Game Theoretic Approaches to Analyzing Wireless Networks : Research Issues

1. Game Theoretic Approaches to Analyzing Wireless Networks:Research Issues Debashis Saha, PhD Professor, MIS & CS Group, Indian Institute of Management (IIM), Calcutta, India Joka, D. H. Road, Calcutta 700 104, India ds@iimcal.ac.in

2. Outline • Overview of Game Theory • Players, actions, payoffs • Representations of games • Wireless Networks (WiNets) • A case for applying game theory? • Literature review • Game Theoretic Models of WiNets • Four sample problems • Analyses • Concluding Remarks • Research Issues

3. Overview of Game Theory

4. What is a game? • A game is a structured or semi-structured activity, usually undertaken for enjoyment and sometimes also used as educational tools. • Key components of games are goals, rules, challenge, and interactivity. • [Ref: http://en.wikipedia.org/wiki/Game] • A game is an interactive decision problem • “A game is a form of art in which participants, termed players, make decisions in order to manage resources through game tokens in the pursuit of a goal.” (Greg Costikyan)

5. Is this a game?

6. Is this a game?

7. Is this a game? • When is it a game?

8. What is Game Theory ? • A Branch of Applied Mathematics • Itdescribes and studies interactive decision problems • Studies strategic interactions among rational players, where players choose different actions in order to maximize their returns.

9. Game + Theory • Various types of games exist (e.g. card, board, sport, war, etc.) • Game Theory deals with games having the following properties: • Two or more players • Choice of action involves a strategy • One or more outcomes • Outcome depends on the chosen strategies: i.e., strategic interaction • Rules out: • Games of pure chance • Games without strategic interaction

10. Five Elements of a Game • Set of Players • Set of Actions • Set of Strategies • Set of Outcomes • Payoff or Utility • The basic notions of game theory include: • players (decision makers) • choices (feasible actions) • payoffs (benefits, prizes, rewards, etc) • preferences over payoffs (objectives) • Game theory is concerned with determining when one choice is better than another choice for a particular player (strategy).

11. Structure of game is taken from the algorithm and the environment [Laboratoire de Radiocommunications et de Traitement du Signal] Modeling Wireless Network as Game Wireless Network Game Nodes Players Power Levels Actions Algorithms Utility Functions+ Learning

12. Example Two Player Action Space A1 = A2 = [0 ) A=A1 A2 A2 = A-1 a a2 = a-1 b b2 = b-1 b1 = b-2 A1= A-2 a1 = a-2 Actions Ai – Set of available actions for player i ai – A particular action chosen by i, ai  Ai A – ActionSpace, Cartesian product of all Ai A=A1 A2· · ·  An a – Action tuple – a point in the Action Space A-i – Another action space A formed from A-i =A1 A2· · · Ai-1  Ai+1· · ·  An a-i – A point from the space A-i A = Ai A-i

13. Utility Function Also known as Objective Function or Payoff Functions Maps action space to set of real numbers. # Quantifying actions brings the problem into the domain of conventional mathematics # After quantification, all sorts of valuable mathematical operations can be introduced.

14. Example Jack prefers Apples to Oranges a) uJack(Apples) = 1, uJack(Oranges) = 0 b) uJack(Apples) = -1, uJack(Oranges) = -7.5 Utility Functions [contd.] # Note that the quantification operation is not unique as long as relationships are preserved. # Many people map relationships to [0,1].

15. Example Game • Matching Numbers • Allen and Brian (A & B) • Each can choose to put out one finger or two fingers. • If they match, Allen gives Brian a dollar. If they’re different, Brian gives Allen a dollar. • Components • Player Set: N = {A,B} • Action Sets: AA = AB= {1,2}, A = {{1,1},{1,2},{2,1},{2,2}} • Utility Functions:

16. How to solve a game ? • We would like to solve a game… • Solving a game consists of • trying to predict the strategy of each player, • considering the information the game offers and • assuming that the players are rational. • Several ways to solve a game. • Simplest way relies on strict dominance.

17. Iterated Dominance/Strict Dominance • Strategy B strictly dominates A: choosing B always gives a better outcome than choosing A, no matter what the other player(s) do. • Iteratively eliminate dominated strategies. • Solution is always unique.

18. Iterated Dominance/Weak Dominance • Strategy B weakly dominates A: There is at least one set of opponents' action for which B is superior, and all other sets of opponents' actions give A and B the same payoff. • Solution depends on the sequence of eliminating weakly dominated strategies (due to multiple best responses). • Might not result in a single solution profile. • Nevertheless, useful as to reduce the size of the strategy space.

19. Prisoners' Dilemma (PD) Game • Players:2 Prisoners • Actions:Prisoner 1: Confess, DenyPrisoner 2: Confess, Deny • Strategies:Choose action simultaneously, without knowing each other’s actions. • Outcomes:Quantified in prison years • Payoff:Fewer years == Better payoff Deny? Confess?

20. Matrix Representation • A matrix which shows the players, strategies, and payoffs. • Presumed that players act simultaneously. • PD example:

21. Multi-access Protocols & PD Games • MAC protocols (carrier sense) • Multiple nodes contend for the shared medium • Can it be modeled as PD game? • Say there are 2 stations • Like 2 prisoners • Both sense channels • If both transmits together, there is collision • Equal loss for both • If one only transmits, it gets maximum payoff • Transmission is successful • If both back off, channel remains idle

22. Matrix Representation • A matrix for 2 nodes, their strategies and payoffs. • Presumed that nodes act simultaneously.

23. Basic game theory models • Normal form game • Also known as Strategic form game • Do not capture sequencing • Simultaneous moves by all players • Extensive form game • Game tree • Captures “time”; players move in sequence • Information is available to latter players • Repeated game • Evolutionary game

24. Normal Form Games (Strategic Form Games) In normal form, a game consists of three primary components (3-tuple) N – Set of Players Ai – Set of Actions Available to Player i A – Action Space {ui} – Set of Individual Objective Functions

25. Representation of Games: Normal (Strategic) Form • A matrix which shows the players, strategies, and payoffs. • Presumed that players act simultaneously. • PD example:

26. Nash Equilibrium • To identify best responses, John Nash introduced the concept of Nash Equilibrium. • In a Nash equilibrium, none of the users can unilaterally change their strategy to increase their utility. • Note: Any solution derived by iterated strict dominance is Nash equilibrium.

27. B A a 1,-1 0,2 b -1,1 2,2 Normal form game: NE Components • A set of 2 or more players • A set of actions for each player • A set of utility functions that describe the players’ preferences over the action space Player 1 Actions {a, b} Player 2 Actions {A, B} States from which no player can unilaterally deviate and improve are Nash Equilibriums

28. Nash Equilibrium Solution Methods • Direct Application of Definition • Improvement Deviations • Iterative Elimination of Dominated Strategies • Best Response Analysis

29. Game Tree/Extensive Form • The games can be static or dynamic. • In dynamic games the order of the moves/choices is important • Here a game tree is a better representation than normal form • Consider a simple game as this: • Player 1 chooses H or T • Player 2 chooses H or T (not knowing what Player 1 chooses). • If both choose the same Player 2 wins $1 from Player 1. • If they are different, Player 1 wins $1 from Player 2. • We can draw this in extensive form as shown next • It can be shown in normal form too (like PD game)

30. Extensive Form

31. Extensive form games Components • A set of players. • The actions available to each player at each decision moment (state). • A way of deciding who is the current decision maker. • Outcomes on the sequence of actions. • Preferences over all outcomes. Backward induction is a technique to solve a game tree of perfect information. It first considers the moves that are the last in the game, and determines the best move for the player in each case. Then, taking these as given future actions, it proceeds backwards in time, again determining the best move for the respective player, until the beginning of the game is reached.

32. Types of Games • Symmetric and Asymmetric • Zero Sum and Non-Zero Sum • Simultaneous and Sequential • Perfect Information and Imperfect Information

33. Symmetric and Asymmetric • Any game in which the identity of the player does not change the resulting game facing that player, is symmetric. • E.g. prisoners’ dilemma, game of chicken, and battle of the sexes. • General form:

34. Zero Sum and Non-Zero Sum • In zero sum game the total benefit to all players in the game, for every combination of strategies, always adds to zero. • i.e. A player benefits only at the expense of others. • E.g. poker, chess, matching pennies.

35. Simultaneous and Sequential • Simultaneous (a.k.a static games) games are games where both players move simultaneously. • Sequential games (a.k.a dynamic games) are games where later players have some knowledge about earlier actions.

36. Information Perfection • A game is one of perfect information if all players know the moves previously made by all other players. • E.g., chess • Head/Tail game: • If Player 1 reveals his choice before Player 2 chooses • Here Player 2 wins always

37. Information Completeness • Often confused with Information Perfection • Complete information means that the players know each element in the game definition: • Who the other players are • What their possible strategies are • What payoff will result for each player for any combination of moves

38. Wireless Networks &Game Theory

39. Related Works • Game theory approaches have been used for optimization and control of wireless networks [1]-[2] • as an alternative to traditional network optimization [1] T. Alpcan, T. Basar, R. Srikant, E. Altman, ‘CDMA uplink power control as a noncooperative game’, in Proc. 40th IEEE Conf. Decision and Control, 197-202, 2001. [2] C. Saraydar, N. B. Mandayam, and D.J. Goodman, ‘Efficient power control via pricing in wireless data networks’, IEEE Trans. Communication, vol. 50 (2), 291-303, 2002.

40. Related Works [contd.] • In wireless communications literature, the tragedy of commons problem has typically been addressed by modeling power consumption (explicit energy) as a direct cost to the users [2] • A Nash equilibrium solution is obtained when users have an interest in maximizing their own utility, defined as the ratio of rate to power • This approach, however, results in a sub-optimal resource allocation. • This work has recently been extended [3] to capture the affect of receiver design where the authors show that receiver design can be used to induce a more efficient Nash equilibrium [2] C. Saraydar, N. B. Mandayam, and D.J. Goodman, ‘Efficient power control via pricing in wireless data networks’, IEEE Trans. Communication, vol. 50 (2), 291-303, 2002. [3] F. Meshkati, H. V. Poor, S. Schwartz, and N. Mandayam, “An energy efficient approach to power control and receiver design in wireless data networks,” IEEE Transactions on Communications, vol. 53, no. 11, pp. 1885–1894, November 2005.

41. Physical Layer Model M – the set of decision making radios Ei– the set of possible energy levels available to radio i ei – the energy level chosen by i e - the tuple of chosen energy levels of all radios in the network i – the set of signature waveforms available to radio i i – the chosen waveform of i - the tuple of chosen waveforms of all radios in the network Ni – noise power at node i ij - the correlation between the signature waveform sequences of radios i and j. Note that ij necessarily equals ji.

42. Radio Model vi – node of “interest” for node i i,j – path loss from i to j v1

43. Some Wireless Network Games • Consider 2 rational players (i.e., nodes) • Games correspond to the protocol stack • The four games: • Forwarder’s Dilemma • Joint Packet Forwarding • Multiple Access • Jamming

44. Forwards’ Dilemma • Symmetric nonzero-sum

45. Joint Packet Forwarding • Asymmetric, nonzero-sum

46. Multiple Access • P1 and P2 are in the same transmission range, and will interfere if they were to transmit at the same time. • Should they transmit during a timeslot or stay quite? • Symmetric, nonzero-sum

47. Jamming • Two channels • Sender tries to send a packet on one of the channels • Jammer tries to jam the sender • Asymmetric, zero-sum

48. Game Theory Analyses

49. Steps in application of game theory • Develop a game theoretic model • Solution of game’s Nash equilibrium yields information about the steady state and convergence of the network • Does a steady state exist? • Uniqueness of Nash equilibrium • Is it optimal? • Do nodes converge to it? • Is it stable? • Does the steady state scale?

50. Iterated Dominance/Strict Dominance • Dropping strategy dominates Forwarding for both players. • (D,D) is the solution!