Moral expectation

Moral expectation is an estimate of the lot, first introduced by the Swiss mathematician Daniel Bernoulli . Unlike mathematical expectation ( expected return ), moral expectation depends on the player’s state and implicitly takes into account the risk factor. The term “moral expectation” itself belongs to the French mathematician Pierre Simon Laplace .

Content

Definition

Let in some game the gain value takes values $x_{i}$ ${\ displaystyle x_ {i}}$ with probabilities $p_{i}$ ${\ displaystyle p_ {i}}$ where $i=1,2,...,n$ ${\ displaystyle i = 1,2, ..., n}$ , C - player state before the start of the game. Then the moral expectation is determined by the equality:

 $Mr(x,C)={\bar {\bar {x}}}=\prod _{i=1}^{n}(x_{i}+C)^{p_{i}}-C.$           $M$       $r$       $($       $x$       $,$       $C$       $)$       $=$                 $x$       $¯$           $¯$             $=$         $\prod$         $i$       $=$       $one$           $n$           $($         $x$         $i$           $+$       $C$         $)$           $p$         $i$               $-$       $C$       $.$           ${\ displaystyle Mr (x, C) = {\ bar {\ bar {x}}} = \ prod _ {i = 1} ^ {n} (x_ {i} + C) ^ {p_ {i}} - C.}$

The moral expectation we will denote

{\bar {\bar {x}}}

{\ displaystyle {\ bar {\ bar {x}}}}

or

Mr(x,C)

{\ displaystyle Mr (x, C)}

when we want to emphasize its dependence on the state.

Properties

Moral expectation $Mr(x,C)$ ${\ displaystyle Mr (x, C)}$ strictly monotonously increases with increasing state C.
In state C, tending to infinity, the limit of moral expectation is equal to the mathematical expectation:

\lim _{C\to +\infty }Mr(x,C)=M(x).

{\ displaystyle \ lim _ {C \ to + \ infty} Mr (x, C) = M (x).}

The moral expectation is strictly less than the mathematical one: $Mr(x,C)<M(x)$ ${\ displaystyle Mr (x, C) <M (x)}$ .
$Mr(x+a,C)=Mr(x,C+a)+a$ ${\ displaystyle Mr (x + a, C) = Mr (x, C + a) + a}$ , where a is an arbitrary real constant.
$Mr(a*x,C)=a*Mr(x,{\frac {C}{a}})$ ${\ displaystyle Mr (a * x, C) = a * Mr (x, {\ frac {C} {a}})}$ , where a is an arbitrary positive real constant.
$\lim _{k\to +\infty }k*Mr({\frac {x}{k}},C)=M(x)$ ${\ displaystyle \ lim _ {k \ to + \ infty} k * Mr ({\ frac {x} {k}}, C) = M (x)}$ where k is a natural number.

Here $M(x)=\sum _{i=1}^{n}p_{i}*x_{i}$ ${\ displaystyle M (x) = \ sum _ {i = 1} ^ {n} p_ {i} * x_ {i}}$ - mathematical expectation of a random variable $x$ ${\ displaystyle x}$ .

Basic Information

The player does not always evaluate the lot by mathematical expectation, that is, he does not always evaluate it as the average gain. Otherwise, insurance companies would have been out of work for a long time. Indeed, in the tasks of risk insurance, the amount of the insurance premium exceeds the expected damage. Consider an example:
Let you have lots, which with equal probability can bring 40 thousand euros of income or nothing. By mathematical expectation, this lot is worth 20 thousand. However, many will agree to sell it for 18 thousand. The latter means that these people value lots less than 18 thousand. But there are those who want to buy this lot for more than 18 thousand. Buyers, therefore, evaluate lots for more than 18 thousand. It can also be assumed that buyers of lots are richer than sellers.

Bernoulli suggested that an elementary increment of state C gives an increment of the utility of state Z by a value proportional to this increment and inversely proportional to the magnitude of the state:

$dZ=k*{\frac {dC}{C}}$ ${\ displaystyle dZ = k * {\ frac {dC} {C}}}$ where $k>0$ ${\ displaystyle k> 0}$ . From this we immediately obtain the logarithmic function of the utility of money $Z=ln(C)$ ${\ displaystyle Z = ln (C)}$ . Then the expectation of utility will take the form: $\ln {({\bar {\bar {x}}}+C)}=\sum _{i=1}^{n}p_{i}*\ln {(x_{i}+C)}$ ${\ displaystyle \ ln {({\ bar {\ bar {x}}} + C)} = \ sum _ {i = 1} ^ {n} p_ {i} * \ ln {(x_ {i} + C )}}$ , whence the equality that determines the moral expectation is obtained. Bernoulli published the results in 1738 in an article entitled “The Experience of a New Theory of Measurement of the Lot”. Thus, Bernoulli built the utility function for such a good as money, long before Jeremy Bentham introduced the concept of utility into economic theory. Evaluation of lots by moral expectation often allows you to build mathematical models that are adequate to the behavior of real economic entities.

Example

The author of the task is considered Nikolai Bernoulli .

The merchant Kaus bought in Amsterdam a product that he could sell in St. Petersburg for 10,000 rubles. The goods will be sent to St. Petersburg by sea. It is known that at this time of the year, out of 100 vessels, 5 crashes. The merchant could not find anyone who would agree to insure the goods for less than 800 rubles. By agreeing to insure the goods on the proposed terms, the merchant changes his lot for a guaranteed 9,200 rubles. It is proposed, based on moral expectation, to answer the questions:

What condition should a merchant (seller of lots) have to agree to insure his goods on the proposed conditions?
What is the condition of the one who undertook to insure the cargo (buyer of the lot)?

The mathematical expectation of income in this problem is 9500 rubles. And what will change if the merchant distributes the cargo equally on two ships. The mathematical expectation of the lot is still 9500. But intuitively we feel that such a lot is more expensive. And, indeed, it turns out that the estimate of the lot by moral expectation increases significantly.

Generalization of the concept of moral expectation

A generalization naturally arises for the case when an elementary increment of the state gives an increase in the utility of the state by an amount inversely proportional to a certain degree of the state. Then we come to the class of money utility functions of the form $f(C)=C^{s}$ ${\ displaystyle f (C) = C ^ {s}}$ where $s\in {(0;+\infty ]}$ ${\ displaystyle s \ in {(0; + \ infty]}}$ . In this case $s\in {(0;1)}$ ${\ displaystyle s \ in {(0; 1)}}$ corresponds to the classical utility function, i.e., increasing and convex upward, and the case $s\in {(1;+\infty ]}$ ${\ displaystyle s \ in {(1; + \ infty]}}$ - areas of convexity down the Friedmann function. Then the generalized moral expectation can be defined as follows. The moral expectation of order s of a random variable x in state C is the quantity $Mr^{(s)}(x,C)={\bar {\bar {x}}}=(\sum _{i=1}^{n}p_{i}*(x_{i}+C)^{s})^{\frac {1}{s}}-C.$ ${\ displaystyle Mr ^ {(s)} (x, C) = {\ bar {\ bar {x}}} = (\ sum _ {i = 1} ^ {n} p_ {i} * (x_ {i } + C) ^ {s}) ^ {\ frac {1} {s}} - C.}$ notice, that $\lim _{s\to 0}Mr^{(s)}(x,C)=\prod _{i=1}^{n}(x_{i}+C)^{p_{i}}-C.$ ${\ displaystyle \ lim _ {s \ to 0} Mr ^ {(s)} (x, C) = \ prod _ {i = 1} ^ {n} (x_ {i} + C) ^ {p_ {i }} - C.}$ The moral expectation can also be generalized to the case when the random variable has a continuous distribution.

Literature

Bernoulli D. Experience of a new theory of measuring lots // Theory of consumer behavior and demand. - St. Petersburg: School of Economics, 1993. - 380 p. - 25,000 copies. - ISBN 5-900428-02-8 .
Bely E.K., Belaya M. E. The theory of the utility of money. Diversification of the securities portfolio and other tasks with the assessment of income by moral expectation. - Saarbrucken, Germany: LAB LAMBERT Academic Publishing, 2012 .-- 84 p. - ISBN 978-3-8484-2665-2 .
Bely E.K. Moral expectation and the task of diversifying a portfolio of securities // Uchen. app. Petrozavodsk state un-that. Ser. Social and human sciences. No. 1 (106). - Petrozavodsk: PetrSU, 2010 .-- 500 copies.