Expected utility Flashcards
What are rational preferences?
Preferences that are:
Complete
- Total
- Reflexive
Transitive
Total: We can compare lotteries with each others, we can rank them.
Reflexive: Each lottery can be compared with it self. A preference relation is at least as good as itself.
This kind of a bi product from compleatness. If x = y, then x is at least as good as it self.
Transitive if a > b, b > c, then a > c.
How do you prove reflexivity?
Reflexive preferences kind of follows from totality. If we have $L_1 \succsim L_2$ and can make $L_1 = L_2$ by construction. Then we can substitute $L_1$ for $L_2$ in the relation and get $L_1 \succsim L_1$. This is at least how Chris proves that safety first preferences are reflexiv in Problemset 1.
What is the conceptual/intuitive difference between FOSD and SOSD?
FOSD defines a relationship between lotteries, or equivalently their CDFs, such that regardless of their risk attitude, rational agents should prefer one lottery to another for all increasing utility functions.
SOSD on the other hand defines a relationship between lotteries, or equivalently their CDFs, such that risk averse agents prefer one lottery to another for all concave utility functions.
What is the Arrow.Pratt measure of absolut risk aversion?
A(x) = - u’‘/u’
Whit constant riskaversion, this off course is a constant!
What is a singelton?
A unique NE
What are the strategies that survies I.E.S.D?
The set of strategies survived I.E.S.D = R
Where R = rationalizable set (the set of rationalizible strategies)
If we isolate a NE using, I.E.S.D, we can be sure that it is the only NE, since we have eliminated all the non-rationalizible strategies.
What is a best reply set?
The responses in each set should be best responses to some in the other set. See assignment 2.
All strategies included are best responses under some believes.
What is the proof that the union of two bestt reply sets is a best reply set?
Suppose the union of two best reply sets is not a best reply set, it means certain elements of the original best reply sets are strictly dominated strategies. However, it contradicts the definition of best reply sets that they include only strategies which are best responses under some beliefs. Therefore the union of best reply sets is indeed a best reply set.
What is a coin flip eq?
In essence, if players can jointly
observe some publicly observable random variable (referred to as ‘sunspots’ in the literature) that
occur with some independent probabilities before play, they can achieve any payoffs inside of the
convex hull of the Nash payoffs.
This is basically mixing between two NE. Pay off is then the average of both NEs.
If (1,5) and (5,1), then the coin flip EQ yields (3,3)
What is the strategy that always holds in the second price auction?
You bid your evaluation! This is a weakly dominant strategy.
In the second price auction it also exists a collusion EQ.
What is a Self-confirming EQ?
This regards the case when players believes do are not correct and coincide.
“Ingame theory,self-confirming equilibrium is a generalization ofNash equilibrium forextensive form games, in which players correctly predict the moves their opponents make, but may have misconceptions about what their opponentswould do atinformation sets that are never reached when the equilibrium is played. Informally, self-confirming equilibrium is motivated by the idea that if a game is played repeatedly, the players will revise their beliefs about their opponents’ play if and only if theyobserve these beliefs to be wrong.” Wikipedia
What are the ABIDE and DEVIDE payoffs in tit-for-tat
ABIDE:
This is just the CC payoff infinitly. C+Cd^+Cd^2+…. = C/(1+d)
DEVIATE:
Here you get DC, CD, DC, CD…….
Thus example
6-1d+6d^2-1d^3…….. = 6/(1-d^2) - 1d/(1-d^2)
what is an gap and what is an atom in bid distributions?
Gap is when the CDF flattens out and then starts again (like sadel point)
Atom is when it jumps in probability. (this is the case if the CDF starts > 0.)
