GAME THEORY BASICS 1ST EDITION
BERNHARD VON STENGEL SOLUTION
MANUAL ALL CHAPTERS 100% ORIGINAL
VERIFIED A+ FINAL STUDY GUIDE 2026
SOLVED QUESTIONS FULLY CORRECT
⫸ What does the set S represent in decision making? Answer: The
choices available to an agent.
⫸ What does the set Ω represent in decision making? Answer: The set
of possible outcomes resulting from choices.
⫸ What is an outcome function? Answer: A function g: S → Ω that
maps each choice to an outcome.
⫸ What is the difference between decision-making under certainty and
uncertainty? Answer: Under certainty, the consequences of choices are
known; under uncertainty, multiple possible consequences exist with
associated probabilities.
⫸ What are the three properties of a preference relation? Answer:
Reflexive, Total, and Transitive.
⫸ What does ω ⪰ ω′ signify? Answer: An agent prefers outcome ω at
least as much as outcome ω′.
,⫸ What does ω ∼ ω′ indicate? Answer: The agent is indifferent between
outcomes ω and ω′.
⫸ What does ω ≻ ω′ mean? Answer: The agent strictly prefers outcome
ω over outcome ω′.
⫸ What is a utility function? Answer: A mapping u: Ω → R that
represents a preference relation.
⫸ How does a utility function relate to preferences? Answer: A utility
function represents preferences if ω ⪰ ω′ iff u(ω) ≥ u(ω′).
⫸ What is the significance of utility values? Answer: They are a
numerical representation of preferences, but their relative order is more
important than the actual values.
⫸ What is the formal representation of decision-making? Answer: A
tuple ⟨S, Ω, g, ⪰, u⟩ where S is choices, Ω is outcomes, g is the outcome
function, ⪰ is the preference relation, and u is the utility function.
⫸ What defines a rational decision maker? Answer: A rational decision
maker is one who maximizes their utility by making the best possible
choice.
,⫸ What is a lottery in the context of utility theory? Answer: A
probability distribution over a set of outcomes Ω.
⫸ What is a compound lottery? Answer: A lottery of lotteries,
represented as a probability distribution over a set of lotteries.
⫸ What is expected utility? Answer: The average utility expected from
a lottery, calculated as EU(L) = Σ u(ω)µ(ω).
⫸ What is the connection between preferences over lotteries and
expected utility? Answer: The link discovered by von Neumann and
Morgenstern connects ordinal representation of preferences with the
numerical concept of expected utility.
⫸ What is the role of probability distributions in decision-making under
uncertainty? Answer: They represent the likelihood of different
outcomes occurring from a choice.
⫸ What is a degenerate lottery? Answer: A lottery where one outcome
has a probability of 1, and all other outcomes have a probability of 0.
⫸ What does the notation L = [0.2(chocolate), 0.5(vanilla),
0.3(strawberry)] represent? Answer: A lottery with specified
probabilities for each outcome.
, ⫸ What does the term 'maximizing utility' mean? Answer: Choosing the
outcome that provides the highest utility according to the agent's
preferences.
⫸ What is the significance of the axioms related to preferences over
lotteries? Answer: They establish the foundational rules that preferences
must satisfy to be consistent with expected utility theory.
⫸ What is the difference between utility and money? Answer: Utility
measures individual preferences, while money is an objective numerical
scale.
⫸ What is the importance of the relative order of utility values?
Answer: It determines how agents rank their preferences, rather than the
absolute values themselves.
⫸ What is the purpose of using numerical techniques in utility theory?
Answer: To facilitate the mathematical modeling of decision-making
processes.
⫸ How do agents make decisions in the presence of uncertainty?
Answer: By considering preferences over lotteries and expected utilities
rather than deterministic outcomes.
⫸ What does the Axiom of Continuity state in VNM theory? Answer:
For every lotteries L1 ≻ L2 ≻ L3, there exist α, β ∈ ]0, 1[ such that
[αL1, (1 − α)L3] ≻ L2 ≻ [βL1, (1 − β)L3].
BERNHARD VON STENGEL SOLUTION
MANUAL ALL CHAPTERS 100% ORIGINAL
VERIFIED A+ FINAL STUDY GUIDE 2026
SOLVED QUESTIONS FULLY CORRECT
⫸ What does the set S represent in decision making? Answer: The
choices available to an agent.
⫸ What does the set Ω represent in decision making? Answer: The set
of possible outcomes resulting from choices.
⫸ What is an outcome function? Answer: A function g: S → Ω that
maps each choice to an outcome.
⫸ What is the difference between decision-making under certainty and
uncertainty? Answer: Under certainty, the consequences of choices are
known; under uncertainty, multiple possible consequences exist with
associated probabilities.
⫸ What are the three properties of a preference relation? Answer:
Reflexive, Total, and Transitive.
⫸ What does ω ⪰ ω′ signify? Answer: An agent prefers outcome ω at
least as much as outcome ω′.
,⫸ What does ω ∼ ω′ indicate? Answer: The agent is indifferent between
outcomes ω and ω′.
⫸ What does ω ≻ ω′ mean? Answer: The agent strictly prefers outcome
ω over outcome ω′.
⫸ What is a utility function? Answer: A mapping u: Ω → R that
represents a preference relation.
⫸ How does a utility function relate to preferences? Answer: A utility
function represents preferences if ω ⪰ ω′ iff u(ω) ≥ u(ω′).
⫸ What is the significance of utility values? Answer: They are a
numerical representation of preferences, but their relative order is more
important than the actual values.
⫸ What is the formal representation of decision-making? Answer: A
tuple ⟨S, Ω, g, ⪰, u⟩ where S is choices, Ω is outcomes, g is the outcome
function, ⪰ is the preference relation, and u is the utility function.
⫸ What defines a rational decision maker? Answer: A rational decision
maker is one who maximizes their utility by making the best possible
choice.
,⫸ What is a lottery in the context of utility theory? Answer: A
probability distribution over a set of outcomes Ω.
⫸ What is a compound lottery? Answer: A lottery of lotteries,
represented as a probability distribution over a set of lotteries.
⫸ What is expected utility? Answer: The average utility expected from
a lottery, calculated as EU(L) = Σ u(ω)µ(ω).
⫸ What is the connection between preferences over lotteries and
expected utility? Answer: The link discovered by von Neumann and
Morgenstern connects ordinal representation of preferences with the
numerical concept of expected utility.
⫸ What is the role of probability distributions in decision-making under
uncertainty? Answer: They represent the likelihood of different
outcomes occurring from a choice.
⫸ What is a degenerate lottery? Answer: A lottery where one outcome
has a probability of 1, and all other outcomes have a probability of 0.
⫸ What does the notation L = [0.2(chocolate), 0.5(vanilla),
0.3(strawberry)] represent? Answer: A lottery with specified
probabilities for each outcome.
, ⫸ What does the term 'maximizing utility' mean? Answer: Choosing the
outcome that provides the highest utility according to the agent's
preferences.
⫸ What is the significance of the axioms related to preferences over
lotteries? Answer: They establish the foundational rules that preferences
must satisfy to be consistent with expected utility theory.
⫸ What is the difference between utility and money? Answer: Utility
measures individual preferences, while money is an objective numerical
scale.
⫸ What is the importance of the relative order of utility values?
Answer: It determines how agents rank their preferences, rather than the
absolute values themselves.
⫸ What is the purpose of using numerical techniques in utility theory?
Answer: To facilitate the mathematical modeling of decision-making
processes.
⫸ How do agents make decisions in the presence of uncertainty?
Answer: By considering preferences over lotteries and expected utilities
rather than deterministic outcomes.
⫸ What does the Axiom of Continuity state in VNM theory? Answer:
For every lotteries L1 ≻ L2 ≻ L3, there exist α, β ∈ ]0, 1[ such that
[αL1, (1 − α)L3] ≻ L2 ≻ [βL1, (1 − β)L3].
