Decision making under uncertainty

Lotteries

Varian notation to describe a prospect (also called a lottery or an asset)

pBx Å (1-p)By is the notation used by Varian

Other common notation

Common notation used in other sources lists the vector of possible outcomes or prizes followed by vector of probabilities--(x,y;p,(1-p)), (x,x',x'';p,p',p'') or, more generally, (x;f(x)) where x is a vector and its elements are the support of the probability distribution f(x).

Some aspects of prospect or lottery

Note 1: When uncertainty is resolved, the owner wins a bundle of goods and services or an amount of money available at a specified date. Most frequently the prizes are in money terms.

Note 2: The concept is defined to include the certainty of realizing x, which in the Varian notation would be written as 1 Bx Å 0By ((x,y;1,0) in the other notation).

Note 3: Complex lotteries are identical in consumer's mind to the lottery of the net probabilities on outcomes. For example, consider the lottery with a probability of .8 of obtaining the lottery A and .2 of obtaining the lottery B, where A is .6B10 Å .4B1000 and B is .5B10 Å .5B1000. The consumer considers this complex lottery to be the same as the lottery .58B10 Å .42B1000.

Note 4: Consider a situation in which prizes are money income. With a set of “reasonable” assumptions about the preference ordering of an individual over lotteries one can break down any lottery into a derived lottery over two outcomes, the worst outcome possible, x_w, and the best outcome possible, x_b. Giving the worst outcome with certainty a utility level of 0 and the best a utility level of 1 and assuming continuity, there will be a p between 0 and 1 for which the lottery pBx_b Å (1-p)Bx_w is indifferent to x with certainty where x lies in the range between x_w and x_b. The probability p provides an index of the utility of all prizes x, with x_w ≤ x ≤ x_b. A lottery with any prize structure can then be reconstructed as a lottery of lotteries with x_w and x_b as prizes and simplified to p*Bx_b Å (1-p*)Bx_w, which will have a utility measure of p*.

Note 5: The probabilities p can be “objective,” as in a gambling game based on fair die, or subjective, created by the individual decision-maker. If the probabilities are subjective, the probabilities and a utility function are jointly ascribed to a set of lottery prizes (each of which may be a lottery with objective probabilities) in order to represent the preferences of an individual by expected utility. The subjective approach dates from a paper by a young scholar, Frank Ramsey, in the 1920s. Ramsey died at the age of 26. Leonard Savage developed the thesis comprehensively in The Foundations of Statistics (1954).

Assumptions about preferences over lotteries (see Ch 11 or class notes)

Assumption set defined by Varian (See Ch 11)

Theorem 1: Existence of an expected utility representation of preferences

Theorem 2: Uniqueness up to an affine transformation of this expected utility representation of preferences

Note that expected utility is linear in the probabilities

Risk aversion

Definition:

An individual with an expected utility function u(x) defined over income, is risk averse if for any prospect (x, f(x)), u(E(x)) > E(u(x)). An individual for whom the opposite equality holds is a risk lover. If for any prospect (x, f(x)), u(E(x)) = E(u(x)), the individual is risk neutral.

Certainty equivalent (CE) and a risk premium

The certainty income is the income received for certain that has the same utility as a risky prospect. u(CE) = E(u(x)). A comparison of the certainty equivalent and the expected income gives a measure of risk called the risk premium, defined in this context as RP = E(x) – CE or implicitly by the relationships u(CE) = u(E(x)-RP) = E(u(x)).

Concavity, convexity and the attitude to risk

Strict concavity of the utility function implies u(p₁x₁ + p₂x₂) > p₁u(x₁)+p₂u(x₂) and therefore implies strict risk aversion. Strict convexity implies risk loving. If the utility function is linear, the individual is risk neutral.

Diagrammatic representations of expected utility and attitude to risk

Risk measures

Arrow-Pratt measure of absolute risk aversion

The Arrow-Pratt measure of absolute risk aversion is r(w) = -(u''(w))/u'(w)

Arrow-Pratt measure of relative risk aversion is r (w)=-(wu''(w))/u'(w).

Global risk aversion

Pratt's theorem

The following are all equivalent and signal global risk aversion

1. Utility function of A is more risk averse than that of B if r^A(w)> r^B(w) for all w.

2. Utility function of A is more risk averse than that of B if u^A(w)=G(u^B(w)) where G is a strictly increasing concave function.

3. RP_A(e)>RP_B(e) for all w, where RP_A(e) and RP_B(e) are the maximum amounts that A and B are willing to give up to avoid the random variable e (with mean 0). For example, RP_A(e) is defined by u^A(w- RP_A(e))=E(u^A(w+e)).

Variance as a risk measure

Weaknesses

Lafont illustrative problem (See problem 7 in class handout)

Uncertainty paradoxes

"As the evidence against the model mounts, this has lead to a growing tension between those who view economic analysis as the description and prediction of what they consider to be rational behavior and those who view it as the description and prediction of observed behavior." Machina, 1987, 127. Other researchers consider that the evidence, which has largely been generated from experiments, is not convincing because the incentives to come up with a rationally best answer in the experiments is very low. (See, e.g., Harrison, 1989)

Prescription of Eu defining rationality v EU being a positive theory of behavior.

Evidence

There has been considerable experimental work done to test the “rationality” of decision makers under uncertainty that has resulted in questioning the predictive behavior of expected utility theory. In addition, some have questioned the “rationality” of financial markets and some gaming structures. The following comments on betting on horse racing are drawn from Richard H. Thaler and William T. Ziemba, “Parimutuel Betting Markets: Racetracks and Lotteries,” Journal of Economic Perspectives, 4,2: 161-74 (T and Z).

Note: Parimutuel means that each event has a separate pool which is divided among winning ticket holders after taking off taxes and the track=s percentage.

If one interprets the proportion of the win pool that is accounted for by wagers on each horse as the subjective probabilities of winning, the correlation between these probabilities and the objective ones is high. However, stricter efficiency criteria, that no bet have a positive expected value and that all bets have the same expected value are not met.

Summarizing previously published studies T and Z report that the expected return falls as the odds on the horse get longer. It exceeds 1 for odds of less than 3-10. The punster receives a fair odds share of the distributed part of the pool, when the odds are 9-2. The curve becomes steeper as higher odds are approached, so that high odds bets are very poor ones.

Daily doubles, picking the winners of two races, should have the same odds as a parlay, where a bettor bets to win on one race and places all the winnings, if he or she wins, on a horse in the next race. That is p_ij = p_ip_j.This relationship is not observed.

In an exacta bet (picking first and second in order), the formula (called the Harville formula) is that p_ip_j/(1-p_i). This formula can be used to calculate the implicit odds of a place or show bet. Significant returns are possible in the place and show markets using this formula according to Ziemba and Hausch.

Does late money reveal inside information? Asch and Quandt test this hypothesis and find supportive evidence. They use a logit model to estimate the probability of winning with the regressor being the change in odds during the last few minutes of betting. Found profitable bets on the show and winning side using this approach.

Lack of arbitrage between tracks: A... in the 1986 Kentucky Derby, the winner, Ferdinand, paid $16.80 for $2 at Hollywood park in California where he had run often and was well known. He paid $37.40 at Aqueduct in New York, $79.60 at Woodbine in Toronto, $63.20 at Hialeah in Florida, and $90.00 at Evangeline in Louisiana.@