Imagine that Alden asks Michael to play a lottery with two possible outcomes after a coin flip:
After his tedious hours in the classroom, Michael realizes that he can map his utility with the following function:
\begin{equation} u(x) = \sqrt{x} \end{equation}where $x$ is the points he gets from the lottery
where $p_i$ are the respective probability to the outcomes $x_i$ such that $p\in[0,1]$ and $\sum p_i = 1$
A situation of risk arises when the probs. are objectively known. In contrast, when prob. are not objectively known, then we have a situation of uncertainty.
For example, $L_A$ = $(30,.5; 50,.5)$
Von-Neumann-Mortegenstern expected utility A utility function has the *expected utility form* if we can assign a real number, $u(x)$ to each possible outcome, $x_i$ such that for the lottery L, we can write $U(L)=\sum p u(x)$
Expected utility has theoretical foundations. For example, it satisfies the independence axiom.
Independence axiom For all loteries, $L_A, L_B, L$ and all p, $L_A \succsim L_B \Leftrightarrow (L_A,p; L, 1-p)\succsim(L_B,p; L, 1-p)$
(From Problem 3 and 4 in Kahneman and Tversky, 1979). Consider a decision maker with initial wealth 0, and a utility function $u$.
Game 1 Choose between two lotteries
$b_1 = (0,0; 3000,1)$ or $b_2=(0,.2; 4000,.8)$
What is your choice?
Now, imagine another gamble
Game 2 Choose between two lotteries
$b_3 = (.75,0; 3000,.25)$ or $b_4=(0,.8; 4000,.2)$
Were you consistent according to EU?
Denoting by $\prec_{EU} $ the binary EU preference relation, the EU prediction is
$b_1 \prec_{EU} b_2 \Leftrightarrow b_3 \prec_{EU} b_4$ Why?
$b_1 \prec_{EU} b_2$ implies that $u(3000)< .8 u(4000)$. Dividing both sides by 4, we get $.25 u(3000)< .2 u(4000)$. Thus, $b_3 \prec_{EU} b_4$.
One way to interpret the new gamble is that now there is only 25$\%$ prob. that you get a positive payoff after playing the first gamble.
it turns out that most play $b_2 \prec b_1$ and $b_4 \prec b_3$.
there is some evidence that professional traders (Chicago
Board of Trade) also exhibit Allais-type violations but to a lesser extent than students (List and Haigh, PNAS 2005).
Consider the lotery $(C_L,1)$. If a solution exits to the equation $\bar{U}(C_L) = \bar{U}(L)$ where $\bar{U}$ is the one considered under EU, then $C_L$ is known as the *certainty equivalent of the lottery L*.
Risk aversion A decision maker is risk neutral if $C_L = EV(L)$ (where $EV = \sum p x$), risk adverse if $C_L<EV$ and risk loving if $C_L>EV$. Under risk aversion, a player would rather accept a sure amount that is lower than the expected value of the lottery, in order to avoid playing the lottery.
Under EU, the properties for risk neutrality, risk aversion, and risk seeking are equivalent to linear, concave and convex utility, respectively.
Multi Price List (MPL) due to Holt and Laury (2002)
Source: Habib et al (JESA, 2017)
BRET, due to Crosetto and Filippin (2013)
how many lotteries does this task implies?
What's the lottery $L_k$, where $k$ denotes the number of boxes collected?
Assuming a power utility $u(x)=x^r$, what's the optimal $k$, or number of boxes collected?
What's the optimal $k$ for a risk-neutral player?
*Case 1.* Assume yourself richer by $300 than you are today. You are offered a choice between
*Case 2.* Assume yourself richer by $500 than you are today. You are offered a choice between
Under PT (due to Kahneman and Tversky 1979) the carriers of utility are not final levels of wealth, but deviations of actual wealth levels from a reference point, and idea that is well established in psychology. (e.g. evidence indicates when exposed to external stimuli, individuals are more sensitive to changes rather than levels).
If the outcomes turns out to better (respectively worse) as compared to reference point, then the decision maker is said to be in the domain of gains (losses).
The idea is that people are risk-adverse in gains and hate losing and therefore are willing to take the chance to avoid.
According to many financial experimentalists the best paper that they ever read has been Kahneman and Tversky (1979).
People form probabilistic beliefs in the absence of known probabilities, transforming uncertainty into risk.
Draw a ball from a jar containing 90 total balls
30 balls are red .
the rest some unknown combination of blue and black
Game 1:
which option do you pick?
Game 2:
which option do you pick?
there are three envelopes, two with happy faces =), =) and one with an unhappy face =(.
You pass this class if you have an envelope with the happy face =)
you will pick one envelope, and after i reveal one of two left (i know where the unhappy face is) you have to decide whether to stay with your envelope or switch.
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
w = np.arange(-5,5.5,.1)
llama = 2.5
gamma = .5
uloss = - llama *(-w[w<0])**gamma
ugains= w[w>=0]**gamma
u = np.append(uloss,ugains)
plt.plot(w,u,linewidth=3)
plt.axhline(ls='--', color='r')
plt.axvline(ls='--', color='r')
plt.title('the utility function under PT')
Text(0.5,1,'the utility function under PT')
from IPython.core.display import HTML
def css_styling():
import os
styles = open(os.path.expanduser("custom.css"), "r").read()
return HTML(styles)
css_styling()