Game theory for wireless networks static games; dynamic games; - PowerPoint PPT Presentation

Game theory for wireless networks static games; dynamic games; repeated games; strict and weak dominance; Nash equilibrium; Pareto optimality; Subgame perfection; … Game theory for wireless networks Georg-August University Göttingen

Outline 1 Introduction 2 Static games 3 Dynamic games 4 Repeated games Game theory for wireless networks Georg-August University Göttingen 2

Brief introduction to Game Theory  Proper operation of wireless networks requires appropriate rule enforcement mechanisms to prevent or discourage malicious or selfish behavior – Discouraging malicious or selfish behavior can benefit from game theoretic modeling  Classical applications: economics , but also politics and biology  Example: should a company invest in a new plan, or enter a new market, considering that the competition may make similar moves?  Most widespread kind of game: non-cooperative (meaning that the players do not attempt to find an agreement about their possible moves) Game theory for wireless networks Georg-August University Göttingen 3

Classification of games Non-cooperative Cooperative Cooperative Static Dynamic (repeated) Strategic-form Extensive-form Perfect information Imperfect information Imperfect information Complete information Incomplete information Incomplete information Game theory for wireless networks Georg-August University Göttingen 4

Classification of games non-cooperative games: individuals cannot make binding agreements and the unit of analysis is the individual. In cooperative game theory, binding agreements between players are allowed and the unit of analysis is the group or coalition. Static games: each player makes a single move and all moves are made simultaneously. Perfect info: each player has a perfect knowledge of the whole previous actions of other players at any moment it has to make a new move. Complete info: each player knows who the other players are, what are their possible strategies and what payoff will result of each player for any combination of moves. Game theory for wireless networks Georg-August University Göttingen 5

Cooperation in self-organized wireless networks D 2 D 1 S 2 S 1 Usually, the devices are assumed to be cooperative. But what if they are not? Game theory for wireless networks Georg-August University Göttingen 6

Assumptions and terminology  We will consider two players and non-cooperative games in four simple game examples related to wireless network operations: • Forwarder’s dilemma • Joint packet forwarding game • Multiple access game • Jamming game  The players wish to transmit or receive data packets coping with limited transmission resources which can make them to have selfish or malicious behavior.  S i : Strategy of player i (e.g. forward the packet, drop the packet, etc.)  S -i : Strategy of the opponents of player i (as we will have only two players i and j, S -i will be S j )  Having any strategy S i the players would get some benefit and have to pay some cost  U i : Utility or payoff of player i expresses the benefit of him given a strategy minus the cost it has to incur: U i = b i - c i  Users controlling the devices are rational, i.e. try to maximize their benefit. Game theory for wireless networks Georg-August University Göttingen 7

Outline 1 Introduction 2 Static games 3 Dynamic games 4 Repeated games Game theory for wireless networks Georg-August University Göttingen 8

Example 1: The Forwarder’s Dilemma ? R2 R1 Green Blue ?  Forwarder’s Dilemma: • Two players: Blue and Green • Green has one data packet to send to its receiver (R1) via Blue and also blue has one data packet to send to its receiver (R2) via Green. • If Blue forwards the packet for Green to R1, it must pay the transmission cost of c and Green will get the benefit of 1. • The dilemma: • each player is tempted to drop the other player’s packet to save its energy • but the other player may reason in the same way -- > they could do better by forwarding for each other Game theory for wireless networks Georg-August University Göttingen 9

From a problem to a game game formulation: G = (P,S,U)  – P: set of players – S: set of strategy functions – U: set of payoff functions • Benefit for the source of a packet reaching the destination: 1 • Cost of packet forwarding for the forwarder: c (0 < c << 1) strategic-form representation of the game  (each cell corresponds to one possible combination of strategies of the players and is presented as (U blue , U green )) Green Forward Drop Blue (1-c, 1-c) (-c, 1) Forward (1, -c) (0, 0) Drop Game theory for wireless networks Georg-August University Göttingen 10

Solving the Forwarder’s Dilemma (1/2) Strict dominance: strictly best strategy, for any possible strategy of the other player(s) s Strategy strictly dominates strategy S’ i if i      ' ' u s s ( , ) u s s ( , ), s S , s S     i i i i i i i i i i  u U where: payoff function of player i i  s S   strategies of all players except player i i i  In this game, Drop strictly dominates Forward from the point of view of both players:  Because 1-c<1 and – c<0  Therefore the solution of the game is (D,D), i.e. both players choose the strategy drop Green Forward Drop Blue (1-c, 1-c) (-c, 1) Forward (1, -c) (0, 0) Drop Game theory for wireless networks Georg-August University Göttingen 11

Solving the Forwarder’s Dilemma (2/2) Solving the game by iterative strict dominance: iteratively eliminate strictly dominated strategies Green Forward Blue Drop (1-c, 1-c) (-c, 1) Forward (1, -c) (0, 0) Drop Drop strictly dominates Forward } Dilemma BUT (F,F) would result in a better outcome for both players This is the lack of trust between the players that leads to this suboptimal solution (green is never sure that Blue will forward for it, if it forwards for blue (as they play simultaneously) and vice versa.) Game theory for wireless networks Georg-August University Göttingen 12

Example 2: The Joint Packet Forwarding Game ? ? Source Green Dest Blue • A sender sends its packets to its receiver and two players (Blue and Green) should forward the packets for it. • Reward for packet reaching the destination: 1 for both players • Cost of packet forwarding: c (0 < c << 1) Green Forward Drop Blue (1-c, 1-c) (-c, 0) Forward (0, 0) (0, 0) Drop For this game there is no strictly dominated strategies ! Game theory for wireless networks Georg-August University Göttingen 13

Weak dominance Weak dominance: Strategy s’ i is weakly dominated by strategy s i if    ' u s s ( , ) u s s ( , ), s S     i i i i i i i i with strict inequality for at least one s -i ? ? Source Green Dest Blue Solving the game by Iterative weak dominance: iteratively eliminate weakly dominated strategies Green  For Green, Drop is weakly dominated Forward Blue Drop by Forward (so the second column is eliminated); then in the remaining parts (1-c, 1-c) (-c, 0) Forward of the table Blue would do better to Forward. (0, 0) (0, 0)  The result of the iterative weak Drop dominance is not unique in general. Game theory for wireless networks Georg-August University Göttingen 14

Nash equilibrium (1/2) Nash Equilibrium: a game strategy such that no player can increase its payoff by deviating unilaterally Green Forward Drop Blue E1: The Forwarder’s (1-c, 1-c) (-c, 1) Forward Dilemma: (1, -c) (0, 0) (Drop, Drop) is a Nash Drop Equilibrium; if Green deviates to (D,F) its payoff decreases (from 0 to – c) and similarly if blue deviates to (F,D) its payoff decreases (from 0 to – c) Green Forward Drop Blue E2: The Joint Packet (1-c, 1-c) (-c, 0) Forward Forwarding game: There are 2 Nash Equilibria for (0, 0) (0, 0) Drop this game: (F,F) and (D,D) Game theory for wireless networks Georg-August University Göttingen 15

Nash equilibrium (2/2) Strategy profile s * constitutes a Nash equilibrium if, for each player i ,    * * * u s s ( , ) u s s ( , ), s S   i i i i i i i i  u U where: payoff function of player i i  s S strategy of player i i i The best response of player i to the profile of strategies s -i is a strategy s i such that:  b s ( ) argmax u s s ( , )   i i i i i  s S i i Nash Equilibrium = Mutual best responses Caution! Many games have more than one Nash equilibrium Game theory for wireless networks Georg-August University Göttingen 16

Example 3: The Multiple Access game  Two players have packets to send to their receivers but through the same channel; if in a given time slot both transmit packets there will be a collision and no packet will be delivered to its destination Time-division channel  If one player transmits and the other stays quiet in that time slot, the packet will be received by its receiver and its source will get some benefit Green Benefit for the source of a Quiet Blue Transmit successful transmission: 1 (0, 0) (0, 1-c) Cost of transmission: c Quiet (0 < c << 1) (1-c, 0) (-c, -c) Transmit • For this game: o There is no strictly dominating strategy o There are two Nash equilibria Game theory for wireless networks Georg-August University Göttingen 17