Repeated Game Theory as a Framework for Algorithm Development in Communication Networks

Transcription

1 1 Repeated Game Theory as a Framework for Algorthm evelopment n Communcaton Networks J. Combra, N. Correa Abstract Ths artcle presents a tutoral on how to use repeated game theory as a framework for algorthm development n communcaton networks. The artcle starts by ntroducng the bass of one-stage games and how the outcome of such games can be predcted, through teratve elmnaton and Nash equlbrum. In communcaton networks, however, not all problems can be modeled usng one-stage games. Some problems can be better modeled through mult-stage games, as many problems n communcaton networks consst of several teratons or decsons that need to be made over tme. Of all the mult-stage games, the nfnte-horzon repeated games were chosen to be the focus n ths tutoral, snce optmal equlbrum settngs can be acheved, contrarly to the suboptmal equlbra acheved n other types of game. Wth the theoretcal concepts ntroduced, t s then shown how the developed game theoretcal model, and devsed equlbrum, can be used as a bass for the behavor of an algorthm, whch s supposed to solve a partcular problem and wll be runnng at specfc network devces. Index Terms Game Theory, Repeated Game Theory, Communcaton Networks, Algorthm evelopment. I. INTROUCTION GAME theory s a mathematcal tool that ams to study and predct the outcome of stuatons where two or more agents have conflctng nterests [1]. The feld of game theory has ts roots n decson theory and, n fact, t can be thought as a generalzaton of decson theory for multple agents [1]. As a feld on ts own, game theory was poneered by John von Neumann and Morgenstern n [2], layng the foundatons of current game theory. A general formal descrpton of games was presented and several zero-sum games were analyzed and solutons to the games were devsed. Followng the concepts publshed n [2], many other contrbutons were publshed, such as the frst mathematcal dscusson of the prsoner s dlemma n [3] and the Nash equlbrum n [4], probably one of the most relevant contrbutons. Nash equlbrum was qute mportant because t s applcable to a wde varety of game types [1], [4]. The feld kept evolvng wth the research and analyss of several types of games such as extensve form and repeated games, whch wll be presented n ths tutoral [5], [6]. Game theory also lad down the foundatons for modern dscplnes, whch are very actve nowadays, such as algorthmc game theory and mechansm desgn [7], [8]. J. Combra (jcombra@ualg.pt) and N. Correa (ncorrea@ualg.pt) are wth the Research Center for Electronc, Optoelectronc and Telecommuncatons (CEOT), Unversty of Algarve, Faculty of Scence and Technology, Faro, Portugal. Snce ts ncepton, game theory has been researched and used manly for economcal purposes, but other felds started to use t as well. For nstance, game theory was extensvely appled to bology, manly due to the work from John Maynard Smth who developed the evolutonary stable strategy [9]. Other felds lke poltcal and socal scences followed and started usng game theory [10]. Computer scence and communcaton networks are no excepton and much research emerged n the fronter between computer scence and game theory. Most of the research n computer scence and game theory has been related to complexty theory, where several algorthms to compute Nash equlbrum have been proposed and studed [11]. As for communcaton networks, game theory has been used manly for analytcal purposes, where devces, such as routers, are players wth selfsh nterests. Nonetheless, there are also some works where game theory s used n communcaton networks as a framework for algorthm development [12], [13]. In ths tutoral, t s shown how repeated game theory can be used as a framework for algorthm development n communcaton networks, nstead of usng t just as an analytcal tool. No pror knowledge on game theory s assumed. That s, the tutoral starts by ntroducng the bass of one-stage games and, wth such knowledge, contnues onto dynamc and repeated games. It s shown how optmal Nash equlbrum strateges can be obtaned wth nfnte-horzon repeated games, whle the equvalent one-stage verson have suboptmal Nash equlbra and how that can be used as a support for the development of an algorthm to be run at devces n the network. Ths tutoral also exemplfes, wth a smplfed model taken from [12], the use of game theory to model a problem, devse an equlbrum strategy and develop an algorthm that mmcs such equlbrum strategy. The rest of ths paper s organzed as follows. The next secton ntroduces one-stage games, pure and mxed strateges and Nash equlbrum. In Secton III, mult-stage games are presented together wth Nash equlbrum and backward nducton. Repeated games are then presented n Secton IV, as well as the Nash equlbrum n nfnte-horzon repeated games and the folk theorem. An example s then shown n Secton V, from the model to the development of the algorthm. The tutoral fnalzes wth some conclusons n Secton VI. II. ONE-STAGE GAMES Game theory can be used to model and study stuatons where agents have conflctng nterests. One example of such Preprnt submtted to Internatonal Journal of Communcaton Systems, John Wley & Sons, Ltd, n 2015.

2 2 d 2 Fg. 1. Forwarder s dlemma representaton. d 1 TABLE I FORWARER S ILEMMA IN NORMAL FORM. F F (1 C,1 C) ( C,1) (1, C) (0, 0) a stuaton s the wdely known prsoner s dlemma, whch s usually presented as follows [1]. Two men are arrested, but the polce does not have enough nformaton for them to be convcted. Both prsoners are then nterrogated separately and at the same tme. Each prsoner can choose to stay slent or betray the other. If both stay slent, both wll go to prson for just 1 month. If one prsoner betrays the opponent, whle the opponent stays slent, then the slent prsoner goes to prson for 12 months and the betrayer goes free for cooperatng wth the polce. Fnally, f both prsoners betray each other smultaneously, then both wll go to prson for 3 months. The queston s what wll prsoners do, assumng that none of them can be sure f the other wll betray or stay slent. If both stay slent, both get a mnor sentence of one month. owever, each prsoner may feel tempted to betray the other n order to be freed. As a result, both may end up betrayng each other. ence the dlemma. In [14], a conceptually smlar verson of the prsoner s dlemma s presented. It s called the forwarder s dlemma and wll be used throughout ths artcle to help explanng some defntons. The game can be explaned as follows. There are two players, router and router, that want to send a packet to d 1 and d 2, respectvely. As shown n Fgure 1, for d 1 to receve the packet from, wll have to cooperate and forward the packet. Conversely, the same apples for the packet from sent to d 2. If a packet reaches ts destnaton, then the player who sent t receves a payment of 1. A player that chooses to forward the packet of the opponent s ncurred a cost of C, where 0 < C << 1. Ths cost represents the consumpton of resources to forward foregn traffc. The queston s whether or not players n the forwarder s dlemma wll cooperate wth each other by forwardng packets. If both players cooperate, then both wll receve a payoff of 1 C. owever, a player mght feel tempted to defect n order to receve a payoff of 1, whch s the hghest payoff n ths game, leavng a payoff of C for the opponent. In non-cooperatve one-stage games, t s assumed that players decde at the same tme what wll be ther actons wthout communcatng ther preferences beforehand. ere, the term preference refers to the acton that a player feels tempted to choose, forward or drop the packet n the case of the forwarder s dlemma. owever, even f players n the forwarder s dlemma communcate ther preferences beforehand and agree to cooperate, both players wll stll be tempted to le and drop the packet belongng to the opponent n order to receve the hghest payoff of 1. As a safe precauton, both players wll defect by not forwardng the packet of the opponent. Ths way, both players wll play and, as a result, wll receve a payoff of 0, even though they could receve a better payoff of 1 C. ence the dlemma. A. Normal and Strategc Form Representatons Games can be represented n many dfferent forms. One of the most common s the normal form representaton, whch s very useful for smple games wth two players and only a few avalable actons to each player [1], [15]. Ths representaton conssts of a table, where the lnes represent the strateges of one player and the columns represent the strateges of the other player. The cell that results from the ntersecton of a row and a column contans the payoffs that both players wll receve. Consderng the just presented forwarder s dlemma, there are two players, and, whch can forward or drop a packet, represented by F and respectvely. The normal form representaton of the forwarder s dlemma s shown n Table I. The rows represent the actons avalable to, whle the columns represent the actons avalable to. As already told, the cell resultng from the chosen lne and column contans the payoffs that players wll receve. For nstance, f forwards and drops the packet, then the resultng cell contans ( C,1), whch means that receves a payoff of C and receves a payoff of 1. The tuple ncludng the strategy chosen by each player s called strategy profle. In the example just used, where forwards and drops, the strategy profle s (F,). The normal form representaton s good for smple examples, however, for games wth many players and multple strateges, t s mpossble to use the normal representaton. For those cases, the strategc form s the most suted. In ths form, a game s represented by G = {P,S,U}, where P represents the set of players, S represents the set of all strategy profles and U represents the set of utlty functons, explaned next [1], [14], [15]. The set of all strategy profles can be obtaned by S = P S, where S s the set of all strateges avalable to player. 1 In game theory lterature, for convenence, the set of all players except s denoted by. Ths way, one can represent a strategy profle (s,s ) that s composed of a specfc strategy from, s S, and any combnaton of strateges from all other players, s S. As for the set of utlty functons, U = {u P}, t ncludes the payoffs that each player receves as a result from the chosen strategy profle,.e. u : S R [1], [14], [15]. Players n a game can have complete or ncomplete nformaton. In a complete nformaton game, every player P knows everythng about the game he s nvolved n. More specfcally, every player P knows all the other players, ther avalable strateges and the respectve payoffs. Moreover, every player knows that the opponents also have that nformaton. Ths knowledge can be used to ntellgently 1 The symbol represents the Cartesan product. ence, P S = {( s1,s 2,...,s P ) s1 S 1 s 2 S 2... s P S P }.

3 3 TABLE II EXAMPLE OF A GAME WIT ONE STRONGLY OMINATE STRATEGY, TAKEN FROM [1]. z 2 x (2,3) (3,0) (0,1) 1 y 1 (0,0) (1,6) (4,2) TABLE III GAME FROM TABLE II WIT TE STRONGLY OMINATE STRATEGY ELIMINATE. x 1 (2,3) (3,0) y 1 (0,0) (1,6) choose strateges that provde the hghest possble payoffs [1], [14], [15]. On the other hand, n a game wth ncomplete nformaton, players do not know whch strateges are avalable to the opponents, nether the resultng payoffs. Certan belefs mght be known about the opponents but those are not accurate and, as such, the behavor of players can be dfferent. In ths thess, only complete nformaton games wll be used. efnton 1 (Complete Informaton Game) A game wth complete nformaton s a game where every player P knows all the other players, ther avalable strateges and all payoffs that they receve as result from the chosen strategy profles. B. omnated Strateges In game theory, players choose ther strateges n order to receve the hghest possble payoff. Thus, t can be expected that strateges that never lead to hgh payoffs wll never be chosen. Consderng the game from Table II, taken from [1], player wll never choose strategyz 2. That s because greater payoffs can be obtaned by, ether by choosng x 2 or y 2, no matter how hs opponent plays. In ths case, t s sad that strategy z 2 s strongly domnated [1], [14], [15]. efnton 2 (Strong omnance) Strategy s of player s strongly domnated f for any strategy profle adopted by the opponents of, s S, there exsts at least one s s such that u (s,s ) < u (s,s ). Strongly domnated strateges can be removed from the game, snce ntellgent players would never choose them. In the case of the game from Table II, f strategy z 2 s elmnated, then the resultng game wll be the one n Table III. Note that n the resultng game, after elmnaton of z 2, strategy y 1 of also becomes strongly domnated and, therefore, can be removed. Ths elmnaton process of strongly domnated strateges s called teratve elmnaton [1], [14], [15]. At the end, for the gven example, only one strategy for each player wll reman, x 1 for and x 2 for. Snce strategy profle (x 1,x 2 ) s expected to be chosen, wll receve a payoff of 2 and wll receve a payoff of 3. TABLE IV EXAMPLE OF A GAME WIT WEAKLY OMINATE STRATEGIES, TAKEN FROM [16]. y 1 (2,0) x 1 (1,0) (1,1) (2,1) Strateges can also be weakly domnated [1], [14], [15]. efnton 3 (Weak omnance) Strategy s of player s weakly domnated f for any strategy profle adopted by the opponents of, s S, there exsts at least one s s such that u (s,s ) u (s,s ), wth strct nequalty for at least one s S. Removng weakly domnated strateges by teratve elmnaton can also be done, however, t can lead to unexpected results. Consderng the game from Table IV, taken from [16], has two weakly domnated strateges, x 1 and y 1. In Fgure 2, t s possble to see how elmnatng x 1 or y 1 frst can lead to dfferent results. That s, the order n whch weakly domnated strateges are elmnated can lead to dfferent outcomes. Such stuaton does not happen wth strongly domnated strateges, because elmnaton does not cause strongly domnated strateges to cease beng strongly domnated. On the other hand, a weakly domnated strategy can cease beng domnated f other strateges are removed. C. Nash Equlbrum It s not always possble to predct the outcome of a game through teratve elmnaton. For nstance, the game n Table V, taken from [1], has no domnated strateges. Nevertheless, t s stll possble to predct what wll be the outcome of the game. For that, the noton of best response needs to be ntroduced [1], [14], [15]. efnton 4 (Best Response) The best response of player s a functon br (s ) that outputs whch strategy should choose n order to receve the hghest possble payoff, gven that the opponents wll play s. That s, br (s ) = argmax s S u (s,s ). In the game from Table V, the strategy x 1 from s the best response to strategy x 2 from. Strategy x 2, n ts turn, s the best response to strategy. One nterestng strategy profle s the one where plays y 1 and plays y 2, wth the payoff (1,1). In ths case, y 1 s the best response to y 2 and, smlarly, y 2 s the best response to y 1. Ths strategy profle s actually the expected outcome of ths game, snce none of the players has any ncentve to unlaterally choose a dfferent strategy. That s, f plays x 1 or y 1, ts payoff wll decrease, consderng that does not change ts strategy. Smlarly, wll also not change to x 2 or z 2 because ts payoff wll decrease, snce s playng y 1. Ths type of strategy profles, where no player has any ncentve to devate, s termed Nash equlbrum [1], [14], [15].

4 4 x 1 y 1 (1,0) (1,1) (2,0) (2,1) Remove x 1 Remove y 1 y 1 x 1 (1,1) (2,0) (2,1) (1,0) (1,1) (2,1) Remove y 2 Remove x 2 y 1 x 1 x 2 (1,1) y 2 (2,1) Remove y 1 Remove x 1 x 2 (1,1) y 2 (2,1) Fg. 2. fference between two alternatve teratve elmnatons of weakly domnated strateges. TABLE V EXAMPLE OF A GAME WITOUT OMINATE STRATEGIES, TAKEN FROM [1]. z 2 y 1 (2,0) (1,1) (2,0) x 1 (3,0) (0,3) (0,3) (3,0) efnton 5 (Nash Equlbrum) ( A( strategy profle s s a Nash equlbrum fu s,s ) u s,s ), s S,s s, P, wth at least one strct nequalty. It s possble to have more than one Nash equlbrum n one game. In the example from Table IV, both strateges obtaned through teratve elmnaton of weakly domnated strateges, (,x 2 ) and (,y 2 ), are actually Nash equlbrum strateges. Indeed, strategy profles obtaned by teratve elmnaton are always Nash equlbrum profles. Note, however, that n the case of teratve elmnaton of weakly domnated strateges, the resultng profles are a subset of the Nash equlbrum profles, meanng that there mght be more Nash equlbrum profles [1]. As for teratve elmnaton of strongly domnated strateges, the resultng profle s the only Nash equlbrum, as n the game from Table II [1]. Nash equlbrum, as shown, predcts what wll be the outcome of a game. For example, n the forwarder s dlemma from Table I, the Nash equlbrum profle s (, ). Note that ths outcome s not the most effcent, snce both players could receve greater payoffs f the profle(f,f) was played nstead. owever, snce any player mght feel tempted do defect n order to receve the hghest payoff of 1, both players, as a precauton, end up choosng n order to avod recevng C. In ths case, the Nash equlbrum strategy s not the most effcent outcome, snce players receve the payoff (0, 0), and a greater payoff (1 C,1 C) could be earned f the profle (F,F) was chosen nstead. In fact, Nash equlbrum only predcts what wll be the natural choces of ntellgent players that do not trust each other and, n many games, t s not the most effcent outcome. The challenge resdes n desgnng systems where players have ncentves to cooperate, forward traffc from each other n the case of the forwarder s dlemma, n order for effcent Nash equlbra to be reached [7], [8], [17]. In game theory, the strategy profle(f,f) of the forwarder s dlemma s sad to be Pareto superor to other profles. efnton 6 (Pareto Superor) A strategy profle s S s Pareto superor to s S f u (s,s ) u ( s,s ), P, wth at least one strct nequalty. The most effcent outcome n a game would be one wth the hghest payoffs for every player, (F,F) n the case of the forwarder s dlemma. Such effcent outcome s sad to be Pareto optmal and no other profle s Pareto superor to t [1], [14], [15]. efnton 7 (Pareto Optmal) A strategy profle s S s Pareto optmal f there s no other strategy that s Pareto superor to s. There are cases where Nash equlbrum s Pareto optmal. In such cases, t s sad that Nash equlbrum s Pareto effcent. Naturally, the most desred Nash equlbrum s the Pareto effcent one, snce payoffs are hgher.. Mxed Strateges Untl now, n ths chapter, t has been assumed that players choose one specfc strategy to be played and the expected outcome of the game s a Nash equlbrum profle. owever, n some games, Nash equlbrum may not exst, as shown n the example from Table VI, taken from [18]. Instead of choosng whch specfc strategy should be played, players can defne a probablstc dstrbuton over ther avalable strateges. In the example from Table VI, a Nash equlbrum would exst f both players defne a probablty of 1/2 over each of ther strateges, as t wll become clear next. Such dstrbuton s termed mxed strategy [18]. efnton 8 (Mxed Strategy) A mxed strategy σ s a dstrbuton over the strateges of, S. The set of all mxed strateges from a player P s denoted by Σ (captal of σ). Smlarly to strategy profles, s S, mxed strategy profles can be defned by Σ = P Σ. From here on, to avod confuson, the set of profles n S wll be called pure strategy profles, whle the profles n Σ wll be termed mxed strategy profles. Snce mxed strateges defne probabltes over the set of avalable pure strateges, the utlty functon n ths case reveals the expected payoff based on the chosen mxed profle σ [1], [15], [18]:

5 5 TABLE VI EXAMPLE OF A GAME WITOUT PURE NAS EQUILIBRIUM, TAKEN FROM [18]. x 1 (1, 1) ( 1,1) y 1 ( 1,1) (1, 1) F F F (1-C,1-C) (-C,1) (1,-C) (0,0) Fg. 3. Two stage verson of the forwarder s dlemma, where s the frst player to move. u (σ) = s S u (s) j Pσ j (s j ), (1) where s j s the strategy of j n profle s and, σ j (s j ) represents the probablty of s j beng chosen. ence, u (s) j P σ j (s j ) represents the expected payoff of f s s chosen. Regardng the game from Table VI, and assumng that chooses x 1 wth probablty q x1 and chooses x 2 wth probablty q x2, then the expected payoff for can be calculated by: u p1 (σ) = [1q x1 q x2 ]+[ 1q x1 (1 q x2 )]+ + [ 1(1 q x1 )q x2 ]+ + [1(1 q x1 )(1 q x2 )], where 1 and 1 are the payoffs that would receve accordng to the dfferent strategy profles. As aforementoned, Nash equlbrum wll exst, n ths case, for q x1 = q x2 = 1/2, meanng that n ths mxed Nash equlbrum both players wll receve a payoff of 0. Accordng to [1], [15], every game wth a fnte set of strateges has at least one pure or mxed Nash equlbrum. Note that a mxed Nash equlbrum profle s never Pareto optmal. That s because a mxed profle s, n fact, a lnear combnaton of pure strateges and, as such, could not result n hgher payoffs than the ones obtaned by pure strateges. III. YNAMIC GAMES One-stage games can only model stuatons where all players take ther decsons at the same tme. owever, many stuatons may be better modeled wth games composed of several stages [1], [15], [19]. For nstance, n the forwarder s dlemma, players may not have packets to send at the same tme. Let us assume that s the frst player wth a packet to be sent, whch may or may not forward. Immedately after, also sends a packet, whch can choose to forward or not. Such games are termed dynamc games or mult-stage games. In ths chapter, and throughout the thess, only dynamc games wth perfect nformaton are consdered. efnton 9 (Perfect Informaton) A dynamc game wth perfect nformaton s one where every player P knows all the actons taken n prevous stages by all opponents. In the prevous mult-stage forwarder s dlemma example, can decde whether or not to forward based on the acton of n the prevous stage. Naturally, dynamc games need a dfferent representaton that must be capable of showng the order n whch players make ther moves [1]. The extensve form representaton, shown n Fgure 3 for the forwarder s dlemma, s the most suted for these stuatons. The extensve form conssts of a tree structure where the root node represents the frst decson n the game. In the prevous example, the frst decson belongs to. The lnes wth labels (F and ) represent the actons avalable to the players. Player, at the frst stage, can decde to forward, F, or to drop,, the packet from. The leafs of the tree contan the payoffs that players wll receve accordng to ther decsons. In mult-stage games, the player to move n the frst stage has a set of strateges equal to the ones avalable n the one-stage game. For nstance, from Fgure 3 has the followng strateges avalable: S p1 = {F,}. The players n the subsequent stages can take ther decsons based on the actons from prevous stages. Player from Fgure 3, can decde ts acton based on the move of n the prevous stage. For ths reason, strateges for wll be dfferent n the mult-stage game. In ths case, has the followng strateges avalable: S p2 = {FF,F,F,}. The frst character n a strategy from represents the acton that takes f choosesf n the frst stage and the second character represents the acton that takes f chooses. For nstance, strategy (F) means that wll forward f has forwarded n the prevous stage and wll drop f has dropped. As prevously explaned, the leafs represent the payoffs that players wll receve accordng to the decsons taken. A. Nash Equlbrum and Backward Inducton The concept of Nash equlbrum n dynamc games s not dfferent from one stage games. That s, a strategy profle s a Nash equlbrum f no player can ncrease ts payoff by unlaterally devatng. Consderng the example from Fgure 4, taken from [18], the pure Nash equlbrum strategy profles are: (,), (,) and (,). These Nash equlbra can be more easly extracted from the normal form equvalent game shown n Table VII. Note that the rows of the table nclude the current possble moves for ( and ) and the columns nclude the current possble moves for that are based on the prevous acton of (,, and ). In strategy profle (,), whch s one of the Nash equlbrum profles, threatens to play regardless of the move from. Player s aware of ths threat and, as a result, could play hs best response to, whch s strategy. owever, lookng more closely at Fgure 4, f chooses n the frst stage, then s really not wllng to choose n the second stage, snce would gve a better payoff. Ths

6 6 (-2,-2) (6,0) (0,6) (3,3) Fg. 4. Example of a dynamc game taken from [18]. TABLE VII NORMAL FORM EQUIVALENT GAME FROM FIGURE 4 ( 2, 2) ( 2, 2) (6,0) (6,0) (0, 6) (3, 3) (0, 6) (3, 3) (-2,-2) (6,0) (0,6) (3,3) Fg. 5. Backward nducton technque appled to the example from Fgure 4. The contnuous thck lne from the root node to the leaf represents the predcted outcome of the game. knd of threats are termed empty threats, snce s actually bluffng and does not represent a real threat [15], [18]. Fndng Nash equlbrum strategy profles n mult-stage games can lead to empty threats and ther removal can be done through backward nducton [15], [18]. Ths technque starts by analyzng the most proftable acton n the last stage and then, based on the most proftable actons at the last stage, t s analyzed whch s the most proftable acton at the penultmate stage. Ths analyss keeps proceedng upward n the tree structure untl the root node s reached. To exemplfy ths, let us assume the game from Fgure 4. Frst, the acton that results n the hghest payoff for s determned, consderng all the possble prevous actons of. If played, then the best choce s for to choose. On the other hand, f chose, then wll be better wth. Gven the best moves of, t s possble to decde whch acton results n the hghest payoff for. Clearly, wll be better by playng, snce t wll gve hm a hgher payoff. In Fgure 5, t s possble to see the result of the backward nducton, where the thck lnes mark the best actons at every stage. The contnuous route of thck lnes from the root to the leaf represents the predcted outcome of the game. ence, (, ) and (, ) are the predcted outcomes, snce both these strateges lead to the actons chosen by backward nducton. IV. REPEATE GAMES Repeated games are a specfc type of dynamc games where players face the same one-stage game repeatedly [1], [15], [18], [19]. An example of a repeated game would be the repeated forwarder s dlemma, where the game n Table I s played repeatedly over several stages. At every stage t, every player P has to choose hs acton, a (t). The profle of all actons chosen by all players at a stage t s represented by a(t) = ( a p1 (t),a p2 (t),...,a p P (t) ). In ths paper, only repeated games wth perfect nformaton are consdered. As such, at every stage, all players have access to the hstory of all prevous actons, h(t) = (a(0),a(1),...,a(t 1)). Such hstory s taken as nput to the strategy of every player to decde what wll be the next acton: s (h(t)) = a (t). Naturally, the strategy profle outputs the next profle of actons, s(h(t)) = a(t). A. Fnte-orzon Games and Nash Equlbrum Repeated games can be fnte-horzon, whch means that the number of stages s lmted, or nfnte-horzon, whch means that players nteract over an nfnte or unknown number of stages [1], [18]. The payoff attrbuted to every player P of fnte-horzon games can be calculated by summng the stage payoffs of all stages: T U (s) = u (s(h(t))), (2) t=0 where T s the last stage, u s the stage payoff of player and U s the total payoff. Consderng the repeated forwarder s dlemma as an example, and assumng that both players are usng a strategy that chooses F at every stage, the payoff attrbuted to both players would be T t=0 (1 C) = (T +1)(1 C) n ths case. To understand Nash equlbrum n fnte-horzon games, let us keep consderng the repeated forwarder s dlemma. If both players played F untl stage T 1, then one of the players could devate to at the last stage T to ncrease hs payoff. The opponent knows that, and to avod recevng C at the last stage, he can also play. Moreover, snce t s predcted that both players wll play at the last stage, then players can also devate at the penultmate stage n order to ncrease ther payoff. Followng ths reasonng, the strategy profle that chooses the acton (,) at every stage s a Nash equlbrum of the fnte-horzon repeated forwarder s dlemma. Note that ths method, used to fnd Nash equlbrum, s smlar to the backward nducton ntroduced n Secton III-A. ence, any strategy profle that produces the outcome predcted by the backward nducton s a Nash equlbrum. B. Infnte-orzon Repeated Games and Nash Equlbrum Infnte-horzon repeated games, as aforementoned, are played on forever or for an unknown number of stages. As such, the payoff functon of the fnte-horzon game, shown n equaton 2, can not be used for nfnte-horzon games because t could result n nfnte payoffs. Instead, a weghted sum, termed dscounted payoff, s used [1], [18], [19]: U (s) = (1 δ) δ t u (s(h(t))), (3) t=0 where δ s the weghtng factor, termed dscountng factor, and accepts only values between 0 and 1, 0 < δ < 1. As for

7 7 (1 δ), t s responsble for normalzng the payoffs, allowng the comparson between dscounted payoffs and the payoffs receved at every stage. For nstance, n an nfnte-horzon repeated game where player receves a stage payoff of 1 at all stages, the dscounted payoff for wll be 1. Note that as t grows, δ t decreases. ence, stage payoffs become less mportant as t grows, snce the stage payoff s beng multpled by δ t. The actual value attrbuted to δ t n a game wll nfluence how fast the stage payoffs loose mportance, alterng the behavor of players and the Nash equlbrum profles, as wll be demonstrated next. In nfnte-horzon games, smlarly to one-stage games, a strategy profle s S s a Nash equlbrum f: U ( s,s ) U ( s,s ), s S, P. (4) owever, n the case of nfnte-horzon games, Nash equlbrum s greatly nfluenced by the dscountng factor and by the fact that the game s played on forever [1], [18], [19]. To exemplfy t, let us ntroduce the grm trgger strategy, whch s wdely used n game theoretcal lterature [1], [18], [19]. A player usng ths strategy wll exert effort at every stage, as long as the opponent also cooperates. If the opponent shrks even only once, then wll stop cooperatng from thereafter. Applyng ths strategy to the forwarder s dlemma, and labelng one of the players by and the opponent by j (f = then j =, f = then j = ), grm trgger strategy can be defned by the followng equaton: s (h(t)) = F, f (a (t 1) = F) and (a j (t 1) = F), otherwse If both players n the nfnte-horzon repeated forwarder s dlemma play ths strategy, then they wll play F at all stages. The outcome of such strategy profle for both players wll be: (1 δ) [ (1 C)δ 0 +(1 C)δ ] = = (1 δ)[(1 C) [ ] t=0 δt 1 ] = (1 δ) (1 C) 1 δ = (1 C). If player devates at some stage t, then wll receve a hgher payoff n that stage but wll receve zero thereafter. The resultng dscounted payoff for the devatng player can be calculated by: (1 δ) [ (1 C)δ 0 +(1 C)δ δ t +0δ t +1 +0δ t ] = [ = (1 δ) (1 C) t 1 t=0 δt +δ t ] = = (1 δ) [(1 C) 1 δt 1 δ +δ t ] = = (1 C) ( 1 δ t ) +(1 δ)δ t = = (1 C) δ t (1 C)+δ t (1 δ) = = (1 C)+δ t C δ t δ = = (1 C)+δ t (C δ). For the profle, where both players use grm trgger, to be Nash equlbrum, the devaton can not be proftable for. That s: 1 C 1 C +δ t (C δ) 0 δ t (C δ). = (5) Snce 0 < δ < 1, then δ C for the nequalty to hold. As long as the cost of forwardng a packet, C, s lower than the dscountng factor, δ, t s more proftable to follow the grm trgger than devatng from t. Ths results n a Nash equlbrum profle where both players exert effort by forwardng packets from each other. As aforementoned and exemplfed, the dscountng factor s an mportant pece n the Nash equlbrum of nfntehorzon repeated games. If δ s close to 0, then the mportance of the successve stage payoffs wll decrease rapdly, and as a result, the relevance of the frst stage s much greater than the subsequent payoffs. As such, players wll care mostly wth the frst stage and wll try to earn the hghest possble mmedate payoff. Ths mmcs the behavor of an mpatent player manly nterested wth the current stage payoff. On the other hand, f δ s close to 1, the mportance of the successve stage payoffs decreases slowly, oblgatng players to be more patent and cooperate to avod severe punshments n future stages. Relatng ths reasonng to the necessary condton for the grm trgger strategy to be a Nash equlbrum n the repeated forwarder s dlemma, δ C, mpatent routers wll devate from the strategy and drop the packets, whle patent routers wll follow the strategy and forward the packets. An alternatve meanng for δ s that t can represent the probablty of the game endng n the current stage. That s, hgh values for δ represent a hgh probablty that there wll exst more stages, whle a low value of δ represents a low probablty that the game wll contnue to a next stage. Therefore, f the probablty of the game beng played for many stages s hgh, then players wll be patent and cooperate. Otherwse, f there s a hgh probablty that the game wll be played only for a few stages, then players wll not care about cooperatons nor possble punshments, and wll try to earn the hghest possble mmedate payoffs. As shown, equlbrum strateges where players cooperate are possble n repeated games, as long as the dscountng factor s hgh enough. In communcaton networks, generally, t can be consdered that the dscountng factor s close to 1. The reasonng s that networks are supposed to operate for very long perods of tme and t s unknown when wll a network cease operaton [14]. That s, the probablty that the network wll operate for many stages s hgh and therefore δ can be assumed to be close to 1. 1) Folk Theorem: Many Nash equlbrum strategy profles exst n nfnte-horzon repeated games that do not exst n one-stage games. Ths allows for certan payoffs to be obtaned that would not be possble n Nash equlbrum of one-stage games. To understand whch payoff values are possble, let us ntroduce the mn-max payoff [18], [19]. efnton 10 (Mn-Max Payoff) The mn-max payoff for player s defned as: u = mn s max s u (s,s ). That s, the mn-max payoff s the lowest payoff that some player can receve, provded that all opponents wll choose a strategy to mnmze the payoff of and wll choose the best response to such strategy to maxmze hs payoff. In the case of the forwarder s dlemma, ths corresponds to

8 8 (-C,1) (1-C,1-C) between players, as wll be shown n the next secton through an example. (0,0) (1,-C) Fg. 6. Illustraton of the mn-max and feasble payoffs. The thck lnes represent the mn-max payoffs, whle the gray area represents all feasble payoffs. the payoff (0, 0), earned when both players play. Any payoff greater or equal than the mn-max s possble to obtan by a Nash equlbrum strategy profle n an nfnte-horzon repeated game. In Fgure 6, the mn-max payoff n the nfntehorzon forwarder s dlemma s shown wth thck lnes. The gray area represents all feasble payoffs by Nash equlbrum strateges [18], [19]. efnton 11 (Feasble Payoffs) The set of feasble payoff profles s gven by: {u = (u 1, u 2,..., u P ) : u u, P}. Provded that δ s hgh enough, then any payoff n the feasble area can be obtaned by a Nash equlbrum strategy profle. Theorem 1 (Folk Theorem) For every feasble payoff profle u {u = (u 1,u 2,..., u P ) : u u, P }, there exsts a dscountng factor δ < 1 such that for all δ ]δ,1[, there s a Nash equlbrum profle wth payoffs u. From all the possble outcomes n equlbrum, the ones wth the hghest payoff,.e. the ones obtaned by a Pareto effcent equlbrum profle, are the most desred from the network pont of vew. An algorthm could be developed that mmcs the behavor of such a Pareto effcent equlbrum strategy. For nstance, n the case of the repeated forwarder s dlemma, an algorthm could be developed that leads every player to cooperate at every stage, as long as the opponent also cooperates. If t happens that the opponent defects, then the harmed player can punsh the defectng opponent by not forwardng ts packets for a certan amount of stages. The punshment needs to last for enough stages n order to make the punshment sever enough to deter any devatons. That s, t has to be clear for the alleged defectng opponent that defectng wll not be more productve. The general dea s to model the problem n queston and seek for equlbrum profles wth the hghest possble payoffs. Wth such knowledge, an algorthm can then be developed that wll mmc the behavor of the equlbrum strategy profle. The example of the forwarder s dlemma represents a game where players have clear conflctng objectves. Ths could be appled, for nstance, to border routers that belong to dfferent autonomous networks wth selfsh nterests. owever, game theory can also be used n settngs where the conflct arses n certan stuatons only or when there s a lack of coordnaton V. ALGORITM EVELOPMENT IN A COMMUNICATION NETWORK CONTEXT As aforementoned, devces/players n a network do not need to have persstent conflctng nterests for game theory to be useful as a mean to develop an algorthm. In ths secton, a smplfed model from [12] wll be used to show how an algorthm can be developed under such assumpton. In ths case, several wreless routers, deployed by a servce provder, have the objectve of forwardng as much traffc as possble, whle avodng wastage of resources. A. Fber-Wreless Access Network Scenaro Fber-Wreless access networks use a mxture of optcal and wreless technologes to provde Internet access to users. They are composed of two sectons: an optcal back end secton, whch brngs fber from the central offce to near the users; and a wreless front end secton, whch provdes wreless Internet access to the users. ere, t s consdered that the wreless front end s composed of wreless routers n a mesh topology, as shown n Fg. 7. Some of those wreless routers are gateways responsble for the fronter between the optcal and wreless envronments. A user wllng to send/receve traffc to/from the Internet can connect to the nearest wreless router or gateway. Traffc may need to travel through several hops n the wreless mesh secton and, as such, one of the key ssues to address s the allocaton of resources throughout the mesh secton n order to serve all users n a far manner. At the wreless secton, every wreless router/gateway wants to send/receve traffc belongng to ts users to/from the Internet, through the optcal secton. Lke n [12], t s assumed that tree structures are already formed as a result from the path selecton done by a routng algorthm. Such tree formaton s exemplfed n Fg. 7, where the establshed connectons are shown. The set of all wreless routers wll be denoted by W. Snce a tree structure s used, Γ + s used to represent all descents of a wreless router W, whle the set of ancestors s denoted by Γ. Every wreless router wll forward traffc belongng to ts drectly connected users, both downstream and upstream traffc. Besdes traffc belongng to ts drectly connected users, every wreless router W also forwards downstream and upstream traffc belongng to users connected to wreless routers n Γ +. The queston s how much bandwdth should every wreless router allocate for traffc belongng to ts own users, and how much should be allocated for the traffc belongng to users connected to wreless routers n Γ +. In ths model, every wreless router and gateway s a player wth the followng objectves: Forward as much traffc as possble, ether belongng to ts drectly connected users or belongng to users connected to Γ +. ave the least possble amount of packet drops. Such packet losses lead to unfrutful use of resources, snce these packets may have traveled through several hops and

9 9 Wreless gateway Wreless router Reachable wreless connecton Establshed wreless connecton Optcal connecton Fg. 7. Example of a wreless mesh front end from a Fber-Wreless access network. Wreless connectons are establshed accordng to the routng protocol n use. drop component wll be close to 0. On the other hand, f too many drops occur, then the drop component value wll be too negatve, η << 0. In computer networks, the reasonng behnd ths s that a few packet drops can be recoverable, whle too many packets beng dropped may end up n unrecoverable servce falures. In TCP/IP networks, for example, a low number of packet losses can easly be solved by the fast retransmsson mechansms that TCP offers, whle a hgh number of packet losses wll cause TCP to enter nto slow start [20]. The expresson (7) calculates only the payoff of one stage. The dscounted payoff that every player receves n the nfnte-horzon repeated game s calculated wth the dscounted payoff functon n (3), whch uses expresson (7) for the payoff of every stage. As for the strateges, they decde the values for B and B,j, j Γ+ to be used at every stage, accordng to the hstory of values B and B,j, j Γ+ chosen n prevous stages. consumed resources from the wreless routers along the the hops. Resumng, every wreless router, whch was deployed by a servce provder, wants to assure the best qualty of servce possble and to use resources n a useful manner. The amount of bandwdth that users connected to a wreless router, or gateway, W need to send/receve ther traffc at tme t s represented by B (t). Such wreless router wll then dedcate the amount B (t) of bandwdth for ths traffc. Also, every node W wll dedcate B,j (t) of bandwdth to downstream and upstream traffc belongng to users connected to every wreless router j Γ +, totalng B = Γ + j Γ + B,j. The actual bandwdth that a wreless router/gateway wll have avalable for traffc belongng to ts users, equal to the mnmum bandwdth made avalable to t throughout all wreless routers/gateway n the route to the optcal lnk, s denoted by B A (t). That s: [ B A (t) = mn B, mn j Γ ( B,j ) ]. (6) Consderng the objectves of the wreless routers/gateways, the stage payoff of every player can be calculated by the followng expresson: { η (t)+b A (t)b A (t),γ + u (t) = Γ + η (t)+b A (t),γ + =, (7) where B A (t) = Γ + j Γ + Bj A(t) and η (t) represents the amount of resources that are wasted due to packets beng dropped, termed drop component. The followng expresson can be used to calculate such component: ( ) ( B A log (t) B B )+log A Γ + (t) η (t) = (t) B Γ + (t),γ + ( ). (8) B A log (t) B (t),γ + = Note that the drop component has a logarthmc nature, meanng that f only a few packets are dropped, then the B. Nash Equlbrum, Pareto Effcency and Algorthm evelopment As already explaned n Secton II-C and Secton IV-B, Nash equlbrum represents the expected outcome of the game. From the network pont of vew, the most desred outcome would be a Pareto effcent outcome, where all wreless routers forward as much traffc as possble wth as few packet drops as possble. In a one stage verson of the game descrbed, the Nash equlbrum s for every wreless router W to set B and B,j, j Γ+, wth values near zero. That s because t s not known beforehand what wll be the bandwdth needs of the other wreless routers n Γ and n Γ +. Ths way, routers can rest assured that they wll not receve a payoff due to the logarthmc nature ofη. owever, f the bandwdth needs were known beforehand, then every router W could set B and B,j, j Γ+ to hgher values,.e. as close tob andb j, j Γ +, as possble. Note that n the forwarder s dlemma case, havng players communcate or coordnate ther preferences does not lead to a Pareto effcent Nash equlbrum because players stll feel compelled to le and devate by droppng the packet of the opponent. In the game presented n ths secton however, players have no ncentves to le to ther opponents because they wll receve a hgher payoff f they do not le nor devate. In game theory lterature, ths type of game s called a coordnaton game. That s, players have ncentves to cooperate, as long as they can coordnate ther actons. In these games, strateges where players communcate ther preferences n order to coordnate ther actons are Nash equlbrum and Pareto effcent [21], [22]. In coordnated repeated games, players try to coordnate ther actons at every stage. In games where the set of actons s small and well known, such coordnaton can be reached by havng players randomze ther actons at every stage untl the desred coordnaton s reached and, once coordnaton s reached, players wll keep usng the coordnated actons. Another alternatve, whch reaches mmedate coordnaton s to have players communcate ther preferences at every stage

10 10 [21]. In the example shown here, the bandwdth needs of the users connected to every wreless router, B (t), may change from stage to stage and, as such, the actons, B and B,j, j Γ+, need to change at every stage n order for coordnaton to be possble. As such, randomzng actons wll most lkely not lead to coordnaton. If players communcate ther current preferences at every stage, then ther actons can be coordnated. Ths s, n fact, a Pareto effcent Nash equlbrum n coordnated repeated games [21]. Note that n the forwarder s dlemma, havng players communcate ther preferences at every stage wll not be benefcal to acheve a Pareto effcent equlbrum. For nstance, f players n the forwarder s dlemma communcated that they prefer to cooperate, then they would stll fell compelled to le and defect n order to receve the hghest payoff, unless the dscountng factor s hgh enough to deter devatons. In our coordnaton game, however, players have no ncentve to le about ther needs, snce lyng would result n a lower payoff. As shown, n ths game, players do not have ntrnscally conflctng nterests. Instead, players want to coordnate ther actons. owever, f traffc congeston s to hgh, then some wreless routers wll end up havng less bandwdth than what s needed. That s, a conflct of nterests arses because there s a shortage of bandwdth to cover all requests. Let us consder the hgh traffc congeston scenaro n Fg. 8, where each wreless connecton has a capacty of 100 Mbps. In ths stuaton, wreless router x s usng all ts bandwdth capacty to forward traffc belongng to ts users and traffc belongng to users connected to wreless routers y and z. That s, Bx = 20, Bx,y = 50 and Bx,z = 30. owever, wreless router w can not forward that much traffc because w also needs 20 Mbps for ts users and may choose to dedcate ts resources the followng way: Bw = 20, Bw,x = 15, Bw,y = 40 and Bw,z = 25. As a result x wll be reservng more bandwdth for tself, y and z than wll be actually used, leavng x wth η x << 0 due to unfrutful utlzaton of resources. Wreless router x, to safeguard tself, can lower Bx,y or Bx,z, or both, to ncrease η x and ts own payoff. Ths would leave y and z wth smaller payoffs due to η y << 0 and η z << 0. These two nodes, y and z, have now two optons: ) each node lowers ts dedcated bandwdth, By and Bz respectvely; ) they le to x, by nflatng ther requests, n order to match as much as possble By A to By and Bz A to Bz. Note that, accordng to the developed model, x does not care whch Bx,y or Bx,z s reduced, snce B A s obtaned by Γ + x summng By A and Bz A. owever, f, alternatvely, B A (t) = Γ + j Γ + Bj A(t) and B (t) = j Γ + B,j (t), then x wll be more nterested n decreasng Bx,y, snce that wll lead to a greater η x and payoff 2. Ths alternatve would lead to a farer bandwdth allocaton among wreless routers. Thus the model can be adjusted accordng to the objectves. The dscussed model, the equlbrum strategy wth communcaton of preferences and detecton of over demandng wreless routers, provde the ground for the development of an allocaton algorthm. Ths algorthm could nclude the 2 For any α and β, f α > β and 1 has to be subtracted from ether α or β, then (α 1)β > α(β 1). y w x Wreless connecton z Wreless router Capacty of each wreless connecton: 100 Mbps Bandwdth needs: w: 20 Mbps x: 20 Mbps y: 50 Mbps z: 30 Mbps Fg. 8. Example of a congeston scenaro wth unfrutful bandwdth utlzaton at one of the wreless routers. followng steps: Ste: Communcaton of the bandwdth needs among all wreless routers and gateway. Ste: etecton of over-demandng routers/gateway. Step 3: ecson, by every wreless router/gateway W, on the values B and B,j, j Γ+, wth the objectve of ncreasng as much as possble the payoff at every stage. Ths represents only an example of a possble algorthm for the bandwdth allocaton problem n Fber-Wreless Access Networks. Note that the concepts explaned here can be appled to other problems related to any of the layers of the OSI model. Moreover, other types of games, such as coalton games or network formaton games, can also be used to model exstng problems and develop algorthms. For nstance, n [13], a coalton game s used to develop a mechansm where unmanned aeral vehcles collect messages from certan data sources, scattered throughout a feld, to be then delvered to a common recever. In [23], a network formaton game s used to develop an energy effcent routng algorthm for the mesh front end of Fber-Wreless Access Networks. VI. CONCLUSION The development of algorthms to solve exstng problems n communcaton networks s a complex task. The purpose of ths tutoral s to showcase how to use repeated game theory as a tool for algorthm development n communcaton networks. It starts by gvng the bass of game theory and repeated game theory, ncludng how ther outcome can be predcted through Nash equlbrum and how certan Nash equlbra can be possble n nfnte-horzon repeated games whle not beng possble n one stage games. Wth the bass ntroduced, an example s then gven where a model s developed and necessary condtons are devsed for the exstence of Nash equlbrum where all players cooperate, whch can be used as a bass for the development of an algorthm. ACKNOWLEGMENT Ths work was supported by FCT (Foundaton for Scence and Technology) of Portugal wthn CEOT (Center for Electronc, Optoelectronc and Telecommuncatons), and by J. Combras Ph.. grant SFR/B/37808/2007.

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