Mathematical Economics - PhD in Economics

- PhD in Part 1: Supermodularity and complementarity in the one-dimensional and Paulo Brito ISEG - Technical University of Lisbon November 24, 2010

1 2 - Supermodular optimization 3 one-dimensional 4 Supermodular Games

Supermodular optimization What? It is a methodology developed by Topkis (1968, 1978) for conducting comparative statics or sensitivity analysis. In other words, it determines how changes in exogenous parameters affect endogenous variables in optimizing models. Where? This methodology is extremely useful in economics as the conclusions that it aims to achieve are often the main motivation behind the construction of the models.

Supermodular optimization Why? The main characteristic of this methodology is that it relies essentially on critical assumptions for the desired monotonicity conclusions and it dispenses with most of the assumptions that traditionally are necessary by the use of the classical method (smoothness, interiority and concavity). The required assumptions are related with complementarity of the decision variables and the parameters.

Supermodular optimization How If, in a maximization problem, having an increase in the parameter increases the marginal return to an increase in the choice variable, then the optimal value of the former will be increasing in the latter. If we have multiple choice variables, all of them must be complements, in order to have mutually reinforcing increments.

Supermodular Games Other breakthroughs Supermodular games or games with strategic complementarities have monotonic relationship between the actions of the players. Hence, with an appropriate fixed-point theorem, it is possible to guarantee the existence of Pure Strategy Nash Equilibria. Also, supermodularity ensures comparability of equilibrium points, again one of the objectives of economic modelling.

Supermodularity and complementarity in the one-dimensional Consider the following problem in economics: max a F (s, a) s S R parameter a A s action A s A R feasible action set when the parameter is s a (s) = argmax {F (s, a) : a A s }

Supermodularity and complementarity in the one-dimensional Traditional Method

Supermodularity and complementarity in the one-dimensional Traditional Method 1 Compute the First Order Condition: F (s,a) a = 0

Supermodularity and complementarity in the one-dimensional Traditional Method 1 Compute the First Order Condition: Assumption 1: Interiority F (s,a) a = 0

Supermodularity and complementarity in the one-dimensional Traditional Method F (s,a) 1 Compute the First Order Condition: a = 0 Assumption 1: Interiority 2 Ensure that the Second Order Condition holds: 2 F (s,a) < 0 Assumption 2: Concavity a 2 3 For the derivatives to exist and be continuous we need to assume that the function is smooth Assumption 3: Smoothness 4 Use the Implicit Function Theorem: Since can compute 2 F a s + 2 F a 2 a (s) = 0 a (s) = Which is than 0 if and only if 2 F a s 0 F (s,a) a 2 F a s 2 F a 2 = 0 we

Supermodularity and complementarity in the one-dimensional New Method - Topkis (1979) We will only need Assumption 4: Complementarity and some assumption on A s. Definition 1: Increasing Differences A function F : S A R has Increasing Differences (ID) in (s, a) if F (s, a ) F (s, a) F (s, a ) F (s, a), a > a, s > s. Definition 2: Strictly Increasing Differences A function F : S A R has Strictly Increasing Differences (SID) in (s, a) if F (s, a ) F (s, a)>f (s, a ) F (s, a), a > a, s > s. Note: ID and supermodularity are equivalent for functions on R 2.

Supermodularity and complementarity in the one-dimensional New Method - Topkis (1979) (cont.) For smooth functions supermodularity/id can be admit a convenient test: Lemma 1: Cross Partial Test If F is C 2, increasing differences is equivalent to 2 F a s 0 for all a and s. Proof:

Supermodularity and complementarity in the one-dimensional Definition 3: Ascending set A set A s : S 2 A is ascending if A(s) = [g(s), h(s)], where h, g : S R are increasing functions with g < h.

Supermodularity and complementarity in the one-dimensional Simplified Version of Topkis s Theorem Assume that: F has increasing differences; A s is ascending; then, a (s) has a minimal and maximal selections ā(s), and a(s), are increasing functions.

Supermodularity and complementarity in the one-dimensional Proof of the Simplified Version of Topkis s Theorem

Supermodularity and complementarity in the one-dimensional Versions of the Topkis s Theorem Strictly Increasing Differences Assume that: F has strictly increasing differences; A s is ascending; then, every selection of a (s) is increasing. Proof - Homework

Supermodularity and complementarity in the one-dimensional Versions of the Topkis s Theorem (cont.) Dual Version Assume that: F has (strictly) decreasing differences; A s is descending; then, every selection of a (s) is decreasing. Proof - Homework

Supermodularity and complementarity in the one-dimensional Example 1 Consider the following problem: max x1,x 2 U(x 1, x 2 )s.t.p 1 x 1 + p 2 x 2 = m Question: is x increasing in m? First transform the problem into a one dimensional problem - x 2 = 1 p 2 (m p 1 x 1 ) max 0 x1 m p 1 U(x 1, 1 p 2 (m p 1 x 1 )) A m is ascending in m U has increasing differences: U 2 U x 1 m = 1 p 2 2 (p 2 U 12 p 1 U 22 ) 0 x 1 = U 1 U 2 p 1 p 2 and

Supermodularity and complementarity in the one-dimensional Example 1 We dont need concavity or quasi-concavity of U. We do not need smoothness of U parameters and decisions can be discrete x 1 needs not be interior

Supermodularity and complementarity in the one-dimensional Example 2 Consider the following problem: max p 0 Π(p, c) = (p c)d(p) Question: When is p increasing in c? p [0, ] is ascending in c Do we have increasing differences: Π p = Dp + (p c)d (p) and 2 Π p c = D (p)) 0 We need to assume that D 0 We are looking at comparative statics. How is existence guaranteed? By the Weierstrass Theorem (a continuous function on a compact set has a max and a min). But we do not need continuity - the theorem extends to upper semi-continuous functions

Supermodularity and complementarity in the one-dimensional Example 2 (cont.) Consider the again the problem of the monopolist as in example 2: Make a monotonic transformation in the profit function which does not alter the maximizer: max p 0 log Π(p, c) = log(p c) + log D(p) Question: When is p increasing in c? p [c, ] is ascending in c Do we have increasing differences: log Π and 2 log Π p c = 1 2 p c 0 We need NO assumptions on D p = 1 p c + D (p) D(p)

Supermodularity and complementarity in the one-dimensional Example 3 Consider the again the problem of the monopolist as in example 2: Question: When is cost pass through larger than 100%? We need a change of variable: m = p c max m [0, ] Π(p, c) = md(m + c) m [0, ] is ascending in c Do we have increasing differences: and 2 log Π p m log Π m = D (m+c)d(m+c) D 2 (m+c) D 2 (m+c) 0 = 1 m + D (m+c) D(m+c) Case 1: 2 log Π m c 0 DD D 2 0 D log-convex Case 2: 2 log Π m c 0 DD D 2 0 D log-concave Remark: Most D are logconcave - linear D for instance.

Supermodularity and complementarity in the one-dimensional Example 4 Growth Theory Two-period Model max 1 δ t u(c t ), s.a.s t+1 = f (s t c t ) t=0 Question 1: When are optimal savings s c nondecreasing in δ? We start by constructing the value function: V 2 (s) = max c [0,s] u(c) + δu[f (s c)] Now, it is useful to do the following change of variable:y = s c V 2 (s) = max y [0,s] u(s y) + δu[f (y)]

Supermodularity and complementarity in the one-dimensional Example 4 - (cont.) A(δ) is ascending and descending since it does not depend on δ, only on s. We need the objective function to have increasing differences in (y, δ) Because we have continuity, we can use the second derivative test: 2 {u(s y) + δu[f (y)]} s δ = u f 0 So, as long as u and f are both increasing, we have that savings are nondecreasing with δ (no need for concavity assumptions)

Supermodularity and complementarity in the one-dimensional Example 4 - (cont.) Question 2: When are optimal savings y = s c nondecreasing in s? A(s) is ascending on s. We need the objective function to have increasing differences in (y, s) Because we have continuity, we can use the second derivative test: 2 {u(s y) + δu[f (y)]} s y = u 0 So, as long as u is concave, we have that savings are nondecreasing with capital, s.

Supermodularity and complementarity in the one-dimensional Example 4 - (cont.) Question 3: When is optimal consumption c nondecreasing or nonincreasing in s? A(s) is ascending on s. We need the unmodified objective function to have increasing differences in (c, s) Because we have continuity, we can use the second derivative test: 2 {u(c) + δu[f (s c)]} s c = δ[u f 2 + u f ] 0 We observe that f < 0 is not necessary for c to be nondecreasing in s. (This is the assumption of the classical model).

Games Definition A Game is a situation of social interaction, characterized by number and identities of players; sets of actions available for each player; payoff functions that map the joint action of players to an outcome for each player; timing of the play. When analysing a game, the main objective is to obtain a prediction of what players choose and what will be their outcome. How can we predict the outcome?

Games (cont.) Set of undominated strategies Nash Equilibria (Pure and Mixed) Cooperative Equilibria Trembling-hand Equilibria Correlated Equilibria

Games (cont.) What are the main concerns when predicting equilibria? Existence of Equilibria Uniqueness of Equilibrium Comparative statics of equilibrium points

Games (cont.) In general games Existence of Equilibria - we can guarantee the existence of mixed-strategy Nash Equilibria Uniqueness of Equilibrium - some games (like global games) have unique equilibria Comparative statics of equilibrium points - we need strong assumptions to use the envelope theorem and do comparative statics.

Games (cont.) Supermodular Games Existence of Equilibria - Pure strategy Nash Equilibria exist Uniqueness of Equilibrium - Even when there exist more than one equilibria, they are Pareto ordered Comparative statics of equilibrium points - under some conditions, maximal and minimal equilibria are monotone in the parameters

Supermodular Games Consider a game two-player game Γ = (X, Y, F, G), where, X and Y are the strategy spaces of the players and F : X Y R and G : X Y R are the payoff functions of each player. Γ is a Supermodular Game if: 1 F and G have increasing differences in x X and y Y ; 2 F and G are upper semi-continuous in own action; 3 X and Y are compact sets.

Submodular Games Consider a game two-player game Γ = (X, Y, F, G), where, X and Y are the strategy spaces of the players and F : X Y R and G : X Y R are the payoff functions of each player. Γ is a Submodular Game if: 1 F and G have decreasing differences in x X and y Y ; 2 F and G are upper semi-continuous in own action; 3 X and Y are compact sets.

Supermodular and Submodular Games Properties of Supermodular and Submodular Games Let r 1 (y) = argmax x X F (x, y) r 2 (x) = argmax y Y G(x, y) Be the reaction functions or correspondences (best replies) of the players. Notice that these always exist as we are optimizing a upper semi-continuous function on a compact set - Assumptions 2 and 3 of the Supermodular Game Definition.

Supermodular Games Property 1 (Supermodular games) In a Supermodular Game, the reaction functions (correspondences) of the players have maximal and minimal selections that are nondecreasing in the strategy of the opponent. Proof For each player, the opponent s strategy is considered a parameter. We can apply Topkis s Theorem to conclude the above result. Namely: The strategy spaces are independent of the opponent s action, hence are ascending (and descending) The objective function has ID in the action and the opponent s action Hence, we have that the argmax has maximal and minimal selections that are nondecreasing.

Submodular Games Property 1 can be generalized easily for submodular games: Property 1 (Submodular games) In a Submodular Game, the reaction functions (correspondences) of the players have maximal and minimal selections that are nonincreasing in the strategy of the opponent. Proof For each player, the opponent s strategy is considered a parameter. We can apply Topkis s Theorem to conclude the above result. The strategy spaces are independent of the opponent s action, hence are descending (and ascending) The objective function has DD in the action and the opponent s action

Strategic complementarity and substitutability Hence, we have that the argmax has maximal and minimal selections that are nonincreasing. In the language of Bulow, Geanakoplos and Klemperer (1985): Supermodular Games are games of strategic complementarities Submodular Games are games of strategic substitutabilities

Examples of Supermodular and Submodular Games Cournot Competition Model Two firms 1,2 choose quantities of a homogeneous product They face a common demand function P(q 1 + q 2 ) Cost function is C(q i ) Payoff function is given by the profit function: Π i (q 1, q 2 ) = q i P(q 1 + q 2 ) C(q i ), i = 1, 2 Question: When is this game supermodular and when is it submodular? The strategy space is constant in the opponent s action The payoff function has increasing (decreasing) differences if 2 Π i q 1 q 2 = q i P + P ( )0 Notice that it is more likely, for decreasing demand functions that we have decreasing differences and hence nonincreasing reaction functions. (it is sufficient a concave demand function)

Examples of Supermodular and Submodular Games Bertrand Competition Model with differentiated product Two firms 1,2 choose prices of differentiated productos They face different demand functions: namely D i (p 1, p 2 ) = a p i + p j, i = 1, 2 and i j Cost function is C(q i ) = cq i Payoff function is given by the profit function: Π i (p 1, p 2 ) = (p i c i )D i i = 1, 2 Question: When is this game supermodular and when is it submodular? The strategy space is constant in the opponent s action The payoff function has increasing (decreasing) differences if D 1,2 + D 1,21 (p 1 c 1 ), D 2,1 + D 2,12 (p 2 c 2 ) Notice that with convex demands that we have increasing differences and hence nondecreasing reaction functions.

Supermodular and Submodular Games Property 2 - Existence of Equilibria From Property 1, in a supermodular game, reaction functions have extremal selections which are increasing in the opponent s action. We can, then, apply construct a mapping which is increasing and use Tarski s Fixed Point Theorem to conclude that this mapping is a fixed point. It turns out that the fixed point corresponds to an Equilibrium of the game.

Supermodular and Submodular Games Proof of the Existence of Equilibria using Tarski s Fixed Point Theorem Tarski s Fixed Point Theorem Let A be a compact set and F : A A be increasing. Then, the set of fixed points of F is nonempty and has coordinate-wise largest and smallest elements. We kow, by property 1, that r 1 (x) and r 2 (y) are increasing functions. Construct the following increasing mapping: B(y) = r 1 (r 2 (y)). A fixed point of the mapping is a Nash Equilibrium :B(y) = y y = r 1 (r 2 (y)) Due to Tarski, we know that a fixed point of this mapping exists.

Supermodular and Submodular Games Existence of Nash Equilibria in general games Brower s Fixed Point Theorem Let S R n be convex and compact. If T : S S is continuous, then there exists a fixed point i.e., there exists x S such that T (x ) = x.

Supermodular and Submodular Games Consider a 2-player game with and all possible mixed strategies of the players. Each player chooses the mixed strategy that maximizes her expected payoff for each mixed strategy of the oponent. The maximum of a continuous function on a parameter is continuous in that parameter. Hence: the best replies are continuous. We can construct a mapping as before that will be also continuos in the compact set of the mixed strategies. Using Brower, this mapping has a fixed point.

Supermodular and Submodular Games Coincidence of Solution Concepts U denotes the set of strategies that survive iterated elimination of strictly dominated strategies. This set contains all NE points. Only strategies in U can have positive mass at any mixed strategy equilibrium or correlated equilibrium. Because the best response map of a supermodular game is increasing, it is easy to see that there exist coordinate-wise smallest and largest Nash Equilibria. We can denote these as ā and a

Supermodular and Submodular Games Property 3 - Coincidence of Solution Concepts For any supermodular game we have that supu = ā and infu = a Proof The argument follows the iterative elimination of strictly dominated strategies.

Supermodular and Submodular Games Corollary A supermodular game with a unique pure strategy Nash equilibrium is dominance solvable and it has also a unique mixed strategy and correlated equilibrium.

Supermodular and Submodular Games Consider a game Γ = (X, Y, F, G, S), where S is the parameter space. We may want to know how does the equilibrium change with changes in s S Property 4 - Comparative statics of Equilibrium Points Milgrom and Roberts 1990 Assume that for each s S, the game is supermodular each player s payoff function has increasing differences in own action and parameter (i, s), i = x X, y Y Then, the extremal equilibria of the game are increasing functions of s.

Supermodular and Submodular Games Let B s (.) : A A denote the best-reply correspondence when the parameter is s. For each s S, there are smallest and largest Nash equilibria, by the supermodularity assumption. For each i, player is reaction correspondence ri s shifts out as s increases, by the increasing differences assumption and Topkis s Theorem. The mapping B also shifts out, and the same happens to the extremal fixed points.