Behavioral Choice Under Risk

Prospect theory is a behavioral alternative to Expected utility theory.

Notice that “prospect” has the same meaning as “lottery”. Under prospect theory, the utility of a prospect $\ell=(\mathbf{x}; \mathbf{p})$ is: $$ U^P(\ell)=\pi(p_1)\cdot \nu(x_1)+\ldots+\pi(p_n)\nu(x_n),$$ where:

• $\nu(x)$ is the utility of obtaining the prize $x$ with certainty;
• $\pi$ is a function that “distorts” probabilities in a special way.

Under Prospect theory, $$ U^P(\ell)=\pi(p_1)\cdot \nu(x_1)+\ldots+\pi(p_n)\nu(x_n),$$ we assume the following:

• $\nu$ is continuous, increasing, and for some reference prize ($r$), $\nu^{\prime\prime}(x)\gt 0$ for $x\lt r$ (risk-loving behavior for losses) and $\nu^{\prime\prime}(x)\lt 0$ for $x\gt r$ (risk-averse behavior for gains).
• For small $p$, $\pi(p)\gt p$ and $\pi(p)+\pi(1-p)\le 1$. That is, $\pi$ overestimates the probability of unlikely events and underestimates the probability of likely events.

Under Prospect theory, $$ U^P(\ell)=\pi(p_1)\cdot \nu(x_1)+\ldots+\pi(p_n)\nu(x_n),$$ in particular, the following functions are commonly used. They satisfy all the assumptions in the previous slide. $$\nu(x)=\begin{cases}(x-r)^\alpha&\text{if } x\ge r,\\ -\lambda\cdot (r-x)^\alpha&\text{if } x\le r.\end{cases}$$ $$\pi(p)=\dfrac{p^\gamma}{\bigl(p^\gamma+(1-p)^\gamma\bigr)^{1/\gamma}},$$ in which $0\lt \alpha,\gamma\lt 1$ and $\lambda\gt 1$. With the same exponent $\alpha$ on both sides of $r$, the condition $\lambda\gt 1$ is exactly loss aversion, $-\nu(r-x)=\lambda\,x^{\alpha}\gt x^{\alpha}=\nu(r+x)$ for every $x\gt 0$, and $\gamma\lt 1$ makes low probabilities overestimated and high probabilities underestimated.

For instance, with reference point $r=0$, $\alpha=1/2$ and $\lambda=2$: $$\nu(x)=\begin{cases}\sqrt{x}&\text{if } x\ge 0,\\[2pt] -2\sqrt{-x}&\text{if } x\lt 0.\end{cases}$$ The value function is concave over gains ($x\gt 0$, risk-averse) and convex over losses ($x\lt 0$, risk-loving). It is also steeper over losses: a loss of $x$ hurts more than a gain of $x$ pleases, $$-\nu(-x)\gt\nu(x)\qquad\text{(loss aversion).}$$

Under Prospect Theory, the utility for money is referred to as the valuation. As seen before, the utility displays:

• loss aversion;
• risk-loving behavior in the loss domain;
• risk-averse behavior in the gain domain.

Prospect Theory weights events using decision weights $\pi(p)$ instead of probabilities $p$. We saw that decision weights:

• underestimate high probability events and overestimate low probability events;
• $\pi(p)\gt p$ for $p\lt \hat p$;
• $\pi(p)+\pi(1-p)\le 1$;
• $\pi(p)=p$ only for $p=0$, $\hat p$ and $1$.

Notice that $\hat p(\gamma)$ is given.

Consider the following experiment: an urn has 90 balls; 30 balls are red and each of the remaining 60 balls is either blue or green, but we do not know the fraction of green (or blue) balls in the urn. One ball is randomly selected from the urn.

There are four lotteries:

lottery	prize	event
R	$\$100$	the ball is red
B	$\$100$	the ball is blue
RG	$\$100$	the ball is either red or green
BG	$\$100$	the ball is either blue or green

lottery	prize	event
R	$\$100$	red
B	$\$100$	blue
RG	$\$100$	red or green
BG	$\$100$	blue or green

Max Min Expected Utility Theory is an alternative theory to Expected Utility Theory that allows ambiguity aversion.

To use max min utility theory:

1. Consider an alternative (lottery).
2. For each possible probability distribution: compute the expected utility of the alternative.
3. The minimum of all expected utilities in (2) is the value of the alternative.
4. Choose the alternative that has the highest value.

Example:

• A coin is either fair or both sides display “heads”.
• There are two possible probability distributions: $p=1/2$ or $p=1$ where $p$ is the probability of “heads”.
• Lottery #1 (L1) pays $\$500$ if “tails”.
• Lottery #2 (L2) pays $\$100$ if “heads”.

$$ U(L1)=\min_{p\in \{ \frac12,1\} } p\,u(0)+(1-p)u(500)=u(0)$$ $$ U(L2)=\min_{p\in \{ \frac12,1\} } p\,u(100)+(1-p)u(0)=\tfrac12 u(100)+\tfrac12 u(0)$$

So lottery #2 is always better for someone who is ambiguity averse and uses the max min rule.