Theory of Individual and Strategic Decisions – Summary
Maastricht University – Human Decision Science –
2024/2025
CHAPTER 1 PREFERENCES AND UTILITY....................................................................................2
1.1 PREFERENCES................................................................................................................ 2
1.2 PREFERENCE FORMATION.................................................................................................2
1.3 UTILITY FUNCTIONS........................................................................................................ 5
1.4 EXERCISES.................................................................................................................... 5
CHAPTER 2 CHOICE.........................................................................................7
SIGNS................................................................................................................................ 7
2.1 CHOICE AND RATIONAL CHOICE........................................................................................7
2.2 RATIONALISING CHOICE................................................................................................... 7
2.3 PROPERTY Α.................................................................................................................. 8
2.4 SATISFICING.................................................................................................................. 9
2.5 THE MONEY PUMP......................................................................................................... 9
2.6 EVIDENCE OF CHOICES INCONSISTENT WITH RATIONALITY.....................................................10
2.7 EXERCISES.................................................................................................................. 11
CHAPTER 3 PREFERENCES UNDER UNCERTAINTY............................................12
3.1 LOTTERIES.................................................................................................................. 12
3.2 PREFERENCES OVER LOTTERIES.......................................................................................12
3.3 EXPECTED UTILITY........................................................................................................ 14
3.4 RISK AVERSION........................................................................................................... 15
3.5 EXERCISES:................................................................................................................. 16
CHAPTER 15 STRATEGIC GAMES....................................................................18
15.1 STRATEGIC GAMES AND NASH EQUILIBRIUM....................................................................18
15.2 EXAMPLES................................................................................................................ 18
15.3 NASH EQUILIBRIUM VS NASH EQUILIBRIUM......................................................................19
15.4 FIRST-PRICE AND SECOND-PRICE AUCTION......................................................................20
15.5 STRICTLY COMPETITIVE GAMES.....................................................................................20
15.6 KANTIAN EQUILIBRIUM................................................................................................. 21
15.7 MIXED STRATEGIES..................................................................................................... 21
15.8 EXERCISES................................................................................................................ 23
CHAPTER 16 EXTENSIVE GAMES....................................................................24
16.1 EXTENSIVE GAMES AND SUBGAME PERFECT EQUILIBRIUM...................................................24
16.2 WHAT IS STRATEGY?................................................................................................... 25
16.3 BACKWARD INDUCTION................................................................................................ 25
16.4 BARGAINING.............................................................................................................. 28
16.5 EXERCISES................................................................................................................ 29
CHAPTER 17 MECHANISM DESIGN..................................................................32
17.1 DECIDING ON A PUBLIC PROJECT...................................................................................32
17.2 STRATEGY-PROOF MECHANISMS....................................................................................32
17.3 THE VICKREY-CLARKE-GROVES MECHANISM....................................................................33
CHAPTER 18 MATCHING................................................................................34
18.1 THE MATCHING PROBLEM............................................................................................ 34
18.2 THE GALE-SHAPLEY ALGORITHM...................................................................................35
18.3 THE GALE-SHAPLEY ALGORITHM AND STABILITY...............................................................37
Alex Verhaar Human Decision Science 2024/2025 Maastricht University
,Chapter 1 Preferences and Utility
1.1 Preferences
In the first part of the book, we discuss models of individuals. These models are of interest in their
own right, but we discuss them mainly to prepare for studying interactions between individuals. The
model of preference contains information only about an individual’s ranking of the alternatives, not
about the intensity of her feelings.
x y if both x ≥ y an y ≥ x
x > y if x ≥ y but not y ≥ x
We interpret the relation as “indifference” and the relation ¿ as “strict preference”.
The property that for all alternatives x and y , distinct or not, either x ≥ y or y ≥ x , is called
completeness. The property is transitive if when x ≥ y∧ y ≥ z then x ≥ z . A preference relation on the
set X is a complete and transitive binary relation on X. A binary relation is an equivalence relation if it
is reflexive, symmetric, and transitive. Completeness implies x x for every x (reflexivity).
Antisymmetric: The preference relation ≥ is antisymmetric if for any two x ; y such that x ≠ y , the
following holds: if x ≥ y , then y ≱ x
1.2 Preference Formation
This section explores how preferences can be formed or conceptualised. Several methods are
discussed:
1.2.1 Value Function
A value function is a straightforward and common way of deriving preferences. The idea is that each
alternative x in the set X is assigned a numerical value v ( x), which represents how much the
individual "values" that alternative. The individual’s preference between two alternatives, x and y , is
determined by comparing their assigned values:
If v ( x) ≥ v ( y) , then the individual prefers x at least as much as y (i.e., x ≥ y ).
The value function creates a complete and transitive preference relation:
Completeness arises because their numerical values can always be compared for any two
alternatives.
Alex Verhaar Human Decision Science 2024/2025 Maastricht University
, Transitivity holds because the comparison of numerical values is inherently transitive: if
v ( x) ≥ v ( y) and v ( y) ≥ v (z ), then v ( x) ≥ v (z).
This is the simplest form of preference formation, and it's often used in models where individuals
have well-defined preferences and can rank alternatives based on their utility or value.
1.2.2 Distance Function
In this approach, individuals have an ideal alternative in mind, and their preference for other
alternatives is determined by their "distance" from this ideal. The distance is measured by a function
d ( x ), where:
The closer an alternative x is to the ideal, the more it is preferred.
Formally, the individual prefers x to y if d ( x )≤ d ( y ).
This method is essentially a variation of the value function, where the value is negative and
represents the distance from the ideal. For example, suppose an individual has an ideal temperature
in mind for a vacation destination. In that case, their preference will be based on how close the actual
temperature of each destination is to their ideal.
Like the value function, this approach generates complete and transitive preferences since distances
between alternatives can always be compared, and the comparisons maintain transitivity.
1.2.3 Lexicographic Preferences
Lexicographic preferences are modelled when individuals consider multiple features of alternatives
but with a strict priority hierarchy between these features. Individuals first compare alternatives
based on the most important feature, and only if the alternatives are tied to that feature do they
proceed to consider the second feature.
For example, when comparing computers, an individual may first compare them based on memory
size, and only if two computers have the same memory size will they compare the screen resolution.
Formally:
x ≥ y if x is better than y on the most important feature, or if x and y are equal on the most
important feature, and x is better on the second feature.
Lexicographic preferences are complete because the individual can always rank alternatives based on
the most important feature. They are transitive because comparisons within each feature are
transitive, and tie-breaking follows a strict hierarchy.
Alex Verhaar Human Decision Science 2024/2025 Maastricht University
, Key points of lexicographic preferences:
The individual gives absolute priority to one feature over the others.
Once the first feature is evaluated, others only come into play if alternatives are tied.
1.2.4 Unanimity Rule
In the unanimity rule, an individual's preference is derived from multiple considerations or criteria,
represented by several complete and transitive binary relations ≥ 1, ≥ 2 ,... , ≥ n . The individual's
preference for an alternative x over y depends on whether all n criteria agree:
x ≥ y if and only if x ≥ y i for all i=1 , ..., n .
This method results in transitive preferences, as transitivity holds within each criterion. However, it
does not always lead to complete preferences:
Completeness can fail if some criteria rank x higher and others rank y higher, leading to cases where
the individual cannot definitively prefer one over the other. Thus, the overall preference relation will
be incomplete if there’s disagreement among the criteria (say, x ≥ y 1 but y ≥ x 2 ¿ .
1.2.5 Majority Rule
The majority rule is similar to the unanimity rule, but it requires only a majority instead of requiring
all criteria to agree. For instance, if there are three criteria ≥ 1, ≥ 2 ,≥ 3 :
x ≥ y if at least two criteria rank x higher than y .
Unlike the unanimity rule, the majority rule leads to complete preferences, as a decision can always
be made based on the majority of criteria. However, transitivity may not hold under majority rule.
This phenomenon is known as the Condorcet paradox. For example, with three alternatives, a , b ,∧c ,
and three criteria, it’s possible that:
a ≥ 1 b≥ 1 c ,
b≥2c≥2a,
c ≥3 a ≥ 3 b ,
Leading to a cycle where a is preferred to b , b is preferred to c , and c is preferred to a , which violates
transitivity.
Alex Verhaar Human Decision Science 2024/2025 Maastricht University
