Expected Utility Theory
The Expected Utility Theory (EUT) is one of the most important pillars that constitute the base of economics and finance theory. The principle of maximizing the individual’s Expected Utility allows indeed building the framework of decision making under uncertainty. The elementary and commonsensical starting point of the Expected Utility Theory is to consider that a decision maker decides between risky prospects, i.e. lotteries, by comparing their utility values weighted by the probabilities of occurrence of these lotteries, i.e.
where x and p are lottery outcomes and their respective probabilities of occurrence.
The introduction of the Utility Theory took place in the early 18th century, when Bernoulli (1738), the French mathematician, underlined that the maximization of the mathematical expected value (as opposed to the expected utility that he will introduce later) is not sufficient to explain observed decision making behaviors for particular lotteries. The historical famous example is the St Petersburg Paradox:
In the St. Petersburg game, the participants where asked to provide a price for a ticket to participate to this lottery: In a coin-tossing game, if tails comes out of the first toss, the participant receives nothing and stops the game, and in the complementary case, he receives two guilders and stay in the game; then the scenario is realized again and gain, so for example for the second toss, if tails comes out, the participant receives nothing and stops the game, and in the complementary case, he receives four guilders and stays in the game; and so on ad infinitum. Basic calculations of the expected monetary value of this prospect allows to obtain an “infinite” price to participate to such lottery as
Σn(2n *(1/2)n) = infinite. Obviously, no one will be willing to pay an “infinite” amount to participate in this lottery…The resolution of this paradox by Bernoulli was achieved thanks to the concept of utility function, suggesting to use instead the Log function to calculate the lottery expected utility value, i.e. Σn(Log(2n) *(1/2)n).
Then, two centuries later, in 1944, when John von Neumann and Oskar Morgenstern had their paper “Theory of Games and Economic Behavior “ published by Princeton University Press, the Expected Utility theory started to know the wide range of applications that it knows today.
Now, why are we covering here the background of the Expected Utility Theory? What is the relation with the Behavioral Finance Theory as this latter constitutes our primary focus?
As mentioned in the beginning of this article, the Expected Utility theory has been the base to several important results in finance. Now, when considering the axioms on which this theory has been built, we will quickly realize that as commonsensical as they seem, several objections/criticisms has been issued with regard to the axiomatization framework of the Expected Utility Theory. In particular, behavioral finance has produced several examples and studies to account for instances for situations where people's choice deviate from those predicted by the EUT and also for the cases where there have been violation or deviation of the axioms of this latter. These deviations are described as "irrational" because they can lead to situations where the decision-makers realize incoherent choices or contort the objective probabilities of occurrence of the lotteries and hence do not base their decisions on the actual costs, rewards, or probabilities involved.
Here, we quickly go through the 5 main axioms of the Expected Utility Theory:
1.Preference Axiom (comparability axiom):
The player is always able to classify any set of 2 lotteries. Let’s assume that these 2 lotteries are named “Lottery a” and “lottery b”: La, and Lb. Then the participant prefers either lottery a to lottery b (i.e. La ≥Lb), or he is indifferent between both lotteries (i.e. La ≈Lb), or he prefers lottery b to a (i.e. La ≤ Lb).
2.Transitivity axiom:
This axiom is also known as the coherence axiom:
This axiom typically says that if lottery a is preferred to lottery b, which is preferred to lottery c, then lottery a will be preferred to lottery c.
3.Non-saturation axiom:
This axiom implies that any individual is willing to increase infinitely his utility.
4.Continuity axiom:
This axiom implies that the a minor change in the probabilities of state of the nature should not induce a modification of the order of preference of the participant:
5. Independence axiom:
This crucial axiom means that if a third lottery is introduced to a set of 2 lotteries, then the preference of the participants should not be modified by this introduction, i.e:
In words, preference inequalities are preserved when the initial two lotteries are mixed in a given proportion with a third lottery.
This latter axiom, the independence condition, has come to be discussed widely in the EUT context, mainly after the paradox underlined by Allais (1953) and his famous lotteries example. A brief description of the experiment undertaken by Allais is here stated:
The participants are asked to choose one of the two lotteries in these two schemas:
Schema 1:
Here, the majority of the participants choose certain amount in option 1, though the expected grain from lottery 2 is higher (1 140 000 against the certain amount 1 000 000 in the first choice).
Schema 2:
Here, the majority of the participants choose lottery 4 to lottery 3.
While in schema 1, the majority of the participants choose lottery 1, in the schema 2, the participants choose lottery 4, violating hence the independence axiom.
Also, with regard to the axiom of transitivity, several violation examples has been provided by a large number of researchers, and in particular in 1969, through the work of Tversky, the vanguard researcher with his work on the introduction of behavioral biases in economics in collaboration with Khaneman with whom the Nobel price was shared in clearly stated the violation of the transitivity axiom with his experience described above:
The participants to this experiment are asked to choose between these 5 lotteries and realize a ranking of them, the output of this experiment underlines that in a situation where 2 lotteries have very close probabilities of occurrence of outcomes, the lottery with the highest outcome is preferred, nevertheless when the difference of the probabilities are high, the choice is set for the lotteries with the highest probabilities. So, this type of behavior has led the participants to select lottery A over B, B over C, C over D and finally D over E. However when lottery A et lottery E are isolated, the participants choose lottery A over E!
Lotteries | Probabilities | Outcome | Expected value |
A | 7/24 | 5 | 1.46 |
B | 8/24 | 4.75 | 1.58 |
C | 9/24 | 4.5 | 1.69 |
D | 10/24 | 4.25 | 1.77 |
E | 11/24 | 4 | 1.83 |
As a conclusion, far from having the goal to provide an exhaustive list of all the illustrations underlying the violation of the axioms of the Expected Utility Theory, the main point here to underline is that despite the width use of EUT and its solid and commonsensical axiomatization, empirical experiments still underline several violations that leads us to think that individuals’ behavior can not necessarily be built within rationality and coherence assumptions. Hence, it is in this perspective that Behavioral Theory comes as a complement to the classical theoretical Framework.
References
[1] Broihanne, M.H., M. Merli, and P. Roger, Finance Comportementale, Economica, p 18-19, p67-77, 2004.
[2] Mongin, P., Expected Utility Theory, Handbook of Economic Methodology (J.Davis, W.Hands, and U.Maki, eds. London, Edward Elgar, p. 342-350, 1997.
