Decision theory (0LSUC0)
Contents
Decision theory (0LSUC0)........................................................................................................................ 1
Chapter 1: Introduction .................................................................................................................. 2
Chapter 2: The decision matrix ....................................................................................................... 2
Chapter 3: Decisions under ignorance ............................................................................................ 4
Chapter 4: Decisions under risk ...................................................................................................... 6
Chapter 5: Utility ............................................................................................................................. 9
Chapter 6: The mathematics of probability .................................................................................. 12
Chapter 7: The philosophy of probability ..................................................................................... 13
Chapter 8: Why should we accept the preference axioms? ......................................................... 15
Chapter 9: Causal vs. evidential decision theory .......................................................................... 16
Chapter 10: Bayesian vs. non-Bayesian decision theory .............................................................. 17
Chapter 11: game theory 1; basic concepts and zero-sum games ............................................... 18
Chapter 12: nonzero-sum and cooperative games....................................................................... 20
Chapter 14: Overview of descriptive decision theory................................................................... 23
Heuristics and biases..................................................................................................................... 25
1
,Chapter 1: Introduction
Decision theory is about rational decision making. A decision maker chooses from a set of
alternatives. The outcome depends on the real state of the world.
Right decision: if and only if its actual outcome is at least as good as that of every other possible
outcome.
Rational decision: if and only if the decision maker chooses to do what she has most reason to do at
the point in time at which the decision is made.
Descriptive: how do people actually make decisions and why?
Normative: what is a rational decision? What decision do I ought to take?
Common ground: action: belief + desire (David Hume)
There’s a distinction between rational decisions and right decisions. In right decisions, the outcome
is optimal. In rational decisions the outcome of the decision does not matter.
The first step in decision making is to decide the decision problem. There are three types of decision
problems:
- Decision under risk: the decision maker knows the probability of the possible outcomes.
- Decisions under ignorance: the probabilities are either unknown or non-existent.
- Decision under uncertainty: This is either a synonym for ignorance, or a broader term that
refers to both risk and ignorance.
Social choice theory and game theory
- Social choice theory: how should decisions involving more than one decision maker be
made? (“democratic elections”). However, not every action performed by a group is a social
choice (e.g. choices of a government once it is selected).
- Individual decision making: a single decision maker, not taking into account what other
decision makers are doing.
- Game theory: the outcome of your decision depends on what others do.
It can make a big difference whether the gameplay is iterative; that is, how many times the
game is played.
Chapter 2: The decision matrix
We are dealing with tree levels of abstraction:
1. Determine the decision problem: what’s happening in the real world?
2. Formalization of the decision problem: define states, acts, outcomes.
The probabilities of the states should be independent of the acts. Decision theory is
concerned with particular acts – that are carried out by specific agents at specific time
intervals – rather than generic acts which can be instantiated by different agents at different
time intervals (e.g. a particular voyage versus sailing in general).
3. A visualization of the formalization: visualize the formalization in a matrix, a decision tree or
alternative visualizations. Rival formalizations arise if two or more formalizations are equally
2
, reasonable and strictly better than all alternative formalizations. These can occur because of
the principle of insufficient reason and merger of states.
State 1 State 2
Act A Outcome A1 Outcome A2
Act B Outcome B1 Outcome B2
Figure 1: a decision tree. A square represents a choice made, circles represent chance nodes.
Scales
1. Ordinal scale: qualitative comparison of objects allowed; no information about
differences or ratios. Something outcome is better or worse than another outcome.
f(x) ≥ f(y) if and only if x ≥ y
2. Cardinal scale
Interval scale: quantitative comparison of objects; accurately reflects
differences between objects. The distance between the numbers is the
same. Preserves differences, but nothing more. E.g. temperature.
f’(x) = k * f(x) + m
Ratio scale: quantitative comparison of objects; accurately reflects ratios
between object. E.g. time, length.
f’(x) = k*f(x)
3
Contents
Decision theory (0LSUC0)........................................................................................................................ 1
Chapter 1: Introduction .................................................................................................................. 2
Chapter 2: The decision matrix ....................................................................................................... 2
Chapter 3: Decisions under ignorance ............................................................................................ 4
Chapter 4: Decisions under risk ...................................................................................................... 6
Chapter 5: Utility ............................................................................................................................. 9
Chapter 6: The mathematics of probability .................................................................................. 12
Chapter 7: The philosophy of probability ..................................................................................... 13
Chapter 8: Why should we accept the preference axioms? ......................................................... 15
Chapter 9: Causal vs. evidential decision theory .......................................................................... 16
Chapter 10: Bayesian vs. non-Bayesian decision theory .............................................................. 17
Chapter 11: game theory 1; basic concepts and zero-sum games ............................................... 18
Chapter 12: nonzero-sum and cooperative games....................................................................... 20
Chapter 14: Overview of descriptive decision theory................................................................... 23
Heuristics and biases..................................................................................................................... 25
1
,Chapter 1: Introduction
Decision theory is about rational decision making. A decision maker chooses from a set of
alternatives. The outcome depends on the real state of the world.
Right decision: if and only if its actual outcome is at least as good as that of every other possible
outcome.
Rational decision: if and only if the decision maker chooses to do what she has most reason to do at
the point in time at which the decision is made.
Descriptive: how do people actually make decisions and why?
Normative: what is a rational decision? What decision do I ought to take?
Common ground: action: belief + desire (David Hume)
There’s a distinction between rational decisions and right decisions. In right decisions, the outcome
is optimal. In rational decisions the outcome of the decision does not matter.
The first step in decision making is to decide the decision problem. There are three types of decision
problems:
- Decision under risk: the decision maker knows the probability of the possible outcomes.
- Decisions under ignorance: the probabilities are either unknown or non-existent.
- Decision under uncertainty: This is either a synonym for ignorance, or a broader term that
refers to both risk and ignorance.
Social choice theory and game theory
- Social choice theory: how should decisions involving more than one decision maker be
made? (“democratic elections”). However, not every action performed by a group is a social
choice (e.g. choices of a government once it is selected).
- Individual decision making: a single decision maker, not taking into account what other
decision makers are doing.
- Game theory: the outcome of your decision depends on what others do.
It can make a big difference whether the gameplay is iterative; that is, how many times the
game is played.
Chapter 2: The decision matrix
We are dealing with tree levels of abstraction:
1. Determine the decision problem: what’s happening in the real world?
2. Formalization of the decision problem: define states, acts, outcomes.
The probabilities of the states should be independent of the acts. Decision theory is
concerned with particular acts – that are carried out by specific agents at specific time
intervals – rather than generic acts which can be instantiated by different agents at different
time intervals (e.g. a particular voyage versus sailing in general).
3. A visualization of the formalization: visualize the formalization in a matrix, a decision tree or
alternative visualizations. Rival formalizations arise if two or more formalizations are equally
2
, reasonable and strictly better than all alternative formalizations. These can occur because of
the principle of insufficient reason and merger of states.
State 1 State 2
Act A Outcome A1 Outcome A2
Act B Outcome B1 Outcome B2
Figure 1: a decision tree. A square represents a choice made, circles represent chance nodes.
Scales
1. Ordinal scale: qualitative comparison of objects allowed; no information about
differences or ratios. Something outcome is better or worse than another outcome.
f(x) ≥ f(y) if and only if x ≥ y
2. Cardinal scale
Interval scale: quantitative comparison of objects; accurately reflects
differences between objects. The distance between the numbers is the
same. Preserves differences, but nothing more. E.g. temperature.
f’(x) = k * f(x) + m
Ratio scale: quantitative comparison of objects; accurately reflects ratios
between object. E.g. time, length.
f’(x) = k*f(x)
3
