipping point" between being
1340:
is either preferred over or viewed with indifference relative to
1023:
In the theorem, an individual agent is faced with options called
5697:
4868:, because it gives him a larger chance to win the best outcome.
5296:
5014:
934:
involves making decisions that take the form of maximizing the
854:
3259:{\displaystyle A_{1}\preceq A_{2}\preceq \cdots \preceq A_{n}}
5432:
4990:
Thus, the content of the theorem is that the construction of
3137:
The proof is constructive: it shows how the desired function
6155:
5844:
EconPort, "Von
Neumann–Morgenstern Expected Utility Theory"
3125:
will obey axioms 1–4. Such a function is called the agent's
6084:
5478:
for the agent's own VNM-utility, a contradiction in terms.
5209:{\displaystyle 20\%u(\$ 10\,000)+80\%u(\$ 0)<u(\$ 1000)}
5002:
5942:
Oxford, Oxford
University Press. 1993 reprinted 1995, 2002
5519:. Von Neumann and Morgenstern recognized this limitation:
5013:
It is often the case that a person, faced with real-world
5951:. Huntington, NY. Robert E. Krieger Publishing Co. 1970.
3805:{\displaystyle 0=q_{1}\leq q_{2}\leq \cdots \leq q_{n}=1}
5607:) are not canonically defined, nor are comparisons like
5835:, so it may be surveyed in brief for intuitive meaning.
2038:(with the "irrelevant" part of the lottery underlined):
5698:"Back to Bentham? Explorations of Experienced Utility"
4717:{\displaystyle M'=u(M)\cdot A_{n}+(1-u(M))\cdot A_{1}}
5886:
5820:
5794:
5762:
5391:
5368:
5345:
5325:
5138:
5034:
4994:
is possible, and they claim little about its nature.
4909:
4882:
4854:
4834:
4828:, a rational decision maker would prefer the lottery
4790:
4758:
4733:
4637:
4487:
4371:
4348:
4335:{\displaystyle q_{i}\cdot A_{n}+(1-q_{i})\cdot A_{1}}
4270:
4243:
4216:
4189:
4136:
3983:
3923:
3871:
3841:
3821:
3747:
3692:
3662:
3635:
3590:
3548:
3466:
3443:
3405:
3360:
3318:
3275:
3213:
3170:
3143:
2951:
2815:
2730:
2605:
2535:
2485:
2437:
2401:
2323:
2303:
2229:
2203:
2180:
2160:
2046:
2006:
1986:
1883:
1838:
1805:
1717:
1676:
1643:
1597:
1570:
1543:
1491:
1464:
1444:
1424:
1350:
1313:
1276:
1250:
1185:
1131:
1056:
2591:{\displaystyle {\text{if}}\qquad L\sim qL'+(1-q)N',}
1520:
preference or indifference). Note that this implies
5695:
5966:Some problems and developments in decision science
5827:
5806:
5780:
5747:. Westview Press (May 12, 1988), chapters 2 and 5.
5681:. Princeton, NJ. Princeton University Press, 1953.
5421:
5377:
5354:
5331:
5208:
5110:
4985:VNM 1953, § 3.1.1 p.16 and § 3.7.1 p. 28
4960:
4895:
4860:
4840:
4820:
4773:
4744:
4716:
4622:
4472:
4354:
4334:
4256:
4229:
4202:
4175:
4119:
3962:
3906:
3854:
3827:
3804:
3727:
3675:
3648:
3618:
3576:
3532:{\displaystyle L(p)=p\cdot A_{n}+(1-p)\cdot A_{1}}
3531:
3449:
3429:
3388:
3346:
3301:
3258:
3196:
3149:
3088:
2915:
2776:
2711:
2590:
2515:
2471:
2413:
2387:
2317:. With the "only if" direction, we can argue that
2309:
2289:
2215:
2186:
2166:
2143:
2030:
1992:
1957:
1863:
1824:
1756:
1701:
1662:
1610:
1583:
1556:
1504:
1477:
1450:
1430:
1365:
1328:
1288:
1262:
1198:
1166:
1080:
5648:
5536:
7267:
5252:Implications for the expected utility hypothesis
3629:By the Continuity axiom, for every sure outcome
5850:
5515:where people choose to avoid the perception of
5219:where the dollar amounts here really represent
6171:
5999:
5632:
891:
5856:
5297:Implications for ethics and moral philosophy
2284:
2230:
2154:In other words, the probabilities involving
5314:possible with various VNM-utility functions
2429:Axiom 4′ (Reduction of compound lotteries):
981:, which can then be defined as the agent's
6178:
6164:
6105:Heuristics in judgment and decision-making
6006:
5992:
5433:Distinctness from other notions of utility
5129:. In this example, we could conclude that
4892:
2784:in the expression in Axiom 4, and expand.
898:
884:
5940:Foundations of Rational Choice Under Risk
5882:Specimen theoriae novae de mensura sortis
5857:Keeney, Ralph L.; Raiffa, Howard (1993).
5821:
5723:
5157:
5050:
5017:with money, does not act to maximize the
2856:
2843:
2833:
2724:To see how Axiom 4 implies Axiom 4', set
2119:
2079:
2063:
1924:
1920:
1916:
1912:
1884:
1839:
1806:
1750:
1746:
1718:
1677:
1644:
1598:
1571:
1544:
1492:
1465:
1400:(whereas continuity is used to set aside
5979:. Cambridge: Cambridge University Press.
5003:Automatic consideration of risk aversion
2297:, and strictly prefers all of them over
2290:{\displaystyle \{pA+(1-p)B:p\in [0,1)\}}
1380:The four axioms of VNM-rationality, are
1101:) = 25% is the probability of
5737:
5735:
1864:{\displaystyle \,\varepsilon \in (0,1)}
7268:
2421:, thus excluding this counterexample.
1296:. If the agent is indifferent between
6159:
5987:
5691:
5689:
5687:
5678:Theory of Games and Economic Behavior
5664:
5662:
5660:
5658:
4176:{\displaystyle M=\sum _{i}p_{i}A_{i}}
3963:{\displaystyle M=\sum _{i}p_{i}A_{i}}
3127:von Neumann–Morgenstern (VNM) utility
1757:{\displaystyle \,pL+(1-p)N\,\sim \,M}
1663:{\displaystyle \,L\preceq M\preceq N}
5896:
5732:
973:there exists a real-valued function
6255:Agent-based computational economics
6013:
4961:{\displaystyle E(u(L))<E(u(M)).}
4781:, and the worst outcome otherwise.
3835:, the utility function for outcome
13:
5949:Utility Theory for Decision Making
5900:(1950). "The Bargaining Problem".
5859:Decisions with Multiple Objectives
5750:
5684:
5655:
5486:
5197:
5179:
5170:
5151:
5142:
5099:
5093:
5081:
5069:
5063:
5044:
5038:
3728:{\displaystyle L(q_{i})\sim A_{i}}
2860:
2857:
2853:
2850:
2847:
2844:
2840:
2837:
2834:
2830:
2827:
2223:, and the agent is indifferent on
1832:, then there exists a probability
1670:, then there exists a probability
1167:{\displaystyle L=\sum p_{i}A_{i},}
14:
7317:
4473:{\displaystyle M'=\sum _{i}p_{i}}
3457:and the worst outcome otherwise:
2804:such that for any two lotteries,
1825:{\displaystyle \,L\prec M\prec N}
957:proved that any individual whose
6713:neoclassical–Keynesian synthesis
5696:Kahneman; Wakker; Sarin (1997).
3917:so the utility of every lottery
3302:{\displaystyle A_{1}\prec A_{n}}
3197:{\displaystyle A_{1}\dots A_{n}}
1797:Axiom 3′ (Archimedean property):
1781:is received with probability (1–
865:
853:
38:
4997:
3132:
2864:
2825:
2611:
2541:
2194:is the same in both lotteries.
2062:
2056:
140:Concepts, theory and techniques
5970:Revista Matematica Complutense
5875:
5838:
5703:Quarterly Journal of Economics
5649:References and further reading
5549:, their VNM-utility functions
5537:Incomparability between agents
5481:
5413:
5401:
5310:agent-focused to agent-neutral
5203:
5194:
5185:
5176:
5161:
5148:
5105:
5096:
5075:
5066:
5054:
5041:
4952:
4949:
4943:
4937:
4928:
4925:
4919:
4913:
4815:
4809:
4800:
4794:
4768:
4762:
4698:
4695:
4689:
4677:
4658:
4652:
4599:
4580:
4467:
4451:
4432:
4403:
4316:
4297:
4081:
4068:
3993:
3987:
3907:{\displaystyle u(A_{i})=q_{i}}
3888:
3875:
3709:
3696:
3619:{\displaystyle L(1)\sim A_{n}}
3600:
3594:
3577:{\displaystyle L(0)\sim A_{1}}
3558:
3552:
3513:
3501:
3476:
3470:
3424:
3412:
3377:
3364:
3335:
3322:
3080:
3067:
3042:
3029:
3010:
2958:
2907:
2904:
2898:
2892:
2883:
2880:
2874:
2868:
2796:which assigns to each outcome
2787:
2763:
2751:
2700:
2688:
2674:
2662:
2633:
2621:
2574:
2562:
2510:
2498:
2281:
2269:
2254:
2242:
2116:
2104:
2076:
2064:
2025:
2013:
1946:
1934:
1897:
1885:
1858:
1846:
1740:
1728:
1696:
1684:
1:
6649:Critique of political economy
6185:
5745:Notes on the Theory of Choice
5541:Since for any two VNM-agents
2777:{\displaystyle M=qL'+(1-q)N'}
1773:is received with probability
1516:(the individual must express
1375:
1081:{\displaystyle L=0.25A+0.75B}
5530:VNM 1953 § 3.7.1, p. 28
5525:specific utility of gambling
5503: + (1 −
5468:greatest happiness principle
5291:generalized expected utility
5284:VNM 1953 § 3.5.2, p. 25
5233:can exhibit properties like
4821:{\displaystyle u(M)>u(L)}
3113: + (1 −
3109:, meaning those of the form
1611:{\displaystyle \,L\succeq N}
1584:{\displaystyle \,M\succeq N}
1557:{\displaystyle \,L\succeq M}
1505:{\displaystyle \,M\succeq L}
1478:{\displaystyle \,L\succeq M}
961:satisfied four axioms has a
7:
6029:Expected utility hypothesis
5639:expected utility hypothesis
5272:expected utility hypothesis
5258:Expected utility hypothesis
4973:
4362:and the following lottery:
1366:{\displaystyle L\preceq M.}
10:
7322:
6787:Real business-cycle theory
6145:Evidential decision theory
5781:{\displaystyle L\prec M=N}
5633:Applicability to economics
5439:utilitarian moral theories
5300:
5255:
5006:
4896:{\displaystyle L\prec M\;}
3389:{\displaystyle u(A_{n})=1}
3347:{\displaystyle u(A_{1})=0}
2414:{\displaystyle A\succeq B}
2031:{\displaystyle p\in [0,1)}
1240:is preferred over lottery
7227:
7185:
6827:
6561:
6310:
6275:
6193:
6080:Principle of indifference
6042:
6021:
5975:Peterson, Martin (2009).
5422:{\displaystyle pM+(1-p)0}
3656:, there is a probability
2472:{\displaystyle L,L',N,N'}
1093:denotes a scenario where
1018:
5807:{\displaystyle L\prec N}
5276:expected utility theorem
4183:, which selects outcome
1329:{\displaystyle L\sim M.}
1289:{\displaystyle L\prec M}
1263:{\displaystyle M\succ L}
128:JEL classification codes
6427:Industrial organization
6250:Computational economics
6140:Emotional choice theory
5828:{\displaystyle \,\sim }
5716:10.1162/003355397555235
5247:monetary risk aversion.
2516:{\displaystyle p,q\in }
1978:Axiom 4 (Independence):
1702:{\displaystyle \,p\in }
1630:a given middle option:
944:expected utility theory
916:von Neumann–Morgenstern
314:Industrial organization
171:Computational economics
7301:Rational choice theory
6632:Modern monetary theory
6297:Experimental economics
6267:Pluralism in economics
6240:Mathematical economics
6135:Causal decision theory
6100:St. Petersburg paradox
6050:Decision-matrix method
5829:
5808:
5782:
5534:
5423:
5379:
5356:
5333:
5287:
5210:
5112:
4988:
4962:
4897:
4862:
4842:
4822:
4775:
4746:
4718:
4624:
4474:
4356:
4336:
4258:
4231:
4204:
4177:
4121:
3970:is the expectation of
3964:
3908:
3856:
3829:
3806:
3729:
3677:
3650:
3620:
3578:
3533:
3451:
3431:
3399:For every probability
3390:
3348:
3303:
3260:
3198:
3151:
3090:
2917:
2778:
2713:
2592:
2517:
2473:
2415:
2389:
2311:
2291:
2217:
2188:
2168:
2145:
2032:
1994:
1959:
1865:
1826:
1758:
1703:
1664:
1612:
1585:
1558:
1535:Axiom 2 (Transitivity)
1506:
1479:
1452:
1432:
1416:Axiom 1 (Completeness)
1367:
1330:
1290:
1264:
1200:
1168:
1082:
1000:in that it is used to
166:Experimental economics
6115:Bayesian epistemology
5830:
5809:
5783:
5521:
5424:
5380:
5357:
5334:
5280:
5211:
5113:
4981:
4963:
4898:
4863:
4843:
4823:
4776:
4747:
4719:
4625:
4475:
4357:
4337:
4259:
4257:{\displaystyle A_{i}}
4232:
4230:{\displaystyle p_{i}}
4205:
4203:{\displaystyle A_{i}}
4178:
4122:
3965:
3909:
3857:
3855:{\displaystyle A_{i}}
3830:
3807:
3730:
3678:
3676:{\displaystyle q_{i}}
3651:
3649:{\displaystyle A_{i}}
3621:
3579:
3534:
3452:
3432:
3430:{\displaystyle p\in }
3391:
3349:
3304:
3261:
3199:
3152:
3091:
2918:
2779:
2714:
2593:
2518:
2474:
2416:
2390:
2312:
2292:
2218:
2189:
2169:
2146:
2033:
1995:
1960:
1866:
1827:
1759:
1704:
1665:
1635:Axiom 3 (Continuity):
1613:
1586:
1559:
1507:
1480:
1453:
1433:
1368:
1331:
1306:indifference relation
1291:
1265:
1201:
1199:{\displaystyle p_{i}}
1169:
1083:
6506:Social choice theory
6262:Behavioral economics
6245:Complexity economics
6130:Social choice theory
6034:Intertemporal choice
5818:
5792:
5760:
5523:"...concepts like a
5499:are lotteries, then
5446:1) VNM-utility is a
5389:
5366:
5343:
5323:
5136:
5032:
4907:
4880:
4852:
4832:
4788:
4774:{\displaystyle u(M)}
4756:
4731:
4635:
4485:
4369:
4346:
4268:
4241:
4214:
4187:
4134:
3981:
3921:
3869:
3839:
3819:
3745:
3690:
3660:
3633:
3588:
3546:
3464:
3441:
3403:
3358:
3316:
3273:
3211:
3168:
3141:
2949:
2813:
2728:
2603:
2533:
2483:
2435:
2399:
2321:
2301:
2227:
2201:
2178:
2158:
2044:
2004:
1984:
1881:
1836:
1803:
1790:Archimedean property
1715:
1674:
1641:
1595:
1568:
1541:
1489:
1462:
1442:
1422:
1348:
1311:
1274:
1248:
1183:
1179:with the sum of the
1129:
1054:
393:Social choice theory
6590:American (National)
6290:Economic statistics
6065:Strategic dominance
2216:{\displaystyle A,B}
1398:Dutch book theorems
1270:, or equivalently,
860:Business portal
181:Operations research
161:National accounting
6110:Probability theory
5946:Fishburn, Peter C.
5825:
5804:
5778:
5673:Morgenstern, Oskar
5419:
5378:{\displaystyle 1N}
5375:
5355:{\displaystyle pM}
5352:
5329:
5319:If the utility of
5206:
5108:
4958:
4893:
4858:
4838:
4818:
4771:
4745:{\displaystyle M'}
4742:
4714:
4620:
4569:
4513:
4470:
4392:
4352:
4332:
4254:
4227:
4200:
4173:
4152:
4117:
4096:
4054:
4016:
3960:
3939:
3904:
3852:
3825:
3802:
3725:
3673:
3646:
3616:
3574:
3529:
3447:
3427:
3386:
3344:
3299:
3256:
3194:
3160:Suppose there are
3147:
3086:
2932:, or more briefly
2913:
2774:
2709:
2588:
2513:
2469:
2431:For any lotteries
2411:
2385:
2307:
2287:
2213:
2184:
2164:
2141:
2139:
2099:
2028:
1990:
1955:
1861:
1822:
1754:
1699:
1660:
1608:
1581:
1554:
1502:
1475:
1448:
1428:
1418:For any lotteries
1363:
1326:
1286:
1260:
1196:
1164:
1078:
1031:mutually exclusive
926:demonstrates that
191:Industrial complex
186:Middle income trap
7263:
7262:
6794:New institutional
6153:
6152:
5957:978-0-471-26060-8
5898:Nash, John F. Jr.
5669:Neumann, John von
5618:) <
5517:risks about risks
5332:{\displaystyle N}
4861:{\displaystyle L}
4848:over the lottery
4841:{\displaystyle M}
4560:
4504:
4383:
4355:{\displaystyle M}
4210:with probability
4143:
4087:
4045:
4007:
3930:
3828:{\displaystyle i}
3450:{\displaystyle p}
3150:{\displaystyle u}
2609:
2539:
2380:
2364:
2348:
2332:
2310:{\displaystyle A}
2187:{\displaystyle M}
2167:{\displaystyle M}
2127:
2087:
2060:
1993:{\displaystyle N}
1451:{\displaystyle M}
1431:{\displaystyle L}
1229: + 0.50
1225: + 0.25
1217:) + 0.5
996:VNM-utility is a
955:Oskar Morgenstern
908:
907:
7313:
7291:John von Neumann
6467:Natural resource
6302:Economic history
6228:Mechanism design
6180:
6173:
6166:
6157:
6156:
6095:Ellsberg paradox
6060:Expected utility
6008:
6001:
5994:
5985:
5984:
5935:
5888:
5879:
5873:
5872:
5854:
5848:
5842:
5836:
5834:
5832:
5831:
5826:
5813:
5811:
5810:
5805:
5787:
5785:
5784:
5779:
5754:
5748:
5739:
5730:
5729:
5727:
5693:
5682:
5666:
5513:Ellsberg problem
5448:decision utility
5428:
5426:
5425:
5420:
5384:
5382:
5381:
5376:
5361:
5359:
5358:
5353:
5338:
5336:
5335:
5330:
5315:
5303:Consequentialism
5264:Daniel Bernoulli
5215:
5213:
5212:
5207:
5117:
5115:
5114:
5109:
4967:
4965:
4964:
4959:
4902:
4900:
4899:
4894:
4867:
4865:
4864:
4859:
4847:
4845:
4844:
4839:
4827:
4825:
4824:
4819:
4780:
4778:
4777:
4772:
4751:
4749:
4748:
4743:
4741:
4723:
4721:
4720:
4715:
4713:
4712:
4673:
4672:
4645:
4629:
4627:
4626:
4621:
4619:
4618:
4606:
4602:
4598:
4597:
4579:
4578:
4568:
4551:
4550:
4538:
4534:
4533:
4532:
4523:
4522:
4512:
4495:
4479:
4477:
4476:
4471:
4466:
4465:
4450:
4449:
4428:
4427:
4415:
4414:
4402:
4401:
4391:
4379:
4361:
4359:
4358:
4353:
4341:
4339:
4338:
4333:
4331:
4330:
4315:
4314:
4293:
4292:
4280:
4279:
4264:and the lottery
4263:
4261:
4260:
4255:
4253:
4252:
4236:
4234:
4233:
4228:
4226:
4225:
4209:
4207:
4206:
4201:
4199:
4198:
4182:
4180:
4179:
4174:
4172:
4171:
4162:
4161:
4151:
4126:
4124:
4123:
4118:
4116:
4115:
4106:
4105:
4095:
4080:
4079:
4064:
4063:
4053:
4041:
4037:
4036:
4035:
4026:
4025:
4015:
3969:
3967:
3966:
3961:
3959:
3958:
3949:
3948:
3938:
3913:
3911:
3910:
3905:
3903:
3902:
3887:
3886:
3861:
3859:
3858:
3853:
3851:
3850:
3834:
3832:
3831:
3826:
3811:
3809:
3808:
3803:
3795:
3794:
3776:
3775:
3763:
3762:
3734:
3732:
3731:
3726:
3724:
3723:
3708:
3707:
3682:
3680:
3679:
3674:
3672:
3671:
3655:
3653:
3652:
3647:
3645:
3644:
3625:
3623:
3622:
3617:
3615:
3614:
3583:
3581:
3580:
3575:
3573:
3572:
3538:
3536:
3535:
3530:
3528:
3527:
3497:
3496:
3456:
3454:
3453:
3448:
3436:
3434:
3433:
3428:
3395:
3393:
3392:
3387:
3376:
3375:
3353:
3351:
3350:
3345:
3334:
3333:
3308:
3306:
3305:
3300:
3298:
3297:
3285:
3284:
3265:
3263:
3262:
3257:
3255:
3254:
3236:
3235:
3223:
3222:
3203:
3201:
3200:
3195:
3193:
3192:
3180:
3179:
3156:
3154:
3153:
3148:
3107:simple lotteries
3095:
3093:
3092:
3087:
3079:
3078:
3063:
3062:
3041:
3040:
3025:
3024:
3009:
3008:
2999:
2998:
2980:
2979:
2970:
2969:
2922:
2920:
2919:
2914:
2863:
2783:
2781:
2780:
2775:
2773:
2747:
2718:
2716:
2715:
2710:
2684:
2655:
2610:
2607:
2597:
2595:
2594:
2589:
2584:
2558:
2540:
2537:
2522:
2520:
2519:
2514:
2478:
2476:
2475:
2470:
2468:
2451:
2420:
2418:
2417:
2412:
2394:
2392:
2391:
2386:
2381:
2373:
2365:
2357:
2349:
2341:
2333:
2325:
2316:
2314:
2313:
2308:
2296:
2294:
2293:
2288:
2222:
2220:
2219:
2214:
2193:
2191:
2190:
2185:
2173:
2171:
2170:
2165:
2150:
2148:
2147:
2142:
2140:
2135:
2100:
2095:
2061:
2058:
2037:
2035:
2034:
2029:
1999:
1997:
1996:
1991:
1964:
1962:
1961:
1956:
1870:
1868:
1867:
1862:
1831:
1829:
1828:
1823:
1763:
1761:
1760:
1755:
1708:
1706:
1705:
1700:
1669:
1667:
1666:
1661:
1617:
1615:
1614:
1609:
1590:
1588:
1587:
1582:
1563:
1561:
1560:
1555:
1511:
1509:
1508:
1503:
1484:
1482:
1481:
1476:
1457:
1455:
1454:
1449:
1437:
1435:
1434:
1429:
1372:
1370:
1369:
1364:
1335:
1333:
1332:
1327:
1295:
1293:
1292:
1287:
1269:
1267:
1266:
1261:
1213: + 0.5
1205:
1203:
1202:
1197:
1195:
1194:
1173:
1171:
1170:
1165:
1160:
1159:
1150:
1149:
1087:
1085:
1084:
1079:
998:decision utility
963:utility function
951:John von Neumann
940:cardinal utility
900:
893:
886:
872:Money portal
870:
869:
868:
858:
857:
354:Natural resource
146:Economic systems
42:
19:
18:
7321:
7320:
7316:
7315:
7314:
7312:
7311:
7310:
7306:Decision theory
7266:
7265:
7264:
7259:
7256:Business portal
7223:
7222:
7221:
7181:
6945:von Böhm-Bawerk
6833:
6832:
6823:
6595:Ancient thought
6573:
6572:
6566:
6557:
6556:
6555:
6306:
6271:
6223:Contract theory
6208:Decision theory
6189:
6184:
6154:
6149:
6055:Decision matrix
6038:
6017:
6015:Decision theory
6012:
5982:
5916:10.2307/1907266
5892:
5891:
5880:
5876:
5869:
5855:
5851:
5843:
5839:
5819:
5816:
5815:
5793:
5790:
5789:
5761:
5758:
5757:
5755:
5751:
5742:Kreps, David M.
5740:
5733:
5694:
5685:
5667:
5656:
5651:
5635:
5623:
5612:
5601:
5590:
5579:
5568:
5561:
5554:
5539:
5489:
5487:Nested gambling
5484:
5435:
5390:
5387:
5386:
5367:
5364:
5363:
5344:
5341:
5340:
5324:
5321:
5320:
5313:
5305:
5299:
5260:
5254:
5137:
5134:
5133:
5033:
5030:
5029:
5011:
5005:
5000:
4976:
4908:
4905:
4904:
4903:if and only if
4881:
4878:
4877:
4853:
4850:
4849:
4833:
4830:
4829:
4789:
4786:
4785:
4757:
4754:
4753:
4734:
4732:
4729:
4728:
4708:
4704:
4668:
4664:
4638:
4636:
4633:
4632:
4614:
4610:
4593:
4589:
4574:
4570:
4564:
4559:
4555:
4546:
4542:
4528:
4524:
4518:
4514:
4508:
4503:
4499:
4488:
4486:
4483:
4482:
4461:
4457:
4445:
4441:
4423:
4419:
4410:
4406:
4397:
4393:
4387:
4372:
4370:
4367:
4366:
4347:
4344:
4343:
4326:
4322:
4310:
4306:
4288:
4284:
4275:
4271:
4269:
4266:
4265:
4248:
4244:
4242:
4239:
4238:
4221:
4217:
4215:
4212:
4211:
4194:
4190:
4188:
4185:
4184:
4167:
4163:
4157:
4153:
4147:
4135:
4132:
4131:
4111:
4107:
4101:
4097:
4091:
4075:
4071:
4059:
4055:
4049:
4031:
4027:
4021:
4017:
4011:
4006:
4002:
3982:
3979:
3978:
3954:
3950:
3944:
3940:
3934:
3922:
3919:
3918:
3898:
3894:
3882:
3878:
3870:
3867:
3866:
3846:
3842:
3840:
3837:
3836:
3820:
3817:
3816:
3790:
3786:
3771:
3767:
3758:
3754:
3746:
3743:
3742:
3719:
3715:
3703:
3699:
3691:
3688:
3687:
3667:
3663:
3661:
3658:
3657:
3640:
3636:
3634:
3631:
3630:
3610:
3606:
3589:
3586:
3585:
3568:
3564:
3547:
3544:
3543:
3523:
3519:
3492:
3488:
3465:
3462:
3461:
3442:
3439:
3438:
3404:
3401:
3400:
3371:
3367:
3359:
3356:
3355:
3329:
3325:
3317:
3314:
3313:
3293:
3289:
3280:
3276:
3274:
3271:
3270:
3250:
3246:
3231:
3227:
3218:
3214:
3212:
3209:
3208:
3188:
3184:
3175:
3171:
3169:
3166:
3165:
3164:sure outcomes,
3142:
3139:
3138:
3135:
3074:
3070:
3058:
3054:
3036:
3032:
3020:
3016:
3004:
3000:
2994:
2990:
2975:
2971:
2965:
2961:
2950:
2947:
2946:
2826:
2814:
2811:
2810:
2790:
2766:
2740:
2729:
2726:
2725:
2677:
2648:
2606:
2604:
2601:
2600:
2577:
2551:
2536:
2534:
2531:
2530:
2484:
2481:
2480:
2461:
2444:
2436:
2433:
2432:
2400:
2397:
2396:
2372:
2356:
2340:
2324:
2322:
2319:
2318:
2302:
2299:
2298:
2228:
2225:
2224:
2202:
2199:
2198:
2179:
2176:
2175:
2159:
2156:
2155:
2128:
2126:
2088:
2086:
2057:
2045:
2042:
2041:
2005:
2002:
2001:
1985:
1982:
1981:
1882:
1879:
1878:
1837:
1834:
1833:
1804:
1801:
1800:
1716:
1713:
1712:
1675:
1672:
1671:
1642:
1639:
1638:
1596:
1593:
1592:
1569:
1566:
1565:
1542:
1539:
1538:
1490:
1487:
1486:
1463:
1460:
1459:
1443:
1440:
1439:
1423:
1420:
1419:
1378:
1349:
1346:
1345:
1312:
1309:
1308:
1304:, we write the
1275:
1272:
1271:
1249:
1246:
1245:
1190:
1186:
1184:
1181:
1180:
1155:
1151:
1145:
1141:
1130:
1127:
1126:
1118:
1055:
1052:
1051:
1021:
928:rational choice
924:utility theorem
912:decision theory
904:
866:
864:
852:
845:
844:
815:
805:
804:
803:
802:
566:von Böhm-Bawerk
454:
443:
442:
204:
196:
195:
151:Economic growth
141:
133:
132:
74:
72:classifications
17:
12:
11:
5:
7319:
7309:
7308:
7303:
7298:
7293:
7288:
7283:
7278:
7261:
7260:
7258:
7253:
7248:
7243:
7238:
7233:
7228:
7225:
7224:
7220:
7219:
7214:
7204:
7199:
7193:
7192:
7191:
7189:
7183:
7182:
7180:
7179:
7172:
7167:
7162:
7157:
7152:
7147:
7142:
7137:
7132:
7127:
7122:
7117:
7112:
7107:
7102:
7097:
7092:
7087:
7082:
7077:
7072:
7067:
7062:
7057:
7052:
7047:
7042:
7037:
7032:
7027:
7022:
7017:
7012:
7007:
7002:
6997:
6992:
6987:
6982:
6977:
6972:
6967:
6962:
6957:
6952:
6947:
6942:
6937:
6932:
6927:
6922:
6917:
6912:
6907:
6902:
6897:
6892:
6887:
6882:
6877:
6872:
6867:
6862:
6857:
6852:
6847:
6842:
6836:
6834:
6828:
6825:
6824:
6822:
6821:
6816:
6811:
6806:
6801:
6796:
6791:
6790:
6789:
6779:
6778:
6777:
6767:
6762:
6757:
6756:
6755:
6745:
6740:
6735:
6734:
6733:
6732:
6731:
6721:
6716:
6701:
6696:
6691:
6686:
6681:
6676:
6671:
6666:
6661:
6659:Disequilibrium
6656:
6651:
6646:
6641:
6636:
6635:
6634:
6624:
6619:
6614:
6609:
6608:
6607:
6597:
6592:
6587:
6582:
6576:
6574:
6562:
6559:
6558:
6554:
6553:
6548:
6543:
6538:
6533:
6528:
6523:
6518:
6513:
6508:
6499:
6494:
6489:
6484:
6479:
6474:
6472:Organizational
6469:
6464:
6459:
6454:
6449:
6444:
6439:
6434:
6429:
6424:
6419:
6414:
6409:
6404:
6399:
6394:
6389:
6384:
6379:
6374:
6369:
6364:
6359:
6354:
6349:
6344:
6339:
6334:
6329:
6324:
6318:
6317:
6316:
6314:
6308:
6307:
6305:
6304:
6299:
6294:
6293:
6292:
6281:
6279:
6273:
6272:
6270:
6269:
6264:
6259:
6258:
6257:
6247:
6242:
6237:
6235:Macroeconomics
6232:
6231:
6230:
6225:
6220:
6215:
6210:
6203:Microeconomics
6199:
6197:
6191:
6190:
6183:
6182:
6175:
6168:
6160:
6151:
6150:
6148:
6147:
6142:
6137:
6132:
6127:
6122:
6117:
6112:
6107:
6102:
6097:
6092:
6090:Allais paradox
6087:
6082:
6077:
6072:
6067:
6062:
6057:
6052:
6046:
6044:
6040:
6039:
6037:
6036:
6031:
6025:
6023:
6019:
6018:
6011:
6010:
6003:
5996:
5988:
5981:
5980:
5973:
5959:
5943:
5936:
5910:(2): 155–162.
5893:
5890:
5889:
5874:
5867:
5849:
5837:
5824:
5803:
5800:
5797:
5777:
5774:
5771:
5768:
5765:
5749:
5731:
5710:(2): 375–406.
5683:
5653:
5652:
5650:
5647:
5643:Allais paradox
5634:
5631:
5621:
5610:
5599:
5596:) −
5588:
5577:
5566:
5559:
5552:
5538:
5535:
5488:
5485:
5483:
5480:
5456:
5455:
5451:
5434:
5431:
5418:
5415:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5374:
5371:
5351:
5348:
5328:
5312:behaviors are
5301:Main article:
5298:
5295:
5256:Main article:
5253:
5250:
5217:
5216:
5205:
5202:
5199:
5196:
5193:
5190:
5187:
5184:
5181:
5178:
5175:
5172:
5169:
5166:
5163:
5160:
5156:
5153:
5150:
5147:
5144:
5141:
5119:
5118:
5107:
5104:
5101:
5098:
5095:
5092:
5089:
5086:
5083:
5080:
5077:
5074:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5049:
5046:
5043:
5040:
5037:
5023:dollar assets.
5019:expected value
5007:Main article:
5004:
5001:
4999:
4996:
4975:
4972:
4971:
4970:
4969:
4968:
4957:
4954:
4951:
4948:
4945:
4942:
4939:
4936:
4933:
4930:
4927:
4924:
4921:
4918:
4915:
4912:
4891:
4888:
4885:
4857:
4837:
4817:
4814:
4811:
4808:
4805:
4802:
4799:
4796:
4793:
4770:
4767:
4764:
4761:
4740:
4737:
4725:
4724:
4711:
4707:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4671:
4667:
4663:
4660:
4657:
4654:
4651:
4648:
4644:
4641:
4630:
4617:
4613:
4609:
4605:
4601:
4596:
4592:
4588:
4585:
4582:
4577:
4573:
4567:
4563:
4558:
4554:
4549:
4545:
4541:
4537:
4531:
4527:
4521:
4517:
4511:
4507:
4502:
4498:
4494:
4491:
4480:
4469:
4464:
4460:
4456:
4453:
4448:
4444:
4440:
4437:
4434:
4431:
4426:
4422:
4418:
4413:
4409:
4405:
4400:
4396:
4390:
4386:
4382:
4378:
4375:
4351:
4329:
4325:
4321:
4318:
4313:
4309:
4305:
4302:
4299:
4296:
4291:
4287:
4283:
4278:
4274:
4251:
4247:
4224:
4220:
4197:
4193:
4170:
4166:
4160:
4156:
4150:
4146:
4142:
4139:
4128:
4127:
4114:
4110:
4104:
4100:
4094:
4090:
4086:
4083:
4078:
4074:
4070:
4067:
4062:
4058:
4052:
4048:
4044:
4040:
4034:
4030:
4024:
4020:
4014:
4010:
4005:
4001:
3998:
3995:
3992:
3989:
3986:
3957:
3953:
3947:
3943:
3937:
3933:
3929:
3926:
3915:
3914:
3901:
3897:
3893:
3890:
3885:
3881:
3877:
3874:
3862:is defined as
3849:
3845:
3824:
3813:
3812:
3801:
3798:
3793:
3789:
3785:
3782:
3779:
3774:
3770:
3766:
3761:
3757:
3753:
3750:
3736:
3735:
3722:
3718:
3714:
3711:
3706:
3702:
3698:
3695:
3670:
3666:
3643:
3639:
3613:
3609:
3605:
3602:
3599:
3596:
3593:
3571:
3567:
3563:
3560:
3557:
3554:
3551:
3540:
3539:
3526:
3522:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3495:
3491:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3446:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3397:
3396:
3385:
3382:
3379:
3374:
3370:
3366:
3363:
3343:
3340:
3337:
3332:
3328:
3324:
3321:
3296:
3292:
3288:
3283:
3279:
3267:
3266:
3253:
3249:
3245:
3242:
3239:
3234:
3230:
3226:
3221:
3217:
3191:
3187:
3183:
3178:
3174:
3146:
3134:
3131:
3099:
3098:
3097:
3096:
3085:
3082:
3077:
3073:
3069:
3066:
3061:
3057:
3053:
3050:
3047:
3044:
3039:
3035:
3031:
3028:
3023:
3019:
3015:
3012:
3007:
3003:
2997:
2993:
2989:
2986:
2983:
2978:
2974:
2968:
2964:
2960:
2957:
2954:
2940:) is given by
2926:
2925:
2924:
2923:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2862:
2859:
2855:
2852:
2849:
2846:
2842:
2839:
2836:
2832:
2829:
2824:
2821:
2818:
2800:a real number
2789:
2786:
2772:
2769:
2765:
2762:
2759:
2756:
2753:
2750:
2746:
2743:
2739:
2736:
2733:
2722:
2721:
2720:
2719:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2683:
2680:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2654:
2651:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2598:
2587:
2583:
2580:
2576:
2573:
2570:
2567:
2564:
2561:
2557:
2554:
2550:
2547:
2544:
2525:
2524:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2467:
2464:
2460:
2457:
2454:
2450:
2447:
2443:
2440:
2410:
2407:
2404:
2384:
2379:
2376:
2371:
2368:
2363:
2360:
2355:
2352:
2347:
2344:
2339:
2336:
2331:
2328:
2306:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2212:
2209:
2206:
2183:
2163:
2152:
2151:
2138:
2134:
2131:
2125:
2122:
2118:
2115:
2112:
2109:
2106:
2103:
2098:
2094:
2091:
2085:
2082:
2078:
2075:
2072:
2069:
2066:
2059:if and only if
2055:
2052:
2049:
2039:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
1989:
1968:
1967:
1966:
1965:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1923:
1919:
1915:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1873:
1872:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1821:
1818:
1815:
1812:
1809:
1767:
1766:
1765:
1764:
1753:
1749:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1659:
1656:
1653:
1650:
1647:
1620:
1619:
1607:
1604:
1601:
1580:
1577:
1574:
1553:
1550:
1547:
1514:
1513:
1501:
1498:
1495:
1474:
1471:
1468:
1447:
1427:
1377:
1374:
1362:
1359:
1356:
1353:
1325:
1322:
1319:
1316:
1285:
1282:
1279:
1259:
1256:
1253:
1206:s equal to 1.
1193:
1189:
1177:
1176:
1175:
1174:
1163:
1158:
1154:
1148:
1144:
1140:
1137:
1134:
1116:
1105:occurring and
1091:
1090:
1089:
1088:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1020:
1017:
1013:utilitarianism
971:if and only if
967:interval scale
936:expected value
906:
905:
903:
902:
895:
888:
880:
877:
876:
875:
874:
862:
847:
846:
843:
842:
837:
827:
822:
816:
811:
810:
807:
806:
801:
800:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
638:
633:
628:
623:
618:
613:
608:
603:
598:
593:
588:
583:
578:
573:
568:
563:
558:
553:
548:
543:
538:
533:
528:
523:
518:
513:
508:
503:
498:
493:
488:
483:
478:
473:
468:
463:
457:
456:
455:
449:
448:
445:
444:
441:
440:
435:
430:
425:
420:
415:
410:
405:
400:
395:
386:
381:
376:
371:
366:
361:
359:Organizational
356:
351:
346:
341:
336:
331:
326:
321:
316:
311:
306:
301:
296:
291:
286:
281:
276:
271:
266:
261:
256:
251:
246:
241:
236:
231:
226:
221:
216:
211:
205:
203:By application
202:
201:
198:
197:
194:
193:
188:
183:
178:
173:
168:
163:
158:
153:
148:
142:
139:
138:
135:
134:
131:
130:
125:
120:
115:
110:
105:
96:
91:
86:
81:
75:
69:
68:
65:
64:
63:
62:
57:
52:
44:
43:
35:
34:
28:
27:
15:
9:
6:
4:
3:
2:
7318:
7307:
7304:
7302:
7299:
7297:
7294:
7292:
7289:
7287:
7284:
7282:
7279:
7277:
7274:
7273:
7271:
7257:
7254:
7252:
7249:
7247:
7244:
7242:
7239:
7237:
7234:
7232:
7229:
7226:
7218:
7215:
7212:
7208:
7205:
7203:
7200:
7198:
7195:
7194:
7190:
7188:
7184:
7178:
7177:
7173:
7171:
7168:
7166:
7163:
7161:
7158:
7156:
7153:
7151:
7148:
7146:
7143:
7141:
7138:
7136:
7133:
7131:
7128:
7126:
7123:
7121:
7118:
7116:
7113:
7111:
7108:
7106:
7103:
7101:
7098:
7096:
7093:
7091:
7088:
7086:
7083:
7081:
7078:
7076:
7073:
7071:
7068:
7066:
7063:
7061:
7058:
7056:
7053:
7051:
7048:
7046:
7043:
7041:
7038:
7036:
7033:
7031:
7028:
7026:
7023:
7021:
7018:
7016:
7013:
7011:
7008:
7006:
7003:
7001:
6998:
6996:
6993:
6991:
6988:
6986:
6983:
6981:
6978:
6976:
6973:
6971:
6968:
6966:
6963:
6961:
6958:
6956:
6953:
6951:
6948:
6946:
6943:
6941:
6938:
6936:
6933:
6931:
6928:
6926:
6923:
6921:
6918:
6916:
6913:
6911:
6908:
6906:
6903:
6901:
6898:
6896:
6893:
6891:
6888:
6886:
6883:
6881:
6878:
6876:
6873:
6871:
6868:
6866:
6863:
6861:
6858:
6856:
6853:
6851:
6848:
6846:
6843:
6841:
6840:de Mandeville
6838:
6837:
6835:
6831:
6826:
6820:
6817:
6815:
6812:
6810:
6807:
6805:
6802:
6800:
6797:
6795:
6792:
6788:
6785:
6784:
6783:
6782:New classical
6780:
6776:
6773:
6772:
6771:
6768:
6766:
6763:
6761:
6758:
6754:
6751:
6750:
6749:
6746:
6744:
6741:
6739:
6738:Malthusianism
6736:
6730:
6727:
6726:
6725:
6722:
6720:
6717:
6714:
6710:
6707:
6706:
6705:
6702:
6700:
6699:Institutional
6697:
6695:
6692:
6690:
6687:
6685:
6682:
6680:
6677:
6675:
6672:
6670:
6667:
6665:
6662:
6660:
6657:
6655:
6652:
6650:
6647:
6645:
6642:
6640:
6637:
6633:
6630:
6629:
6628:
6625:
6623:
6620:
6618:
6615:
6613:
6610:
6606:
6603:
6602:
6601:
6598:
6596:
6593:
6591:
6588:
6586:
6583:
6581:
6578:
6577:
6575:
6570:
6565:
6560:
6552:
6549:
6547:
6544:
6542:
6539:
6537:
6534:
6532:
6529:
6527:
6524:
6522:
6519:
6517:
6514:
6512:
6509:
6507:
6503:
6502:Public choice
6500:
6498:
6495:
6493:
6490:
6488:
6485:
6483:
6480:
6478:
6477:Participation
6475:
6473:
6470:
6468:
6465:
6463:
6460:
6458:
6455:
6453:
6450:
6448:
6445:
6443:
6440:
6438:
6437:Institutional
6435:
6433:
6430:
6428:
6425:
6423:
6420:
6418:
6415:
6413:
6410:
6408:
6405:
6403:
6400:
6398:
6395:
6393:
6390:
6388:
6387:Expeditionary
6385:
6383:
6380:
6378:
6377:Environmental
6375:
6373:
6370:
6368:
6365:
6363:
6360:
6358:
6355:
6353:
6350:
6348:
6345:
6343:
6340:
6338:
6335:
6333:
6330:
6328:
6325:
6323:
6320:
6319:
6315:
6313:
6309:
6303:
6300:
6298:
6295:
6291:
6288:
6287:
6286:
6283:
6282:
6280:
6278:
6274:
6268:
6265:
6263:
6260:
6256:
6253:
6252:
6251:
6248:
6246:
6243:
6241:
6238:
6236:
6233:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6209:
6206:
6205:
6204:
6201:
6200:
6198:
6196:
6192:
6188:
6181:
6176:
6174:
6169:
6167:
6162:
6161:
6158:
6146:
6143:
6141:
6138:
6136:
6133:
6131:
6128:
6126:
6123:
6121:
6120:Risk aversion
6118:
6116:
6113:
6111:
6108:
6106:
6103:
6101:
6098:
6096:
6093:
6091:
6088:
6086:
6083:
6081:
6078:
6076:
6073:
6071:
6068:
6066:
6063:
6061:
6058:
6056:
6053:
6051:
6048:
6047:
6045:
6041:
6035:
6032:
6030:
6027:
6026:
6024:
6020:
6016:
6009:
6004:
6002:
5997:
5995:
5990:
5989:
5986:
5978:
5974:
5972:11(1):113–41.
5971:
5967:
5963:
5960:
5958:
5954:
5950:
5947:
5944:
5941:
5938:Anand, Paul.
5937:
5933:
5929:
5925:
5921:
5917:
5913:
5909:
5905:
5904:
5899:
5895:
5894:
5887:
5883:
5878:
5870:
5868:0-521-44185-4
5864:
5860:
5853:
5847:
5841:
5822:
5801:
5798:
5795:
5775:
5772:
5769:
5766:
5763:
5753:
5746:
5743:
5738:
5736:
5726:
5721:
5717:
5713:
5709:
5705:
5704:
5699:
5692:
5690:
5688:
5680:
5679:
5674:
5670:
5665:
5663:
5661:
5659:
5654:
5646:
5644:
5640:
5630:
5628:
5624:
5617:
5613:
5606:
5602:
5595:
5591:
5584:
5580:
5573:
5569:
5562:
5555:
5548:
5544:
5533:
5531:
5526:
5520:
5518:
5514:
5510:
5506:
5502:
5498:
5494:
5479:
5477:
5473:
5469:
5465:
5461:
5452:
5449:
5445:
5444:
5443:
5440:
5430:
5416:
5410:
5407:
5404:
5398:
5395:
5392:
5372:
5369:
5349:
5346:
5326:
5317:
5311:
5304:
5294:
5292:
5286:
5285:
5279:
5277:
5273:
5269:
5265:
5259:
5249:
5248:
5244:
5240:
5236:
5232:
5228:
5227:
5222:
5200:
5191:
5188:
5182:
5173:
5167:
5164:
5158:
5154:
5145:
5139:
5132:
5131:
5130:
5128:
5124:
5102:
5090:
5087:
5084:
5078:
5072:
5060:
5057:
5051:
5047:
5035:
5028:
5027:
5026:
5024:
5020:
5016:
5010:
5009:Risk aversion
4995:
4993:
4987:
4986:
4980:
4955:
4946:
4940:
4934:
4931:
4922:
4916:
4910:
4889:
4886:
4883:
4876:
4875:
4874:
4873:
4872:
4869:
4855:
4835:
4812:
4806:
4803:
4797:
4791:
4782:
4765:
4759:
4738:
4735:
4709:
4705:
4701:
4692:
4686:
4683:
4680:
4674:
4669:
4665:
4661:
4655:
4649:
4646:
4642:
4639:
4631:
4615:
4611:
4607:
4603:
4594:
4590:
4586:
4583:
4575:
4571:
4565:
4561:
4556:
4552:
4547:
4543:
4539:
4535:
4529:
4525:
4519:
4515:
4509:
4505:
4500:
4496:
4492:
4489:
4481:
4462:
4458:
4454:
4446:
4442:
4438:
4435:
4429:
4424:
4420:
4416:
4411:
4407:
4398:
4394:
4388:
4384:
4380:
4376:
4373:
4365:
4364:
4363:
4349:
4327:
4323:
4319:
4311:
4307:
4303:
4300:
4294:
4289:
4285:
4281:
4276:
4272:
4249:
4245:
4222:
4218:
4195:
4191:
4168:
4164:
4158:
4154:
4148:
4144:
4140:
4137:
4112:
4108:
4102:
4098:
4092:
4088:
4084:
4076:
4072:
4065:
4060:
4056:
4050:
4046:
4042:
4038:
4032:
4028:
4022:
4018:
4012:
4008:
4003:
3999:
3996:
3990:
3984:
3977:
3976:
3975:
3973:
3955:
3951:
3945:
3941:
3935:
3931:
3927:
3924:
3899:
3895:
3891:
3883:
3879:
3872:
3865:
3864:
3863:
3847:
3843:
3822:
3799:
3796:
3791:
3787:
3783:
3780:
3777:
3772:
3768:
3764:
3759:
3755:
3751:
3748:
3741:
3740:
3739:
3720:
3716:
3712:
3704:
3700:
3693:
3686:
3685:
3684:
3668:
3664:
3641:
3637:
3627:
3611:
3607:
3603:
3597:
3591:
3569:
3565:
3561:
3555:
3549:
3524:
3520:
3516:
3510:
3507:
3504:
3498:
3493:
3489:
3485:
3482:
3479:
3473:
3467:
3460:
3459:
3458:
3444:
3421:
3418:
3415:
3409:
3406:
3383:
3380:
3372:
3368:
3361:
3341:
3338:
3330:
3326:
3319:
3312:
3311:
3310:
3294:
3290:
3286:
3281:
3277:
3251:
3247:
3243:
3240:
3237:
3232:
3228:
3224:
3219:
3215:
3207:
3206:
3205:
3189:
3185:
3181:
3176:
3172:
3163:
3158:
3144:
3130:
3128:
3124:
3120:
3116:
3112:
3108:
3104:
3083:
3075:
3071:
3064:
3059:
3055:
3051:
3048:
3045:
3037:
3033:
3026:
3021:
3017:
3013:
3005:
3001:
2995:
2991:
2987:
2984:
2981:
2976:
2972:
2966:
2962:
2955:
2952:
2945:
2944:
2943:
2942:
2941:
2939:
2935:
2931:
2910:
2901:
2895:
2889:
2886:
2877:
2871:
2865:
2822:
2819:
2816:
2809:
2808:
2807:
2806:
2805:
2803:
2799:
2795:
2785:
2770:
2767:
2760:
2757:
2754:
2748:
2744:
2741:
2737:
2734:
2731:
2706:
2703:
2697:
2694:
2691:
2685:
2681:
2678:
2671:
2668:
2665:
2659:
2656:
2652:
2649:
2645:
2642:
2639:
2636:
2630:
2627:
2624:
2618:
2615:
2612:
2599:
2585:
2581:
2578:
2571:
2568:
2565:
2559:
2555:
2552:
2548:
2545:
2542:
2529:
2528:
2527:
2526:
2507:
2504:
2501:
2495:
2492:
2489:
2486:
2465:
2462:
2458:
2455:
2452:
2448:
2445:
2441:
2438:
2430:
2427:
2426:
2425:
2422:
2408:
2405:
2402:
2382:
2377:
2374:
2369:
2366:
2361:
2358:
2353:
2350:
2345:
2342:
2337:
2334:
2329:
2326:
2304:
2278:
2275:
2272:
2266:
2263:
2260:
2257:
2251:
2248:
2245:
2239:
2236:
2233:
2210:
2207:
2204:
2195:
2181:
2161:
2136:
2132:
2129:
2123:
2120:
2113:
2110:
2107:
2101:
2096:
2092:
2089:
2083:
2080:
2073:
2070:
2067:
2053:
2050:
2047:
2040:
2022:
2019:
2016:
2010:
2007:
1987:
1979:
1976:
1975:
1974:
1971:
1952:
1949:
1943:
1940:
1937:
1931:
1928:
1925:
1921:
1917:
1913:
1909:
1906:
1903:
1900:
1894:
1891:
1888:
1877:
1876:
1875:
1874:
1855:
1852:
1849:
1843:
1840:
1819:
1816:
1813:
1810:
1807:
1798:
1795:
1794:
1793:
1791:
1786:
1784:
1780:
1776:
1772:
1751:
1747:
1743:
1737:
1734:
1731:
1725:
1722:
1719:
1711:
1710:
1693:
1690:
1687:
1681:
1678:
1657:
1654:
1651:
1648:
1645:
1636:
1633:
1632:
1631:
1629:
1625:
1605:
1602:
1599:
1578:
1575:
1572:
1551:
1548:
1545:
1536:
1533:
1532:
1531:
1529:
1525:
1523:
1519:
1499:
1496:
1493:
1472:
1469:
1466:
1445:
1425:
1417:
1414:
1413:
1412:
1409:
1407:
1406:infinitesimal
1403:
1402:lexicographic
1399:
1395:
1391:
1387:
1383:
1373:
1360:
1357:
1354:
1351:
1343:
1339:
1323:
1320:
1317:
1314:
1307:
1303:
1299:
1283:
1280:
1277:
1257:
1254:
1251:
1243:
1239:
1234:
1232:
1228:
1224:
1220:
1216:
1212:
1207:
1191:
1187:
1161:
1156:
1152:
1146:
1142:
1138:
1135:
1132:
1125:
1124:
1123:
1122:
1121:
1119:
1112:
1108:
1104:
1100:
1096:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1050:
1049:
1048:
1047:
1046:
1044:
1040:
1036:
1032:
1029:. Given some
1028:
1027:
1016:
1014:
1010:
1006:
1003:
999:
994:
992:
988:
984:
980:
976:
972:
968:
964:
960:
956:
952:
947:
945:
941:
937:
933:
929:
925:
921:
917:
913:
901:
896:
894:
889:
887:
882:
881:
879:
878:
873:
863:
861:
856:
851:
850:
849:
848:
841:
838:
835:
831:
828:
826:
823:
821:
818:
817:
814:
809:
808:
799:
798:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
574:
572:
569:
567:
564:
562:
559:
557:
554:
552:
549:
547:
544:
542:
539:
537:
534:
532:
529:
527:
524:
522:
519:
517:
514:
512:
509:
507:
504:
502:
499:
497:
494:
492:
489:
487:
484:
482:
479:
477:
474:
472:
469:
467:
464:
462:
461:de Mandeville
459:
458:
453:
447:
446:
439:
436:
434:
431:
429:
426:
424:
421:
419:
416:
414:
411:
409:
406:
404:
401:
399:
396:
394:
390:
389:Public choice
387:
385:
382:
380:
377:
375:
372:
370:
367:
365:
364:Participation
362:
360:
357:
355:
352:
350:
347:
345:
342:
340:
337:
335:
332:
330:
327:
325:
324:Institutional
322:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
285:
282:
280:
277:
275:
274:Expeditionary
272:
270:
267:
265:
264:Environmental
262:
260:
257:
255:
252:
250:
247:
245:
242:
240:
237:
235:
232:
230:
227:
225:
222:
220:
217:
215:
212:
210:
207:
206:
200:
199:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
147:
144:
143:
137:
136:
129:
126:
124:
121:
119:
116:
114:
111:
109:
106:
104:
100:
97:
95:
94:International
92:
90:
87:
85:
82:
80:
77:
76:
73:
70:Branches and
67:
66:
61:
58:
56:
53:
51:
48:
47:
46:
45:
41:
37:
36:
33:
30:
29:
25:
21:
20:
7251:Publications
7207:Publications
7174:
6770:Neoclassical
6760:Mercantilism
6669:Evolutionary
6531:Sociological
6504: /
6402:Geographical
6382:Evolutionary
6357:Digitization
6322:Agricultural
6285:Econometrics
6213:Price theory
5976:
5969:
5948:
5939:
5907:
5903:Econometrica
5901:
5885:
5881:
5877:
5858:
5852:
5840:
5752:
5744:
5707:
5701:
5676:
5636:
5626:
5619:
5615:
5608:
5604:
5597:
5593:
5586:
5582:
5575:
5571:
5564:
5557:
5550:
5546:
5542:
5540:
5529:
5524:
5522:
5516:
5508:
5504:
5500:
5496:
5492:
5490:
5475:
5471:
5459:
5457:
5447:
5436:
5318:
5306:
5288:
5283:
5281:
5275:
5271:
5267:
5261:
5242:
5238:
5234:
5230:
5224:
5220:
5218:
5126:
5122:
5120:
5022:
5012:
4998:Consequences
4991:
4989:
4984:
4982:
4977:
4870:
4783:
4727:The lottery
4726:
4129:
3971:
3916:
3814:
3737:
3628:
3541:
3398:
3268:
3161:
3159:
3136:
3133:Proof sketch
3126:
3122:
3118:
3114:
3110:
3106:
3102:
3100:
2937:
2933:
2929:
2927:
2801:
2797:
2793:
2791:
2723:
2428:
2423:
2196:
2153:
1977:
1972:
1969:
1796:
1787:
1782:
1778:
1774:
1770:
1768:
1634:
1627:
1623:
1621:
1534:
1528:Transitivity
1526:
1517:
1515:
1415:
1410:
1408:utilities).
1394:independence
1393:
1389:
1386:transitivity
1385:
1382:completeness
1381:
1379:
1341:
1337:
1305:
1301:
1297:
1241:
1237:
1235:
1230:
1226:
1222:
1218:
1214:
1210:
1208:
1178:
1120:, we write:
1114:
1110:
1106:
1102:
1098:
1094:
1092:
1042:
1038:
1025:
1022:
1004:
1001:
997:
995:
990:
989:, only that
986:
982:
978:
974:
970:
948:
923:
919:
915:
909:
830:Publications
795:
418:Sociological
391: /
289:Geographical
269:Evolutionary
244:Digitization
209:Agricultural
113:Mathematical
84:Econometrics
7281:Game theory
7045:von Neumann
6814:Supply-side
6799:Physiocracy
6743:Marginalism
6432:Information
6372:Engineering
6352:Development
6347:Demographic
6218:Game theory
6195:Theoretical
6125:Game theory
5482:Limitations
5472:correlation
3683:such that:
2788:The theorem
1624:better than
1522:reflexivity
1344:, we write
1244:, we write
1236:If lottery
1221:= 0.25
1035:probability
983:VNM-utility
959:preferences
932:uncertainty
666:von Neumann
319:Information
259:Engineering
239:Development
234:Demographic
176:Game theory
118:Methodology
7270:Categories
7202:Economists
7075:Schumacher
6980:Schumpeter
6950:von Wieser
6870:von Thünen
6830:Economists
6729:Circuitism
6694:Humanistic
6689:Historical
6664:Ecological
6654:Democratic
6627:Chartalism
6617:Behavioral
6580:Mainstream
6541:Statistics
6536:Solidarity
6457:Managerial
6422:Humanistic
6417:Historical
6362:Ecological
6327:Behavioral
5962:Sixto Rios
5725:1765/23011
5454:required).
4784:Hence, if
3815:For every
3542:Note that
1709:such that
1628:worse than
1390:continuity
1376:The axioms
825:Economists
696:Schumacher
601:Schumpeter
571:von Wieser
491:von Thünen
452:economists
428:Statistics
423:Solidarity
344:Managerial
309:Humanistic
304:Historical
249:Ecological
214:Behavioral
108:Mainstream
7296:Economics
7120:Greenspan
7085:Samuelson
7065:Galbraith
7035:Tinbergen
6975:von Mises
6970:Heckscher
6930:Edgeworth
6809:Stockholm
6804:Socialist
6704:Keynesian
6684:Happiness
6644:Classical
6605:Mutualism
6600:Anarchist
6585:Heterodox
6482:Personnel
6442:Knowledge
6407:Happiness
6397:Financial
6367:Education
6342:Democracy
6277:Empirical
6187:Economics
6022:Decisions
5932:153422092
5823:∼
5799:≺
5767:≺
5491:Since if
5476:disregard
5460:E-utility
5458:The term
5408:−
5262:In 1738,
5198:$
5180:$
5171:%
5152:$
5143:%
5121:However,
5100:$
5094:%
5082:$
5070:$
5064:%
5045:$
5039:%
5021:of their
4887:≺
4702:⋅
4684:−
4662:⋅
4608:⋅
4587:−
4562:∑
4540:⋅
4506:∑
4455:⋅
4439:−
4417:⋅
4385:∑
4320:⋅
4304:−
4282:⋅
4145:∑
4089:∑
4047:∑
4009:∑
3932:∑
3784:≤
3781:⋯
3778:≤
3765:≤
3713:∼
3604:∼
3562:∼
3517:⋅
3508:−
3486:⋅
3410:∈
3287:≺
3244:⪯
3241:⋯
3238:⪯
3225:⪯
3182:…
3101:As such,
3049:⋯
2985:⋯
2820:≺
2758:−
2695:−
2669:−
2640:∼
2628:−
2569:−
2546:∼
2496:∈
2406:⪰
2354:⪰
2267:∈
2249:−
2137:_
2111:−
2102:⪯
2097:_
2071:−
2051:⪯
2011:∈
1944:ε
1941:−
1926:ε
1922:≺
1914:≺
1907:ε
1895:ε
1892:−
1871:such that
1844:∈
1841:ε
1817:≺
1811:≺
1748:∼
1735:−
1682:∈
1655:⪯
1649:⪯
1603:⪰
1576:⪰
1549:⪰
1497:⪰
1470:⪰
1458:, either
1355:⪯
1318:∼
1300:and
1281:≺
1255:≻
1139:∑
1026:lotteries
1005:decisions
949:In 1947,
741:Greenspan
706:Samuelson
686:Galbraith
656:Tinbergen
596:von Mises
591:Heckscher
551:Edgeworth
369:Personnel
329:Knowledge
294:Happiness
284:Financial
254:Education
229:Democracy
123:Political
89:Heterodox
32:Economics
7276:Theorems
7231:Category
7211:journals
7197:Glossary
7150:Stiglitz
7115:Rothbard
7095:Buchanan
7080:Friedman
7070:Koopmans
7060:Leontief
7040:Robinson
6925:Marshall
6775:Lausanne
6679:Georgism
6674:Feminist
6622:Buddhist
6612:Austrian
6511:Regional
6487:Planning
6462:Monetary
6392:Feminist
6337:Cultural
6332:Business
6043:Concepts
5241:($ 1) ≠
5221:outcomes
4974:Reaction
4739:′
4643:′
4493:′
4377:′
2771:′
2745:′
2682:′
2653:′
2582:′
2556:′
2479:and any
2466:′
2449:′
2395:implies
1980:For any
1002:describe
993:exists.
938:of some
834:journals
820:Glossary
771:Stiglitz
736:Rothbard
716:Buchanan
701:Friedman
691:Koopmans
681:Leontief
661:Robinson
546:Marshall
450:Notable
398:Regional
374:Planning
349:Monetary
279:Feminist
224:Cultural
219:Business
24:a series
22:Part of
7286:Utility
7246:Outline
7217:Schools
7209: (
7170:Piketty
7165:Krugman
7030:Kuznets
7020:Kalecki
6995:Polanyi
6885:Cournot
6880:Bastiat
6865:Ricardo
6855:Malthus
6845:Quesnay
6748:Marxian
6639:Chicago
6569:history
6564:Schools
6551:Welfare
6521:Service
6312:Applied
6075:Leximin
6070:Minimax
5964:(1998)
5924:1907266
5464:Bentham
5015:gambles
4871:Hence:
2930:E(u(L))
1591:, then
1009:Bentham
840:Schools
832: (
791:Piketty
786:Krugman
651:Kuznets
641:Kalecki
616:Polanyi
506:Cournot
501:Bastiat
486:Ricardo
476:Malthus
466:Quesnay
438:Welfare
408:Service
79:Applied
55:Outline
50:History
7155:Thaler
7135:Ostrom
7130:Becker
7125:Sowell
7105:Baumol
7010:Myrdal
7005:Sraffa
7000:Frisch
6990:Knight
6985:Keynes
6960:Fisher
6955:Veblen
6940:Pareto
6920:Menger
6915:George
6910:Jevons
6905:Walras
6895:Gossen
6819:Thermo
6497:Public
6492:Policy
6447:Labour
6412:Health
5955:
5930:
5922:
5865:
5585:) and
5237:($ 1)+
5223:(cf. "
2928:where
1392:, and
1019:Set-up
930:under
914:, the
776:Thaler
756:Ostrom
751:Becker
746:Sowell
726:Baumol
631:Myrdal
626:Sraffa
621:Frisch
611:Knight
606:Keynes
581:Fisher
576:Veblen
561:Pareto
541:Menger
536:George
531:Jevons
526:Walras
516:Gossen
384:Public
379:Policy
334:Labour
299:Health
156:Market
7241:Lists
7236:Index
7187:Lists
7160:Hoppe
7145:Lucas
7110:Solow
7100:Arrow
7090:Simon
7055:Lange
7050:Hicks
7025:Röpke
7015:Hayek
6965:Pigou
6935:Clark
6850:Smith
6765:Mixed
6724:Post-
6546:Urban
6526:Socio
6516:Rural
5928:S2CID
5920:JSTOR
5788:then
5437:Some
5226:value
813:Lists
781:Hoppe
766:Lucas
731:Solow
721:Arrow
711:Simon
676:Lange
671:Hicks
646:Röpke
636:Hayek
586:Pigou
556:Clark
471:Smith
433:Urban
413:Socio
403:Rural
103:Macro
99:Micro
60:Index
7176:more
6900:Marx
6890:Mill
6875:List
6753:Neo-
6709:Neo-
6085:Risk
5953:ISBN
5863:ISBN
5671:and
5637:The
5574:) +
5556:and
5545:and
5495:and
5385:and
5201:1000
5189:<
5103:1000
5088:>
5085:2000
4932:<
4804:>
3738:and
3584:and
3354:and
2887:<
2802:u(A)
2608:then
2000:and
1777:and
1626:and
1564:and
1518:some
1438:and
1073:0.75
1064:0.25
1041:and
953:and
797:more
521:Marx
511:Mill
496:List
7140:Sen
6860:Say
6719:New
6452:Law
5912:doi
5884:or
5720:hdl
5712:doi
5708:112
5466:'s
5339:is
5159:000
5091:100
5052:000
1799:If
1785:).
1637:If
1537:If
1485:or
1404:or
1336:If
1011:'s
920:VNM
910:In
761:Sen
481:Say
339:Law
7272::
5968:,
5926:.
5918:.
5908:18
5906:.
5861:.
5734:^
5718:.
5706:.
5700:.
5686:^
5675:,
5657:^
5645:.
5501:pL
5316:.
5168:80
5155:10
5140:20
5123:if
5061:80
5048:10
5036:20
3974::
3626:.
3129:.
3111:pA
2934:Eu
2538:if
1524:.
1388:,
1384:,
1233:.
1045:,
1015:.
946:.
922:)
101:/
26:on
7213:)
6715:)
6711:(
6571:)
6567:(
6179:e
6172:t
6165:v
6007:e
6000:t
5993:v
5934:.
5914::
5871:.
5802:N
5796:L
5776:N
5773:=
5770:M
5764:L
5728:.
5722::
5714::
5627:L
5625:(
5622:Y
5620:u
5616:L
5614:(
5611:X
5609:u
5605:L
5603:(
5600:Y
5598:u
5594:L
5592:(
5589:X
5587:u
5583:L
5581:(
5578:Y
5576:u
5572:L
5570:(
5567:X
5565:u
5560:Y
5558:u
5553:X
5551:u
5547:Y
5543:X
5532:.
5509:M
5507:)
5505:p
5497:M
5493:L
5417:0
5414:)
5411:p
5405:1
5402:(
5399:+
5396:M
5393:p
5373:N
5370:1
5350:M
5347:p
5327:N
5268:u
5243:u
5239:u
5235:u
5231:u
5204:)
5195:(
5192:u
5186:)
5183:0
5177:(
5174:u
5165:+
5162:)
5149:(
5146:u
5127:u
5106:)
5097:(
5079:=
5076:)
5073:0
5067:(
5058:+
5055:)
5042:(
4992:u
4956:.
4953:)
4950:)
4947:M
4944:(
4941:u
4938:(
4935:E
4929:)
4926:)
4923:L
4920:(
4917:u
4914:(
4911:E
4890:M
4884:L
4856:L
4836:M
4816:)
4813:L
4810:(
4807:u
4801:)
4798:M
4795:(
4792:u
4769:)
4766:M
4763:(
4760:u
4736:M
4710:1
4706:A
4699:)
4696:)
4693:M
4690:(
4687:u
4681:1
4678:(
4675:+
4670:n
4666:A
4659:)
4656:M
4653:(
4650:u
4647:=
4640:M
4616:1
4612:A
4604:)
4600:)
4595:i
4591:q
4584:1
4581:(
4576:i
4572:p
4566:i
4557:(
4553:+
4548:n
4544:A
4536:)
4530:i
4526:q
4520:i
4516:p
4510:i
4501:(
4497:=
4490:M
4468:]
4463:1
4459:A
4452:)
4447:i
4443:q
4436:1
4433:(
4430:+
4425:n
4421:A
4412:i
4408:q
4404:[
4399:i
4395:p
4389:i
4381:=
4374:M
4350:M
4328:1
4324:A
4317:)
4312:i
4308:q
4301:1
4298:(
4295:+
4290:n
4286:A
4277:i
4273:q
4250:i
4246:A
4223:i
4219:p
4196:i
4192:A
4169:i
4165:A
4159:i
4155:p
4149:i
4141:=
4138:M
4113:i
4109:q
4103:i
4099:p
4093:i
4085:=
4082:)
4077:i
4073:A
4069:(
4066:u
4061:i
4057:p
4051:i
4043:=
4039:)
4033:i
4029:A
4023:i
4019:p
4013:i
4004:(
4000:u
3997:=
3994:)
3991:M
3988:(
3985:u
3972:u
3956:i
3952:A
3946:i
3942:p
3936:i
3928:=
3925:M
3900:i
3896:q
3892:=
3889:)
3884:i
3880:A
3876:(
3873:u
3848:i
3844:A
3823:i
3800:1
3797:=
3792:n
3788:q
3773:2
3769:q
3760:1
3756:q
3752:=
3749:0
3721:i
3717:A
3710:)
3705:i
3701:q
3697:(
3694:L
3669:i
3665:q
3642:i
3638:A
3612:n
3608:A
3601:)
3598:1
3595:(
3592:L
3570:1
3566:A
3559:)
3556:0
3553:(
3550:L
3525:1
3521:A
3514:)
3511:p
3505:1
3502:(
3499:+
3494:n
3490:A
3483:p
3480:=
3477:)
3474:p
3471:(
3468:L
3445:p
3425:]
3422:1
3419:,
3416:0
3413:[
3407:p
3384:1
3381:=
3378:)
3373:n
3369:A
3365:(
3362:u
3342:0
3339:=
3336:)
3331:1
3327:A
3323:(
3320:u
3295:n
3291:A
3282:1
3278:A
3252:n
3248:A
3233:2
3229:A
3220:1
3216:A
3190:n
3186:A
3177:1
3173:A
3162:n
3145:u
3123:u
3119:B
3117:)
3115:p
3103:u
3084:.
3081:)
3076:n
3072:A
3068:(
3065:u
3060:n
3056:p
3052:+
3046:+
3043:)
3038:1
3034:A
3030:(
3027:u
3022:1
3018:p
3014:=
3011:)
3006:n
3002:A
2996:n
2992:p
2988:+
2982:+
2977:1
2973:A
2967:1
2963:p
2959:(
2956:u
2953:E
2938:L
2936:(
2911:,
2908:)
2905:)
2902:M
2899:(
2896:u
2893:(
2890:E
2884:)
2881:)
2878:L
2875:(
2872:u
2869:(
2866:E
2861:f
2858:i
2854:y
2851:l
2848:n
2845:o
2841:d
2838:n
2835:a
2831:f
2828:i
2823:M
2817:L
2798:A
2794:u
2768:N
2764:)
2761:q
2755:1
2752:(
2749:+
2742:L
2738:q
2735:=
2732:M
2707:.
2704:N
2701:)
2698:p
2692:1
2689:(
2686:+
2679:N
2675:)
2672:q
2666:1
2663:(
2660:p
2657:+
2650:L
2646:q
2643:p
2637:N
2634:)
2631:p
2625:1
2622:(
2619:+
2616:L
2613:p
2586:,
2579:N
2575:)
2572:q
2566:1
2563:(
2560:+
2553:L
2549:q
2543:L
2523:,
2511:]
2508:1
2505:,
2502:0
2499:[
2493:q
2490:,
2487:p
2463:N
2459:,
2456:N
2453:,
2446:L
2442:,
2439:L
2409:B
2403:A
2383:B
2378:2
2375:1
2370:+
2367:B
2362:2
2359:1
2351:B
2346:2
2343:1
2338:+
2335:A
2330:2
2327:1
2305:A
2285:}
2282:)
2279:1
2276:,
2273:0
2270:[
2264:p
2261::
2258:B
2255:)
2252:p
2246:1
2243:(
2240:+
2237:A
2234:p
2231:{
2211:B
2208:,
2205:A
2182:M
2162:M
2133:M
2130:p
2124:+
2121:N
2117:)
2114:p
2108:1
2105:(
2093:M
2090:p
2084:+
2081:L
2077:)
2074:p
2068:1
2065:(
2054:N
2048:L
2026:)
2023:1
2020:,
2017:0
2014:[
2008:p
1988:N
1953:.
1950:N
1947:)
1938:1
1935:(
1932:+
1929:L
1918:M
1910:N
1904:+
1901:L
1898:)
1889:1
1886:(
1859:)
1856:1
1853:,
1850:0
1847:(
1820:N
1814:M
1808:L
1783:p
1779:N
1775:p
1771:L
1752:M
1744:N
1741:)
1738:p
1732:1
1729:(
1726:+
1723:L
1720:p
1697:]
1694:1
1691:,
1688:0
1685:[
1679:p
1658:N
1652:M
1646:L
1618:.
1606:N
1600:L
1579:N
1573:M
1552:M
1546:L
1512:.
1500:L
1494:M
1473:M
1467:L
1446:M
1426:L
1361:.
1358:M
1352:L
1342:L
1338:M
1324:.
1321:M
1315:L
1302:M
1298:L
1284:M
1278:L
1258:L
1252:M
1242:L
1238:M
1231:C
1227:B
1223:A
1219:C
1215:B
1211:A
1192:i
1188:p
1162:,
1157:i
1153:A
1147:i
1143:p
1136:=
1133:L
1117:i
1115:A
1111:B
1109:(
1107:P
1103:A
1099:A
1097:(
1095:P
1076:B
1070:+
1067:A
1061:=
1058:L
1043:B
1039:A
991:u
987:u
979:u
975:u
918:(
899:e
892:t
885:v
836:)
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