C. Hinsley

27 February 2022

Last edited 6 April 2022


Abstract

Expected utility theory extends the notion of outcome preferences in game theory to probabilistic games with uncertain outcomes. Even if two scenarios (“lotteries”) present equal expected utility, a player’s attitude towards risk may result in one scenario being preferred to another. The Arrow-Pratt measure of absolute risk aversion fully characterizes the attitude of a player towards risk given reasonable assumptions about the player’s utility function. This characterization is discussed in detail. Additionally, we discuss for which pairs of lotteries there exists an Arrow-Pratt measure producing a preference relation which is indifferent as to which of the two lotteries is chosen, generalizing a 1969 result.

Real lotteries

In game theory, we often deal with games of a single player, having potential outcomes that can be modeled by real random variables. A classic example is a slot machine in a casino; the change to our balance (i.e., “bankroll”) after a single pull of the slot arm is a number that may be positive, negative, or zero. Different slot machines may have different costs per pull and different probability distributions for their outcomes. As such, the potential change to our balance from a single game is a real random variable which we will refer to as a “lottery;” conventionally, a particular slot machine will be associated with a single fixed lottery. If a slot machine costs five dollars to play and has a potential payout of either zero or twenty dollars, with a twenty-percent chance of getting the second outcome, the associated lottery may be written

$$ L = \left[\frac45 (-\$5), \frac15 (\$15)\right]. \tag{1} $$

In general, a lottery may take the form of any random variable with real-valued outcomes. As this can be a discrete, continuous, or even mixed random variable, we write a general lottery as

$$ L = \left[ p_L(x) (\$x) \right]_{x \in \R} \tag{2} $$

where $p_L$ is the mixed probability density for the lottery. We denote by $P_L(x)$ the cumulative distribution (in the case of a continuous distribution,$\int_{-\infty}^x p_L(y)\ dy$). The space of all such lotteries is denoted $\mathscr{L}$.

Preference relations & utility functions

A player may prefer one lottery over another. For instance, someone who is afraid of risk might prefer to play a slot machine associated with the lottery $\left[ 1 (\$10) \right]$ rather than another slot machine associated with $\left[ \frac12 (\$5), \frac12 (\$20)\right]$ (and from now on, instead of writing “play a slot machine associated with the lottery $L$,” we’ll write “play the lottery $L$”). In this case, we write $[1(\$10)] \succcurlyeq_i \left[\frac12(\$5), \frac12(\$20)\right]$, where the symbol $\succcurlyeq_i$ is read “is preferred by player $i$ over;” in particular, we are referring to the player as if that player is a member of an indexed set of all players, each of which may have different preferences for lotteries. We call the ordering relation on $\mathscr{L}$ associated with a particular player $i$ a preference relation; we will be concerned only with the case where this preference relation is a total order on $\mathscr{L}$.

It will be useful for our purposes to be able to deal with preference relations in a quantitative fashion. We can express any player’s preferences over the set of deterministic lotteries, $\mathscr{O}$, as a function $u_i: \mathscr{O} \to \R$ which admits an overloaded function $u_i: \R \to \R$ so that $u_i(x) = u_i\left([1(\$x)]\right)$ for any $x \in \R$. To make this overloaded $u_i$ correspond in some way with player $i$’s preference relation on $\mathscr{O}$, we say that given any two deterministic lotteries $L_1 := [1(\$x)], L_2 := [1(\$y)]$, we will have $L_1 \succ_i L_2$ if and only if $u_i(L_1) > u_i(L_2)$ and $L_1 \approx_i L_2$ (player $i$ is indifferent as to whether $L_1$ or $L_2$ is played) if and only if $u_i(L_1) = u_i(L_2)$. Of course, we can assume any rational player would prefer to make more money than less, so $u_i$ is a strictly increasing function on $\R$. We call $u_i$ the utility function of the player $i$.

Given a utility function $u_i$ we may extend to a utility function over the space of lotteries $u_i: \mathscr{L} \to \R$ by assuming linearity. That is, given any lottery $L$ we may write

$$ u_i(L) = \int_\R u_i(x) p_L(x)\ dx. \tag{3} $$

This linearity criterion turns out not to be too great an assumption in many cases. Treating $L$ as a mixed random variable, this integral coincides with the utility on deterministic lotteries $u_i \mid \mathscr{O}$. We maintain the same correspondence between $u_i$ and $\succcurlyeq_i$ that we had used for the utility on deterministic lotteries: for any pair of lotteries $L_1, L_2 \in \mathscr{L}$ we have $L_1 \succ_i L_2$ if and only if $u_i(L_1) > u_i(L_2)$ and $L_1 \approx_i L_2$ if and only if $u_i(L_1) = u_i(L_2)$.

Risk attitudes and utility curvature