Choice under Uncertainty

All choices made under some kind of uncertainty. Rather than choosing outcome directly, decision-maker chooses uncertain prospect (or lottery).

Outcomes and Lotteries

We study how a rational consumer should behave in order to maximize his expected utility.

Two basic elements of expected utility theory:

• outcomes: Finite set $C$ of outcomes, with $c_n\in C,n=1,2,\dots ,N$. Outcomes are what the decision maker ultimately cares about. For instance, (win, lose)
• lotteries: the probabilities distribution over outcomes.

Definition A simple lottery $L$ is a list $L=(p_1,\dots ,p_n)$ with $p_n\ge 0$ for all $n$ and $\sum _n{p}_n=1$, where $p_n$ is interpreted as the probability of outcome $c_n$ occurring.

• Denote the set of lotteries as $\mathcal{L}$.
• Consumer does not choose outcomes $c\in C$ directly, but he chooses a lottery. The lotteries are objectively known.

Definition Given $K$ simple lotteries $L_k=(p_1^k,\dots ,p_N^k),k=1,\dots ,K$ and probabilities ${\alpha }_k\ge 0$ with $\sum _k{\alpha }_k=1$, the compound lottery $(L_1,\dots ,L_k;{\alpha }_1,\dots ,{\alpha }_k)$ is the risky alternative that yields the simple lottery $L_k$ with probability ${\alpha }_k$ for $k=1,...,K$.

Expected utility maximization

In expected utility theory, no distinction between simple and compound lotteries: simple lottery and above compound lottery give same distribution over outcomes, so identified with same element of $\mathcal{L}$.

Just like in UMP, assume decision-maker has a rational preference relation $\succsim$ on $\mathcal{L}$.

Preference

Since we know how to model a risky environment, we shall now study how agents make their choice.

In a riskless environment, this is done by assuming the existence of a preference ordering over the possible points of the consumption set.

In a risky setting, we will say that the agents only care about the final results (the outcomes $c\in C$ ) and not about the way those results have been generated (simple lottery, compound lottery,...).

We will therefore assume that the agent has a rational preference (complete, transitive) $\succsim$ over the set of (simple) lotteries $\mathcal{L}$ allowing comparison of any pair of simple lotteries.

Utility

Expected Utility Theory is based on two important axioms.

Definition The preference relation $\succsim$ on the space of simple lotteries $\mathcal{L}$ is continuous if for all $L_1,L_2,L_3\in \mathcal{L}$ with $L_1\succsim L_2\succsim L_3$, there exists $\alpha \in [0,1]$ such that $L_2\sim \alpha L_1+(1-\alpha )L_3$.

Definition (The independence axiom) The preference relation $\succsim$ on the space of simple lotteries $\mathcal{L}$ satisfies the independence axiom if for all $L_1,L_2,L_3\in \mathcal{L}$ and for all $\alpha 2[0;1]$, we have

$$L_1\succsim L_2⟺\alpha L_1+(1-\alpha )L_3\succsim \alpha L_2+(1-\alpha )L_3.$$

Definition (Von-Neumann Morgenstern utility function). The utility function $U:\mathcal{L}\to \mathbb{R}$ has a expected utility form if there is an assignment of numbers $(u_1,\dots ,u_N)$ to the $N$ outcomes (a function $u:C\to \mathbb{R})$ such that for every simple lottery $L=(p_1,\dots ,p_N)\in \mathcal{L}$, we have

$$U(L)=\sum _n{p}_n{u}_n.$$

A utility function $U:\mathcal{L}\to \mathbb{R}$ with the expected utility form is called a Von-Neumann Morgenstern expected utility function.

If preferences over lotteries happen to have an expected utility representation, it’s as if consumer has a “utility function” over outcomes (and chooses among lotteries so as to maximize expected “utility over outcomes”).

The expected utility theorem

Theorem Suppose that the rational preference relation $\succsim$ on the space of simple lotteries L satisfies the continuity and independence axioms. Then $\succsim$ admits a utility representation of the VNM expected utility form. That is, we can assign a number $u_n$ to each outcome $n$ in such a manner that for any two lotteries $L$ and $L^{\prime }$, we have

$$L\succsim L^{\prime }⟺\sum _n{u}_n{p}_n\ge \sum _n{u}_n{p}_n^{\prime }$$

Property of expected utility

Proposition (Linearity in Probabilities). A utility function $U$ has an expected utility form iff it is linear, i.e.,

$$U(\sum _{k=1}^K{\alpha }_k{L}_k)=\sum _{k=1}^K{\alpha }_{k}U(L_k)$$

for any $K$ lotteries $L_k\in \mathcal{L},k=1,\dots ,K$ and probabilities $({\alpha }_1,\dots ,{\alpha }_k)\ge 0,\sum _k{\alpha }_k=1$.

Proposition (Invariant to Affine Transformation). Suppose $U:\mathcal{L}\to \mathbb{R}$ is an expected utility function representing preferences $\succsim$. Then $\tilde{U}:\mathcal{L}\to \mathbb{R}$ is also an expected utility function representing $\succsim ⟺$ there are scalars $\beta >0$ and $\gamma$ such that $\tilde{U}(L)=\beta U(L)+\gamma$ for every $L\in \mathcal{L}$

Money Lotteries

Special case of choice under uncertainty: outcomes are measured in dollars.

Set of outcomes $C$ (or money is a continuous variable x) is subset of $\mathbb{R}$. A lottery is a cumulative distribution function $F$ on $\mathbb{R}$.

Assume preferences have expected utility representation:

$$U(F)=E_F[u(x)]=\int u(x)dF(x).$$

Assume $u(·)$ is the Bernoulli utility function (von Neumann-Morgenstern), increasing and differentiable.

Question: how do properties of the Bernoulli utility function $u$ relate to decision-maker’s attitude toward risk?

Expected value vs. expected utility

Expected value of lottery $F$ is

$$E_F[x]=\int xdF(x)$$

Expected utility of lottery F is

$$E_F[u(x)]=\int u(x)dF(x)$$

Can learn about consumer’s risk attitude by comparing $E_F[u(x)]$ and $u(E_F[x])$.

Risk attitude

A decision-maker is risk-averse if she always prefers the sure wealth level $E_F[x]$ to the lottery $F$: that is,

$$U(F)=\int u(x)dF(x)\le u(\int xdF(x))=U(E_F[x])\text{ for all }F.$$

A decision-maker is strictly risk-averse if the inequality is strict for all non-degenerate lotteries F.

A decision-maker is risk-neutral if she is always indifferent:

$$U(F)=\int u(x)dF(x)=u(\int xdF(x))=U(E_F[x])\text{ for all }F.$$

A decision-maker is risk-loving if she always prefers the lottery:

$$U(F)=\int u(x)dF(x)\ge u(\int xdF(x))=U(E_F[x])\text{ for all }F.$$

Proposition. A decision-maker is (strictly) risk-averse if and only if $u$ is (strictly) concave. A decision-maker is risk-neutral if and only if $u$ is linear. A decision-maker is (strictly) risk-loving if and only if $u$ is (strictly) convex.

Certainty equivalents

Definition (Certainty Equivalents). The certainty equivalent of a lottery $F$ is the sure wealth level that yields the same expected utility as $F$: that is,

$$CE(F,u)=u^{-1}(\int u(x)dF(x))$$

Proposition. A decision-maker is risk-averse $⟺CE(F,u)\le E_F(x)$ for all $F$. A decision-maker is risk-neutral $⟺CE(F,u)=E_F(x)$ for all $F$. A decision-maker is risk-loving $⟺CE(F,u)\ge E_F(x)$ for all $F$.

Measure of risk aversion

Two possibilities:

1. $u_1$ is more risk-averse than $u_2$ if, for every F, $CE(F,u_1)\le CE(F,u_2)$.
2. $u_1$ is more risk-averse than $u_2$ if $u_1$ is “more concave” than $u_2$, in that $u_1=g\circ u_2$ for some increasing, concave $g$.

Definition Given a twice-differentiable Bernoulli utility function $u(·)$ for money, the Arrow-Pratt coefficient of absolute risk aversion at $x$ is defined as $r_A(x)=-u^{″}(x)/u^{\prime }(x)$.

One more, based on local curvature of utility function: $u_1$ is more-risk averse than $u_2$ if, for every $x$,

$$-\frac{u_1^{″}(x)}{u_1^{\prime }(x)}\ge -\frac{u_2^{″}(x)}{u_2^{\prime }(x)}$$

Proposition The following are equivalent:

1. For every $F,CE(F,u_1)\le CE(F,u_2)$.
2. There exists an increasing, concave function $g$ such that $u_1=g\circ u_2$.
3. For every x, $r_A(x,u_1)\ge r_A(x,u_2)$.

Risk Attitude and Wealth Levels

How does risk attitude vary with wealth?

Assumption

Natural to assume that a richer individual is more willing to bear risk: whenever a poorer individual is willing to accept a risky gamble, so is a richer individual.

Captured by decreasing absolute risk-aversion:

Definition A Bernoulli utility function u exhibits decreasing (constant, increasing) absolute risk-aversion if $r_A(x,u)$ is decreasing (constant, increasing) in $x$.

Proposition Suppose $u$ exhibits decreasing absolute risk-aversion. If the decision-maker accepts some gamble at a lower wealth level, she also accepts it at any higher wealth level: that is, for any lottery $F(z)$, if

$$E_F[u(x_1+z)]\ge u(x_1),$$

then, for any $x_2>x_1$,

$$E_{F}u(x_2+z)\ge u(x_2).$$

Relative risk-aversion

What about gambles that multiply wealth, like choosing how risky a stock portfolio to hold? The loss/gain is proportional to current wealth.$\tilde{u}(t)=u(tx)$

Are richer individuals also more willing to bear multiplicative risk?

Depends on increasing/decreasing relative risk-aversion: $r_R(x,u)=-x{u}^{″}/u^{\prime }$.

We have

$$r_R(x)=x{r}_A(x).$$

Proposition Suppose $u$ exhibits decreasing relative risk-aversion. If the decision-maker accepts some multiplicative gamble at a lower wealth level, she also accepts it at any higher wealth level: that is, for any lottery $F(t)$, if

$$E_F[u(t{x}_1)]\ge u(x_1)$$

then, for any $x_2\ge x_1$

$$E_F[u(t{x}_2)]\ge u(x_2)$$