INTRODUCTION
Tobin (1958), in setting, 50 years ago, the microeconomic
foundations of the keynesian liquidity preference theory in the light of the
Markowitz (1952, 1959) portfolio
approach, has shown an important equivalence between the Von
Neumann and Morgenstern (1944) expected utility (VNM) and a preference function
in mean and standard deviation.
His extension of this result to all twoparameter distributions was subsequently
proved incorrect by Samuelson (1967), Borch
(1969) and Feldstein (1969) so that a meanvariance
analysis has been justified for a long time only under the restrictive assumption
of quadratic VNM utility or Gaussian distribution.
Two research areas have therefore attracted considerable interest: one focused on suitable distributional assumptions and one devoted to the most appealing form of the utility function.
Regarding the former, the admissible distributions have been identified by
the elliptical class (Chamberlain, 1983; Owen
and Rabinowitch, 1983), which is closed under linear transformations of
random variables. Elliptical distributions are also known as scalelocation
parameter distributions (as in Tobin’s conjecture) or linear distributions
(Sinn, 1983; Meyer, 1987; Levy,
1989). They include the symmetric stable distributions analyzed by Fama
(1971).
In addition, a large amount of work has been done to justify meanvariance
(EV) analysis just as a secondorder approximation of the expected utility model,
avoiding the absurd assumption of quadratic utility (Hicks,
1962; Pratt, 1964; Arrow, 1965)
and its property of increasing risk aversion and decreasing asset demand as
wealth increases, making risky assets inferior goods.
Early studies concerning EV as an approximation of expected utility were carried
out by Samuelson (1967, 1970),
Tsiang (1972, 1974) and Rubinstein
(1973), also including higherorder moments in a generalized approach.
Levy and Markowitz (1979), Kroll
et al. (1984), Reid and Tew (1986) and Markowitz
(1987) assess the effectiveness of the EV approximation, confirming Markowitz’s
intuition that meanvariance is, in practice, as efficient as expected utility
in selecting optimal portfolios.
Even in more theoretical works, such as Hakansson (1972),
Baron (1977) and Bigelow (1993),
the aim was to investigate the consistency between meanvariance (or moment)
utility and the VNM axioms or utility function, providing restrictions to a
simultaneous validity of both approaches.
However, any attempt to make expected utility and moment utility equivalent does not seem a truly compelling task.
Many authors (Borch, 1969, 1973;
Levy, 1989) are, in fact, well aware of the different
setup of the two approaches, particularly when the arguments of moment utility
are not confined to mean and variance but include all relevant moments of the
probability distributions.
Moreover, during the last century, a number of works, from Knight
(1921) to Keynes (1921), from Hicks
(1935, 1962) to Marschak (1938),
from Lange (1944) to Simpson (1950),
expressed intriguing suggestions on variance, dispersion and higherorder moments
as relevant parameters directly influencing the agent’s decisions under
uncertainty.
In this study, we develop the foundations of an ordinal utility of moments as a rational and autonomous criterion of choice under uncertainty, showing that it dissolves all the best known behavioral paradoxes which are still embarrassing the expected utility theory. This ordinal approach is strongly reminiscent of standard microeconomic theory and it could be used to reset and generalize both assets demand theory and asset pricing models.
FOUNDATIONS
It is well known that the theory of choice under uncertainty assumes that preferences
are defined over the set of probability distribution functions (Savage,
1954; DeGroot, 1970).
Confining ourselves, for ease of exposition, to the case of univariate distributions, we assume that the essential information concerning any distribution F is contained in the mdimensional vector of moments M ≡ (μ, μ^{(2)}, μ^{(3)},...., μ^{(m)}) where, μ is the mean and μ^{(s)} is the sorder central moment in original units, so that (μ^{(s)})^{s} is the usual central moment of order s≥2.
Note that, instead of central moments, noncentral moments could, equivalently, be used. Moreover, scale, location and dispersion parameters can be considered in the case of distributions (e.g., stable) for which moments do not exist.
The existence of an ordinal utility of moments is obtained under the assumption of a preference order satisfying the axioms of (1) asymmetry, (2) transitivity and (3) continuity (Appendix).
Assumption of Preference Order
Let Q⊆R^{m} be a rectangular subset of R^{m} (the cartesian
product of m real intervals), whose elements are the mdimensional vectors of
moments, M∈Q. Let › be a preference order i.e., a binary relation
defined by a subset
of the Cartesian product QxQ, whose elements are the ordered pairs of vectors
(M_{a}, M_{b}).
We write M_{a}›M_{b} instead of (M_{a},M_{b})
and we say that M_{a} is preferred to M_{b}, corresponding to
F_{a} is preferred to F_{b} for distribution functions.
Clearly, or M_{a}›M_{b} or M_{a}M_{b}
and both cannot hold: in fact, or (M_{a}, M_{b})
or (M_{a}, M_{b}) .
We write M_{a}~M_{b} (equivalence) if and only if M_{a}M_{b} and M_{b}M_{a}.
Theorem 1: Of Complete Preferences
The preference order is complete.
Theorem 2: Of Negatively Transitive Preferences
The preference order is negatively transitive.
Theorem 3: Of Equivalence Classes
The equivalence ~ is reflexive, symmetric and transitive.
Theorem 4: Of Ordinal Utility on Moments
Under Axioms I, II, III there is a real function H: Q→R which represents
the preferences ›, i.e., such that, for every M_{a}, M_{b}∈Q:
The function H is unique up to any orderpreserving transformation Ø:
The function H is called an ordinal utility because it just represents the given preference order › in terms of otherwise arbitrary real numbers (utils).
Theorem 5: Of Continuous Utility
Under Axioms I, II, III and the usual relative topology for Q (intersection
of Q with the set of all open rectangles in R^{m}, including arbitrary
unions and finite intersections; Debreu (1959), Rader
(1963)), the utility function H is continuous if and only if, for every
M_{a}∈Q, the sets {M_{b}∈Q: M_{a}›M_{b}}
and {M_{b}∈Q: M_{b}›M_{a}} belong to the
topology.
In the following, we assume that H is continuous with bounded first order partial derivatives. The usual concepts available in the theory of choice under uncertainty can be extended to our approach.
Let F, G be two probability distributions with relevant moment vectors:
M_{F} ≡ (μ_{F}, μ_{F}^{(2)}, μ_{F}^{(3)},...., μ_{F}^{(m)} ) and M_{G}≡(μ_{G}, μ_{G}^{(2)}, μ_{G}^{(3)},...., μ_{G}^{(m)}), respectively. Let H(M) be a differentiable utility function.
Definition of Non Satiation
The utility H is non satiated if, for every δ>0:
In differential terms: .
Definition of Risk Aversion
A utility H is risk averse if, for every F:
Note that for m = 2, risk aversion means ;
for m≥3, a negative marginal utility of volatility
does not imply risk aversion.
Definition of Certainty Equivalent
The certainty equivalent of F is defined as the amount C_{F} such
that:
Definition of Risk
Given two distributions F, G with equal mean, μ_{F} = μ_{G},
we say that F is less risky than G if H(M_{F})> H(M_{G})
for every risk averse utility H.
Definition of Stochastic Dominance
Given two distribution functions, F, G defined over the same support, we
have mthorder stochastic dominance of F over G, F›_{m}G, m≥1,
if F≠G and:
where:
Clearly, if F›_{m}G then F›_{m+1}G.
We recall a well known result linking stochastic dominance and VNM expected utility functions.
Theorem 6 on Stochastic Dominance and Expected Utility
We have:
• 
F›_{1}G ⇔ E_{F} (U(x))≥E_{G}(U(x))
for every U with U’≥0 
• 
F›_{2}G ⇔ E_{F} (U(x))≥E_{G}(U(x))
for every U with U’≥0, U”≤0 
• 
F›_{3}G ⇔ E_{F} (U(x))≥E_{G}(U(x))
for every U with U’≥0, U”≤0, U”’≥0 
Therefore, first order stochastic dominance means increasing VNM utilities; second order stochastic dominance means increasing and concave VNM utilities.
This is no longer true for the moment ordinal utility.
Theorem 7 on Stochastic Dominance and Moment Utility
If F›_{m}G then M_{F} ≡ (μ_{F},
μ_{F}^{(2)}, μ_{F}^{(3)},...., μ_{F}^{(m)})≠(μ_{G},
μ_{G}^{(2)}, μ_{G}^{(3)},...., μ_{G}^{(m)})
≡ M_{G} and (1)^{k1}μ_{F}^{(k)}>
(1)^{k1}μ_{G}^{(k)} for the smallest k for which
μ_{F}^{(k)}≠μ_{G}^{(k)}.
As special cases we have:
• 
If F›_{1}G then μ_{F}>μ_{G} 
• 
If F›_{2}G then (μ_{F}>μ_{G})
or (μ_{F} = μ_{G} and μ_{F}^{(2)}<μ_{G}^{(2)}) 
• 
If F›_{3}G then (μ_{F}>μ_{G})
or (μ_{F} = μ_{G} and μ_{F}^{(2)}<μ_{G}^{(2)})
or (μ_{F} = μ_{G} and μ_{F}^{(2)}
= μ_{G}^{(2)} and μ_{F}^{(3)}>μ_{G}^{(3)}) 
In the first case, in particular, it does not necessarily follow (even if it
is, however, plausible) that H(M_{F})>H(M_{G}) whenever higher
order moments are relevant. In particular, Allais (1953,
1979) assumes that F›_{1}G implies H(F)>H(G)
(axiom of absolute preference). In the expected utility approach it is equivalent
to assume U’>0 (nonsatiation).
INDIFFERENCE PRICING AND THE (DIS)SOLUTION OF PARADOXES
The Von NeumannMorgenstern (1944) theory of choice under
uncertainty has long been the standard approach to model the maximizing behavior
of agents in financial markets. The basic result is the existence of a utility
function U(.) (the VNM utility) describing the optimal decisions of an investor
as those maximizing the expected utility of his or her future wealth E(U( )).
This beautiful result was obtained at the cost of a special and controversial
axiom concerning the structure of preferences in the case of uncertainty, the
socalled independence axiom, inducing a linear dependence between utility and
probabilities.
In fact, the independent axiom assumes that randomization is irrelevant: it
says that if A›B then pA+(1p)C›pB+(1p)C (Machina,
1987; Fishburn, 1982, 1988 and
the seminal work in Von Neumann and Morgenstern, 1944,
especially chapter 1 and Appendix, where, considering their penetrating results,
they ask themselves, p.28, Have we not shown too much?).
The approach followed above replaces the VNM expected utility with the more
fundamental and less demanding ordinal utility H(E( ),
Std ( ),...),
allowing us to avoid the main flaws of the former and providing us with a generalized
framework for optimal behavior in both complete and incomplete markets. In particular,
note that H is a utility of expectations and not an expectation of utilities.
Note also that this approach is different from (and much simpler than) other
generalizations (nonexpected utility approaches) suggested in the literature
after Allais (1953) contribution to the empirical critique
of the VNM expected utility theory.
In fact, in most cases, including Allais (1979), Kahneman
and Tversky (1979), Chew (1983), Fishburn
(1983), Machina (1982) and many others, the utility
function is a function of moments of a Bernoullian utility U, or a nonlinear
function of both Bernoullian utilities and subjective probabilities:
In our case, we obtain a more general result based on more intuitive elements:
Moreover, it is well known that the expected utility can be approximated by a particular function of m central moments:
where, U^{(j)} is the jth derivative of U, but this is only a special (polynomial) case of H and is not always able to account for the observed phenomena.
The utility of moments is perfectly compatible with the empirical observations in all well known behavioral paradoxes, described in terms of games and lotteries.
In order to show this compatibility we use the so called utilityindifference
pricing (Henderson and Hobson, 2009), which can be applied
in all cases of personal valuation of nontraded assets and incomplete markets.
For the sake of simplicity, let us consider the twomoment ordinal utility.
Let W be the current wealth of the decision maker and
be the random variable representing the game, with mean M_{G} and vol
Σ_{G}. Future wealth, in case of a decision to gamble, is given
by:
where, WP_{G} is wealth left after payment for the game, invested at the riskless rate r = 1/P_{0}1 (often set to zero) and the personal indifference price P_{G} is defined as the price at which the agent is indifferent to paying the price and entering the game or paying nothing and avoiding the game:
In the l.h.s., using a Taylor series approximation for small risks (Pratt,
1964), we have:
so that, imposing Eq. 9 and simplifying, we obtain:
where, the term in brackets is the (negative) personal price of the vol, P_{σ} and it is independent of monotonic transformation of H. Therefore, the indifference price, P_{G}, of the game is obtained as moment quantities, M_{G}, Σ_{G}, times subjective moment prices, P_{0}, P_{σ}.
Moreover, Eq. 11 can be easily generalized to higher moments (skewness Γ and kurtosis Ψ):
and this version will be used in the following to show that the behavior of
a momentutility maximizer is perfectly compatible with all the proposed paradoxes,
from St. Petersburg (1713) to Friedman and Savage (1948),
Allais (1953, 1979) and Kahneman
and Tversky (1979).
Table 1: 
The St. Petersburg game 

The St. Petersburg Paradox
The name of the paradox is due to the solution, proposed by Daniel Bernoulli
in 1738 to a question posed by his cousin Nicolas Bernoulli, in a letter dated
September 9th, 1713. Note that at that time, the method to value an uncertain
prospect was established by Christian Huygens in 1657 as the mathematical expectation
of the gain (Hacking, 1975). Peter tosses a coin and continues
to do so until it should land heads when it comes to the ground. He agrees to
give Paul one ducat if he gets heads on the very first throw, two ducats if
he gets it on the second, four if on the third, eight if on the fourth and so
on, so that with each additional throw, the number of ducats he must pay is
doubled.
Paul is the player: if he obtains heads at the first flip he wins 1, at the second flip he wins 2,..., at the nth flip he wins 2^{n1} and so on. The question is to determine a fair price, P(G), to enter the game.
Clearly, the price must be at least 1 (the minimum gain), but the expected gain is infinite (Table 1).
However, as Nicolas Bernoulli observed in stating the paradox, nobody would
pay an arbitrarily large amount to play the game: it has, he said, to be admitted
that any fairly reasonable man would sell his chance, with great pleasure, for
twenty ducats.
The famous Bernoulli solution, in log terms, provided a pathbreaking device, introducing the concept of utility (moral expectation) and reducing expectation to a finite value:
Alternatively, the price, P(G), of the game can be obtained, in our approach,
by considering the sequential game as a oneshot game (a lottery) with an infinity
of tickets, identified by natural numbers (1, 2,..., n,.....) with decreasing
probability of extraction (½, ¼,...,½^{n},....)
and increasing rewards (1, 2,..., 2^{n1},...). This means that the
original game is a portfolio of subgames G_{n} n≥1 or ArrowDebreu
securities (one for each row in the table above), the nth of which implies
a prize of 2^{n1} if we get heads at the nth flip and zero otherwise.
Clearly, each ticket could be sold separately at its price P(G_{n}) and the price of the lottery is the sum of the prices of all tickets.
We show that a twomoment utility approach is sufficient to solve the paradox.
Table 2: 
The first two moments of the St. Petersburg game 

For each ticket n, the expected value is always ½ and the standard deviation
is 0.5 (2^{n}1)^{0.5}:
Considering the first two moments (Table 2) the price of
the nth ticket is, from Eq. 12:
where, the limited liability provision has been applied and the price of the
game is simply the sum of the (non negative) prices of all tickets:
For example, if P_{0} = 1 and P_{σ} = 0.134 then P(G_{n})
= 0 for n>5 and P(G) = 1.507. Coin tosses beyond the fifth have no economic
value. Note that in this gamesituation, P_{σ} is not a proper
market price but just the gambler’s personal price of volatility. Analogously
for higherorder moments; a refined price P(G) could also be obtained using
higher moments: the skewness of the nth ticket is [0.25 (2^{n}1) (2^{n1}1)]^{1/3};
the kurtosis is [(2^{n}1) ((2^{n}1)^{3}+1)/2^{n+4}]^{1/4}.
The Friedman and Savage (1948) Paradox
In their classical article, Friedman and Savage (1948)
observed the difficulty of combining the belief in diminishing marginal utility
and the observation that the same individual buy insurance as well as lottery
tickets. Clearly, the first choice is evidence of risk aversion but the second
one can be rationalized only by a risk loving behavior, being well known that
lotteries are largely unfair games, accepted only by individuals having a strong
preference for risk.
The clever solution they proposed, in the framework of the Von NeumannMorgenstern theory, just published a few years before, was based on a complex hypothesis concerning the shape of the utility function, made by three segments, concaveconvexconcave, where current income is in the initial convex segment (ivi, paragraph IV).
In terms of moment utility the solution is much simpler and it clarify that skewness is the key element behind the observed behavior.
A typical lottery represents a large chance of losing a small amount (the price of the lottery ticket) plus a small change of winning a large amount (a prize).
Viceversa, the game against which you buy insurance, paying the premium π, contains a small chance of a much larger loss and a large chance of no loss. Let L and J be, respectively, the two random variables so that L (lottery) is preferred to 0 as well as π is preferred to J:
Table 3: 
The first three moments of the FriedmanSavage lottery (L)
and insurance (J) 

Let us calculate the first three moments of L and J (Table 3).
Mean and standard deviation are the same but skewness is reversed in sign, with L having a large, positive skewness and J a large, negative one. Assuming the following prices of the three moments: P_{0} = 1, P_{σ} = 0.34, P_{ζ} = 0.1 we obtain the prices P(L) = 5.71>0 and P(J) = 40.70<0 so that the ticket is bought with pleasure (and considered cheap) and up to 40.7 money units could be willingly paid to avoid the insurable risk.
The Allais Paradox
The Allais (1953) paradox was the first factual
evidence against expected utility. In fact, asking people to choose between
games A and B, where A gives 1 million with certainty and B gives 1 million
with 89% probability and 0 or 5 millions with, respectively, 1 and 10% probabilities:
people prefer in large majority A to B: A›B.
Then, asking them to choose between A’ and B’ defined by:
the same people very often prefer B’ to A’: B’›A’.
The paradox stems from the fact that, from A›B, the expected utility approach deduces A’›B’, which is at variance with the experimental evidence (Allais reports 53% of cases of violation of the logical implication).
In fact, A›B means:
but collecting U(1) and adding to both members of the inequality 0.89U(0) you
obtain, algebraically:
i.e., A’›B’, against the empirical evidence.
Table 4: 
The first four moments of the Allais games 

According to Allais, either people in experimental situations do not use the
rational thinking used in real world decisionmaking, or people do not follow
the expected utility paradigm.
In fact, using our approach, the rationality of the actual choices may be easily recognized.
Considering each lottery as an asset, the first four moments are in Table 4.
Assuming the following prices of the four moments: P_{0} = 1, P_{σ} = 0.34, P_{ζ} = 0.01, P_{κ} = 0.001 we obtain the prices of the lotteries: P(A) = 1>P(B) = 0.994 and P(A’) = 0.007< P(B’) = 0.008, in accordance with the Allais experiments.
This means that, using the Marschak triangle as in Machina
(1987), the indifference curves in our approach are nonlinear in the probabilities
and may display a ‘fanning out’ effect from the sure event A, as implied
by actual behavior.
The Kahneman and Tversky (1979) Paradox
In a famous experiment, a systematic violation of the independence axiom
was documented: 80% of 95 respondents preferred A to B where:
65% preferred B’ to A’ where:
and more than 50% of respondents violated the independent axiom, given that,
if Q pays 0 for sure, then:
and B” is considered equal to B’ in terms of outcomes and probabilities.
Note that treating lotteries as assets implies that linear combinations such as 0.75Q+0.25A are meaningful and P(0.75Q+0.25A) = 0.25P(A)≠P(A’).
The point is that, in terms of valuation, B’ and B” are not
the same asset and B” is equivalent to:
Table 5: 
The fist four moments of KahnemanTversky games 

Table 6: 
The first four moments of the TverskyKahneman games 

Using the first four moments in Table 5 and assuming the
following prices of the four moments: P_{0} = 1, P_{σ}
= 0.2, P_{ζ} = 0.1, P_{κ} = 0.001 we obtain the
prices of the lotteries: P(A) = 3000>P(B) = 2694.70 and P(A’) = 624.87<P(B’)
= 661.01, in accordance with the experimental results. Note also that P(B”)
= 561.28<P(A’)<P(B’).
The Tversky and Kahneman (1981) Paradox
Most subjects, confronted with the following alternatives: A versus B and
A’ versus B’, prefer B to A but also A’ to B’ where:
The paradox (reversal or isolation effect) stems from the fact that not only
A = A’ but also B = B’ in terms of ultimate outcomes and probabilities.
However, from the point of view of our theory of valuation and choice, the twostage frame in B’ is not irrelevant: in B, not 0 means 45; in B’, not 0 means a new game B”, which can be sold for a certain price.
Using the first four moments in Table 6 and assuming the
following prices of the four moments: P_{0} = 1, P_{σ}
= 0.2, P_{ζ} = 0.05, P_{κ} = 0.1, we obtain the
prices of the lotteries: P(A) = 3.98 <P(B) = 4.01 and P(A’) = 3.98>P(B’)
= 3.84, being 30>P(B”) = 28.95. This result is in accordance with the
Tversky and Kahneman (1981) experiment, showing that,
in effect, no paradox is implied in the observed behavior.
In particular, note, once again, that in terms of valuation:
and the riskneutral probabilities for B”, for which the personal price
of B” is the expected value:
are given by:
This observation also holds for the Markowitz (1959)
formulation of Allais’s experiment:
CONCLUSIONS
After Tobin (1958), considerable effort has been devoted
to connecting the expected utility approach to a utility function directly expressed
in terms of moments. In contrast with this approach, we have provided the theoretical
foundation of an ordinal utility function of moments which is free of any independence
axiom and compatible with all the behavioral paradoxes documented in recent
and less recent works on decisions under uncertainty. This momentutility can
be used as the starting point for a new formulation of asset demand models and
asset pricing, having more general properties and greater flexibility than existing
expected utility results. Future research could be addressed in this promising
direction.
APPENDIX: BASIC AXIOMS AND PROOFS OF THE THEOREMS
Let (Ω, ,
)
be a standard probability space, Ω being the set of elementary events (states
of the world),
the set (σalgebra) of subsets of Ω (events),
a (subjective) probability measure of the events. Given the set A of all possible
actions or decisions, all couples (ω,a) with ω∈Ω and a∈A,
are mapped onto a real vector of monetary consequences c∈R^{n},
the Euclidean space of ndimensional real vectors, so that X (ω,a) = c
or X_{a} (ω) = c is a random variable and F_{a}∈F
is its probability distribution function. Clearly, the preferences over acts
in A are, equivalently, preferences over the set of random variables X_{a}
as well as preferences over the set F of distribution functions. Let us confine
ourselves, for ease of exposition, to the case of univariate distributions (n
= 1) and assume that the essential information concerning any distribution F
is contained in the mdimensional vector of moments M≡(μ, μ^{(2)},
μ^{(3)},...., μ^{(m)}) where μ is the mean and
μ^{(s)} is the sorder central moment in original units:
Definition of sorder modified central moment:
Note that (μ^{(s)})^{s} is the usual central moment of order s≥2.
Let Q⊆R^{m} be a rectangular subset of R^{m} (the Cartesian product of m real intervals), whose elements are the mdimensional vectors of moments, M∈Q.
Assumption of Preference Order
Let › be a preference order i.e., a binary relation defined by a subset
of
the Cartesian product QxQ, whose elements are the ordered pairs of vectors (M_{a},M_{b}).
We write M_{a}›M_{b} instead of (M_{a},M_{b})
and
we say that M_{a} is preferred to M_{b}, corresponding to F_{a}
is preferred to F_{b}.
Clearly, or M_{a}›M_{b} or M_{a}M_{b}
and both cannot hold: in fact, or (M_{a},M_{b})
or (M_{a},M_{b}) .
I. Axiom of Asymmetric Preferences
We assume that › is asymmetric i.e., that:
The relation is therefore irreflexive and, moreover, if M_{b}M_{a}
then two alternative cases are possible: either M_{a}›M_{b}
or M_{a}M_{b}.
In the latter case we say that M_{a} and M_{b} are equivalent and we write M_{a} ~ M_{b}.
Definition of Equivalence
M_{a} ~ M_{b} if and only if M_{a}M_{b}
and M_{b}M_{a}
Theorem 1 of Complete Preferences
Given M_{a}, M_{b} ∈Q then one and only one case holds:
M_{a}›M_{b} or M_{b}›M_{a} or M_{a}~M_{b}.
Proof: It is easy to show that any two cases are a contradiction. ^{ }
II. Axiom of Transitive Preferences
We assume that › is transitive:
Definition of Weak Order
The preference order › is a weak order if it is asymmetric and transitive.
Definition of Negatively Transitive Preferences
If M_{a}M_{b} and M_{b}M_{c}
then M_{a}M_{c}.
Lemma 1
› is negatively transitive if and only if, for every M_{a},
M_{b}, M_{c}∈Q, M_{a}›M_{b} implies
M_{a}›M_{c} or M_{c}›M_{b}.
Proof: Under negative transitivity, if M_{a}›M_{b} but M_{a}M_{c} and M_{c}M_{b} then by negative transitivity M_{a}M_{b} against the assumption.
Viceversa, if M_{a}›M_{b} implies M_{a}›M_{c}
or M_{c}›M_{b} and negative transitivity is false we have
from M_{a}M_{c} and M_{c}M_{b}
that M_{a}›M_{b} so that M_{a}›M_{c}
or M_{c}›M_{b}, in both cases a contradiction.^{ }
Theorem 2 of Negatively Transitive Preferences
Asymmetric and transitive preferences are equivalent to asymmetric and negatively
transitive.
Proof: Under transitivity if M_{a}›M_{b} and M_{b}›M_{c} then M_{a}›M_{c}; therefore, by asymmetry, if M_{b}M_{a} and M_{c}M_{b} then M_{c}M_{a} which is negative transitivity.
Viceversa, under negative transitivity, if M_{a}›M_{b} and M_{b}›M_{c} then, from Lemma 1, (M_{a}›M_{c} or M_{c}›M_{b}) and (M_{b}›M_{a} or M_{a}›M_{c}). But, by asymmetry, M_{b}M_{a} and M_{c}M_{b} so that M_{c}›M_{b} and M_{b}›M_{a} are false. Therefore M_{a}›M_{c} which means transitivity.^{ }
Theorem 3 of Equivalence Classes
The equivalence ~ is reflexive, symmetric and transitive and we have:
Moreover, › on Q~ (the set of equivalence classes of Q under ~)
is a strict order i.e., it is a weak order and for every equivalence class M_{A},
M_{B} ∈Q~ one and only one case holds: M_{A}›M_{B}
or M_{B}›M_{A} (weak connectedness).
Proof: The equivalence is clearly reflexive and symmetric. Suppose it is not transitive: M_{a}~M_{b} and M_{b}~M_{c} but M_{a} ~ M_{c} is false. Then, by definition, either M_{a}›M_{c} or M_{c}›M_{a}. From Lemma 1, in the first case, M_{a}›M_{b} or M_{b}›M_{c}; in the second case M_{c}›M_{b} or M_{b}›M_{a}, in contradiction with the hypothesis.
If M_{a}›M_{b} and M_{a}~M_{c} then by Theorem 1 and Lemma 1 we have M_{c}›M_{b}. If M_{a}›M_{b} and M_{b}~M_{c} then by Theorem 1 and Lemma 1 we have M_{a}›M_{c}.
Finally, › on Q~ is a weak order, being asymmetric and negative
transitive: under symmetry, if M_{A}›M_{B} and M_{B}›M_{A}
then exist M_{a}, M_{a’} in M_{A} and M_{b},
M_{b’} in M_{B} such that M_{a}~M_{a’},
M_{b}~M_{b’} and M_{a}›M_{b} and M_{b’}›M_{a’}.
From (A.4) M_{a}›M_{b’} and M_{b’}›M_{a}
which is a contradiction; for negative transitivity, if M_{A}›M_{B}
then M_{a}›M_{b} for any M_{a }in M_{A}
and M_{b }in M_{B} and if M_{c }is any vector in M_{C}
then we have, from Lemma 1, M_{c}›M_{b} or M_{a}›M_{c}.
Therefore, M_{C}›M_{B} or M_{A}›M_{C}.
For weak connectedness, given that M_{A} and M_{B} are disjoint, from Theorem 1, either M_{a}›M_{b} or M_{b}›M_{a} for every M_{a} in M_{A} and M_{b} in M_{B}. Therefore, either M_{A}›M_{B} or M_{B}›M_{A}.^{ }
III. Axiom of Continuity
There is a countable subset D⊆Q~ that is›dense in Q~
i.e., for every M_{A}, M_{C} ∈Q~\D, M_{A}›M_{C}
there is M_{B}∈D such that:
Note that the subset of rational numbers is >dense and <dense in the
set of real numbers.
Theorem 4 of Ordinal Utility on Moments
Proof: The proof follows the steps as in Fishburn (1970).^{}
Theorem 5 of Continuous Utility
Proof: See Debreu (1964).^{}
Theorem 6 on Stochastic Dominance and Expected Utility
Proof: (i) using integration by parts:
Therefore, if F(x)≤G(x) then E_{F}(U(x))≥E_{G}(U(x)).
Viceversa, if E_{F}(U(x))≥E_{G}(U(x)) let I be an interval
in which F(x)>G(x) and let χ_{I} be the indicator function of
I. Define:
so that U’(x)≡χ_{I}(x)≥0 and inserted into the above
equation gives a contradiction. For (ii) and (iii) Fishburn
and Vickson (1978) and Whitmore (1970).^{}
Theorem 7 on Stochastic Dominance and Moment Utility
Proof: Apply Fishburn (1980, theorem 1) and the
relation between central, μ and non central, v, moments: