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Decisions and Games

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Decisions and Games DECISIONS AND GAMES. PART I 1. Preference and choice 2. Demand theory 3. Uncertainty 4. Intertemporal decision making 5. Behavioral decision theory DECISIONS AND GAMES. PART II 6. Static Games of complete information 7. Dynamic games of complete information 8. Static games of incomplete information 9. Dynamic games of incomplete information 10. Behavioral game theory References Gibbons, R. 1992. Game theory for applied economists. Princeton University Press. Princeton, New Jersey. Varian, H.1992. Microeconomic analysis. Norton & Company. New York. Chapters 7, 8, 9, 10, 11, 19. Camerer, C. , Loewenstein, G. , and Rabin, M. 2004. Advances in Behavioral Economics. Princeton University Press. Mas-Colell, A., Whinston, M., and Green, J. 1995. Microeconomic theory. Oxford University Press. Chapters 1, 2, 3, 4, 6. 1 Two approaches to analyze behavior of decision makers. 1. From an abstract construction generate choices that satisfy rationality restrictions. 2. Directly from choice, ask for some structure to these choices and derive properties. Relation between the two approaches. 2 1. Preference and choice Consider a set of alternatives (mutually exclusives) X and we consider a preference relation on X. Preferences The preference relation “at least as good as” is a binary relation − over the set of alternatives X. Preferences express the ordering between alternatives resulting from the objectives of the decision maker. Properties: Rational Preference Relation (or Weak order, or Complete preorder) if -Completeness, Transitivity (and Reflexivity) A relation is complete if ∀∈x,,yXxy or yx. A relation is transitive if and only if ∀∈x,yz, X, if x≺≺y and y z, then x≺z. 3 Transitiveness is more delicate than completeness. Cases where transitiveness may not hold are: -Just perceptible differences -Framing: Choice depends on how the problem is framed. Kahneman & Tversky. From can also be deduced the indifference relation ~ and the strict − preference relation and the corresponding properties when is − rational. ~ is reflexive, transitive and symmetric. is irreflexive and transitive. Utility A function u : X → ℜ is a utility function representing a preference relation if: − x y ⇔ u()x ≥ u(y) − -Any monotone (strictly increasing) transformation of u(x) will be also a utility function representing the same preferences. That is, if f is an increasing function, v(x) = f(u(x)) represents the same preferences as u(x). 4 -A preference relation can be represented by a utility function only if it is rational. -Not all rational preference relations can be represented by a utility function. (It’s enough if X is finite). Choice Another way to analyze decisions is via choices that we suppose satisfy some minimal properties to understand behavior. -We define a budget set Β∈ X as a subset of X . Β is a family of nonempty sets of X. - A choice rule C() is a correspondence that selects a subset of B for every B defined in the problem. Since any choice is possible, we impose a “reasonable” restriction on choices that we call the weak axiom of revealed preference. If an individual chooses x when she can choose between x and y, then she shouldn’t choose y when faced with x, y, z. 5 Weak Axiom of Revealed Preference If for some B ∈Β, with x, y∈ B we have x∈CB( ) , then ∀∈Bx′Β with , y∈B′ and y∈C(B′) we must also have x∈CB( ′) . Examples: Xx= { ,,yz} Β={}{}xy,,{xy,,z},{x} Possible Choice Structure: ()Β⋅,C1 () ⎫ Cx1 (){}, y= x⎪ ⎬ → It does not satisfy the Weak axiom. Cx,,yz y 1 (){}= ⎭⎪ Xx= { ,,yz} Β={}{}xy,,{xy,,z},{x} Possible Choice Structure: ()Β⋅,C2 () ⎫ Cx2 (){}, y= x ⎪ ⎬ → It does not satisfy the Weak axiom Cx,,yz x,y 2 (){}= {}⎭⎪ Revealed preference relation xy* ⇔∃ B∈Β such that x, y∈B and x∈C(B) . x * yB ⇔∃ ∈Β such that x, y∈B and x∈C()B and y∉C(B) WARP: If x is revealed at least as good as y, then y cannot be revealed preferred to x. 6 Relation between preference approach and choice approach If an agent has a rational preference relation , her decisions − generate a choice structure satisfying the Weak Axiom of Revealed Preference. CB* (), =∈{xB: x y for every y∈B} Suppose CB* (,) is nonempty. Proof: if we have a rational preference relation , we must show that (Β⋅,C* (,)) satisfies the weak axiom. Suppose we have x,yB∈∈ and xC* (B,) ⇒x y. Consider now Bx′ with , y∈∈B′′ and yC* (B,) ⇒yz,∀z∈B′. By transitivity, since xy and yz, then xz,∀∈zB′ ⇒x∈C* ()B′,. 7 If choices from a choice structure satisfy the weak axiom, is there a rational preference relation consistent with them? Given a choice structure (Β,C (⋅)) , we say that a rational preference relation rationalizes C() relative to Β if CB( ) = C* (B,) The Weak Axiom is not sufficient to guarantee that there exists a preference relation that rationalizes choices. Example: Xx= { ,,yz} Β={}{}xy,,{y,z},{x,z} Cx(){},,y==={}xC(){y,z} {}y,Cx(){,z} {}z WARP is satisfied. Rationalizable preferences do not exist. Cx(){}, y=⇒{}x x y⎪⎫ ⎬transitivity : x z but C (){}x, z = {}z Cy(){}, z=⇒{}y y z⎭⎪ 8 If a choice structure satisfies: -Weak axiom - Β includes all subsets of X of up to three elements. Then there is a rational preference relation that rationalizes C() relative to Β and it is unique. (the revealed preference relation). This result is positive but too restrictive for many situations that economists care about. The result needs choice defined over all subsets of X and many economic situations (consumer problem) define choice for certain subsets. Hence, in consumer theory the WARP is not enough to guarantee the equivalence. More conditions on choice will be needed. 9 Consumer choice Commodities: a commodity vector x (commodity bundle) is a list of the amount of the L commodities. R L is the commodity space. We will use commodity vectors also as consumption vectors. ⎡⎤x1 ⎢⎥ ⎢⎥⋅ x = ⎢⎥⋅ ⎢⎥ ⎢⎥⋅ ⎢⎥ ⎣⎦xL Consumption set: X ∈ RL . The consumption set incorporates physical or institutional restrictions on the commodity space. Leisure hours 24 8 Bread x2 L X = R+ LL Xx==RR+ { ∈: xl ≥0,l=1,..., L} x1 10 Competitive budgets: Consumers must choose facing economic constraints. 1. L commodities traded in the market at monetary prices publicly quoted (principle of completeness or universality of markets). We further assume, for convenience, that p 0, that is pl >0 2. Consumers are price-taking. Affordability depends on prices, p, and consumer’s wealth, w. L xp∈⋅R+ is affordable if x=p11x+p2x2+⋅⋅⋅pLLx≤w. L Walrasian ()competitive budget :Bpw, =∈{}x R+ :p ⋅x ≤w x2 L {x ∈⋅R+ : px=w} p slope =− 1 Bpw, p2 x1 The Walrasian budget set is convex. (Show it) 11 Demand functions The Walrasian (market, ordinary) demand correspondence x()pw, assigns a set of consumption bundles from each pair ( pw, ) . Assumptions: 1. Homogeneity of degree zero. x(ααp, w) = x( pw, ),∀>pw, and α0. Homogeneity of degree zero allows us normalizations 2. Walras’ Law: x()pw, satisfies Walras' Law if ∀>p 0 and w 0, then p⋅x=w,∀x∈x()p, w Notice that we have a choice structure (ΒW , x(⋅)) that does not include all possible subsets of X. W Β={Bp,w :0 ,w>0} . By homogeneity of degree zero, x() depends only on the budget set. 12 Comparative statics Wealth Wealth effects: Given p , x( pw, ) is the consumer’s Engel Function. L Its image in R+ ,,Exp ={ ()pw:w>0} is the Wealth Expansion Path. We ∂xpl (, w) call ∂w the wealth effect for good l. x2 Ep x1 ∂xp, w l ( ) ≥ 0 - Commodity l is normal if ∂w . Demand is normal if every commodity is normal. ∂xp, w l ( ) < 0 - Commodity l is inferior if ∂w . It is common to assume normality for aggregates. Specific goods usually become inferior at some point by substitution. 13 Prices Offer curve p2 p2 ↓ x2 x2 x1 p2 ↓ x2 w p2′ x2 is Giffen at p2′ x1 A good can be Giffen only if it is inferior. 14 -If the Walrasian demand function is homogeneous of degree zero, ( x()ααp, w =∀x(pw, ), pw, and α>0.) then for all p and w. L ∂∂xpll(),,w x()pw ∑ pk +=wl0 for =1,..., L. k=1 ∂∂pwk We define the elasticities as the percentage change in demand per marginal percentage change in prices or wealth: ∂xpl (), w pk εlk ()pw, = ∂pklxp(), w ∂xpl (), w w εlw ()pw, = ∂wxl ()p, w L ∑εεlk ()p, wp+=lw (), w0 for l=1,..., L. k=1 15 - If Walras’ law is satisfied x()pw, satisfies Walras' Law if ∀>p 0 and w 0, then p⋅x=w,∀x∈x()p, w L ∂xpl (), w ∑ plk+=xp(), w 0 for k =1,..., L. l=1 ∂pk Total expenditure cannot change in response to a change in price. L ∂xpl (), w ∑ pl =1 . l=1 ∂pw If we define the budget share of expenditure in commodity l as px⋅ p, w bp, w= ll(), we can also derive: l () w L ∑bpll(),,wε k()pw+=bk(p,w)0 l=1 L ∑bpll(),,wε w(pw)=1. l=1 16 Weak axiom of Revealed Preference Assume that x( pw, ) is single valued, homogeneous of degree zero and satisfies Walras’ Law. x(pw, ) satisfies the WARP if the following condition holds for any two pairs ()p, wp and (′, w′): If px()p′′, w ≤≠w and x(p′′, w)x( pw, ) then p′x( pw, ) >w′ This means that if x(pw′, ′) was affordable at ( p, w) and was not chosen, then x( pw, ) must be preferred and will not be chosen at (p′, w′) because is not affordable.
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