Why is randomiazion important in game theory?
else -> opponent can estimate our strategy
-> adapt to it and have advantage over us
What is an agent?
autonomous entity
ability to interact with its environment
What are prerequisities for making rational decisions?
perferences over a set of alternatives A
How are preference relations usually modeled?
binary relations
What is a weak preference relation?
x at least as good as y
In what parts can a relation be factorized?
assymetric part (strict preference relation >)
symmetric part (indifference relation ~)
How is a strict preference mathematically defined?
x is preferred over y
if and only if
x is at least as good as y
and y is not at least as good as x
How is a indifference relation mathematically defined?
x is equally preffered as y
x is at least as preffered as y
y is at least as preffered as x
What is rationality w.r.t. preference relations?
preference is rational if:
complete
transitive
What means complete w.r.t rational preferences?
any two pairs of alternatives can be compared with each other
What is transitive w.r.t trational preference relations?
preferences fulfill transitivity
-> if x is preffered over y
-> if y is preffered over z
=> x is preffered over z
What do preference relations not capture w.r.t. comparison of alternatives?
the intensity of preferences
When is an agent called rational?
it chooses the most desirable
among all feasible alternatives
What follows from rational preference relations w.r.t. ordering?
the preferences are weakly ordered
How can one find out about preferences in rational agents?
throuth observintg their behavior
What does transitivity allow for?
it is sufficient to ensure
that every finite non-empty set of alternatives
admits a most desirable alternative
=> allows us to make rational decisions, as it is clear what yields the maximum utility…
What is the concept behind money pump?
have no transitive preference relation
e.g. apple < banana < chocolate < apple
A has banana and chocolate
B has apple
A exchanges banana with apple
with additional e.g. monetary compensaiton as apple < banana
A exchanges chocolate with banana
again with positive monetary compensation as before
A exchanges apple with chocolate
again monetary positive
same situation as in 1…
-> restart and “pump money”
Is transitivity always given?
no
humans sometimes exhibit intransitive preferences
i.e.
aggregation of multiple criteria
indistinguishability
What is aggregation of multiple criteria?
assume: agent has rational preferences when
considering each criterion in isolation
do majority comparison (i.e. object is better than other it is better in at least n/2 criterias better thatn the other)
-> can yield cyclical preference relations
SUV > sports car
sedan > SUV
sports car > sedan
What is an example for indistinguishability?
i.e. have several indifferences
example color intensity
1 ~ 2 as not distinguishable by eye
2 ~ 3
3 ~ 4
but 1 < 4
=> transitivity broken!!!
sorites paradox…
What is a utility function?
u: A -> R
assigns our alternatives a numerical (utility) value
When does a utility represent a preference relation?
not unique representation of utility funciton, only certain things must hold
How can we create a new utility function from a given one (that still represents the same preference relation)?
for every strict increasing function f
f(u(.)) is a new utility function
Why are utility functions so useful although we could as well simply use preference relations?
naturalness
increasing / decreasing utility
utility maximization
=> numerical…
analytical convenience
differential calculus…
When can a preference relation be represented by a utility function (proposition from lecture)?
contable number of alternatives
preference relation representable by utility funciton
<=>
preference relation is rational
When does the proposition of preference relation being representable by utility funciton iff it is rational hold for uncountable number of alternatives?
some form of continuity (stetigkeit) is required
Last changed2 years ago
