Decision Theory

• ordinal - which options are better, without quantifying the preference
• binary relation - given a set of economic objects X, a binary relation is a description of pairs of elements in X that are related - xBy
• reflexivity - for every object in X, xBx must be true
• Completeness - for every x, y in X, either xBy, yBx or both must be true
• Connectedness - every pair of distinct elements can be related in one way or another
• transitivity - for every x, y, z in X, whenever xBy and yBx it must also be true the xBz
• A rational preference is a complete and transitive binary relation
• given a rational preference, its strict part is always transitive, may or may not be connected, is never reflexive and is always antisymmetric (no two distinct elements are both related to each other)
• given a rational preference, its indifference part is an equivalence relation - is reflexive and symmetric (x~y implies y~x)
• given a rational preference we say that the real-valued function u is a utility representation or function whenever x is preferred to y, u(x)>=u(y)
• if X is finite, every rational preference can be represented
• if a rational preference has a finite number of equivalence classes, the preference can be represented
• uniform consumption equivalent - a bundle of the form (x,x) that every bundle is indifferent to. Assign utility x to (x1,x2)
• Lexicographic preferences over R^2 cannot be represented
• representations of a rational preference are not unique - the preference is ordinal so if u represents this preference, any monotone transformation of u represents the same preference
• a choice problem is a subset of alternatives over which an analyst can observe the behaviour of the individuals (often called menu)
• choice data is a map describing what is chosen in each of the menus, with the obvious assumption that the choice for each menu must be in that menu
• universal choice data - when we observe the individual choosing from every possible choice problem
• binary choice data - we observe the individual choosing from the collection of all choice problems formed by two alternatives
• given choice data c, we say that x is Directly Revealed Preferred to y distinct from x when x is chosen over y in menu containing y and x
• given choice data c we say that x is indirectly revealed preferred to y distinct from x whenever there is a sequence of elements that show x must be preferred to y even if not shown at the same time
• if X contains n elements, n-1 choice problems may be enough to learn all the preferences
• choice data is said to be rationalizable if there exists a rational preference such that c(A) is always the maximal element in A according to the preference
• 2. Choice data is said to be rationalizable if there exists a utility representation u such that c(A) is always the maximal element in A according to u
• Property alpha: we say that c satisfies property alpha if for any pair of choice problems B as a subset of A, we have c(A) in B implies c(B)=c(A)
• if choice data is rationalizable, it must satisfy property alpha
• Weak axiom of revealed preference: menus A and B cannot reveal x>y and y>x
• Rational choice data satisfies the WARP - if X is finite and the universal choice data c satisfies the WARP, then it is rational
• a preference is strictly monotone if x1>=x2 and y1>=y2, with one of them strict, implies (x1,y1)>(x2,y2)
• strict convexity - for every x,y such that x>y and every a in (0,1) x > ax + (1-a)y > y
• the maximisation of Cobb-Douglas preferences over any budget set predicts that the fraction of income spent in each of the goods is a constant (given by the coefficient of the Cobb-Douglas utility function)
• Quasi-linear preferences satisfy: (x,m) >= (y,m') <=> (x, m+n) >= (y, m'+n)
• discounted utility is a rational preference which also satisfies stationarity: (x,t) >= (y,s) <=> (x, t+r) >= (y, s+r)
• A preference is single-peaked if there exists x* such that: y2 < y1 < x* implies y1 > y2 and x* < y1 < y2 implies y1 > y2
• Jensen's inequality: if h is strictly concave, it is EV(p_h) < h(EV(p))
• F(z) is the total mass of experiencing a result below or equal to z
• a lottery is a finite random variable over the prize space
• Expected utility theorem: a binary relation is rational and satisfies independence and continuity if and only if it can be represented by an expected utility
• the certainty equivalent of a lottery is the amount of money that is indifferent to the lottery - v(CE(p))=EU(p)
• if v is monotone and continuous, every lottery has a certainty equivalent