Chair of Communications Theory, Prof. Dr.-Ing. E. Jorswieck. Übung 5: Supermodular Games

Transcription

1 Chair of Communications Theory, Prof. Dr.-Ing. E. Jorswieck Übung 5: Supermodular Games

2 Introduction Supermodular games are a class of non-cooperative games characterized by strategic complemetariteis A player desires to increase his strategy as a response to an increase in the strategy of the other players. In order to study supermodular games we need dbackground on lattice theory and monotone comparative statics Supermodular games are interesting since They arise in many models We can establish the existence of a pure strategy equilibrium without requiring the quasiconcavity of the payoff functions The equilibrium set has a smallest and a largest element and there exists a simple algorithm to compute these They have nice sensitivity (or comparative statics) properties and behave well under a variety of learning rules. 2

3 Lattice Theory Let be a binary relation on a nonempty set S. The pair (S, ) is a partially ordered set if is Reflexive: x x for all x S Transitive: x y and y z implies that x z Antisymmetric: x y and y x implies that x=y. A partially ordered set (S, ) is (completely) ordered if for x S and y S, either x y or y x. A lattice is a partially ordered set (S, ) in which two elements x, y have a least upper bound (supremum) called the join of x and y, and a greatest lower bound (infimum) called the meet of x and y in the set. 3

4 Lattice Theory A lattice (S, ) is complete if every nonempty subset of S has a supremum and an infimum in S. A subset L of the lattice S is a sublattice of S if the supremum and infimum of any two elements of L (with the supremum and infimum taken with respect to S) belong also to L. A function f: S R is supermodular on S if for all x, y S The supermodularity is automatically ti satisfied if S is single dimensional. i Theorem: Let I be an interval R n. Assume that f:r n R is twice continuously differentiable on some open set containing I. Then f is supermodular on I if and only if for all x I and all i j, 4

5 Lattice Theory Let (S, ) be a partially ordered set. A function f from S to S is increasing if for all x, y S, x y implies f(x) f(y). Theorem: (Tarski) Let (S, ) be a complete lattice and f: S S an increasing function. Then, the set of fixed points of f, denoted by E, is nonempty and (E, ) is a complete lattice. 5

6 Monotone Comparative Statics Let f: X T R, X R, and T is some partially ordered set. We are interested in conditions under which we can establish that x(t), is a nondecreasing function of t. Let Let X R and T be some partially ordered set. A function f: X T R has increasing i differences in (x,t) iff for all x x and dt t t, we have Lemma: Let X R and T R k for some k, a partially ordered set with the usual vector order. Let f: X T R be a twice continuously differentiable function. Then, the following statements are equivalent a) The function f has increasing differences in (x,t) b) For all t t and all x X, we have c) For all x X, t T, and all i = 1,,k, we have 6

7 Main Results Theorem: Let X R be a compact set and T be some partially ordered set. Assume that the function f: X T R is upper semicontinuous in x for all t T and has increasing differences in (x,t). Define Then we have: For all t T, x(t) is nonempty and has a greatest t and least element, denoted d by and respectively. For all t t, we have 7

8 Supermodular Games The strategic form game hn, (S i ), (u i )i is a supermodular game if for all i S i is a compact subset of R (more generally S i is a sublattice of R m ), u i is upper semi continuous in s i, continuous in s -i, u i has increasing differences in (s i, s -i ) (more generally u i supermodular in (s i, s -i )). Corollary: Assume hn, (S i ), (u i )i is a supermodular game. Let Then br_i(s_{-i}) has greatest and least element, denoted by and If s _{-i} \geq s_{-i}, then 8

9 Supermodular Games Theorem: Let hn, (S i ), (u i )i be a supermodular game. Then the set of strategies that survive iterated strict dominance (i.e. iterated elimination of strictly dominated strategies) has greatest and least elements and, which are both pure strategy Nash Equilibria. Corollary: Supermodular games have the following properties: There exists at least one pure strategy NE in the game [Topkis Theorem 4.2.1] The set of Nash equilibria is a complete lattice and there exist a largest element and a least element [Topkis Theorem 4.2.1] A unique NE is globally stable [Vives Result 4] If the utility of a player is increasing in the strategies of the other players, then the largest (resp. smallest) equilibrium is the player's Pareto best (resp. worst) equilibrium [Vives Result 3] 9

10 Example Resource allocation in protected and shared bands Instantaneous channel coefficients α i, β ij, i,j = 1,2. For each BS a total power constraint of P to allocate in its 2 bands Noise power is σ 2. Define ρ = P/σ 2 10

11 Game in Strategic Form The players are the base stations BS 1 and BS 2 Their strategies are their power allocation in the protected and shared bands Define π i [0,1] the strategy for player I Player 1 allocates π 1 P in the protected and (1 - π 1 ) P in the shared band Player 2 allocates (1 - π 2 ) P in the protected and π 2 in the shared band Their utility is the achievable rate at the corresponding receiver 11

12 Our game is a Supermodular Game The game is a Supermodular game The strategy space [0,1] is compact subset of R. The utility functions in are continuous in the strategies of the players The utility functions have increasing differences in the strategies (π 1,π 2 ) since 12

13 Properties This implies that the properties of Supermodular games apply to our game There exist at least on pure strategy Nash equilibrium (NE) in the game The set of NEs is a complete lattice 13

14 Properties If the utility of a player is increasing in the strategies of the other players, then the largest (resp. smallest) equilibrium is the player's Pareto best (resp. worst) equilibrium 14

15 Properties A unique NE is globally stable Interference regions that satisfy a unique NE (β 11 = β 22 = 1, α 1 = 2, α 2 = 1) 21 15