Short course on Comparative Statics and Comparative Information

Short course on Comparative Statics and Comparative Information Course description: This course provides a concise and accessible introduction to some basic concepts and techniques used when comparing information structures or solutions to optimization problems. It will introduce concepts like supermodularity and the single crossing property and show its relevance for guaranteeing that the solutions to an optimization problem are monotonically increasing (or decreasing) with respect to various parameters. Consideration will be given to both decision-making with and without uncertainty. Lehmann s notion of informativeness and its role in valuing information structures will also be examined. Applications to games with strategic complementarities and to statistical decision theory will be discussed. Style: This is a short course and not a seminar. While there will be some discussion of very new results, most of the time will be spent on known results in this area that I consider important. The emphasis is on providing an accessible introduction that will allow anyone interested in the techniques or its applications to do further reading and research. To give a flavor of how the arguments work, I shall at various points be proving results in some detail. In other words, I shall be sacrificing some coverage for the sake of providing greater depth and texture. Readings: AMIR, R. (1996): Cournot Oligopoly and the Theory of Supermodular Games, Games and Economic Behavior, 15, 132-148. AMIR, R. (2005): Supermodularity and Complementarity in Economics: An Elementary Survey, Southern Economic Journal, 71(3), 636660. ATHEY, S. (2001): Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information, Econometrica, 69(4), 861-890. ATHEY, S. (2002): Monotone Comparative Statics Under Uncertainty, The Quarterly Journal of Economics, 117(1), 187-223. ATHEY, S., P. MILGROM, AND J. ROBERTS (1998): (Draft Chapters, available on Athey s webpage.) Robust Comparative Statics. BLACKWELL, D. AND M. A. GIRSHIK (1954): Theory of Games and Statistical Decisions. New York: Dover Publications ECHENIQUE, F. (2002): Comparative Statics by Adaptive Dynamics and the Correspondence Principle, Econometrica, 70(2), 833-844. GOLLIER, C. (2001): The Economics of Risk and Time. Cambridge: MIT Press. KARLIN, S. AND H. RUBIN (1956): The Theory of Decision Procedures for Distributions with Monotone Likelihood Ratio, The Annals of Mathematical Statistics, 27(2), 272-299. LEHMANN, E. L. (1988): Comparing Location Experiments, The Annals of Statistics, 16(2), 521-533. 1

MILGROM, P. (1994): Comparing Optima: Do Simplifying Assumptions Affect Conclusions? Journal of Political Economy, 102(3), 607-615. MILGROM, P. AND C. SHANNON (1994):* Monotone Comparative Statics, Econometrica, 62(1), 157-180. MILGROM, P. AND J. ROBERTS (1990):* Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities, Econometrica, 58(6), 1255-1277. MILGROM, P. AND J. ROBERTS (1996): The LeChatelier Principle, American Economic Review, 86(1), 173-179. QUAH, J. K.-H. (2007):* The Comparative Statics of Constrained Optimization Problems, Econometrica, Vol. 75, No. 2, 401-431. QUAH, J. K.-H. AND B. STRULOVICI (2007):* Comparative Statics, Informativeness, and the Interval Dominance Order, Nuffield College Working Paper No. 4. TOPKIS, D. M. (1998):* Supermodularity and Complementarity. Princeton: Princeton University Press. VIVES, X. (1990): Nash Equilibrium with Strategic Complementarities, Journal of Mathematical Economics, 19(3), 305-321. VIVES, X. (1999): Oligopoly Pricing. Cambridge: MIT Press. ZHOU, L. (1994): The Set of Nash Equilibria of a Supermodular Game is a Complete Lattice, Games and Economic Behavior, 7(2), 295-300. The closest thing to a textbook for the material covered in this course is the book by Topkis. The survey by Amir and the incomplete notes by Athey et al. are also relevant. The books by Gollier and Vives are not books on mathematical techniques as such, but they do make use of the concepts and techniques covered in the course. The book by Blackwell and Girschik on statistical decision theory is very old, but it is beautifully written and provides an introduction to the topic that is congenial to economists. The readings most closely related to the lectures are indicated with asterisks*. 2

Short Course oncomparative Statics and Comparative Information p. 1/2 Short Course on Comparative Statics and Comparative Information John Quah

Short Course oncomparative Statics and Comparative Information p. 2/2 One-dimensional comparative statics Let X R and f, g : X R. We are interested in comparing argmax x X f(x) with argmax x X g(x). Standard approach: Assume X is a compact interval and f and g are quasi-concave functions. Let x be the unique maximizer of f. Then f (x ) = 0. Show that g (x ) 0. Then optimum has shifted to the right. This approach makes various assumptions, most notably the quasi-concavity of f and g.

Short Course oncomparative Statics and Comparative Information p. 3/2 One-dimensional comparative statics Assume that f and g are continuous functions and their domain X is compact. Then argmax x X f(x) and argmax x X g(x) are nonempty. But these sets need not be singletons or intervals. First question: how do we compare sets? Definition: Let S and S be subsets of R. S dominates S in the strong set order (S S ) if for any for x in S and x in S, we have max{x, x } in S and min{x, x } in S. Example: {3, 5, 6, 7} {1, 4, 6}

Short Course oncomparative Statics and Comparative Information p. 4/2 One-dimensional comparative statics Definition: g dominates f by the single crossing property (g SC f) if for all x and x such that x > x, the following holds: f(x ) f(x ) (>) 0 = g(x ) g(x ) (>) 0. (2) Let S be a partially ordered set. A family of real-valued functions {f(, s)} s S is an SCP family if the functions are ordered by the single crossing property (SCP), i.e., whenever s > s, we have f(, s ) SC f(, s ). Motivation for the term single crossing : for any x > x, the function : S R defined by (s) = f(x, s) f(x, s) crosses the horizontal axis at most once.

Short Course oncomparative Statics and Comparative Information p. 4/2 One-dimensional comparative statics Definition: g dominates f by the single crossing property (g SC f) if for all x and x such that x > x, the following holds: f(x ) f(x ) (>) 0 = g(x ) g(x ) (>) 0. (3) Let S be a partially ordered set. A family of real-valued functions {f(, s)} s S is an SCP family if the functions are ordered by the single crossing property (SCP), i.e., whenever s > s, we have f(, s ) SC f(, s ). Motivation for the term single crossing : for any x > x, the function : S R defined by (s) = f(x, s) f(x, s) crosses the horizontal axis at most once. Single crossing is an ordinal property: if g SC f, then φ g SC ψ f, where φ and ψ are increasing functions mapping R to R.

Short Course oncomparative Statics and Comparative Information p. 5/2 One-dimensional comparative statics In this case, f(, s ) SC f(, s ) y f (, s) ) f (, s) x x x

Short Course oncomparative Statics and Comparative Information p. 6/2 One-dimensional comparative statics g f f * x ** x g x

Short Course oncomparative Statics and Comparative Information p. 6/2 One-dimensional comparative statics g f f * x ** x g x In the top picture f SC g, but not in the bottom picture.

Short Course oncomparative Statics and Comparative Information p. 7/2 One-dimensional comparative statics Definition: g dominates f by the increasing differences (g IN f) if for all x and x such that x > x, the following holds: g(x ) g(x ) f(x ) f(x ) (4) Clearly, g IN f implies g SC f. Similarly, a family {f(, s)} s S satisfies increasing differences if the functions are ordered by increasing differences, i.e., whenever s > s, we have f(, s ) IN f(, s ). Let S be an open subset of R l and X an open interval. Then a sufficient (and necessary) condition for increasing differences is at every point (x, s) and for all i. 2 f x s i (x, s) 0

Short Course oncomparative Statics and Comparative Information p. 8/2 One-dimensional comparative statics Theorem 1: (MS-94) Suppose X R and f, g : X R; then argmax x Y g(x) argmax x Y f(x) for any Y X if and only if g SC f. Proof: Assume g SC f, x argmax x Y f(x), and x argmax x Y g(x). We have to show that max{x, x } argmax x Y g(x) and min{x, x } argmax x Y f(x). We need only consider the case where x > x. Since x is in argmax x Y f(x), we have f(x ) f(x ) and since g SC f, we also have g(x ) g(x ); thus x is in argmax x Y g(x). Furthermore, f(x ) = f(x ) so that x is in argmax x Y f(x). If not, f(x ) > f(x ) which implies (by the fact that g SC f) that g(x ) > g(x ), contradicting the assumption that g is maximized at x.

Short Course oncomparative Statics and Comparative Information p. 9/2 One-dimensional comparative statics Application: Recall Π(x, c) = xp(x) cx. Then {Π(, c)} c R obey increasing differences, since 2 Π x c = 1. By Theorem 1, argmax x X Π(x, c) is increasing in c. For example, argmax x X Π(x, 3) argmax x X Π(x, 5). In other words, the profit-maximizing output decreases as the marginal cost of output increases.

Short Course oncomparative Statics and Comparative Information p. 10/2 One-dimensional comparative statics Application: Bertrand Oligopoly with differentiated products, with Π a (p a, p a ) = (p a c a )D a (p a, p a ) lnπ a (p a, p a ) = ln(p a c a ) + lnd a (p a, p a ) So {lnπ a (, p a )} a A has increasing differences if 2 p a p a [lnd a ] 0. This is equivalent to p a [ p ] a D a D a p a 0; i.e., firm a s own-price elasticity of demand, ǫ a (p a, p a ) = p a D a D a p a decreases with p a. If this assumption holds, argmax pa P Π a(p a, p a ) increases with p a.

Short Course oncomparative Statics and Comparative Information p. 11/2 Supermodular Games Let X = Π N i=1 X i, where each X i is a compact interval. Theorem 2: (Tarski) Suppose φ : X X is an increasing function. Then the set of fixed points {x X : φ(x) = x} is nonempty. In fact, x = sup{x X : x φ(x)} is a fixed point and is the largest fixed point, i.e., for any other fixed point x, we have x x. Note: φ need not be continuous.

Short Course oncomparative Statics and Comparative Information p. 12/2 Supermodular Games Theorem 3: Suppose φ(, t) : X X is increasing in (x, t). Then the largest fixed point of f(, t) is increasing in t. Proof: By Theorem 2, the largest fixed point x(t) = sup{x X : x φ(x, t)}. Suppose t > t ; since φ(x, t ) φ(x, t ) for all x, Therefore, {x X : x φ(x, t )} {x X : x φ(x, t )}. sup{x X : x φ(x, t )} sup{x X : x φ(x, t )}. By Theorem 2 again, x(t ) x(t ). QED

Short Course oncomparative Statics and Comparative Information p. 13/2 Supermodular Games Bertrand Oligopoly: assume the set of firms is A; the typical firm a chooses its price from the compact interval P to maximize Π a (p a, p a ) = (p a c a )D a (p a, p a ). Recall: if own-price elasticity is decreasing in p a then the family {D a (, p a )} p a forms an SCP family. Define B a (p a ) = argmax pa P Π a(p a, p a ). This is the set of a s best responses to the pricing of other firms. It is nonempty if Π a is continuous. Define B a (p a ) = max [ argmax pa P Π a(p a, p a ) ] ; This is the largest best response to the pricing of other firms. This is an increasing function of p a (by Theorem 1).

Short Course oncomparative Statics and Comparative Information p. 14/2 Supermodular Games By Theorem 1, B is an increasing function. By Tarski s Fixed Point Theorem, a fixed point exists. Specifically, p = sup{p P : p B(p)} is a fixed point of the map B and thus a NE. In fact, this is the largest NE. Why? Let ˆp = (p a ) a A be any other NE. Then ˆp a B a (ˆp a ). So ˆp a B a (ˆp a ), and we obtain ˆp B(ˆp). Thus, ˆp {p P : p B(p)}, which implies ˆp p sup{p P : p B(p)}. In this game, best response function is monotonic but not necessarily continuous. Existence guaranteed by appealing to Tarski s fixed point theorem, rather than Kakutani s. The game has a largest NE. We can do comparative statics exercises on the largest NE...

Short Course oncomparative Statics and Comparative Information p. 15/2 Supermodular Games What happens to the largest NE when firm ã experiences an increase in marginal cost from cã to c ã? Recall Observe that Πã(pã, p ã, cã) = (pã cã)dã(pã, p ã ) lnπã(pã, p ã, cã) = ln(pã cã) + lndã(pã, p ã ) pã cã [lnπã] > 0. By Theorem 1, firm a s best response increase with cã (for fixed p a ). Formally, Bã(p ã, c ã) Bã(p ã, cã). This implies that B(p, c ã ) B(p, cã). So largest fixed point of B(, c ã ) is larger than the largest fixed point of B(, cã) (by Theorem 3). In other words, if firm ã s marginal cost increases from cã to c ã, the largest NE increases: every firm increases its price.

Short Course oncomparative Statics and Comparative Information p. 16/2 Decisions Under Uncertainty Recall that a family {f(, s)} s S is an SCP family (for S R) if for any x > x, the function : S R defined by (s) = f(x, s) f(x, s) crosses the horizontal axis at most once. Definition: A function : S R is said to have property ( ) if, whenever s > s, (s ) (>) 0 = (s ) (>) 0. Definition: Consider the family h(, t) : S R where S R. This family obeys log increasing differences if {ln h(, t)} t T obeys increasing differences. Equivalently, whenever s > s, h(s, t) h(s, t) increases with t.

Short Course oncomparative Statics and Comparative Information p. 17/2 Decisions Under Uncertainty Theorem 4: Suppose S R, the function : S R has property ( ), and {h(, t)} t T obeys log increasing differences. Then φ(t) = (s)h(s, t)ds has property ( ). S Indeed, if crosses the horizontal axis at s = s 0, then, for t 2 > t 1, φ(t 2 ) h(s 0, t 2 ) h(s 0, t 1 ) φ(t 1).

Short Course oncomparative Statics and Comparative Information p. 18/2 Decisions Under Uncertainty Proof: We split the integral φ(t 2 ) = S (s)h(s, t 2)ds into two: φ(t 2 ) = s0 (s)h(s, t 1 ) h(s, t 2) h(s, t 1 ) ds + s 0 (s)h(s, t 1 ) h(s, t 2) h(s, t 1 ) ds. Between [, s 0 ], (s) 0, so the first term on the right is greater than h(s 0, t 2 ) s0 (s)h(s, t 1 ) ds. h(s 0, t 1 ) Between [s 0. ], (s) 0, so the second term is greater than h(s 0, t 2 ) h(s 0, t 1 ) s 0 (s)h(s, t 1 ) ds. Adding up the two lower bounds gives us φ(t 2 ) h(s 0, t 2 ) (s)h(s, t 1 ) ds = h(s 0, t 2 ) h(s 0, t 1 ) h(s 0, t 1 ) φ(t 1). S. QED

Short Course oncomparative Statics and Comparative Information p. 19/2 Decisions Under Uncertainty Application: consider Bernoulli utility functions v(, t) parameterized by t. Assume that the family {v (, t)} t T obeys log increasing differences. This is equivalent to v (z, t 2 ) v (z, t 2 ) (z, t 1 ) v v (z, t 1 ) whenever t 2 > t 1 ; i.e., type t 2 is less risk averse than type t 1. Agent has wealth w and chooses between a safe and risky asset to maximize expected utility V (x, t) = v((w x)r + xs, t) λ(s) ds r and s are payoffs of safe and risky assets, latter with density λ; x [0, w] is the investment in the risky asset.

Short Course oncomparative Statics and Comparative Information p. 20/2 Decisions Under Uncertainty Recall V (x, t) = v((w x)r + xs, t) λ(s) ds. Differentiating w.r.t x, V (x, t) = (s r)λ(s) v ((w x)r + xs, t) ds. s S By Theorem 4 and the fact that {v (, t)} t T obeys log increasing differences, for any t 2 > t 1, So whenever x > x, V (x, t 2 ) v (wr, t 2 ) v (wr, t 1 ) V (x, t 1 ). V (x, t 2 ) V (x, t 2 ) v (wr, t 2 ) v (wr, t 1 ) [V (x, t 1 ) V (x, t 1 )]. Clearly, the family {V (, t)} t T obeys SCP.

Short Course oncomparative Statics and Comparative Information p. 21/2 Decisions Under Uncertainty Definition: A family of density functions {λ(, t)} t T (defined on S R) that obeys log increasing differences is said to respect the monotone likelihood ratio order (MLR). Fact: If {λ(, t)} t T is MLR-ordered then it is ordered by first order stochastic dominance (FOSD), i.e., whenever t 2 > t 1, λ(, t 2 ) first order stochastically dominates λ(, t 1 ). Theorem 5: Suppose {f(, s)} s S (for S R) is an SCP family. Then the family {F(, t)} t T (for T R) given by F(x, t) = f(x, s) λ(s, t) ds s S is also an SCP family if {λ(, t)} t T is MLR-ordered.

Short Course oncomparative Statics and Comparative Information p. 22/2 Decisions Under Uncertainty Theorem 5: Suppose {f(, s)} s S (for S R) is an SCP family. Then the family {F(, t)} t T given by F(x, t) = f(x, s) λ(s, t) ds s S is also an SCP family if {λ(, t)} t T is MLR-ordered. Proof: The function (s) = f(x, s) f(x, s) has property ( ). By Theorem 4, the function φ given by φ(t) = F(x, t) F(x, t) = [f(x, s) f(x, s)] λ(s, t) ds s S = (s) λ(s, t) ds also obeys property ( ). In other words, {F(, t)} t T is an SCP family. QED s S

Short Course oncomparative Statics and Comparative Information p. 23/2 Decisions Under Uncertainty Theorem 6: Suppose {f(, s)} s S (for S R) satisfies increasing differences. Then the family {F(, t)} t T given by F(x, t) = f(x, s) λ(s, t) ds s S also satisfies increasing differences if {λ(, t)} t T is FOSD-ordered. Proof: Since {f(, s)} s S satisfies increasing differences, for x > x, the function (s) = f(x, s) f(x, s) is increasing. By a standard result (Rothschild-Stiglitiz), (s)f(s, t 2 )ds (s)f(s, t 1 )ds if t 2 > t 1 S Re-writing this expression, we obtain S V (x, t 2 ) V (x, t 2 ) V (x, t 1 ) V (x, t 1 ), as required by increasing differences. QED

Short Course oncomparative Statics and Comparative Information p. 24/2 Decisions Under Uncertainty Application: The family {f(, s)} s S given by f(x, s) = v((w x)r + xs) is an SCP family. By Theorem 5, the family {F(, t)} t T given by F(x, t) = v((w x)r + xs) λ(s, t) ds is an SCP family provided {λ(, t)} t T is MLR-ordered. By Theorem 1, argmax x [0,w] F(x, t) increases with t. Interpretation: an MLR shift in the payoff distribution λ(, t) means that higher payoffs for the risky asset are more likely, so investment in the risky asset increases.

Short Course oncomparative Statics and Comparative Information Part II p. 1/2 Short Course on Comparative Statics and Comparative Information Part II John Quah john.quah@economics.ox.ac.uk

Short Course oncomparative Statics and Comparative Information Part II p. 2/2 Recall... One-dimensional comparative statics Definition: Let S and S be subsets of R. S dominates S in the strong set order (S S ) if for any for x in S and x in S, we have max{x, x } in S and min{x, x } in S. Example: {3, 5, 6, 7} {1, 4, 6} but {3, 4, 5, 6, 7} {1, 3, 4, 5, 6}. Note: if S = {x } and S = {x }, then x x. Definition: g dominates f by the single crossing property (g SC f) if for all x and x such that x > x, the following holds: f(x ) f(x ) (>) 0 = g(x ) g(x ) (>) 0. (2)

Short Course oncomparative Statics and Comparative Information Part II p. 3/2 Definitions: Multidimensional comparative statics We endow R l with the product order: x x if x i x i for all i. An element y in R l is an upper bound of S R l if y x for all x in S. It is the least upper bound (or supremum) if it is an upper bound and for any other other upper bound y, we have y y. An element z in R l is a lower bound of S R l if z x for all x in S. It is the greatest lower bound (or infimum) if it is a lower bound and for any other other lower bound z, we have z z.

Short Course oncomparative Statics and Comparative Information Part II p. 4/2 Multidimensional comparative statics We denote the supremum of x and x by x x and their infimum by x x. Easy to check that x x = (max{x 1, x 1, max{x 2, x 2},..., max{x l, x l }) and x x = (min{x 1, x 1, min{x 2, x 2},..., min{x l, x l }). x x x x x x A set S is a sublattice of R l if for any x and x in S, x x and x x are both in S.

Short Course oncomparative Statics and Comparative Information Part II p. 5/2 Multidimensional comparative statics More definitions: Let S and S be subsets of R l. S dominates S in the strong set order (S S ) if for any for x in S and x in S, we have x x in S and x x in S. Note: (1) If S = {x } and S = {x }, then S S if and only if x x. (2) Suppose S and S both contain their suprema; then S S implies that sups sups. Proof of (2): Let x and x be suprema of S and S respectively. Then x x is in S. Since x is sups, we have x x x, which implies s x. QED

Short Course oncomparative Statics and Comparative Information Part II p. 6/2 Multidimensional comparative statics Let X be a sublattice of R l and f : X R. The function f is supermodular (SPM) if f(x x ) f(x ) f(x ) f(x x ). x x x x x x

Short Course oncomparative Statics and Comparative Information Part II p. 7/2 Multidimensional comparative statics Theorem 2.1: Let X be a sublattice of R l and suppose that f : X R is a QSM function. Then argmax x X f(x) is a sublattice. Proof: Suppose x and x are both in argmax x X f(x). Since X is a sublattice, x x and x x are both in is in X. Note that f(x ) f(x x ) since x argmax x X f(x). The QSM property guarantees that f(x x ) f(x ), which implies that x x argmax x X f(x). Etc. QED x x x x x x

g(x ) g(x ) f(x ) f(x ) (5) Short Course oncomparative Statics and Comparative Information Part II p. 8/2 Multidimensional comparative statics Definition: g dominates f by the single crossing property (g SC f) if for all x and x such that x > x, the following holds: f(x ) f(x ) (>) 0 = g(x ) g(x ) (>) 0. (4) x x x x x x Definition: g dominates f by the increasing differences (g IN f) if for all x and x such that x > x, the following holds:

Short Course oncomparative Statics and Comparative Information Part II p. 9/2 Multidimensional comparative statics A family {f(, s)} s S satisfies SCP if the functions are ordered by SCP, i.e., whenever s > s, we have f(, s ) SC f(, s ). Clearly, g IN f implies g SC f. Similarly, a family {f(, s)} s S satisfies increasing differences if the functions are ordered by increasing differences, i.e., whenever s > s, we have f(, s ) IN f(, s ). Fact: Let S be an open subset of R l and X an open interval. Then a sufficient (and necessary) condition for increasing differences is 2 f x i s j (x, s) 0 at every point (x, s) and for all i and j.

Short Course oncomparative Statics and Comparative Information Part II p. 10/2 Multidimensional comparative statics Theorem 2.1: (MS-94) Suppose X R is a sublattice and f, g : X R two QSM functions. Then argmax x Y g(x) argmax x Y f(x) for every sublattice Y contained in X if and only if g SC f. Proof: Let x argmax x Y f(x) and x argmax x Y g(x). Then f(x ) f(x x ). By QSM of f, f(x x ) f(x ). Since g SC f, we obtain g(x x ) g(x ), which implies that x x is in argmax x Y g(x). Etc. QED x x x x x x

Short Course oncomparative Statics and Comparative Information Part II p. 11/2 Application: Multidimensional comparative statics Let x denote the vector of inputs (drawn from X = R l +), p the vector of input prices, and R the revenue function mapping input vector x to revenue (in R). The firm s profit is Note that So there is decreasing differences. Π(x; p) = R(x) p x. 2 Π x i p j (x; p) = 1. If R is supermodular, then Π is supermodular. Conclusion: By Theorem 2.1, argmax x X Π(x; p ) argmax x X Π(x; p ) if p < p.

Short Course oncomparative Statics and Comparative Information Part II p. 12/2 One-dimensional comparative statics - again! Recall that g SC f if, for x > x, we have f(x ) f(x ) (>) 0 g(x ) g(x ) (>) 0. But look at these pictures g f f * x ** x g x g SC f in the upper diagram but not in the lower. This is unsatisfactory since in both cases, the optimum has increased.

Short Course oncomparative Statics and Comparative Information Part II p. 13/2 The interval dominance order Motivation for the interval dominance order: to develop an ordering for functions that captures both situations. Let X R and f, g : X R. Recall: g SC f if, for any pair x > x, we have f(x ) f(x ) (>) 0 g(x ) g(x ) (>) 0. g dominates f by the interval dominance order (g I f) if for any pair x > x satisfying

Short Course oncomparative Statics and Comparative Information Part II p. 14/2 The interval dominance order g I f if for any pair x > x satisfying f(x ) f(x) for x in the interval [x, x ] = {x X : x x x }, the following holds: f(x ) f(x ) (>) 0 g(x ) g(x ) (>) 0. ( ) g f f * x ** x g x Notice that ( ) need not be applied to x and x.

Short Course oncomparative Statics and Comparative Information Part II p. 15/2 The interval dominance order A subset I of X is called an interval of X if for any x and x in I and x in X such that x < x < x, we have x in I. Example: If X = {1, 2, 3, 5, 6}, then {1, 2, } and {3, 5, 6} are intervals of X but {3, 6} is not an interval of X. Theorem 2.2: Suppose that f and g are real-valued functions defined on X R. Then g I f if and only if argmax x I g(x) argmax x I f(x) for any interval I of X. Compare this with

Short Course oncomparative Statics and Comparative Information Part II p. 16/2 IDO with Uncertainty x Basic example of IDO family: for every s, u(, s) is quasiconcave (in x) with argmax u(x, s ) argmax u(x, s ) for s > s. s ' s '' s ''' This family of quasiconcave functions with increasing peaks need not be an SCP family. At each state s there is an optimal action, which increases with s.

Short Course oncomparative Statics and Comparative Information Part II p. 17/2 IDO with Uncertainty We expect the optimal choice of x to increase when higher states are more likely. Recall: a family of density functions {λ(, t)} t T (defined on S R) respects the monotone likelihood ratio order (MLR) if for t > t, λ(s, t ) λ(s, t ) increases with s. Theorem 2.2: (QS-07) Suppose {u(, s)} s S forms an IDO family and {λ(, t)} t T is ordered by MLR. Then {U(, t)} t T, defined by U(x; t) = u(x, s)λ(s, t)ds forms an an IDO family. Consequently, argmax x X U(x, t) is increasing in t. S

Short Course oncomparative Statics and Comparative Information Part II p. 18/2 IDO with Uncertainty Example: a Firm decides on the time x to launch a new product. The more time the Firm gives itself, the more it can improve the quality of the product and its manufacturing process. But there is a Rival who may launch a similar product at time s. If the Firm is anticipated, it suffers a discrete fall in profit. Graphically, the Firm s family of profit functions {π(, s)} s S looks like (, s ) x ( s') x ( s'') ( ''') x s x The firm chooses x to maximize F(x, λ) = s S π(x, s)λ(s)ds. Note that {π(, s)} s S is an IDO, but not necessarily SCP family. By the Theorem 2.2, Firm s optimal launch time increases if λ experiences MLR shift.

Short Course oncomparative Statics and Comparative Information Part II p. 19/2 Statistical Decision Theory Agent chooses action after observing a signal z; z is drawn from an interval Z (Z R), but before realization of state. Agent s decision rule is a map from Z to X R. The distribution over signals at a state s S is H(z s) (with density h(z s)). We refer to the family {H( s)} s S as the information structure H. Expected utility of rule φ in state s is Z u(φ(z), s)dh(z s). How should a decision maker choose amongst decision rules?

Short Course oncomparative Statics and Comparative Information Part II p. 20/2 Statistical Decision Theory Theorem 2.3: (QS-07) Suppose {u(, s)} s S is an IDO family and {h( s)} s S is MLR-ordered. Then there exists an increasing decision rule that maximizes the Bayesian s ex ante utility. Proof: We denote that posterior distribution (over states) conditional on z by H P ( z). The ex ante utility of rule φ is [ ] u(φ(z), s)dh(z s) dp(s) = u(φ(z), s)d H P (s z) dν H S Z z Z where ν H is the marginal distribution of z. It follows that an optimal decision rule can be found by choosing, for each signal z, an action that maximizes argmax x X s S u(x, s)d H P (s, z). s S

Short Course oncomparative Statics and Comparative Information Part II p. 21/2 Statistical Decision Theory Proof: We denote that posterior distribution (over states) conditional on z by H P ( z). The ex ante utility of rule φ is [ ] u(φ H (z), s)dh(z s) dp(s) = u(φ H (z), s)d H P (s z) dν H S Z z Z where ν H is the marginal distribution of z. It follows that an optimal decision rule can be found by choosing, for each signal z, an action that maximizes argmax x X s S u(x, s) dh P (s, z). We know that { h P ( z)} z Z is MLR-ordered because {h( s)} s S is MLR-ordered. By Theorem 2.2, for z > z, we have argmax x X u(x, s) h P (s, z )ds argmax x X u(x, s) h P (s, z )ds. s S ] For each z, choose φ (z) = max [argmax x X s S u(x, s) h P (s, z)ds. Then φ is optimal and increasing in z. QED s S s S

Short Course oncomparative Statics and Comparative Information Part II p. 22/2 Lehmann informativeness Assume prior is P and consider information structure H. If φ H is the optimal decision rule, then agent s ex ante utility is V(H, P) = u(φ H (z), s)dh(z s) dp(s) S Z [ ] = u(φ H (z), s)d H P (s z) dν H z Z s S where ν H is the marginal distribution of z. We wish to compare H with another information structure G = {G( s)} s S. The ex ante utility of G is V(G, P) = u(φ G (z), s)dg(z s) dp(s) where φ G is the optimal decision rule for G. S Z

Short Course oncomparative Statics and Comparative Information Part II p. 23/2 Lehmann informativeness Given information structures H {H( s)} s S and G {G( s)} s S, define T(z, s) by H(T(z, s) s) = G(z s). Definition: H is more informative than G (in the sense of Lehmann) if T(z, ) is increasing in s.

Short Course oncomparative Statics and Comparative Information Part II p. 24/2 Lehmann informativeness Theorem 2.4: (QS-07) Suppose that (i) information structure H is more informative than G (ii) {u(, s)} s S is an IDO family (iii) G = {g( s)} s S is MLR-ordered.

Short Course oncomparative Statics and Comparative Information Part II p. 25/2 Other decision criteria... The risk of a decision rule φ : Z X is given by [ ] u(φ(z), s)dh(z s). Z s S The Bayesian (with prior P ) chooses the rule that maximizes [ ] u(φ(z), s)dh(z s) dp(s). S Z The minimax criterion chooses the rule that maximizes u(φ(z), s)dh(z s). min s S Z

Short Course oncomparative Statics and Comparative Information Part II p. 26/2 Complete Class Theorem Definition: A rule φ is better than ψ if [ ] [ u(φ(z), s)h(z s) dz z Z s S z Z ] u(ψ(z), s)h(z s) dz s S Geometrically... when S is finite, we can think of [ Z u(φ(z), s)dh(z s)] s S as a point in R S. Each rule is associated with a point in R S. s 2 s 2 s 1

Short Course oncomparative Statics and Comparative Information Part II p. 27/2 Complete Class Theorem Definition: A subset of decision rules C form an essentially complete class if for every rule ψ there is a rule φ in C such that φ is better than ψ. Theorem 2.5: (QS-07) Let H be an MLR-ordered information structure and {u(, s)} s S an IDO family. Then the increasing decision rules form an essentially complete class.