Definitions

Expected utility hypothesis

In economics, game theory, and decision theory the expected utility theorem or expected utility hypothesis predicts that the "betting preferences" of people with regard to uncertain outcomes (gambles) can be described by a mathematical relation which takes into account the size of a payout (whether in money or other goods), the probability of occurrence, risk aversion, and the different utility of the same payout to people with different assets or personal preferences. It is a more sophisticated theory than simply predicting that choices will be made based on expected value (which takes into account only the size of the payout and the probability of occurrence).

Daniel Bernoulli described the complete theory in 1738. John von Neumann and Oskar Morgenstern reinterpreted and presented an axiomatization of the same theory in 1944. They proved that any "normal" preference relation over a finite set of states can be written as an expected utility, sometimes referred to as von Neumann-Morgenstern utility.

A very simple economic theory is that outcomes with a higher expected value are always preferred. For example, the expected value of getting a \$100 payment with a 1 in 80 chance is \$1.25. Given the choice between this gamble and a guaranteed payment of \$1, by this simple theory, people should choose the \$100 gamble.

A key insight is that people do not consider real assets and "expected" (in the probabilistic sense) assets to be equivalent. In the example, no one will ever end up with a payment of \$1.25; real payments are made only in the amounts of \$0, \$1, and \$100. (This difference is minimized when such a "lottery" is repeated many times, due to the law of large numbers.)

Bernoulli's formulation

Nicolas Bernoulli described the St. Petersburg paradox (involving infinite expected values) in 1713, prompting two Swiss mathematicians to develop expected utility theory as a solution. The theory can also more accurately describe more realistic scenarios (where expected values are finite) than expected value alone.

In 1728, Gabriel Cramer, in a letter to Nicolas Bernoulli, wrote, "the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it.

In 1738, Nicolas' cousin Daniel Bernoulli, published the canonical 18th Century description of this solution in Specimen theoriae novae de mensura sortis or Exposition of a New Theory on the Measurement of Risk.

Daniel Bernoulli proposed that a mathematical function should be used to correct the expected value depending on probability. This provides a way to account for risk aversion, where the risk premium is higher for low-probability events than the difference between the payout level of a particular outcome and its expected value.

Bernoulli's paper was the first formalization of marginal utility, which has broad application in economics in addition to expected utility theory. He used this concept to formalize the idea that the same amount of additional money was less useful to an already-wealthy person than it would be to a poor person.

von-Neumann Morgenstern formulation

There are four axioms of the expected utility theory that define a rational decision maker. They are completeness, transitivity, independence and continuity. Completeness assumes that an individual has well defined preferences and can decide between two alternatives.

Axiom (Completeness): For every A and B either (this means: A is worse than B, better, or equally good) holds.

Transitivity assumes that, as an individual decides according to the completeness axiom, the individual also decides consistently.

Axiom (Transitivity): For every A, B and C with $A>B$ and $B>C$ we must have $A>C$.

Independence also pertains to well-defined preferences and assumes that the preference order of two gambles mixed with a third one maintains the same preference order as when the two are mixed independently.

Axiom (Independence): Let A and B be two lotteries with $A > B$, and let $t in \left[0, 1\right]$ then $tA+\left(1-t\right)C>t B+\left(1-t\right)C$ .

Continuity assumes that when there are three lotteries (A, B and C) and the individual prefers A to B and B to C, then there should be a possible combination of A and C in which the individual is then indifferent between this mix and the lottery B.

Axiom (Continuity): Let A, B and C be lotteries with $A>B>C$ then there exists a probability p such that B is equally good as $pA+\left(1-p\right)C$.

If all these axioms are satisfied, then the individual is said to be rational and the preferences can be represented by a utility function. In other words: if an individual always chooses his/her most preferred alternative available, then the individual will choose one gamble over another if and only if the expected utility of one exceeds the other; thereby maximizing his/her utility. The utility of any gamble may be expressed as a linear combination involving only the utility of the outcomes and their respective probabilites. Utility functions are also normally continuous functions. Such utility functions are also referred to as von Neumann-Morgenstern (VNM) utility functions. This is a central theme of the expected utility in which an individual chooses not the highest expected value, but rather the highest expected utility. The expected utility individual makes decisions rationally based on the axioms of the theory.

The expected utility theory generally accepts the assumption that individuals are risk averse, meaning that the individual would refuse a fair gamble (a fair gamble has an expected value of zero), and also implying that their utility functions are concave and show diminishing marginal wealth utility. The risk attitude is directly related to the curvature of the utility function: risk neutral individuals have linear utility functions, while risk seeking individuals have convex and risk averse have concave utility functions. The degree of risk aversion can be measured by the curvature of the utility function.

Since the risk attitudes are unchanged under affine transformations of u, this has to be normalized by u'. This leads to the definition of the Arrow-Pratt measure of absolute risk aversion:

$ARA\left(x\right) =-frac\left\{u\text{'}\text{'}\left(x\right)\right\}\left\{u\text{'}\left(x\right)\right\}$

Special classes of utility functions are the CRRA (constant relative risk aversion) functions, where ARA(x)/x is constant, and the CARA (constant absolute risk aversion) functions, where ARA is constant. They are often used in economics for simplification purposes.

Infinite expected value — St. Petersburg paradox

The St. Petersburg paradox (named after the journal in which Bernoulli's paper was published) arises when the potential reward from a very low probability event is infinite. Because of the infinite expected value, a rational person would be expected to pay an infinite amount to take this gamble. In real life, people do not do this.

Bernoulli proposed a solution to this paradox in his paper: the utility function used in real life means that the expected utility of the infinite payout is finite, even if its expected value is infinite. (Hypothesizing diminishing marginal utility of increasingly larger amounts of money.) It has also been resolved differently by other economists by proposing that very low probability events are neglected, by taking into account the finite resources of the participants, or by noting that one simply cannot buy that which is not sold (and that sellers would not produce a lottery whose expected loss to them were unacceptable).

The von-Neumann Morgenstern formulation is important in the application of set theory to economics because it was developed shortly after the Hicks-Allen "ordinal revolution" of the 1930s, and it revived the idea of cardinal utility in economic theory.

Application in poker

The concept of risk-aversion comes into play in many gambling scenarios, such as poker strategy. A risk-neutral stance is generally the best strategy in most situations, as it attempts to maximize the expected value of each bet. However, there are situations where different strategies will be more beneficial. For example, many experts advocate a risk-averse strategy in the early stages of a poker tournament, when there are still many players left. As the tournament advances, a more risk-neutral or even risk-acceptant strategy becomes the more optimal play. This change in strategy is due to the difference between expected value and expected utility in tournament poker. If a player has only a small number of chips remaining, they should begin to make larger and more frequent bets, and consequently take on more risk, because this is the only approach that gives them a chance to quickly amass a large number of chips, which will be necessary in order to have success in the tournament. A risk-averse strategy may appear to be a good way of preserving a player's remaining chips, but due to the rising blinds, this approach generally decreases the player's chances of finishing "in the money" for the tournament. See M-ratio for more information on this concept as it relates to poker theory.

Reality vs. theory

The expected utility model fails to provide a good description of how people make choices in many circumstances because it assumes too much; humans rarely, if ever, have all the information necessary to make a decision. In many cases there is no real way to foresee consequences with any certainty.

Irrational deviations

Behavioral finance has produced several generalized expected utility theories to account for instances where people's choice deviate from those predicted by expected utility theory. These deviations are described as "irrational" because they can depend on the way the problem is presented, not on the actual costs, rewards, or probabilities involved.

Particular theories include prospect theory, rank-dependent expected utility and cumulative prospect theory and SP/A theory

Preference reversals over uncertain outcomes

Starting with studies such as Lichtenstein & Slovic (1971), it was discovered that subjects sometimes exhibit signs of preference reversals with regards to their certainty equivalents of different lotteries. Specifically, when eliciting certainty equivalents, subjects tend to value "p bets" (lotteries with a high chance of winning a low prize) lower than "\$ bets" (lotteries with a small chance of winning a large prize). When subjects are asked which lotteries they prefer in direct comparison, however, they frequently prefer the "p bets" over "\$ bets." Many studies have examined this "preference reversal," from both an experimental (e.g., Plott & Grether, 1979) and theoretical (e.g., Holt, 1986) standpoint, indicating that this behavior can be brought into accordance with neoclassical economic theory under certain assumptions.