# Lottery (probability)

In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature. The elements of a lottery correspond to the probabilities that each of the states of nature will occur.[1] Much of the theoretical analysis of choice under uncertainty involves characterizing the available choices in terms of lotteries.

In economics, individuals are assumed to rank lotteries according to a rational system of preferences, although it is now accepted that people make irrational choices systematically. Behavioral economics studies what happens in markets in which some of the agents display human complications and limitations.[2]

## Choice under risk

According to expected utility theory, people choose among risky alternatives or scenarios using a criterion that combines three features: the subjective estimate of the probabilities of the possible outcomes, the gambling preferences, and the ranking of prizes and penalties. The combination of the last two features is made through a utility attached to them by a utility function. Then this abstract measure is combined with the subjective probabilities through a linear combination of the utilities. The weights in this linear combination are the subjective probabilities.[3]

For example, let there be three states of nature, "beautiful and eventful trip by car", "staying home", and "death by car accident". Their consequences and utility values for a particular individual are:

• Beautiful and eventful trip by car: 16 utils
• Staying home: 9 utils
• Death by car accident: 4 utils

If the person had to choose the best of two scenarios A and B, each of which assigns probabilities to the states of nature, how would they do it? A theory of choice under risk starts by letting people have preferences on the set of lotteries over these kind of states of nature. If preferences over lotteries are complete and transitive, they are called rational.

As a result of computing the expected utility from scenarios A and B, rational people would pick the one with the highest expected utility. Rankings of alternatives made under uncertainty can be represented by cardinal utility, but they are not ordinal.

The assumption about combining linearly the individual utilities and making the resulting number be the criterion to be maximized can be justified of the grounds of the independence axiom. Therefore, the validity of expected utility theory depends on the empirical validity of the independence axiom. The preference relation ${\displaystyle \succsim \!}$ satisfies independence if for any three simple lotteries p, q, r, and any number ${\displaystyle \alpha }$ E(0,1) it holds that

${\displaystyle p\succsim \!q}$ if and only if ${\displaystyle \alpha p+(1-\alpha )r\succsim \!\alpha q+(1-\alpha )r.}$

Indifference maps can be represented in the simplex.

## References

1. ^ Mas-Colell, Andreu, Michael Whinston and Jerry Green (1995). Microeconomic theory. Oxford: Oxford University Press. ISBN 0-19-507340-1
2. ^ Mullainathan, Sendhil., & Thaler, Richard. (2000). 'Behavioral Economics'. NBER Working Paper No. 7948, p. 2.
3. ^ Archibald, G (1959). "Utility, risk, and linearity". Journal of Political Economy. 67 (5): 438. doi:10.1086/258216.