Exam Prep

for every A € E, C(A) <_ A

Weak Axiom of Revealed Preference can be satisfied or unsatisfied

one criteria which needs to be satisifed to ensure that consumer is consistent with their preferences.

for any two A, B € E that have at least two elements in common a,b € A intersection B it must be that if a € C(A) and b € C(B) then b€ C(A) and a € C (B)

question to ask: can we create cycles of two elements?

if its satisfied and the experiment is complete, a utility function can be found for the data/experiment

if its violated, then there is no complete or transitive weak preference that would’ve given the data to construct a utility function.

if a bundle of goods is a is chosen over bundle b when both are affordable, then they reveal their preference a over b —> revealed preference

if a cycle exsits, warp is violated

can have completeness issues

generalized axiom of revealed preference

a >_*c b

a is indirectly revealed preferred to b

it cannot be that a >_*c b and b >_c a

question to ask: can we create cycles of more than 2 elemnets?

If garp is satisfied a utility function can be found for the data/experiment

if garp is violated there is no utility function generating the observed data.

if we can create cycles of more than two elements garp is violated, if garp is violated warp is also necessarily violated —>indirectly contradicting each other, if there is an indirect contradiction, there has to be a direct contradiction

can have transitivity and completeness issues

Lotteries defined by choice domains: p and x

p1 x x1 , p2 x x2 , p3 x x3

20% 0. 50% 5. 30%. 20

multiplication of percentages and utilities (eg payoffs)

calculate the outcomes and see what matches the utility. if preferences and outcomes create contradiction then preferences are inconsistent, find out how they need to be adapted

can be used to test the vNM axioms

vNM Axioms: pleteness and transitivity that we have already discussed, plus two additional ones, (A3) continuity and (A4) independence.

Axiom A2

if x is weakly oreferred to y and y is weakly preferred to z, then x is weakly preferred to z

preferring apples to bananas, bananas to cranberries , resulting in preferring apples to cranberries

Axiom A1

x is weakly preferred over y or y is weakly preferred over x

incompleteness means x is not weakly preferred over y and y is not weakly preferred over x

there cannot be indifference between two choices

also goes for strict preferences

for infinite sets

even if indifference, still complete because we can compare and conclude they're indifferent

consumer prefers a particular bundle over the other bundle if the former consists of at least more of one good and no less of the other good

players, strategies, outcomes identified for each player

rational players think about actions other players might take, players form beliefs sbout one anothers behavior.

Prisoners Dilemma: situation where individual dm always have an incentive to choose in a way that created a less than optimal outcome for the individuals in the group

nash equilibrium: nobody has reason to deviate from strategy given that the other player does not deviate from their strategy

preferences are binary relations of outcomes

weakly preferred means at least as desirable as… <>_

strictly preferred <>

preferences are unobservables

way to infer preferences given the observed choices

ground for warp and garp

x is directly revealed preferred to y whenever we can find A € E such that x, y € (A)

in this case we write x >_c y

strictly direct revealed preferences:

x is strictly directly revealed preferred to y whenever x >_c y and y />_c x

there exists some A € E with x,y € A such that x € C (A) and y /€ C(A)

then we write x >c y

Objects compared in decision making

X x,y,z - set of outcomes

x € X - one outcome within a set

outcomes that are preferred get a higher utility function

U: X—>R

a utility function can be found if the Axioms hold, meaning transitivity and completeness must be satisfied

if x is weakly preferred to y then the utility of x is weakly preferred to the utility of y

Is neighbor of – symmetric

Is older than – antisymmetric

is a collection of solution concepts

any bundle is at least as good as itself

state contingent outcomes

each state has a lottery

given the state what is the lottery that is faced

act is an alternative which can be chosen

are indexed by probabilities

of the world/of nature

definition of all possible outcomes

sequential game of giving and keeping money, here time matters

sequential choice of actions at an order which is commonly known

perfect information:players observe what has happend so far

imperfect Information: past moves are not necessarily known or observed by all players

incomplete information: utility functions of others are not necessarily known by all players

Ann gets 10 can decide to give Bob amount

second whatever Bob go gets doubled and can decide to give Ann some of her money back

backward induction: draw Tree and find subgame perfect equilibrium

start at end and go back in nodes

Choice domain is defined in acts

acts and states need to be matched together

multiplication of utilities and beliefs (mu)

preferences over the set of acts have a subjective expected utility representation for some utility function and some belief mu

match the utility functions with the preferences and calculate the outcome, if there is a contradiction inconsistent choices exist

see what acts dominate each other

can be used to test the hypothesis that there is a SEU function

axioms guaranteeing SEU: all that we have already discussed, plus two new axioms, called state monotonicity and non-triviality (Anscobe Aumann Axiom)

strict dominance

a strategy strictly dominates if for each feasible combination of the other players strategy the payoff is strictly greater than the payoff from playing any other strategy

rational players never play strictly dominated strategies because such strategies can never be best responses to any other strategies of the other players

weak dominance

a strategy which never does worse than another maybe sometimes better than another. they can be a best respinse for a player

best response

selecting a strategy which yields the greatest lexpected payoff given their belief