Tutorials
Session 2 Expected utility theory, thinking and deciding
The first session starts with an overview of rational decision-making. Before we analyze actual
judgment and decision-making, we have to develop a benchmark model for choices in economic
settings. Our benchmark will be the theory of rational choice, the basis for the analysis of decisions in
economics.
Perloff (2012), Sections 17.0-2 (pp. 595 – 605)
These sections introduce uncertainty and expected utility. Also, the concept of risk aversion is
presented.
When making a decision about investments and other matters, you consider the possible outcomes
under various circumstances (states of nature). While we cannot know with certainty what the future
outcome will be, we may know that some outcomes are more likely than others. When this
uncertainty can be quantified, it is sometimes called risk. The likelihood of each possible outcome is
known or can be estimated, and no single possible outcome is certain to occur. Consumers and firms
modify their decisions about consumption and investment as the degree of risk varies.
In this article, two main topics that we have to study are examined: Degree of Risk and Decision
Making Under Uncertainty.
Degree of Risk
Probabilities are used to measure the degree of risk and the likely profit from a risky undertaking.
Before we can analyze decision making under uncertainty, we need a way to describe and quantify
risk. We can use our estimate of how risky each outcome is to estimate the most likely outcome. We
then present measures of risk that reflect how much actual outcomes deviate from the most likely
outcome.
Probability
A probability is a number between 0 and 1 that indicates the likelihood that a particular outcome will
occur. How do we estimate a probability? Frequency, subjective probability & probability distribution.
- Frequency: If the history of outcomes for an event is known, which we can use the frequency
of a particular outcome as an estimate of probability. = n/N, n = occurrence & N = total
number. 13 out of 1000 houses burned down, 13/1000 = 1.3% =
- Subjective probability: When history is not known, a subjective probability has to be formed,
which is the best estimate of the likelihood that an outcome will occur. For this, all available
information is used, conscious and unconscious.
- Probability distribution: This relates the probability of occurrence to each possible outcome.
, -
Outcomes can be mutually exclusive and exhaustive. When outcomes are mutually exclusive only one
outcome can occur at a given time (you are either a citizen of The Netherlands or you are not), and
when they are exhaustive no other outcomes are possible (you can be a citizen of all countries on
earth, but you cannot be a citizen of Mars).
Expected Value
The amount someone expects to gain is called expected value (EV), what someone actually gains is
value (V). The EV is the value of each possible outcome times the probability of that outcome:
EV = (Prpos * Vpos) + (Prneg * Vneg).
Variance and Standard Deviation
If there is no difference in outcomes under different circumstances, there would be no risk. The risk
that is faced can be measured in many ways. One of those is looking at the degree by which actual
outcomes vary from the EV. It is convenient to combine two differences into a single measure of risk.
One such is variance, which measures the spread of the probability distribution. The variance is the
probability-weighted average of the squares of the differences between observed outcome and the
expected value: Variance = (Prpos * (Vpos – EV)2 )+ (Prneg * (Vneg – EV)2 ).
An alternative to reporting the variance is the standard deviation (SD), which is the square root of the
variance. The symbol for the SD is , which means that the symbol for variance is 2.
Decision Making Under Uncertainty
In the face of risk, different people will make different decisions based on their attitude towards risk.
If the EV of a riskier option is higher than that of a less risky option, people might choose the risky
option.
,Expected Utility
If people made choices to maximize expected value, they would always choose the option with the
highest expected value regardless of the risk involved. However, most people care about risk as well
as expected value. Most people dislike risk, are risk averse, and will choose an option with a higher
risk only if the EV is substantially higher. A rational person maximizes expected utility (EU). This is the
probability-weighted average of the utility of each possible outcome: EU = (Prpos * U(Vpos)) + (Prneg
* U(Vneg)). Here the utility function (U) depends on what someone earns from an outcome.
The difference between EV and EU is that the EV is the probability-weighted average of the monetary
value, while the EU is the probability-weighted average of the utility from the monetary value.
We can classify how people react to risky propositions by their willingness to make a fair bet (a wager
with an EV of 0, e.g. flipping a coin). Someone who is unwilling to make a fair bet is risk averse, one
who is indifferent is risk neutral and one who does want to make the bet is risk preferring.
Risk aversion
Someone who is risk averse, has a utility function that is concave to wealth, which means that her
utility rises with wealth but at a diminishing rate, this is called diminishing marginal utility of wealth.
The pleasure from an extra dollar is smaller than that of the previous one. A person whose utility
function is concave picks the less risky choice if both have the same EV.
The risk premium is the amount that a risk-averse person would pay to avoid taking a risk.
Risk neutrality
Someone who is risk neutral has a constant marginal utility of wealth: each extra dollar of wealth
raises utility by the same amount as the previous dollar. With constant marginal utility of wealth, the
utility curve is a straight line in a utility and wealth graph. A risk-neutral person chooses the option
with the highest EV, because maximizing EV maximizes utility.
The risk premium for a risk-neutral person is zero.
Risk preference
An individual with an increasing marginal utility of wealth is risk preferring: willing to take a fair bet.
They have a convex utility function to wealth. A risk-preferring person is willing to pay for the right to
make a fair bet, which is called a negative risk premium.
, Some people are willing to take an unfair bet. This is because they are risk preferring or they have a
compulsion to gamble. This does not explain the non-compulsive gambling by most risk-averse
people. Risk-averse people may make unfair bets for 3 reasons: 1 – they enjoy the game
(entertainment), 2 – their utility curve has both risk-averse and risk-preferring regions (averse to small
gains, preferring large gains), or 3 – they falsely believe the gamble favors them (mistaken confidence
in ability). These 3 are not mutually exclusive.
Hastie and Dawes (H&D), Preface, Ch. 1
The book of Hastie and Dawes primarily deals with violations of expected utility theory. However,
after having read and understood the two chapters described above, you should be able to
understand the reasoning and derivations on pp. 18-19 of Hastie and Dawes. This is important as this
kind of violation of expected utility theory will come back later in the book. This reasoning and these
derivations should be part of the presentation. Make sure that you understand the essential
difference between maximizing expected utility and maximizing monetary value, and the relation of
this difference to risk aversion.
1.1 Decision Making Is a Skill
We dominate this planet today because of our distinctive capacity for good decision making. This
same skill has allowed us to leave the planet, for brief periods; but, of course, the skill has allowed us
to develop technologies and weapons that could render the planet uninhabitable if we make a few
really bad decisions. Human beings have an exceptional ability to choose appropriate means to
achieve their ends. This book is about decision making, but it is not about what to choose; rather, it is
about how we choose.
We have a common set of cognitive skills that are reflected in similar decision habits. But we also
bring with us a common set of limitations on our thinking skills that can make our choices far from
optimal, limitations that are most obvious when we must make judgments and decisions that are not
like those we were “selected” to make in the ancestral environments in which we evolved. Our
decision-making capacities are not simply “wired in,” following some evolutionary design. Choosing
wisely is a learned skill, which, like any other skill, can be improved with experience.
To better understand the decision process and to identify the situations in which our choices are less
than optimal, we introduce a second perspective on decision making, namely analyses of the nature
of rational decision processes by philosophers and mathematicians.
Session 2 Expected utility theory, thinking and deciding
The first session starts with an overview of rational decision-making. Before we analyze actual
judgment and decision-making, we have to develop a benchmark model for choices in economic
settings. Our benchmark will be the theory of rational choice, the basis for the analysis of decisions in
economics.
Perloff (2012), Sections 17.0-2 (pp. 595 – 605)
These sections introduce uncertainty and expected utility. Also, the concept of risk aversion is
presented.
When making a decision about investments and other matters, you consider the possible outcomes
under various circumstances (states of nature). While we cannot know with certainty what the future
outcome will be, we may know that some outcomes are more likely than others. When this
uncertainty can be quantified, it is sometimes called risk. The likelihood of each possible outcome is
known or can be estimated, and no single possible outcome is certain to occur. Consumers and firms
modify their decisions about consumption and investment as the degree of risk varies.
In this article, two main topics that we have to study are examined: Degree of Risk and Decision
Making Under Uncertainty.
Degree of Risk
Probabilities are used to measure the degree of risk and the likely profit from a risky undertaking.
Before we can analyze decision making under uncertainty, we need a way to describe and quantify
risk. We can use our estimate of how risky each outcome is to estimate the most likely outcome. We
then present measures of risk that reflect how much actual outcomes deviate from the most likely
outcome.
Probability
A probability is a number between 0 and 1 that indicates the likelihood that a particular outcome will
occur. How do we estimate a probability? Frequency, subjective probability & probability distribution.
- Frequency: If the history of outcomes for an event is known, which we can use the frequency
of a particular outcome as an estimate of probability. = n/N, n = occurrence & N = total
number. 13 out of 1000 houses burned down, 13/1000 = 1.3% =
- Subjective probability: When history is not known, a subjective probability has to be formed,
which is the best estimate of the likelihood that an outcome will occur. For this, all available
information is used, conscious and unconscious.
- Probability distribution: This relates the probability of occurrence to each possible outcome.
, -
Outcomes can be mutually exclusive and exhaustive. When outcomes are mutually exclusive only one
outcome can occur at a given time (you are either a citizen of The Netherlands or you are not), and
when they are exhaustive no other outcomes are possible (you can be a citizen of all countries on
earth, but you cannot be a citizen of Mars).
Expected Value
The amount someone expects to gain is called expected value (EV), what someone actually gains is
value (V). The EV is the value of each possible outcome times the probability of that outcome:
EV = (Prpos * Vpos) + (Prneg * Vneg).
Variance and Standard Deviation
If there is no difference in outcomes under different circumstances, there would be no risk. The risk
that is faced can be measured in many ways. One of those is looking at the degree by which actual
outcomes vary from the EV. It is convenient to combine two differences into a single measure of risk.
One such is variance, which measures the spread of the probability distribution. The variance is the
probability-weighted average of the squares of the differences between observed outcome and the
expected value: Variance = (Prpos * (Vpos – EV)2 )+ (Prneg * (Vneg – EV)2 ).
An alternative to reporting the variance is the standard deviation (SD), which is the square root of the
variance. The symbol for the SD is , which means that the symbol for variance is 2.
Decision Making Under Uncertainty
In the face of risk, different people will make different decisions based on their attitude towards risk.
If the EV of a riskier option is higher than that of a less risky option, people might choose the risky
option.
,Expected Utility
If people made choices to maximize expected value, they would always choose the option with the
highest expected value regardless of the risk involved. However, most people care about risk as well
as expected value. Most people dislike risk, are risk averse, and will choose an option with a higher
risk only if the EV is substantially higher. A rational person maximizes expected utility (EU). This is the
probability-weighted average of the utility of each possible outcome: EU = (Prpos * U(Vpos)) + (Prneg
* U(Vneg)). Here the utility function (U) depends on what someone earns from an outcome.
The difference between EV and EU is that the EV is the probability-weighted average of the monetary
value, while the EU is the probability-weighted average of the utility from the monetary value.
We can classify how people react to risky propositions by their willingness to make a fair bet (a wager
with an EV of 0, e.g. flipping a coin). Someone who is unwilling to make a fair bet is risk averse, one
who is indifferent is risk neutral and one who does want to make the bet is risk preferring.
Risk aversion
Someone who is risk averse, has a utility function that is concave to wealth, which means that her
utility rises with wealth but at a diminishing rate, this is called diminishing marginal utility of wealth.
The pleasure from an extra dollar is smaller than that of the previous one. A person whose utility
function is concave picks the less risky choice if both have the same EV.
The risk premium is the amount that a risk-averse person would pay to avoid taking a risk.
Risk neutrality
Someone who is risk neutral has a constant marginal utility of wealth: each extra dollar of wealth
raises utility by the same amount as the previous dollar. With constant marginal utility of wealth, the
utility curve is a straight line in a utility and wealth graph. A risk-neutral person chooses the option
with the highest EV, because maximizing EV maximizes utility.
The risk premium for a risk-neutral person is zero.
Risk preference
An individual with an increasing marginal utility of wealth is risk preferring: willing to take a fair bet.
They have a convex utility function to wealth. A risk-preferring person is willing to pay for the right to
make a fair bet, which is called a negative risk premium.
, Some people are willing to take an unfair bet. This is because they are risk preferring or they have a
compulsion to gamble. This does not explain the non-compulsive gambling by most risk-averse
people. Risk-averse people may make unfair bets for 3 reasons: 1 – they enjoy the game
(entertainment), 2 – their utility curve has both risk-averse and risk-preferring regions (averse to small
gains, preferring large gains), or 3 – they falsely believe the gamble favors them (mistaken confidence
in ability). These 3 are not mutually exclusive.
Hastie and Dawes (H&D), Preface, Ch. 1
The book of Hastie and Dawes primarily deals with violations of expected utility theory. However,
after having read and understood the two chapters described above, you should be able to
understand the reasoning and derivations on pp. 18-19 of Hastie and Dawes. This is important as this
kind of violation of expected utility theory will come back later in the book. This reasoning and these
derivations should be part of the presentation. Make sure that you understand the essential
difference between maximizing expected utility and maximizing monetary value, and the relation of
this difference to risk aversion.
1.1 Decision Making Is a Skill
We dominate this planet today because of our distinctive capacity for good decision making. This
same skill has allowed us to leave the planet, for brief periods; but, of course, the skill has allowed us
to develop technologies and weapons that could render the planet uninhabitable if we make a few
really bad decisions. Human beings have an exceptional ability to choose appropriate means to
achieve their ends. This book is about decision making, but it is not about what to choose; rather, it is
about how we choose.
We have a common set of cognitive skills that are reflected in similar decision habits. But we also
bring with us a common set of limitations on our thinking skills that can make our choices far from
optimal, limitations that are most obvious when we must make judgments and decisions that are not
like those we were “selected” to make in the ancestral environments in which we evolved. Our
decision-making capacities are not simply “wired in,” following some evolutionary design. Choosing
wisely is a learned skill, which, like any other skill, can be improved with experience.
To better understand the decision process and to identify the situations in which our choices are less
than optimal, we introduce a second perspective on decision making, namely analyses of the nature
of rational decision processes by philosophers and mathematicians.
