Iii. The doctrine of desired utility
In early decision-making theories, the academic community chose the mathematical expectations of money as a target for people's decision-making. In the expected value guideline, the expected value of a decision variabled is equal to the sum of its relative probability of occurrence in different natural situations (gulf and li bei wei, 2006)。
In formula 1. 1, e(d) is the expected value of the decision variable di, dij is the profit or loss of the decision variable under the state j and p(s) is the probability that the natural state zirconiumj will occur. The expected value criterion is that people are guided in decision-making by the principle of maximizing the desired value。
In 1738, nikolai bernuli of st. Petersburg university in russia presented a “st. Petersburg paradox” that cannot be explained by mathematical expectations. The st. Petersburg experiment allowed the following lottery tickets to be valued:
, the integer and fractions in the lottery represent value and probability respectively. Nikolai bernuli has found a large number of students and professors at st. Petersburg university and their valuation of the lottery is generally between 5 and 7, which is far from the infinitely mathematical expectations of the lottery. Subsequently, the economist daniel bernoully used the term “use” to explain the value of the lottery. Daniel bernoully considers utility to be a function of monetary value, which presents a decreasing pattern of marginal utility。
The early economic definition of utility is the degree of satisfaction that economic agents obtain from possession of a certain number of items (thermal syria, balarus, vitri, etc., 2012). While utility is the underlying concept of economics, the absolute value of utility cannot be measured directly. Economics later developed the concept of scalability utility: it was used to express the preferences of an economic subject for a set of elements of his choice (hot-shu, balarus, vitri, etc., 2012)。
In 1947, von newman and morganstein defined the concept of utility in a rigorous mathematical sense and introduced the theory of desired utility in their great economics book, the game and economic behaviour. According to the expected utility theory, the expected utility of lottery a = (x1, p. 1; x2, p2; ...; xn, pn) is:
The formula (1. 2) is the basic formula for the calculation of expected utility, and its mathematical meaning is that the desired utility value for lottery a is equal to the weighted average of the effects of each value in the lottery。
Based on the findings of the book " game theory and economic behaviour " , and referring to the von newman-morganstein system of justice summarized in the economics dictionary, the book summarizes six relevant principles of the theory of desired utility, namely:
Justice 1: passivity justice (sequence justice). Assuming there are three lottery a1, a2 and a3, if the preferred order of the three lotterys is
Then..
Justice 2: continuing justice. Suppose there are three lottery tickets, if the preferred order of three lottery tickets is
, then there must be a combination of a1 and a3, which is equivalent to a2. It is argued that the preference can be expressed by the continuity figure。
Justice 3: protracted characterization. Different expressions of the same issues do not affect the outcome of decision-making. The relationship between the lottery itself and the lottery is key to decision-making, and the problem is described in a manner that is not related to the outcome of the decision-making。
Justice 4: alternative justice. The value of lottery a1 may be recognized as a numerical solution. In the decision-making process, lottery a1 and numerical xi can be replaced。
Justice 5: independence justice. If the preferred order of the two lotteryes is a1≻a2, assuming that there is a third lottery a3 and if the given value of α∈ is given, the following relationship exists: α a1+(1-α) a3≻a2+(1-α)a3。
Justice 6: first class randomly. If lottery a1 is better than lottery a2 in one respect and lottery a1 is no less than lottery a2 in other respects, then a1 is better than a2。
The utility function of the desired utility theory is a generic function that contains the six hypothetical justices described above and has a model expression of strict mathematical logic and regulation. The doctrine of desired utility is in most cases in line with people's decision-making patterns. The desired utility theory, refined by many economists, is the cornerstone of economics。
