Decision making under
uncertainty

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).

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_{i}p_{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_{i}p_{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.@