# 15The reflection effect

The reflection effect involves an asymmetry in risk preferences in the gain and loss domains.

When people make a risky choice related to gains, they are risk averse. They prefer a certain option with value lower than the expected value of the risky choice. When choosing an option in the loss domain, they become risk-seeking. This phenomenon is called the reflection effect.

The reflection effect might also be thought of as diminishing sensitivity to gains or losses in either direction. This contrasts with expected utility theory, where the pain of losses increases as they grow in size.

## 15.1 The Asian Disease problem

The reflection effect explains the framing effects in the following experiment by Kahneman and Tversky (1984).

One group of experimental subjects were asked the following hypothetical question that would be unlikely to be asked post-Covid-19.

Imagine that the U.S. is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimates of the consequences of the programs are as follows:

If Program A is adopted, 200 people will be saved.

If Program B is adopted, there is a one-third probability that 600 people will be saved and a two-thirds probability that no people will be saved.

Which of the two programs would you favour?

72% of participants chose option A.

Another group of experimental participants were shown the following:

Imagine that the U.S. is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimates of the consequences of the programs are as follows:

If Program C is adopted, 400 people will die.

If Program D is adopted, there is a one-third probability that nobody will die and a two-thirds probability that 600 people will die.

Which of the two programs would you favour?

22% of participants chose option C.

72% of participants chose A and 22% of participants chose option C. Yet these two are equivalent. Option A and B are in the gain domain. Therefore the less risky option A is preferred. Options C and D are in the loss domain. Therefore the more risky option C is preferred.

## 15.2 The reflection effect in the value function

The following value function is an example of a function with diminishing sensitivity to both gains and losses. This function can generate the reflection effect.

\begin{equation*} v(x)= \Bigg\{ \begin{matrix} x^\frac{1}{2} \space \text{where} \space x \geq 0\\ -(-x)^\frac{1}{2} \space \text{where} \space x<0 \end{matrix} \end{equation*}

As x increases in magnitude in either direction, the marginal increase in value from each incremental unit of x decreases.

This value function results in risk-averse behaviour in the gain domain and risk-seeking behaviour in the loss domain. The following plot shows the diminishing effect in each direction.

Code
library(ggplot2)

loss_fun <- function(x){
-(-x)^0.5
}
gain_fun <- function(x){
x^0.5
}

loss <- data.frame(
x=seq(-10,0,0.05),
y=NA
)
loss$y <- loss_fun(loss$x)

gain <- data.frame(
x=seq(0,10,0.05),
y=NA
)
gain$y <- gain_fun(gain$x)

ggplot(mapping = aes(x, y)) +
geom_line(data = loss) +
geom_line(data = gain) +
geom_vline(xintercept = 0, linewidth=0.25)+
geom_hline(yintercept = 0, linewidth=0.25)+
labs(x = "", y = "v(X)")+

# Set the theme
theme_minimal()

In the gain domain, the function is concave, indicating risk aversion. In the loss domain, the convex function indicates risk-seeking behaviour.

## 15.3 An example

The following numerical example illustrates further.

Suppose an agent with the above value function is offered a choice between $10 for certain and a 50:50 bet to win$20 or end up with nothing. The value of each choice is as follows.

\begin{align*} v(\text{certainty})&=v(10) \\[6pt] &=10^{\frac{1}{2}} \\[6pt] &=3.16 \\ \\ v(\text{bet})&=0.5\times v(20)+0.5\times v(0) \\[6pt] &=0.5\times 20^{\frac{1}{2}}+0.5\times 0 \\[6pt] &=2.24 \end{align*}

The $10 for certain has a higher value for the agent. This agent is risk averse in the gain domain and therefore prefers an amount for certain over a bet with the same expected value. The following chart illustrates. Plotted are the value of the certain payment of$10 and the outcome from winning the gamble, $20. A loss results in value of 0. The value of the gamble itself is the probability-weighted average of the two gamble outcomes. Due to diminishing marginal utility in the gain domain, the value of the gamble is less than the value of$10 with certainty. The extra $10 over the certain outcome from winning the bet is less than the value of the first$10. As a result, the agent does not want to risk the bet.

Code
library(ggplot2)

u <- function(x){
ifelse (x >= 0, x^0.5, -(-x)^0.5)
}

df <- data.frame(
x = seq(-25, 25, 0.1),
y = u(seq(-25, 25, 0.1))
)

#Variables for plot (may not match labels as not done to scale)
#Payoffs from gamble
x1<-0 #loss
x2<-20 #win
ev<-10 #expected value of gamble
xc<-10 #certain outcome
px2<-(ev-x1)/(x2-x1)

ggplot(mapping = aes(x, y)) +

#Plot the utility curve
geom_line(data = df) +
geom_vline(xintercept = 0, linewidth=0.25)+
geom_hline(yintercept = 0, linewidth=0.25)+
labs(x = "x", y = "v(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(-25, 25), ylim = c(-5, 5))+

#Add labels 10, v(10) and line to curve indicating each
annotate("text", x = xc, y = 0, label = "10", size = 4, hjust = 0.6, 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 = "v(10)", size = 4, hjust = 1.05, vjust = 0.3)+

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

#Add labels 20, v(20) and line to curve indicating each
annotate("text", x = x2, y = 0, label = "20", 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 = "v(20)", size = 4, hjust = 1.05, vjust = 0.45)+

#Add labels v(bet) and curve indicating position
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(x1)+(u(x2)-u(x1))*px2, label = "v(bet)", size = 4, hjust = 1.05, vjust = 0.45)

Suppose the agent is now offered another choice. They can now have a certain loss of $10 or a 50:50 bet to lose$20 or to lose nothing. The value of each choice is as follows.

\begin{align*} v(\text{certainty})&=v(-10) \\[6pt] &=-10^{\frac{1}{2}} \\[6pt] &=-3.16 \\ \\ v(\text{bet})&=0.5\times v(-20)+0.5\times v(0) \\[6pt] &=-0.5\times 20^{\frac{1}{2}}+0.5\times 0 \\[6pt] &=-2.24 \end{align*}

This bet delivers higher value than the certain loss, despite the bet and the certain loss having the same expected value. The agent is willing to take a risk to avoid a loss. They are risk seeking in the loss domain.

The following chart illustrates. Plotted are the value of the certain payment of -$10 and the outcome from losing the gamble, -$20. A win results in value of 0.

The value of the gamble itself is the probability-weighted average of the two gamble outcomes. Due to diminishing marginal utility in the loss domain, the value of the gamble is greater than the value of -$10 with certainty. The potential loss of another$10 over and above the certain loss is given less weight than the first \$10. As a result, the agent wants to take the risk.

Code
library(ggplot2)

u <- function(x){
ifelse (x >= 0, x^0.5, -(-x)^0.5)
}

df <- data.frame(
x = seq(-25, 25, 0.1),
y = u(seq(-25, 25, 0.1))
)

#Variables for plot (may not match labels as not done to scale)
#Payoffs from gamble
x1<--20 #loss
x2<-0 #win
ev<--10 #expected value of gamble
xc<--10 #certain outcome
px2<-(ev-x1)/(x2-x1)

ggplot(mapping = aes(x, y)) +

#Plot the utility curve
geom_line(data = df) +
geom_vline(xintercept = 0, linewidth=0.25)+
geom_hline(yintercept = 0, linewidth=0.25)+
labs(x = "x", y = "v(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(-25, 25), ylim = c(-5, 5))+

#Add labels -10, v(-10) and line to curve indicating each
annotate("text", x = xc, y = 0, label = "-10", size = 4, hjust = 0.6, vjust = -0.5)+
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 = "v(-10)", size = 4, hjust = -0.1, vjust = 0.3)+

#Add labels -20, v(-20) and line to curve indicating each
annotate("text", x = x1, y = 0, label = "-20", size = 4, hjust = 0.5, vjust = -0.5)+
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 = "v(-20)", size = 4, hjust = -0.1, vjust = 0.6)+

annotate("text", x = 0, y = u(x1)+(u(x2)-u(x1))*px2, label = "v(bet)", size = 4, hjust = -0.1, vjust = 0.45)