0 Wireless Foundations 0July 7, 2004 Distributed Optimization of Power Allocation in Interference Channel Raul Etkin, Abhay Parekh, and David Tse Spectrum Sharing Group - University of California, Berkeley

1 Wireless Foundations 1July 7, 2004 Problem: Spectrum Sharing Can multiple heterogeneous wireless systems coexist and share spectrum in a flexible and efficient manner?

2 Wireless Foundations 2July 7, 2004 Project Goals Find strategies to allow multiple systems to share spectrum efficiently Algorithms must be –Distributed –Simple and require minimum cooperation/communication between systems –Robust to non-complying systems –Fair

3 Wireless Foundations 3July 7, 2004 Interference Channel Model M systems. Will consider M small. Bandwidth W Flat fading assumption: h i,j constant over frequency System m can use total power P m Define Background noise AWGN with PSD N 0 Can have 1Tx-Rx pair per system, or many of them. 1 2 M 1 2 M TxRx

4 Wireless Foundations 4July 7, 2004 Distributed Power Allocation Problem User m allocates its power with power spectral density p m (f) Noise plus interference seen by user m at frequency f: Rate of system m with Gaussian interference assumption: Want to maximize in a distributed way for some utility function f(.)

5 Wireless Foundations 5July 7, 2004 Algorithmic vs. Game Theoretic Formulations Algorithmic formulation: systems cooperate by following a distributed power allocation algorithm. Game Theoretic formulation: systems are selfish and want to maximize their own utility function. –Systems do not cooperate with each other, and are not willing to follow any algorithm that does not maximize their own utility. –Can use game theory to find “good” Nash Equilibria (N.E.). –Depending on assumptions can have different game formulations. Some confusion in literature: papers analyze N.E. of a static game of complete information, and apply dynamic algorithms to achieve them.

6 Wireless Foundations 6July 7, 2004 Algorithmic Approach: Iterative Water-filling Analyzed Iterative Water-filling Algorithm (IWFA): each system updates its power allocation performing water-filling over noise+interference seen. –Algorithm motivated by static Gaussian interference game: water-filling is the best response of each system to the other systems’ strategies in the static game –Advantages: distributed, simple, robust, no feedback across systems. –Disadvantages: may not converge, may converge to different Nash equilibria (when N.E. is not unique), may converge to a configuration with large price of anarchy. Insights gained with IWFA analysis: Algorithms should … –take into account N 0 to achieve a bounded price of anarchy when N 0  0. –take into account interference created over other systems. –“look into the future” instead of greedily maximizing immediate rate.

7 Wireless Foundations 7July 7, 2004 Algorithmic Approach: Other algorithms Orthogonal power allocations have bounded price of anarchy. –Rose and Popescu propose a (flawed) distributed orthogonalization algorithm. Each system gets 1/M of the total bandwidth. –We modified (and fixed) algo. to assign to user m a fraction of the total bandwidth. –Advantages: fast convergence, fair, good performance for small N 0 and or large cross gains c i,j, does not require knowledge of channel gains. –Disadvantages: requires some minimal synchronization/timing, requires constant monitoring of spectrum, needs to start all over again if a new system appears. Future research: find other distributed algorithms with better features than the 2 analyzed. May allow for limited communication across systems to improve performance. Question: how to allow for minimal communication among systems without making the problem trivial.

8 Wireless Foundations 8July 7, 2004 Game Theoretic Approach Regardless of the game formulation, need to find Nash Equilibria. Why ? Nash equilibrium strategies are self enforcing. No need to have the regulating agency monitoring the spectrum everywhere to enforce them. Basic assumption: there is an initial stage where systems agree on following a given N.E. If each system believes that the others will comply with the agreement, its best response is also to comply. Can have different formulations depending on the way the game is defined and the information that each player has. –Static / Dynamic game: 1 stage-simultaneous move / multi-stage game. –Complete / Incomplete information: all parameters are known / some not known to players. –Perfect / Imperfect information: players know the actions taken by other players when they choose their actions / don’t know some of the actions. Literature on spectrum sharing only considers static games of complete information. May need more general formulation for real world problems.

9 Wireless Foundations 9July 7, 2004 Static Game of Complete Information Case M=2: we have complete characterization of N.E. –If N.E. (full spread) is unique. –If there are 1 N.E. (full spread + orthogonal) Case M > 2: –Full spread power allocations are always N.E. –We obtained sufficient condition for uniqueness of full spread N.E.: (more general than Cioffi’s result) –We obtained necessary and sufficient condition for the existence of orthogonal N.E. (nice proof involving network flows): –Many more N.E. besides full spread and full orthogonal

10 Wireless Foundations 10July 7, 2004 Dynamic Game of Complete Information Stage game of static model repeated indefinitely. Sequences of N.E. of static game are N.E. in dynamic game. Can obtain new N.E. through believable threats. –Each systems threatens to apply a punishment if the other system does not comply with a given rule. –For threats to be believable, the punishment should be a “bad” N.E. Analyzed punishment strategies for M=2, c 1,1 =c 2,2 =1. –Strategy: allocate power over a BW W 1 as long as the other system stays orthogonal. Otherwise spread the power over the whole BW forever (punishment). –Questions: when is this strategy a N.E. ? When it is not, do we lose much by not being able to use this kind of strategy ?

11 Wireless Foundations 11July 7, 2004 Punishment Strategies Can apply punishment strategy Spreading better for both systems Region where can apply punishment strategy (green) larger for high SNR, where full-spread N.E. has large price of anarchy. In most cases where orthogonal N.E. is preferable but cannot be achieved through punishment strategies, performance loss is small.

12 Wireless Foundations 12July 7, 2004 Punishment Strategies: what if c i,j unknown ? Complete information formulation assumes knowledge of {c i,j } Can apply punishment strategies when user i does not know c i,j ? –c i,j needed to determine the W 1 that maximizes the minimum rate. –Can set W 1 =W/2 and apply the new strategy: start with orthogonal W/2 P.A.. If full spread allocation results in better rate, or the other system departs from orth. P.A., apply full spread allocation forever. –Resulting region slightly smaller than green region

13 Wireless Foundations 13July 7, 2004 Games of Incomplete Information Each system is randomly assigned a type when the game starts. –Can think of type for system m as {c j,m : j  m} Each system knows its own type, but does not know the types of other systems. Model may allow messaging across systems to inform about types, but systems may lie (cheap talk game). Each system’s best response is computed by maximizing the expected utility over the posterior distribution of the other systems’ types, and some belief about the other systems’ strategies.

14 Wireless Foundations 14July 7, 2004 Static Game of Incomplete Information Analyzed simple model: –M=2 –Binary action space {a 1, a 2 }: a 1 = full-spread power allocation a 2 = orthogonal W/2 power allocation –c 1,1 =c 2,2 =known constant –c i,j ~ Exp(1) Computed expected rates for different values of c 1,1 =c 2,2 When multiple Bayesian N.E. exist, choose the one with largest average rates. Compared performance to that of the equivalent game with complete information.

15 Wireless Foundations 15July 7, 2004 Static Game of Incomplete Information (cont.) N 0 =0.01 N 0 =0.1 N 0 =1 Complete information Incomplete information c 1,1 =c 2,2 E[R] Lack of knowledge of the other system’s type is helpful for small c i,i

16 Wireless Foundations 16July 7, 2004 Dynamic Game of Incomplete Information Want to analyze when communication across systems is beneficial. Structure of Game: –Player 1 sends a message to player 2 about its type –Player 2 chooses an action based on the message and its own type. –Player 1 chooses an action based on player 2’s action and its own type. Fixed c 1,1 =c 2,2 and chose c i,j to have binary and exponential distribution. Limited action space to {a 1,a 2 } as defined before Depending on parameter choices observed situations where communication is beneficial, and situations where Player 1 has no incentive in communicating its type. Player 1 prefers not to send message when user 2 is likely to play orthogonal and Player 1 can exploit its advantage of playing later by playing full-spread.

17 Wireless Foundations 17July 7, 2004 Games of Incomplete Information: what’s next ? All models analyzed so far had –M=2 –Binary action space {a 1,a 2 } –1 or 2 stage game Results depend on parameter choices and modeling assumptions. Changing any of these assumptions would make the models more realistic but math becomes significantly more involved. Seems hard to obtain generalizations that are mathematically tractable and resemble practical systems.

18 Wireless Foundations 18July 7, 2004 Conclusions and Research Direction If we can rely on systems to follow a specified algorithm, then algorithmic approach seems viable. –May not be possible to obtain very good performance unless explicit or implicit communication is allowed. –If we allow for communication, then problem becomes trivial. –Challenge: find interesting problem formulation allowing for some communication among systems. Can obtain good performance in dynamic game of complete information through punishment strategies. Could extend approach for M>2. Analysis of games of incomplete information is hard, and results are very dependent on model assumptions and parameter choices. Hope to get more insight into modeling assumptions and parameter choices by analyzing the case of multiple type systems sharing spectrum. In case: –Each system contains multiple transceivers, which access the medium in a TDMA fashion. –Since each terminal transmits from time to time, it may be possible to use reciprocity to justify complete information assumption. –New problems/challenges may arise due to the larger number of terminals in each system.