Expected Utility Theory - Essay Example

St. Petersburg paradox is a form of paradox mostly applied in economic and it is closely related to probability and decision theory. Its basis can be derived from a particular lottery game with a lead to a random variable with infinite expected payoffs. This paradox is a form of naïve criteria of making decisions with a bias towards only the expected values. Using this it helps predict a course of a course action that presumably no actual person in his or her right senses would be willing to take1. An immediate example is that St. Petersburg is normally played by flipping a fair coin until it comes up tail. In such a case, the total number of flips is represented by, n and this is what determines the prize, which is represented by $4n. Therefore, if the coin is tossed and it comes up tail the first instance, the prize will be determined by $41and the game ends here. On the other hand, if the coin is flipped and the head comes up the first time, it is flipped again and if the tails comes up the second time, the prize becomes $42, which is equal to $162. The prize of such a toss depend on the outcome which is based on how many times the coin is tossed for the tail to appear. From the previous outcomes it is evident that there are an infinite number of possible ‘consequences’ i.e. a given number of run for heads before one tail appears and this is what is used to determine the outcome. Hence, the probability of an occurrence given n number of flips P (n) is given by 1 divided by 2 while the expected payoff will therefore be the probability times the prize. For instance, if n is 1 it means the probability is ½ and given that the prize at this point is $4, therefore the expected outcome will be $2. This is the basis that individuals choosing between lotteries; will depend on in making their choice. Moreover, an individual will normally go for the highest expected value2. Expected utility theory (EUT) has the proposition that any given decision maker (DM) normally has to make a choice between risky and uncertain occurrences and they do this by making a comparison of the expected utility values. The expected values are given by the adding the utility values to obtain the weighted sums and this will then be multiplied by the respective probabilities. This theory has a relationship with St. Petersburg since it is an advancement St. Petersburg1. In St. Petersburg, individuals were questioned of the much they would be willing to pay if a coin is tossed and the talk comes out first of a fair coin, to receive nothing and stop the game. On the contrary, the individuals were given a choice to either receive four guilders and then allowed to stay in the game or the former. As seen above the expected monetary value would be a sum of Xn * 1/Xn and such observations should be to infinity. Research has it those individuals normally base expected monetary value at the lowest based on the St. Petersburg prospect. This creates the relationship with the EUT where Bernoulli argued that this should be based on the utility of money outcomes. In this argument, the log function get an applause since it has the property of quickly decreasing the marginal utilities2. The function is hence represented by Log Xn *1/Xn which is convergent and help in solving the problem created by the paradox. This is the first occurrence of the EUT theory, which was as well later questioned based on the naturalness of the choices that it presents. The difference that the utility theory comes in handy is the fact that it advocated a creation of a clear difference in utility between desirability and satisfaction of the payoff in terms of the monetary value. For example; given n outcomes represented by 2, probability P (n) will be ¼ giving a prize with a quantity of four. The utile in terms of prize will as well be 4 and this would result into an Expected utility of 1, this will continue in this manner at a decreasing rate3. It is a significant challenge to account for gambling and insurance under the theories of decision-making under uncertainty most of if the variables in question keep on changing. The introduction of the no concave segments in the utility function for the sake of using the expected utility theory in explaining gambling and insurance for decisions under uncertain is highly recommended3. This is the concept as first explained by Friedman by using a utility function with a single convex and ended up justifying the shape. The outcome was that the utility function included a section with an increasing marginal utility function. This explained the existence of consumers with the appetite for both insurance and lottery tickets. Therefore, the Expected Utility Theory (EUT) with a nonconcave utility function, that has the capabilities of explaining a behavior under uncertainty. In addition, it can explain gambling and insurance though Martin has argued against this approach. Martin argues that the function is not capable of explaining in their nonconcave utility form uncertainty in principle. The explanation behind this considers Friedman savage utility function of which either is preferred in giving an explanation to gambling. The setbacks being that the pattern for saving or borrowing is only applicable if the income is chosen in the right manner2. It was also not captured in that an agent may at times have three options, ether to borrow, save, or gamble. With gambling in the picture together with saving and borrowing, a demand for gambling would take preference even in the case where the two suffice. This is because gambling has the impact of instantly offering a direct consumption value. The assumptions here are first, the utility of non-monetary activities that are closely related to gambling2. The other assumption creates a modification of the EUT theory by creating and assumption that money values as well as the probabilities in any risky situation have far much more direct value beyond that which is included in the expression for the expected utility. This was as explained by Conlisk (1993) who presented an explanation that an addition of minute functions of money values and probabilities to a concave utility function; an explanation of a risk preferring behavior will be exhibited. The most common example of such a behavior is the purchase of a lottery ticket which s otherwise gambling3. Derived above, we find that Friedman Utility function gives an explanation that the curvature of this utility function will always depend on the amount of wealth that an individual posses. According to the function, the curvature indicates that the more wealth an individual has the more risk loving they become. This makes the highly wealthy be prone to playing the lottery while on the other hand the poorer ones would prefer to buy insurance given they are risk averse3. This explains the origin of gambling in economic aspects; the overstatements of low probabilities and the understatement of the high probabilities was found to be the main cause of the existence of gambling and insurance. Therefore, the utility function that an individual chooses to operate with today should be more sensitive to the future demands and existence of the investment opportunities as they perceive. This is based on the maximization that an individual will affix on their expected utility from their consumption as compared to his remaining life5. Individuals have the chance of making choices among a variety of alternatives depending on the degree of risk they are likely to be exposed to. This difference is clearly given in the case of insurance and gambling. A person has the choice of taking insurance on a house they own; in such a case, they are accepting the loss of a small sum in form of a premium in preference of a large loss in form of the value of the house and a large chance of loss4. This is otherwise referred to as a choice between certainty as opposed to uncertainty. On the other hand, another individual would choose to buy lottery and subject themselves to a chance of a large loss of a small sum and a small chance of winning a large amount. This is in preference of avoiding both these risks. This is called the choice of uncertainty over certainty. The choice among different degrees of risks is quite evident in insurance and gambling is much applicable in broader aspects of economics and economic choices. This is the basis upon which, the more wealthy individuals will decide to buy lottery tickets. On the other hand, the less wealthy will choose to purchase insurance policy3. The common ratio effect is an example of classical systematic violations of the expected utility theory. In the real world individuals normally have the chance of either deciding to get a sure money payoff and a two outcome lottery that results into higher outcomes with a probability of more than half. The sure amount of money is normally selected closer to below the expected value of the lottery so that the majority of the people would have no choice but to pick on the sure amount as opposed to the risky lottery5. The expected utility theory makes an implication that the people who chose sure payoff in the first decision should also choose a safer lottery while making the second decision. Or if they made a choice of a risky lottery in their first decision they should as well make a choice of a riskier lottery in their second decision2. It doe not normally happen like that though since individuals usually prefer a monetary payoff in the first decision and a riskier lottery in the second choice. This is normally with the exception of only a few people whom reveal the opposite tendency. This is what is called the common ratio effect. Under the common ratio effect, an individual who chooses the sure alternative and hence neglects the risky lottery are likely to repeat the same pattern of choice in their second choice1. Those who make a choice of the sure alternative in the scaled up problem will either choose a safer lottery in theory scaled down choice. In common ratio effect perspective, such individuals will choose a riskier alternative. Most of the times the Common ratio effect has been confused to be the same as the Allais Paradox. This is not true as the two are quite distinct; this is because Allais paradox is concerned with making choices between options in two completely independent situations2. The common ration effect on the other hand would involve making choices among related parameters. The Allais paradox has two pairs of choices and each of the outcomes have two other alternative prospects. Expected utility theory has never and can never be made consistent with the observed patterns and this is irrespective of whether a utility function is used. This is because the EUT possesses other approaches and can be presented in other different and unique models. This weakens the independence axiom of the EUT theory. Such other models are referred to as the rank dependent models. Since the well-known paradox of Allais, however, a large body of experimental evidence has been gathered which indicates that individuals tend to violate the assumptions underlying the EUT theory? This empirical evidence has acted as a motivator to the researchers to develop alternative theories of choice under risk and uncertainty. Such alternative theories are able to accommodate the observed patterns of behavior. They are usually termed "non-expected utility" or "generalizations of expected utility.” Another reason for this restriction is the fact that most of the important generalizations of subjective expected utility have an analogous counterpart for choice under risk3. References 1. Quiggin J.P. and Wakker P. (2004). The Axiomatic Basis of Anticipated Utility: A Clarification. Journal of Economic Theory, 64, 486-499. [Correction of Quiggin (1982).] 2. Segal U (1999). Anticipated Utility: A Measure Representation Approach. Annals of Operations Research, 19, 359-373. [Axiomatization of the general rank-dependent model. 3. Segal U. and Spivak A. (2000). First Order versus Second Order Risk Aversion. Journal of Economic Theory, 51, 111-125. [Seminal work on first-order risk aversion.] 4. Tversky A. and Kahneman D. (1992). Advances in Prospect Theory: Cumulative Representations of Uncertainty. Journal of Risk and Uncertainty, 5, 297-323. [Axiomatization of cumulative prospect theory.] 5. Viscusi W.K (2009). Prospective Reference Theory: Toward an Explanation of the Paradoxes, Journal of Risk and Uncertainty 2, 235-264. [Axiomatization of prospective reference theory Read More