1. Introduction

John von Neumann and Oskar Morgenstern developed the Expected Utility Theory (EUT) in 1944. This theory has been extensively used in game theory, industrial organization, financial theory, and other important areas in economic theory. Graduate courses use texts like ^{Mas-Colell et al. (1995)}, ^{Jehle and Reny (2011)}, ^{Maschler et al. (2013)}, ^{Rubinstein (2012)}, among others, to explain it. The axiomatic approach on each text apparently uses different languages and axioms. In this paper, we prove the equivalence among those axiomatic systems, and build the lottery space in a recursive way. The importance of doing this homologation consists on the unification in a common language of properties and systems that look different from a first sight.

Although the equivalence between axiomatizations is almost obvious for specialists, it is not so for most students. That is why we believe in the need to unify seemingly different languages and to offer a formal proof of the equivalence between several axiomatizations. In general, even slight modifications of axioms can generate different theories. A classic example is non-Euclidean geometries.

The expected utility theory has been heavily revised. The well-known Allais paradox showed the inconsistency of the theory with some cases of agents’ choices faced with specific lotteries. This has made it possible to analyse the consequences of weaknesses and variants of the independence axiom. Several authors have proposed alternative schemes to model decisions under risk and uncertainty. Its study is beyond the scope of this note. We recommend consulting ^{Barbera et al. (2004)}, and specially ^{Sudgen (2004)} and ^{Schmidt (2004)}.

The standard proof of the expected utility theorem relies heavily on the axiom of continuity to construct the utility of a lottery. Given an arbitrary initial lottery, we construct its utility considering a special lottery that is indifferent to the initial one. The special lottery is a simple lottery that combines the most preferred with the least preferred result. The utility of the initial lottery is the weight that has the most desired result. The axiom of independence is the key to obtain that the numerical representation is a linear mapping on the probabilities; that is, it has the fundamental property of the expected utility.

^{Cantala (2007)} describes an interesting characterization of Expected Utility. The author translates preferences over lotteries into preferences over ‘‘shifts in probabilities". Changes in probabilities are relatively straightforward. The probability of one outcome is reduced by a fixed feasible amount and the same amount increases the probability of another outcome. The author constructs an independence axiom suitable to represent these types of ‘‘jumps" in lotteries. He obtains a new characterization of the expected utility: continuous preferences over lotteries are represented by an expected utility function if and only if the induced preferences satisfy his version of the independence axiom for preferences over shifts in probabilities. The axiom means that the ordering of shifts in probability is independent of the weight and the original lottery. Cantala’s treatment (2007) can be repeated in any of the axiomatic versions. It is possible to use his proof technique to generate new versions of the expected utility theorem by adapting the new version of the independence axiom on each axiomatic system. In this paper, we are interested in the equivalences among the different axiomatic systems rather than in the constructive proof of the expected utility theorem. The approach of ^{Cantala (2007)} is relevant because ‘‘it makes it clear that the cardinal nature of expect utility is only a behavioural characteristic rooted in the independence axiom and has no normative appeal" (^{Cantala, 2007: 101}).

We show a proof based strongly on the construction of a common language for rewriting the different systems of axioms. This means the specification of a common lottery space where the preferences are defined. The recursive construction of the lottery space is necessary: it is not an exaggeration. It is introduced for technical reasons of consistency, it makes sense, and it is suggested in ^{Jehle and Reny (2011)}.

Compound lotteries can be classified by their level of complexity. Every compound lottery of any level has an expected utility level to compare with any other lottery. However, this comparison requires introducing an axiom that makes it possible to relate a compound lottery with a simple one of the first level. This is done in the ^{Jehle and Reny (2011)} axiomatization and ^{Maschler et al. (2013)}. These two axiomatizations are the better ones for explaining the economic content of the axioms, in addition to the axioms of rationality of preferences. The axiomatic presentations of ^{Mas-Colell et al. (1995)} and ^{Rubinstein (2012)} focus on the two essential axioms in the proof of the expected utility theorem: independence and continuity. In their works, they consider compound lotteries, but they are artificially reduced to those of the first level. They avoid an axiom of simplification by introducing the convexity of the lottery space.

The simplification axiom from compound to simple lotteries is an empirically unsustainable assumption. It is not obvious that an agent recognizes a fairly complex compound lottery as indifferent to its simplified form. In ^{Mas-Colell et al. (1995)}, a simplex is introduced as the lottery space. The convexity of this space makes it closed under convex combinations. But this is a mathematical reason that artificially simplifies and reduces (by convexity) the meaning of a compound lottery of a second or more level. The simplification axiom is strong but important because it considers the levels of complexity in the construction of lotteries.

In summary, the presentation by ^{Mas-Colell et al. (1995)} is more classical, but its continuity axiom is apparently more complex. We could say that it abuses mathematics somewhat and sacrifices part of the economic meaning. On the other hand, the presentations by Jehle and Reny (1911) and ^{Maschler et al. (2013)} are more careful. Their economic content has greater detail. The formulation of the continuity axiom is clearer. The first one even breaks down independence into two apparently less strong axioms: substitution and reduction to simple games.

The order of the paper is as follows: in section 2, we analyze the similarities in notation and concepts of the four axiomatic systems, establishing a common language regarding the concept of compound lotteries and the simplification action. Then, we compare two pairs of axiomatic systems due to the similarities between them: ^{Maschler et al. (2013)} and ^{Jehle and Reny (2011)} with four axioms each, and ^{Mas-Colell et al. (1995)} and ^{Rubinstein (2012)}, each one with a system of two axioms. Finally, we show the equivalence of the four systems by means of a theorem in section 3 of this article.

2. Model

An agent must take a decision before a random event occurs. The final consequences of his decision are represented as possible outcomes in the set X=x1,…,xK . The objects that each agent values are called lotteries and are listed such that L=p1x1,…, pKxK is a simple lottery with pk≥0 and ∑k=1Kpk=1 . L denotes the lottery that assigns probability p _k to each consequence xk∈X . For simplicity, we will denote the lottery αx1,0x2,…,0xK-1,1-αxK just by αx1,1-αxK for α∈0,1 .

Let L1={p1x1,…,pKxK|x1,…,xK∈X;pk≥0 and ∑k=1Kpk=1} be the set of simple lotteries, and Ln+1=LnU {[q1L1,q2L2,…,qHLH]|Lh∈Ln;qh≥0 and ∑h=1Hqh=1,  ∀h=1, 2,  …,H} . Then, the space of lotteries is given by L=Un=1∞Ln The lotteries that have the particularity of granting a consequence xk∈X with probability equal to one will be called degenerated lotteries and we denote them by Lxk=1xk . Lotteries that have the characteristic of having other lotteries as consequences will be called compound lotteries, that is, elements in Ln with n≥2 .

Example: In order to illustrate the previous ideas, let us consider the next lotteries:

Notice that L,LA,LB,LD∈L1,LC∈L2,LE∈L3 . We can represent the process of composing the lotteries with the tree scheme given in figure 1.

Source: Authors’ elaboration.

Figure 1 Scheme of the composing of lotteries over a set of outcomes (the terminal nodes) 

A classical axiom about the simplification of lotteries says that each compound lottery must be indifferent to a simple lottery. In this case, lottery L is indifferent to lottery L _E , because the probability of each outcome is the same in both lotteries. A strong axiom about individual preferences is avoided if we assume convexity of the lottery space. However, its economic meaning is important.

For comparing the four axiomatic systems, we define a general binary relation ∼≻ on L , which allows us to compare lotteries. Such relation is called a preference relation.

A preference relation ∼≻ is rational if:

1. For every L,L'∈L×L we have L∼≻L' or L'∼≻L .

2. For any L,L',L″∈L , if L∼≻ L' and  L'∼≻L″ , then L∼≻L″ , where L∼≻ L' means that L is at least as good as L' .¹

According to the definition of compound lottery, it is possible to have uncertain decision problems where the consequences of the alternatives are uncertain objects. For example, we have a raffle such that one of the possible prizes is a chance to participate in another raffle. So, this idea motivates the concept of simplification of compound lotteries into an equivalent simple lottery, in the sense of the preference relation. As the notion of simplification belongs to the four axiomatic systems under our study, it is convenient to have a common notation for it.

We define the lottery simplification, Simp:L→L1, for n≥2, as follows:

where L=q1L1,…,qMLM with L1,…,LM∈Ui=1n-1⁡Li ; we assume that SimpL1=p11x1,…,pK1xK , SimpL2=p12x1,…,pK2xK, SimpLM=p1Mx1,…,pKMxK and we have, for every k∈1,…,K , rk=q1pk1+q2pk2+…+qMpkM .

As we said before, each axiomatic system under our study assumes that a compound lottery can be simplified. In ^{Jehle and Reny (2011)} and ^{Maschler et al. (2013)}, this fact is assured by axioms G6 and S, respectively. In ^{Mas-Colell et al. (1995)} and ^{Rubinstein (2012)}, the definition of compound lottery allows to make this kind of simplification.

3. Axiomatic equivalencies

First, we analyze the Simplification axiom; notice that this property is part of the ^{Maschler et al. (2013)}, and ^{Jehle and Reny (2011)} axiomatic systems (S and G6, respectively), and the equivalence between them is straightforward. In the ^{Mas-Colell et al. (1995)} and ^{Rubinstein (2012)} axiomatic systems, this property is implicit in the definition of a compound lottery. In ^{Mas-Colell et al. (1995)}, a compound lottery is denoted by L1,…,LM;α1,…,αM and its simplification is given by q1,…,qK where, for every k=1,…,K , qk=α1pk1+α2pk2+…+αMpkM denotes the probability under L1,…,LM;α1,…,αM of the consequence x _k . That construction implies the generalization for any level of composition as long as the probability distribution of the compound lottery is well defined. The construction of compound lotteries in Rubinstein (2011) is given in a similar way.

On the other hand, the axiomatic system of ^{Jehle and Reny (2011)} has six axioms. Two of them are included in our definition of preference relation: the completeness axiom G1 and the transitivity axiom G2. Also, the simplification axiom G6 was rewritten according to our notation and the function Simp. The equivalence between S and G6 is straightforward. Henceforth, we conclude that the simplification axiom is present in the four axiomatic systems.

Now, we show the main result of our work.

Theorem: The following statements are equivalent.

We use the following lemma.²

4. Conclusions

As the main result of this paper, we show the equivalence among four textbook axiomatic systems regarding the expected utility theorem given by four authors. This implies that the objects satisfying the four axiomatic systems coincide. Also, we provide a common language for the space of lotteries, the simplification of a lottery, and their composition. These kinds of results could help the understanding of the logic and objects involved in different treatments of the expected utility theory.