Game Theory: Supermodular Games 1


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1 Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike / 22
2 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22
3 Motivation I several solution concepts: (dominance) rationalizability (iterative deletion of dominated strategies) Nash equilibrium correlated equilibrium Which one to use? Are there games where all solution concepts give (approximately) same prediction? 3 / 22
4 Motivation II comparative statics: how does the equilibrium change if one parameter changes? how do market prices depend on demand elasticity/number of firms in the market/product substitutability? how will the central bank change its interest rate when unemployment rises? how does campaign spending depend on media bias? size of effect has to be estimated but game theoretic models can sometimes determine the sign of effect! are there properties of models under which the sign of the effect is unambiguously clear? 4 / 22
5 Example: Search Example: Simple search market N traders exert effort x i to search for other trader if two traders find each other, each gets benefit 1 probability { that ( trader i finds other trader is )} min 1, αx i j i x j + β where α > 0 and β>0 are such that 0 < α((n 1) + β) < 1 costs of effort are x 2 i Given the effort of the others, how much effort should i exert? If trader j exerts more effort, how does this affect the optimal effort of i? If the search technology improves, will a trader exert more or less effort? Could our answers depend on the solution concept used? 5 / 22
6 Model Setup: Simple smooth supermodular games N players action sets A i = [y i, ȳ i ] simultaneous move game utility functions: u i (a i, a i ): u i is twice continuously differentiable 2 u i a i a j 0 for i j ( supermodularity: this is the important assumption of today!) check: Does the simple search market satisfy the assumptions? 6 / 22
7 What are the effects of 2 u i a i a j 0? Increasing differences : let a 1 > a 1, then u 1(a 1, a 2, a 3,... ) u 1 (a 1, a 2, a 3,... ) = a1 u 1 a 1 a 1 (x, a 2, a 3,... ) dx is increasing in a 2 because of 2 u 1 a 1 a 2 0 check: verify that the simple search market satisfies increasing differences Result Let a i > a i. If u i(a i, a i ) > u i (a i, a i), then u i (a i, a i ) > u i(a i, a i ) for all a i a i. Roughly: Best responses are increasing. 7 / 22
8 Serially undominated strategies (another solution concept... ) same as iterative elimination of strictly dominated strategies BUT only pure strategies are used Which actions of player i are not strictly dominated by pure strategies? Call them A 1 i. Which actions of player i are not strictly dominated by pure strategies if the other players play only actions from A 1 j? Call them A2 i. Which actions of player i are not strictly dominated by pure strategies if the other players play only actions from A 2 j? Call them A3 i.... note that A 1 i A 2 i... U i = A 1 i A 2 i... is the set of serially undominated strategies of player i 8 / 22
9 Example: serially undominated strategies Difference between serially undominated strategies and iterative elimination of strictly dominated actions L C R T 6,5 4,2 3,3 M 3,0 3,5 6,1 B 4,1 3,0 2,16 9 / 22
10 Main result let u i denote the minimum of U i and ū i the maximum of U i Theorem (Supermodularity Result 1) The strategy profile (u 1, u 2,... ) is a Nash equilibrium. The strategy profile (ū 1, ū 2,... ) is also a Nash equilibrium. Why is this important? U i gives a range in which all rationalizable actions have to lie in (why?) all Nash equilibria have to be in (why?) the support of all correlated equilibria has to be in (why?) if u i and ū i are close, all these solution concepts give similar predictions! 10 / 22
11 Proof of Supermodularity Result 1 We show that (u 1, u 2,... ) is NE. Can a deviation to a i < u i or to a i > ū i be profitable? Can a deviation to a i (u i, ū i ] be profitable? Suppose it was, i.e. u i (a i, u i) u i (u i, u i ) > 0. Then,... Similar proof for (ū 1, ū 2,... ) (exercise) 11 / 22
12 Extending the result Result 1 is only useful if ū i and u i are close if u i = ū i, then there is a unique rationalizable action which is the unique Nash equilibrum which is a unique correlated equilibrium the game can be solved by iterative elimination of strictly dominated strategies note that result implies the following: Corollary If a supermodular game either has a unique pure strategy NE or is symmetric and has a unique pure strategy symmetric NE, then each player has a unique rationalizable action and the game can be solved by iterative elimination of strictly dominated actions. 12 / 22
13 Is the corollary useful? Take simple search market with 2 agents. Determine all pure strategy NE. How many correlated equilibria does this game have? What is the set of rationalizable actions? Does the answer change if there are n agents? 13 / 22
14 Bertrand with differentiated goods Example: price competition Two firms compete by setting prices. Demand for firm i is D i (p 1, p 2 ) = γ p i + βp j with 0 < β < 1 and γ > 0. Costs for firm i are c i D i (p 1, p 2 ). Both firms maximize profits. Is the game supermodular? Does it have a unique pure strategy Nash equilibrium? What are the sets of rationalizable actions and correlated equilibria? 14 / 22
15 Comparative statics I Now we want to ask questions like: how does the equilibrium effort in the search market change if α increases? how do the equilibrium prices in the Bertrand game change if c i changes (or β) changes? now: utility functions depend on actions and a parameter τ u i (a i, a i, τ) hence, lower and upper bounds on serially undominated strategies u i and ū i are functions of τ Theorem (Supermodularity Result 2) If 2 u i a i τ 0 for all i, then u i and ū i are increasing in τ. 15 / 22
16 Comparative Statics II result 2 is especially useful if u i (τ) = ū i (τ) check: how does the equilibrium effort in the search market depend on α? how do the equilibrium prices in the Bertrand game change if c i changes (or β) changes? idea behind result 2: assume 2 players and interior equilibrium (i.e. u i / a i = 0 in equilibrium) assume strictly concave utility functions, i.e. 2 u i / a 2 i < 0 if τ increases, how does this affect marginal utility u i / a i? can both a 1 and a 2 decrease if τ increases? can a 1 increase and a 2 decrease if τ increases? 16 / 22
17 Supermodularizing : Tricks I Example: Cournot Two firms with zero marginal costs set quantities q i. The resulting market price is 1 q 1 q 2. Firm i maximizes its profit q i (1 q i q j ). Is this game supermodular? Any idea what to do? 17 / 22
18 Supermodularizing : Tricks II We use again the price competition model but now we assume a logit demand 1 D i (p i, p j ) = e p i p j + 1 profits are then π i (p i, p j ) = p i e p i p j + 1 which do not satisfy 2 π i / p 1 p 2 0 trick: the price maximizing π i (p i, p j ) is the same that maximizes log(π i (p i, p j )) let firms maximize log profits u i (p i, p j ) = log(π i (p i, p j )) = log(p i ) log(e p i p j + 1) which satisfies 2 u i / p 1 p 2 0 (check!) 18 / 22
19 Extensions What if each player takes several actions? e.g. a firm sets price, quality and warranty length A i = [y 1, ȳ 1 i i ] [y 2, ȳ 2 i i ] [y k i, ȳ k i i i ] hence, an action of player i is now a vector (ai 1, a2 i,..., ak i i ) supermodularity in this setup: 2 u i a m i a n j 2 u i a m i a n i 0 for all i j and 1 m k i and 1 n k j 0 for all i and 1 m < n k i with these assumptions all results still hold 19 / 22
20 Revision questions Why are we interested in supermodular games? How are (simple smooth) supermodular games defined? (what is the crucial assumption?) Explain the increasing differences property. What is the main result for supermodular games? What does it imply for the case that the game has only one pure strategy Nash equilibrium? What do we mean by comparative statics and what kind of comparative static result is special for supermodular games? What kind of tricks can you use to make a game supermodular? 20 / 22
21 Exercises I Exercise 1: 1 Take the standard Bertrand model with homogenous goods: 2 firms have costs 0 and set a price p i. There is one consumer who buys from the firm with the lower price (as long as this price is below his valuation v). Is this game supermodular? 2 Choose a game theoretic model from another course you have taken and check whether the game was supermodular (or could be transformed into a supermodular model using one of the tricks). Exercise 2: Complete the proof of supermodularity result 1 by showing that (ū 1, ū 2,... ) is a Nash equilibrium. 21 / 22
22 Exercises II reading: Milgrom/Roberts paper: introduction+ sections 2+4 (you will probably encounter some terms that you don t know and the proofs might look a bit hard; that s ok: just try to read through it and relate it to the things we did in the lecture; section 4 might be easier to follow than section 2) *Exercise 3: Assume in addition to the assumptions so far that u i is strictly concave: 2 u i / a 2 i < 0. Show that each A k i (in the iterative elimination of dominated actions) is an interval and conclude that U i is an interval. 22 / 22
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