Expected Utility Theory (EUT)
Expected Utility Theory (EUT) has taken over, analysis of decision making under uncertainty. It was originally devised by Daniel Bernoulli in 1738 and later axiomatized by Von Neumann and Morgenstern in 1944 (Tversky 1975, p. 163). After a decade, Savage incorporated subjective probability into EUT according to Tversky (1975, p.163). EUT has been used extensively in economics to explicate various scenarios. Further, it has been put to use to establish the best possible decisions and policies.
The concept of Expected Utility states that ‘a person entitled to make a decision opts to choose among risky or unsure scenes by contrasting their expected utility value.’ The expected utility value is the summation of the products of probability and utility values of the possible outcomes (Davis, Hands & Maki 1997, p. 342).
A person can act rationally or irrationally, and also a person can be termed to be rational if his or her action gets the most out of the outcomes. Indecisiveness crops up if the result of potential action are uncertain. One might think that if the result is uncertain then there is no need to perform the task or act. For example, consider a person who does not run for safety due to uncertainty of whether he has heard a gunshot. Uncertainty is mingled with actions people execute; therefore the concept of the expected value of activity provides a way of choosing how to act if uncertainty is spelled out.
Expected Utility Function
Granger & Machina (2006) provide for the concept of expected value as depicted below (p.15);
As stated earlier, the expected value is the summation of the multiplication between probabilities and values of each possible result.
For instance; Let S1, S2,S3…Sm —————— represent possible results
Let P(Sm) —————————– represent the probability of outcome Sm
And Let V(Sm) —————————– represent the value of the result Sm
Therefore, the expected value of Z = (V (S1)*P (S1)) + (V (S2)*P (S2)) + (V (S3)*P (S3)) + … + (V (Sm)*P (Sm)) ———————- (A)
To properly calculate the expected value, every possible result must be established, and then probabilities and values are then allotted. In reality, numbers cannot be allotted to probabilities or values of any outcome, but for clear understanding, the following scenario is highlighted (Starmer & Chris 2000, p. 159);
The scenario is of a particular person betting a dollar on a reward of 100 dollars if he or she gets a six on a single throw of a six-sided dice. If six do not appear, the person loses his or her dollar.
Critically analyzing the situation, the results are only two; losing or winning the bet. On one hand, if the player loses then the result has a negative value on the dollar and it attains a probability of 5/6. On the other hand, if the player wins then the result has a positive value of 99 dollars ($100 – $ 1- the dollar the player bet) and a probability of 1/6 is assigned.
Therefore, using the derived formula denoted as (A) above the expected value of taking the bet will be:
(-1)(5/6) + (99) (1/6) = 94/6
A rational player might look for another bet with a higher expected utility which his dollar can bet against and forgo the above illustration. This is due to the fact that a bet with a higher reward gives a higher expected utility.
There are two cases of Expected Utility Theory; uncertainty and risk. The cases are well discussed under Bernoulli’s and Von-Neumann Morgenstern Formulations (Anand 1993).
Bernoulli’s formulation
In 1713, Nicolas Bernoulli explained the St. Petersburg paradox which triggered two mathematicians to generate expected utility theory as the answer. Later Daniel Bernoulli- cousin to Nicolas- laid down a suggestion that a mathematical function should be the key to approve the expected value based on probability (p. 89). Therefore, this gives means to factor in risk aversiveness.
Von-Neumann Morgenstern formulation
Von-Neumann and Morgenstern translated again the theory set up by Daniel Bernoulli and came up with four axioms namely; completeness, transitivity, independence and continuity (p. 90).
The above axioms are explained as follows:
- Completeness – this axiom presumes that a person has delineated predilections and that he can choose between two options.
- Transitivity – it presumes that an individual has steady decisions.
- Independence – predilection order between two individuals is maintained even when a third person is incorporated.
- Continuity – it presumes that if an individual prefers 1 to 2 and 2 to 3 then a merge of 1 and 3 should be possible and the individual still acts the same with the merge and option 2.
Therefore, Anand (1993) states that ‘if the axioms are fully contented, then the player or the individual is viewed to be rational and the predilection is signified by a utility function (p. 91).
St. Petersburg paradox – infinite expected value
This paradox states that according to (Sugden 1986, p. 63) ‘a possible recompense from the least probability occasion is infinite. A rational person, therefore, is anticipated to release an infinite sum to take the gamble, as an infinite value is expected.’ Bernoulli came up with a solution as the case above does not occur in real life. Bernoulli stated that according to (Machina 1987, p. 121) ‘utility function, in reality, reflects that the expected utility of the infinite reward should be definite, even when the expected value is unlimited.’
Application in a poker game
In a poker game, a risk-neutral position is the best as it tries to make the most of the expected value. However, diverse approaches are used so as to succeed in any gambling game. For instance, in a poker game, an individual may first be risk aversive at the beginning of the game. As the players reduce in number, an individual becomes risk-neutral and even accepts risky play. The use of different approaches in a single game is due to changes in the expected value and utility within the game (Anand 1993, p. 92).
Critique of Expected Utility Theory
The theory does not substantiate well how choices are made by people due to too much postulation. In most cases, people do not have the right and essential information to make any decision and in reality, there is no definite way of forecasting consequences with any surety. Also, fulfillment level cannot be measured quantitatively if something is consumed or purchased (Anand 1993, p. 93).
To conclude, Expected Utility Theory is a very wide topic and topics like cardinal, marginal and ordinal utility should be studied to expand on the concept of Expected Utility Theory.
References
Anand, P 1993, Foundations of Rational Choice under Risk, Oxford University Press, Oxford.
Granger, CW & Machina, MJ 2006, Structural attribution of observed volatility clustering, Irwin, Sydney.
Machina, M 1987, Choice under uncertainty: problems solved and unsolved.
Starmer, C & Chris, M 2000, Probability and Juxtaposition Effects: An experimental Investigation of the Common Ratio Effect, Wiley, Brisbane.
Sugden, R 1986, New Developments in the theory of choice under uncertainty, Blackwell publishing, Norwich. January.