Why is randomiazion important in game theory?
else -> opponent can estimate our strategy
-> adapt to it and have advantage over us
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
if and only if
x is at least as preffered as y
y is at least as preffered as x
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
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?
humans sometimes exhibit intransitive preferences
aggregation of multiple criteria
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!!!
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?
increasing / decreasing utility
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