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

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