19  Prospect theory examples

19.1 A 50:50 gamble

Suppose an agent has the following reference-dependent value function:

v(x)=\left\{\begin{matrix} x^\frac{1}{2} \qquad &\textrm{where} \space x \geq 0\\[6pt] -2(-x)^\frac{1}{2} \quad &\textrm{where} \space x < 0 \end{matrix}\right.

Where x is the realised outcome relative to the reference point.

Assume that the agent’s reference point is the status quo and that they weight outcomes linearly.

The agent is offered the gamble A:

(0.5, \$110; 0.5, −\$100)

19.1.1 Accept or reject

Will they want to play this gamble?

The weighted value of the gamble is:

\begin{align*} V(A)&=p_1v(x_1)+p_2v(x_2) \\[6pt] &=0.5\times v(110)+0.5\times v(-100) \\[6pt] &=0.5\times (110)^\frac{1}{2}-0.5\times 2\times (100)^\frac{1}{2} \\[6pt] &=-4.76 \end{align*}

They will not want to play this gamble as it has a negative value for the agent. They could receive a weighted value of 0 by simply not playing.

The reason for this negative value is that the agent is loss averse. The loss of $100 is given twice the weight of an equivalent gain.

19.1.2 Accept or reject after loss

Suppose the agent loses their wallet containing $100. They feel bad about it and perceive it as a loss. Their reference point is unchanged at the original status quo, but the amount of money they have after any outcome is $100 less than otherwise. Would they be willing to take gamble A now?

After losing $100 but not changing their reference point, they have two possible outcomes relative to their reference point: a gain of $10 (winning $110 minus the lost money in the wallet) and a loss of $200 (losing $100 and also losing their wallet).

The weighted value of gamble A is now:

\begin{align*} V(A)&=p_1v(x_1)+p_2v(x_2) \\[6pt] &=0.5\times v(110-100)-0.5\times v(-100-100) \\[6pt] &=0.5\times (10)^\frac{1}{2}-0.5\times 2\times (200)^\frac{1}{2} \\[6pt] &=-12.56 \end{align*}

The value of not playing the gamble involves remaining with a loss of $100:

\begin{align*} V(\neg A)&=v(-100) \\[6pt] &=-2\times (100)^\frac{1}{2} \\[6pt] &=-20 \end{align*}

They will now want to play the gamble as it has a greater value than staying with their current loss. The gamble becomes attractive as it allows recovery the loss. The agent is risk seeking in the loss domain. (They would even accept a 50:50 gamble to win $100, lose $100 with an expected value of zero.)

19.1.3 Accept or reject after adaptation to loss

The agent has now adapted to their loss of $100. The new reference point is the new wealth level incorporating the loss wallet. Would they take gamble A now?

We are now back to an identical situation as when they were first offered the gamble with their reference point as the status quo. They will not want to partake in the gamble.

19.1.4 Accept or reject after win

The agent wins $10,000 at the casino. They feel good about their win, so their reference point remains at their wealth excluding the win. Would they take gamble A now?

With the additional $10,000, the value from the gamble is:

\begin{align*} V(A)&=p_1v(x_1)+p_2v(x_2) \\[6pt] &=0.5\times v(10000+110)+0.5\times v(10000-100) \\[6pt] &=0.5\times (10110)^\frac{1}{2}+0.5\times (9900)^\frac{1}{2} \\[6pt] &=100.02 \end{align*}

The value of not playing the gamble is:

\begin{align*} V(\neg A)&=v(10000) \\[6pt] &=10000^\frac{1}{2} \\[6pt] &=100 \end{align*}

The gamble is now attractive. The agent is less risk averse at a higher wealth. Further, the gamble is entirely in the gain domain, meaning that loss aversion does not affect the decision.

19.2 A 60:40 gamble

Paddy makes decisions in accordance with prospect theory, has wealth $300 and value function:

v(x)=\left\{\begin{matrix} x^{\frac{1}{2}} \quad &\textrm{where} \quad x \geq 0 \\[6pt] -2(-x)^{\frac{1}{2}} \quad &\textrm{where} \quad x < 0 \end{matrix}\right.

Assume Paddy weights probabilities linearly.

Paddy is offered the following bet A:

  • a 60% probability to win $150
  • a 40% probability to lose $100.

19.2.1 Accept or reject

Does Paddy accept bet A?

Paddy compares the value of taking versus not taking the bet:

\begin{align*} V(\text{A})&=p_1v(x_1)+p_2v(x_2) \\[6pt] &=0.6\times v(150)-0.4\times v(100) \\[6pt] &=0.6\times (150)^{\frac{1}{2}}-0.4\times 2\times (100)^{\frac{1}{2}} \\[6pt] &=-0.652 \end{align*}

The value of not taking the bet is zero. Paddy would have no change from his reference point.

Paddy rejects the bet as V(A) is less than the V(0)=0 Paddy could get by simply rejecting the bet.

The following figure shows Paddy’s value function, the bets and the value of the bets. The figure illustrates that Paddy’s rejection is caused by both Paddy’s loss aversion and his diminishing sensitivity in the gain domain, which has a larger effect than the diminishing sensitivity in the loss domain due to the larger magnitude of the potential gain.

19.2.2 Accept or reject after loss

Following some bad economic news, Paddy’s wealth declines to $150. Paddy cannot get over the loss, so his reference point remains his former wealth of $300.

Paddy is offered bet A again. Does Paddy accept the bet?

As Paddy is now in the loss domain, the two potential outcomes from the bet are a gain of $0 and a loss of $250. His alternative is remaining at a point $150 below his reference point (L).

Paddy compares the value of taking versus not taking the bet is:

\begin{align*} V(\text{A})&=p_1v(x_1)+p_2v(x_2) \\[6pt] &=0.6\times v(-150+150)+0.4\times v(-150-100) \\[6pt] &=0.6\times (-150+150)^{\frac{1}{2}}-0.4\times 2\times (150+100)^{\frac{1}{2}} \\[6pt] &=-12.649 \\ \\ V(\text{L})&=v(L) \\[6pt] &=-2\times (150)^{\frac{1}{2}} \\[6pt] &=-24.495 \end{align*}

Paddy accepts the bet as V(A) is greater than the value of the certain loss of $150.

The following figure shows Paddy’s value function, the bets and the value of the bets. The figure shows that Paddy accepts the bet as he is risk seeking in the loss domain.

19.3 Insurance

The classical economic explanation for the purchase of insurance is based on the risk aversion of consumers. Insurance has a negative expected value due to the insurer’s profit and administrative costs. However, consumers are willing to buy insurance as the consumer prefers the certainty of the premium payment to the risk of suffering an uninsured loss.

Prospect theory provides an alternative explanation. The purchase of insurance involves a certain loss (the premium) or a gamble involving the possibility of either a large loss or the status quo. As prospect theory has people as risk seeking in the loss domain, we would not expect them to purchase insurance.

However, under prospect theory people also overweight small probabilities. This overweighting of small probabilities can make the purchase of insurance attractive even though it is in the loss domain. This combination of the loss domain but small probabilities is the bottom-right quadrant of the fourfold pattern to risk attitudes generated by prospect theory.

The following numerical example is an illustration.

An agent is considering insurance against bushfire for its $1,000,000 house. The house has a 1 in 1000 (p=0.001) chance of burning down. An insurer is willing to offer full coverage for $1100. (Note: $1000 is the actuarially fair price, the additional $100 might represent profit or administrative costs.)

19.3.1 Expected value

The first question we will ask is whether an expected value maximiser or risk-neutral person would purchase the insurance.

A risk-neutral agent will choose the option with the highest expected value.

First, we calculate the expected value of purchasing insurance policy I. R is the premium paid.

\begin{align*} E[I]&=-R \\ &=-\$1,100 \end{align*}

The expected value of purchasing insurance is the guaranteed loss of the premium.

Then we calculate the expected value of not purchasing the insurance. H is the value of the house.

\begin{align*} E[\neg I]&=p\times -H \\ &=-0.001\times 1000000 \\ &=-\$1000 \end{align*}

The expected value of purchasing insurance is $100 less than the expected value of risking the house burning down. A risk-neutral agent (who maximises expected value) would not purchase this insurance.

19.3.2 Expected utility

Would a risk-averse agent purchase the insurance? Suppose they have a logarithmic utility function (U(x)=ln(x)) and they have $10,000 in cash in addition to their house, giving them wealth (W) of $1,010,000.

First, we calculate the expected utility of purchasing insurance policy I, which is simply the utility of the wealth after the premium is paid.

\begin{align*} E[U(I)]&=\text{ln}(W-R) \\[6pt] &=\text{ln}(1,008,900) \\[6pt] &=13.8244 \end{align*}

Then we calculate the expected utility of not purchasing the insurance.

\begin{align*} E[U(\neg I)]&=0.999\times \text{ln}(W)+0.001\times \text{ln}(W-H) \\[6pt] &=0.999\times \text{ln}(1,010,000)+0.001\times \text{ln}(10,000) \\[6pt] &=13.8208 \end{align*}

The expected utility of purchasing insurance is greater than the expected utility of not purchasing insurance. This agent will insure against the fire despite it being actuarially unfair.

The following diagram illustrates. The agent’s utility function is plotted, with the outcome on the horizontal axis and the utility of each outcome on the vertical axis. Each outcome and the utility of that outcome is marked - wealth after losing the house when uninsured (W-H), wealth after paying the insurance premium (W-R), and wealth if uninsured but the house does not burn down (W).

The expected utility of not purchasing insurance is on the dash-dot line between U(W-H) and U(W). The precise point is p along this line from U(W) (or 1-p along the line from U(W-H)). This point aligns with the expected value of leaving the house uninsured E[\neg I].

The utility of purchasing insurance (U(W-R)) is greater than the expected utility of not purchasing insurance (E[U(\neg I)]). The agent will purchase insurance.

Code 
library(ggplot2)
library(latex2exp)

u <- function(x){
  log(x)
}

df <- data.frame(
  x=seq(1,100,0.1),
  y=NA
)

df$y <- u(df$x)

#Variables for plot (may not match labels as not done to scale)
#Payoffs from gamble
x1<-3 #loss
x2<-90 #win
ev<-77 #expected value of gamble
xc<-70 #certain outcome
px2<-(ev-x1)/(x2-x1)

ggplot(mapping = aes(x, y)) +
    geom_line(data = df) +
    geom_vline(xintercept = 0, linewidth=0.25)+ 
    geom_hline(yintercept = 0, linewidth=0.25)+
    labs(x = "x", y = "U(x)")+

    # Set the theme
    theme_minimal()+

    #remove numbers on each axis
    theme(axis.text.x = element_blank(),
            axis.text.y = element_blank(),
            axis.title=element_text(size=14,face="bold"),
            axis.title.y = element_text(angle=0, vjust=0.5))+

    #limit to y greater than zero and x greater than -8 (need -8 so space for y-axis labels)
    coord_cartesian(xlim = c(-8, 100), ylim = c(0, 5))+

    #Add labels W, U(W) and line to curve indicating each
    annotate("text", x = x2, y = 0, label = "W", size = 4, hjust = 0.4, vjust = 1.5)+
    annotate("segment", x = x2, y = 0, xend = x2, yend = u(x2), linewidth = 0.5, colour = "black", linetype="dotted")+
    annotate("segment", x = 0, y = u(x2), xend = x2, yend = u(x2), linewidth = 0.5, colour = "black", linetype="dotted")+
    annotate("text", x = 0, y = u(x2), label = "U(W)", size = 4, hjust = 1.1, vjust = 0.4)+

    #Add labels W-R, U(W_R) and line to curve indicating each
    annotate("text", x = xc, y = 0, label = "W-R", size = 4, hjust = 0.5, vjust = 1.5)+
    annotate("segment", x = xc, y = 0, xend = xc, yend = u(xc), linewidth = 0.5, colour = "black", linetype="dotted")+
    annotate("segment", x = 0, y = u(xc), xend = xc, yend = u(xc), linewidth = 0.5, colour = "black", linetype="dotted")+
    annotate("text", x = 0, y = u(xc), label = "U(W-R)", size = 4, hjust = 1.05, vjust = 0.45)+

    #Add expected utility line
    annotate("segment", x = x2, xend = x1, y = u(x2), yend = u(x1), linewidth = 0.5, colour = "black", linetype="dotdash")+

    #Add labels W-H, U(W_H) and line to curve indicating each
    annotate("text", x = x1, y = 0, label = "W-H", size = 4, hjust = 0.4, vjust = 1.5)+
    annotate("segment", x = x1, y = 0, xend = x1, yend = u(x1), linewidth = 0.5, colour = "black", linetype="dotted")+
    annotate("segment", x = 0, y = u(x1), xend = x1, yend = u(x1), linewidth = 0.5, colour = "black", linetype="dotted")+
    annotate("text", x = 0, y = u(x1), label = "U(W-H)", size = 4, hjust = 1.05, vjust = 0.45)+

    #Add labels E[not I], E[U(not I)] and curve indicating each
    annotate("text", x = ev, y = 0, label = TeX("E[$\\neg$ I]", output='character'), parse=TRUE, size = 4, hjust = 0.4, vjust = 1.4)+
    annotate("segment", x = ev, y = 0, xend = ev, yend = u(x1)+(u(x2)-u(x1))*px2, linewidth = 0.5, colour = "black", linetype="dashed")+
    annotate("segment", x = 0, y = u(x1)+(u(x2)-u(x1))*px2, xend = ev, yend = u(x1)+(u(x2)-u(x1))*px2, linewidth = 0.5, colour = "black", linetype="dashed")+
    annotate("text", x = 0, y = u(3)+(u(x2)-u(x1))*px2, label = TeX("E[U($\\neg$ I)]", output='character'), parse=TRUE, size = 4, hjust = 1.05, vjust = 0.45)
Figure 19.1: Insurance choice by a risk averse expected utility maximiser

19.3.3 The reflection effect

Consider an agent who is risk seeking in the domain of losses but weights probability linearly. Their value function is:

v(x)=\left\{\begin{matrix} x^{0.8} \qquad &\textrm{where} \space x \geq 0\\ -2(-x)^{0.8} \quad &\textrm{where} \space x < 0 \end{matrix}\right.

Where x is the realised outcome relative to the reference point.

Determination of the reference point can be arbitrary. What if you pay insurance every year? Could the reference point then be wealth minus the insurance payment (meaning the insurance payment is in the gain domain)?

Taking the reference point as current wealth, would this agent purchase the insurance?

\begin{align*} V(purchase)&=v(-1,100) \\ &=-(1,100)^{0.8} \\ &=-271.1 \\ \\ V(don't)&=0.999\times (0)+0.001\times v(-1,000,000) \\ &=0.999\times 0-0.001\times (1,000,000)^{0.8} \\ &=-63.1 \end{align*}

As V(purchase)<V(don't), the agent does not purchase insurance. The diminishing feeling of loss leads to them weigh the certain loss of the premium relatively more heavily than the chance of losing the value of their house.

Including loss aversion in the value function does not change the decision as all possible outcomes are in the loss domain.

19.3.4 Probability weighting

Would a person who is risk seeking in the domain of losses (i.e. the value function with reflection effect above) and applies the decision weights described below purchase the insurance?

They apply decision weights as per the following table:

Probability 0.001 0.01 0.1 0.25 0.5 0.75 0.9 0.99 0.999
Weight 0.01 0.05 0.15 0.3 0.5 0.7 0.85 0.95 0.99

\begin{align*} V(purchase)&=v(-1,100) \\ &=-(1,100)^{0.8} \\ &=-271 \\ \\ V(don't)&=\sum_{i=1}^n \pi(p_i)v(x_i) \\ &=\pi(0.999)\times v(0)+\pi(0.001)\times v(-1,000,000) \\ &=0.99\times 0-0.01\times (1,000,000)^{0.8} \\ &=-631 \end{align*}

Although the diminishing feeling of loss leads to them weigh the certain loss of the premium relatively more heavily than the chance of losing the value of their house, the overweighting of the probability of fire leads them to purchase insurance. Again, if we had included loss aversion it would not have changed the decision as all possible outcomes are in the loss domain.