Distributed Welfare Games

Transcription

1 OPERATIONS RESEARCH Vol. 61, No. 1, Januay Febuay 2013, pp ISSN X (pint) ISSN (online) INFORMS Distibuted Welfae Games Jason R. Maden Depatment of Electical, Compute, and Enegy Engineeing, Univesity of Coloado, Boulde, Coloado 80309, Adam Wieman Computing and Mathematical Sciences, Califonia Institute of Technology, Pasadena, Califonia 91125, Game-theoetic tools ae becoming a popula design choice fo distibuted esouce allocation algoithms. A cental component of this design choice is the assignment of utility functions to the individual agents. The goal is to assign each agent an admissible utility function such that the esulting game possesses a host of desiable popeties, including scalability, tactability, and existence and efficiency of pue Nash equilibia. In this pape we fomally study this question of utility design on a class of games temed distibuted welfae games. We identify seveal utility design methodologies that guaantee desiable game popeties iespective of the specific application domain. Lastly, we illustate the esults in this pape on two commonly studied classes of esouce allocation poblems: coveage poblems and coloing poblems. Subject classifications: esouce allocation; game theoy; distibuted contol. Aea of eview: Revenue Management. Histoy: Received July 2010; evisions eceived August 2011, August 2012; accepted Septembe Published online in Aticles in Advance Febuay 8, Intoduction The cental challenge in any esouce allocation poblem is to allocate a fixed numbe of esouces such that thei utilization is optimized. Successfully dealing with this challenge is essential fo ensuing eliable and efficient pefomance in a host of applications anging fom management of tanspotation netwoks to the outing of infomation though the intenet. Taditionally, eseaches have focused on developing centalized algoithms to detemine efficient allocations. Howeve, in many moden applications, these centalized algoithms ae neithe applicable o desiable. One concete example highlighting the need fo distibuted esouce allocation is the senso coveage poblem, whee the goal is to allocate a fixed numbe of sensos acoss a given mission space so as to maximize the pobability of detecting a paticula event (see, e.g., Li and Cassandas 2005, Matinez et al. 2007). A centalized algoithm fo senso allocation equies that a cental authoity maintain complete knowledge of the envionment and communicate diectly with each senso duing the entie mission. Both equiements might be unealistic in lage and/o hostile envionments. Simila issues aise in many compute netwok esouce allocation poblems, with examples anging fom wieless access point assignment (Kauffmann et al. 2007) to wieless powe management (Campos-Náñez et al. 2008, Li and Cassandas 2005). Thee ae also many examples outside of compute systems that suffe fom simila issues. Fo example, in tanspotation systems a global planne does not have the authoity to assign dives to oads; athe, a global planne must entice dives appopiately to settle on a desiable allocation, e.g., using tolls to minimize aggegate congestion (Sandholm 2002). The need fo distibuted esouce allocation has led to a suge of eseach aimed at undestanding the possibility of decentalizing (localizing) decisions in esouce allocation poblems. This is an extemely divese liteatue whee potocols ae designed using a wide vaiety of tools, e.g., distibuted optimization (Misha et al. 2006, Villegas et al. 2008), distibuted contol (Li et al. 2005, Paag et al. 2010), physics-inspied contol (Kauffmann et al. 2007, Mhate et al. 2007), and game-theoetic contol (Alpcan et al. 2009, Zou and Chakabaty 2004, Sivastava et al. 2005, Campos-Náñez et al. 2008). In this pape we focus on game-theoetic contol, which is a pomising new appoach fo distibuted esouce allocation. A game-theoetic appoach to distibuted esouce allocation equies two distinct design steps. Fist, a system designe must model the inteaction famewok of the decision-making entities as a stategic fom game. This involves specifying the decision makes, thei espective choices, and a local utility function fo each agent. Second, a system designe must specify a local behavioal o leaning ule fo each agent that specifies how an individual agent pocesses available infomation to fomulate a decision. The oveaching goal is to complete the two design steps, efeed to as utility design and leaning design, espectively, to ensue that the emegent global behavio is desiable (Aslan et al. 2007, Maden et al. 2009, Gopalakishnan et al. 2011). Thee ae wide-anging advantages to the gametheoetic appoach, including obustness to failues and 155

2 Maden and Wieman: Distibuted Welfae Games 156 Opeations Reseach 61(1), pp , 2013 INFORMS envionmental distubances, minimal communication equiements, impoved scalability, and eal-time adaptation (Aslan et al. 2007). Accodingly, game-theoetic esouce allocation designs ae inceasingly popula in a vaiety of wieless and senso netwok applications, e.g., channel access contol in wieless netwoks (Altman et al. 2004, Kauffmann et al. 2007), coveage poblems in senso netwoks (Cassandas and Li 2005), and powe contol in both (Altman and Altman 2003, Campos-Náñez et al. 2008, Falomai et al. 1999). A compehensive suvey of applications can be found in Altman et al. (2006). Howeve, nealy all these designs ae highly applicationspecific, with both the utility and leaning designs cafted caefully fo the specific setting. Ou Contibutions The goal of this pape is to establish a geneal famewok fo studying utility design. To that end, we intoduce a class of games temed distibuted welfae games, which epesents a game theoetic model fo esouce allocation poblems with sepaable objective/welfae functions. Hee, we focus on the design of local agent utility functions only whee local means that an agent s utility function is able to depend only on the esouces selected, the welfae at each esouce, and the othe agents that selected the same esouces. Consequently, the utility design question can be viewed as a welfae shaing poblem whee an agent s utility is defined as some faction of the welfae ganeed at each of the agent s selected esouces. Theefoe, designing local utility functions is equivalent to defining a distibution ule that depicts how the welfae ganeed fom each esouce is distibuted to the playes at that paticula esouce. The fist set of esults ( 3) illustates that cost shaing methodologies can be used effectively fo utility design. In paticula, we identify two cost-shaing methodologies, the maginal contibution and the Shapley value, that povide valuable methodologies fo utility design. The maginal contibution distibution ule systematically povides local utility functions that always guaantee the existence of a (pue) Nash equilibium iespective of the application domain. The Shapley value distibution ule systematically povides local utility functions that ae budget balanced and always guaantee the existence of a Nash equilibium iespective of the application domain. Howeve, computing the Shapley value is often intactable. These esults highlight an inheent tension between budgetbalanced utility functions and tactability. Ou second set of esults seeks to ovecome the limitations of the Shapley value by establishing tactable budget-balanced distibution ules, ( 4). We identify thee sufficient conditions on distibution ules, see Conditions , that guaantee the existence of a Nash equilibium in any distibuted welfae game whee playes ae esticted to selecting a single esouce. These sufficient conditions can be viewed fom two pespectives. The fist pespective is as a check fo whethe a given set of distibution ules guaantees the existence of an equilibium. The second pespective, which is ou motivation fo this wok, is as a design guideline fo distibution ules, i.e., if a global planne can design a distibution ule to satisfy these conditions, then an equilibium is guaanteed to exist. We illustate these developments in 6. Ou thid set of esults petains to the efficiency of the esulting Nash equilibia. When esticting attention to submodula welfae functions, which is a common attibute to many esouce allocation poblems (Vetta 2002, Kause and Guestin 2007), we identify distibution ules that guaantee that all esulting Nash equilibia obtain a welfae within 50% of the welfae associated with the optimal assignment. This compaes favoably with the best known esults of centalized appoximations fo esouce allocation poblems with submodula welfae functions, which guaantee welfae within 1 1/e of the optimal (Feige and Vondak 2006, Ageev and Sviidenko 2004, Ahuja et al. 2004). Supisingly, this compaison demonstates that the inefficiency esulting fom localizing decisions in esouce allocation poblems is elatively small when the welfae functions ae submodula. Futhemoe, we demonstate that these 50% guaantees can be significantly shaped in many settings. Lastly, in 6 we povide an illustation of the theoetical contibutions contained in this pape on two impotant classes of esouce allocation poblems: coveage poblems and coloing poblems. 2. Model Oveview: Resouce Allocation Poblems We conside a class of esouce allocation poblems whee thee exists a set of playes N and a finite set of esouces R that ae to be shaed by the playes. Each playe i N is assigned an action set i 2 R whee 2 R denotes the powe sets of R; theefoe, a playe may have the option of selecting multiple esouces. Let = 1 n denote the set of joint action pofiles. A key featue of esouce allocation poblems is the existence of a global welfae function W that the system designe seeks to optimize. In this pape, we estict ou attention to the class of sepaable welfae functions of the fom W a = R W a whee W 2 N + is the welfae function fo esouce and a = i N a i is the set of playes using esouce in the allocation a. Note that when esticting attention to sepaable welfae functions, the welfae geneated at a paticula esouce depends only on which playes ae cuently using that esouce. While sepaable welfae functions cannot model the global objective fo all esouce allocation poblems, they can model the global objective fo seveal impotant classes of esouce allocation poblems

3 Maden and Wieman: Distibuted Welfae Games Opeations Reseach 61(1), pp , 2013 INFORMS 157 including outing ove a netwok (Roughgaden 2005), vehicle taget assignment poblem (Muphey 1999), content distibution (Goemans et al. 2004), gaph coloing (Panagopoulou and Spiakis 2008), and netwok coding (Maden and Effos 2012), among many othes. Thoughout this pape we fequently estict ou attention to submodula welfae functions. A welfae function W is submodula if fo any playe sets S T N, W S i W S W T i W T fo any i N. Submodulaity coesponds to a notion of deceasing maginal etuns and is a common featue of many objective functions fo engineeing applications anging fom content distibution (Goemans et al. 2004) to coveage poblems (Kause and Guestin 2007). We now intoduce the class of distibuted welfae games as the game-theoetic model fo the class of esouce allocation poblems defined above. We conside a finite stategic-fom game whee each playe i N has a finite action sets i and a utility function U i. In a distibuted welfae game, each agent s utility is defined as some faction of the welfae ganeed at each esouce the agent is using. Moe fomally, the utility of agent i fo any joint action pofile a is defined as U i a = a i f i a (1) whee f N 2 N is efeed to as the distibution ule at esouce. We define a distibuted welfae game by the tuple G = N R i i N W R f R. Fo bevity, we often omit the subscipts on the sets and denote a game as puely G = N R i W f. A few comments ae in ode egading the stuctue imposed on the utility functions in (1). Fist, this stuctue imposes a natual notion of locality as each agent s utility function depends only on the esouces the agent selected and the othe agents that selected the same esouces. Second, defining a distibution ule fo each esouce f esults in a well-defined game iespective of the stuctue of the playes action sets i. This allows fo a degee of scalability in utility design because we can emove the dependence between utility design, o equivalently distibution ule design, and the stuctue of the playes action sets. 3. Utility Design fo Distibuted Welfae Games In this section, we exploe seveal appoaches fo utility design in distibuted welfae games. We discuss these design methodologies in the fom of distibution ules as opposed to utility functions fo a moe diect pesentation. Many of these appoaches ae deived fom methodologies in the cost-shaing liteatue, e.g., Shapley value, as ou context can be viewed as the invese of a cost-shaing poblem (Young 1994, Shapley 1953, Hat and Mas-Colell 1989). Howeve, while cost-shaing methodologies can be effective as distibution ules, we also discuss seveal issues that limit thei applicability Pefomance Citeia Befoe poceeding with the discussion of distibution ules, we fist fomally define ou pefomance citeia. We assume thoughout that the set of playes N, esouces R, and welfae functions W ae all known; howeve, the action sets i ae unknown. We equie that the distibution ule f fo each esouce R depend only on the welfae function W. Theefoe, distibution ules ae completely defined using only local infomation. We gauge the quality of a design by the following metics. Existence and efficiency of pue Nash equilibium. A well-known equilibium concept that emeges in stategic fom games is that of a pue Nash equilibium, which we will efe to simply as an equilibium. An action pofile a is called a equilibium if fo all playes i N, U i a i a i = max U i a i a i (2) a i i whee we adopt the convention that a i denotes the pofile of playe actions othe than playe i, i.e., a i = a 1 a i 1 a i+1 a n With this notation, we sometimes wite a pofile a of actions as a i a i. Similaly, we may wite U i a as U i a i a i. A pue Nash equilibium, as defined in (2), epesents a scenaio fo which no playe has a unilateal incentive to deviate. Does the distibution ule f guaantee the existence and efficiency of an equilibium iespective of the stuctue of the playes action sets i? Potential game. The class of potential games (Mondee and Shapley 1996) imposes a estiction on the agents utility functions. In a potential game, the change in a playe s utility that esults fom a unilateal change in stategy equals the change in a global potential function. Specifically, thee is a potential function such that fo evey playe i N, fo evey a i i, and fo evey a i a i i, U i a i a i U i a i a i = a i a i a i a i (3) When this condition is satisfied, the game is called a potential game with the potential function. It is easy to see that in potential games any action pofile maximizing the potential function is an equilibium, hence evey potential game possesses at least one such equilibium. Does the distibution ule f guaantee that the esulting game is a potential game iespective of the stuctue of the playes action sets i? Budget balance. A distibution ule f is budget balanced if, fo any esouce R and any playe set S N, i S f i S = W S. Recent esults demonstate that having budget-balanced utility functions (o in this case distibution ules), o some faction theeof, is impotant fo poviding desiable efficiency guaantees (Vetta 2002, Roughgaden 2009, Gaiing 2009). Futhemoe, having budget-balanced distibution ules is also impotant fo the contol (o influence) of social systems whee thee is a

4 Maden and Wieman: Distibuted Welfae Games 158 Opeations Reseach 61(1), pp , 2013 INFORMS cost o evenue that needs to be completely absobed by the paticipating playes, e.g., netwok fomation (Chen et al. 2008) and content distibution (Goemans et al. 2004). Infomational dependencies. This metic seeks to measue the infomational dependencies between a distibution ule f and the associated the welfae function W. Fo any esouce R, playe i N and playe set S N such that i S, we expand the notation of a distibution ule fom f i S to f i S to explicitly highlight the infomation dependencies, denoted by, needed to compute the distibution to playe i. We categoize infomational dependencies as follows: High: The distibution to playe i given the playe set S is conditioned on infomation egading the welfae associated with all playe set T S, i.e., f i S W T T S. Medium: The distibution to playe i given the playe set S is conditioned on infomation egading the welfae associated with the playe sets S and S\i, i.e., f i S W S W S\i. Low: The distibution to playe i given the playe set S is conditioned on infomation egading the welfae associated with the set S, i.e., f i S W S Distibution Rules In this section we will intoduce fou distibution ules motivated by methodologies fom the cost-shaing liteatue (Young 1994). These ules ae summaized in Table Equally Shaed. The equally shaed distibution ule takes on the following fom: fo any esouce R, playe set S N, and playe i N, f ES i S = W S (4) S It is staightfowad to veify that this ule is budgetbalanced and has a low infomation dependency. Howeve, in geneal such a design does not guaantee the existence of an equilibium (Aslan et al. 2007). Howeve, if playes ae anonymous with egad to thei impact on the welfae functions, i.e., fo any esouce R and any playe sets S T N such that S = T the welfae satisfies W S = W T, then the equally shaed utilities in (4) guaantee the existence of an equilibium. We efe to such welfae functions as anonymous. Poposition 1. If G is a distibuted welfae game with anonymous welfae functions and the equally shae distibution ule in (4), then an equilibium is guaanteed to exist. Poof. Define a = a as the numbe of playes that chose esouce in the allocation a. It is staightfowad to show that any distibuted welfae game with anonymous playes is a congestion game (Rosenthal 1973, Mondee and Shapley 1996), with the following specification: (a) esouces R; (b) cost functions of the fom c k = W k/k k > 0, whee k is the numbe of playes utilizing esouce ; and (c) utility functions of the fom U i a = a i c a. Because any congestion game is a potential game, this completes the poof Maginal Contibution. The maginal contibution distibution ule takes on the following fom: fo any esouce R, playe set S N, and playe i N, f MC i S = W S W S\i (5) Note that the maginal contibution distibution ule equies a medium infomational dependency as each agent is equied to compute the playe s maginal contibution to the welfae. This design is sometimes efeed to as the wondeful life utility (WLU) (Wolpet and Tumo 1999). It is well known that distibuting the welfae as in (5) esults in a potential game with potential function W ; hence any action pofile that maximizes the global welfae is an equilibium. Howeve, othe equilibia might also exist. Futhemoe, the maginal contibution distibution ule may distibute moe (o less) welfae than is gatheed; hence, it is not budget-balanced Shapley Value. The Shapley value distibution ule (Shapley 1953, Hat and Mas-Colell 1989, Haeinge 2006) takes on the following fom: fo any esouce R, playe set S N, and playe i N f SV i S = T S\i T!S T 1! W S! T i W T (6) Utilizing the Shapley value as in (6) equies a high infomational dependency; howeve, it ectifies the budgetbalanced poblems associated with the maginal contibution distibution ule as shown in the following poposition. Poposition 2. If G is a distibuted welfae game with the Shapley value distibution ule in (6) then the playes utility functions ae budget balanced and the game is a potential game with potential function SV a = ( ) 1 1 S T W S T (7) R S a T S Poof. Budget-balanced utility functions follow diectly fom Hat and Mas-Colell (1989) and Haeinge (2006). We pove the second pat of this poposition using the potential function deived in Hat and Mas-Colell (1989). Fist, we expess the Shapley value distibution ule in (6) as a weighted sum of unanimity games (Hat and Mas-Colell 1989, Haeinge 2006), which takes on the fom f SV i S = T S i T ( ) 1 1 T R W T R (8) R T

5 Maden and Wieman: Distibuted Welfae Games Opeations Reseach 61(1), pp , 2013 INFORMS 159 Let T = R T 1 T R W R and SV a = T a T /T be the esouce-specific potential function (Hat and Mas-Colell 1989). Futhemoe, let a be any allocation and a 0 i = be the null action fo playe i. Using (6), the maginal utility of playe i is U i a U i a 0 i a i = f SV i a a i = a i ( T a i T ) T T = ( T a i T T a = a i SV T a \i a SV a 0 i a i = SV a SV a 0 i a i ) T T Theefoe, fo any playe i, actions a i a i i, and allocation a i i we have U i a i a i U i a i a i = SV a i a i SV a i a i which completes the poof. Thee ae two limitations of the Shapley value utility design that could pevent it fom being applicable. Fist, thee is a high infomational equiement as each playe must be able to compute his maginal contibution to all action pofiles in ode to evaluate his utility. Second, in geneal, computing a Shapley value is intactable in games with a lage numbe of playes. This is highlighted explicitly in eithe (6) o (8), whee computation of the Shapley value equies a weighted summation ove all subsets of playes, of which thee might be exponentially many. Howeve, it should be noted that this computational cost is lessened damatically fo special classes of welfae functions, e.g., Conitze and Sandholm (2004). Fo example, if playes ae anonymous, then the Shapley value is equivalent to the equal shae distibution ule in (4) The Weighted Shapley Value. A genealization of the Shapley value is the weighted Shapley value (Shapley 1953, Hat and Mas-Colell 1989, Haeinge 2006). Define w i + as the weight of playe i. Let w = w i i N be the associated weight vecto. The weighted Shapley value distibution ule (Shapley 1953, Hat and Mas-Colell 1989, Haeinge 2006) takes on the following fom: fo any esouce R, playe set S N, and playe i N, f WSV is= T Si T w i j T w j ( ) 1 T R W R R T (9) Note that the Shapley value distibution ule in (6) is ecaptued if w i = 1 fo all playes i N. We state the following poposition without poof to avoid edundancy. Poposition 3. If G is a distibuted welfae game with the Shapley value distibution ule in (9), then the playes utility functions ae budget balanced and the game is a (weighted) potential game. The weighted Shapley value does not esult in as clean a closed-fom expession fo the potential function as the Shapley value in (7). Howeve, as with the Shapley value, the potential function can be computed ecusively and is of the fom WSV a = R WSV a, whee (Hat and Mas-Colell 1989) WSV = 0 WSV S = 1 i S w i [ W S + i S ] w i WSV S\i 3.3. Compaison of Distibution Rules S N S The following table summaizes the featues of the fou distibution ules intoduced above. Note that thee is an inheent tade-off between the desiable featues of a distibution ule and the computational and infomational equiement needed to obtain such a ule. 4. Single Selection Distibuted Welfae Games The esults of the pevious section suggest that deiving budget-balanced distibution ules that always guaantee the existence of an equilibium equies a high infomation equiement and is often intactable. In this section we exploe whethe this appaent tade-off is a limitation of cost-shaing methodologies o utility design in geneal. To study this question we focus on a simplified setting whee playes ae allowed to select only a single esouce, i = R, as opposed to multiple esouces, i 2 R. Although this setting is simplified, thee ae still a wide vaiety of esouce allocation poblems that typically ae modeled as single selection esouce allocation poblems, e.g., task allocation Sufficient Conditions fo Existence of an Equilibium In this section, we identify thee sufficient conditions on distibution ules that guaantee the existence of an equilibium in any single selection esouce allocation game. These sufficient conditions tanslate to paiwise compaisons of playes distibuted shaes. Condition 4.1. Let i j N be any two playes. If f i S i j > f j S i j fo some esouce R and playe set S N \i j, then f i S i j f j S i j

6 Maden and Wieman: Distibuted Welfae Games 160 Opeations Reseach 61(1), pp , 2013 INFORMS Table 1. Summay of distibution ules fo distibuted welfae games. Distibution Existence of Potential Budget Infomational ule equilibium game balanced Tactable equiement Equally shaed Yes Yes Yes Yes Low (anonymous) Equally shaed No No Yes Yes Low WLU Yes Yes No Yes Medium Shapley Yes Yes Yes No High Weighted Shapley Yes Yes Yes No High fo any esouce R and playe set S N \i j. Fo this situation we say that playe i is stonge than playe j. Futhemoe, note that stengths ae tansitive, i.e., if playe i is stonge than playe j who is stonge than playe k, then playe i is also stonge than playe k. Condition 4.2. If playe i is stonge than playe j, then fo any esouce R and playe set S N \i j we have f i S i f i S i j Condition 4.3. If playe i is stonge than playe j, then fo any esouce R and playe set S N \i j we have f j S j f max j S i j f i S i R f i S i j Theoem 1. If G is a distibuted welfae game whee the action sets satisfy i = R fo all agents i N and the distibution ule satisfies Conditions , then an equilibium exists. Poof. We begin by enumbeing the playes in ode of stengths, with playe 1 being the stongest playe. This is possible because of Condition 4.1. We constuct an equilibium by letting each playe select a esouce one at a time in ode of stength. The geneal idea of the poof is that once a playe selects a esouce, the playe will neve seek to deviate egadless of the othe playes selections. Fist, playe 1 selects a esouce 1 accoding to 1 ag max f 1 1 (10) R Denote the action pofile a 1 = 1. Note that if thee was only one playe, a 1 would epesent an equilibium. If this is not the case, let playe 2 select a esouce 2 accoding to 2 ag max f 2 a 1 2 R Denote the action pofile a 2 = 1 2. If 1 2, then by (10) and Condition 4.2 we know that f 11 1 f 21 1 f Theefoe, playe 1 can not impove his utility by alteing his selection. If 1 = 2 =, then by Condition 4.3, we know that fo any esouce R, f 2 2 f 1 1 f f Using the above inequality, we can conclude that fo any esouce R, f f 2 2 f f 1 1 Theefoe, playe 1 cannot impove his utility by alteing his selection. Note that if thee wee only two playes, a 2 would epesent an equilibium. Othewise, this agument could be epeated n times to constuct an equilibium. It emains an open question as to whethe Conditions guaantee additional popeties petaining to the stuctue of the game besides existence of an equilibium Compaison with Existing Results The elated wok in Chen et al. (2008) studies cost shaing methodologies in a class of netwok fomation games fo a specific anonymous cost function. A netwok fomation game is simila to a distibuted welfae game whee the diffeence lies in cost minimization vesus welfae maximization; hence the esults contained in that pape do not immediately tanslate to the famewok of distibuted welfae games. The authos focus on a specific anonymous cost function and pove that a distibution ule is budget balanced and guaantees the existence of an equilibium fo any game if and only if the distibution ule can be epesented by a weighted Shapley value. To pove this esult, the authos establish necessay and sufficient paiwise conditions on playe distibuted shaes that ae slightly stonge than the ones in Conditions The authos make no claim as to whethe thei esults also hold fo altenative cost functions. In 6.2, we demonstate that ou weake paiwise conditions on playe cost shaes lead to the constuction of a set of distibution ules that ae budget balanced and guaantees an equilibium in all games whee playes actions ae singletons, i.e., i = R. Futhemoe, the deived distibution ules do not coespond to a weighted Shapley value. This gap can potentially be a esult of the following diffeences in the setup: (i) cost minimization vesus welfae maximization, (ii) stuctue on action set, i.e., i = R vesus i 2 R, o (iii) stuctue of the welfae functions, i.e., anonymous vesus nonanonymous.

7 Maden and Wieman: Distibuted Welfae Games Opeations Reseach 61(1), pp , 2013 INFORMS Efficiency of Equilibia in Distibuted Welfae Games In this section, we focus on bounding the efficiency of equilibia in distibuted welfae games using use the wellknown measues of pice of anachy (PoA) and pice of stability (PoS) (Nisan et al. 2007). In tems of distibuted welfae games, the PoA gives a lowe bound on the global welfae achieved by any equilibium, while the PoS gives a lowe bound on the global welfae associated with the best equilibium fo any distibuted welfae game. Specifically, let denote a set of distibuted welfae games. Fo any paticula game G, let EG denote the set of equilibia, PoAG denote the pice of anachy, and PoSG denote the pice of stability fo the game G, whee PoAG = min a ne EG W a ne (11) W a opt W a ne PoSG = max (12) a ne EG W a opt whee a opt ag max a W a. We define the PoA and PoS fo the set of distibuted welfae games as PoA = inf G PoAG and PoS = inf G PoSG. In geneal, the pice of anachy can be abitaily close to 0 in distibuted welfae games. Howeve, when the welfae function is submodula it is possible to attain a much bette pice of anachy. We can intepet Theoem 3.4 in Vetta (2002) in the context of distibuted welfae games to povide a faily weak condition on the inteaction between the welfae function W and the utility functions, which guaantees that the pice of anachy is at least 1/2. Poposition 4 (Vetta 2002). If G is a distibuted welfae game, whee fo each esouce R: (i) the welfae function W is submodula, (ii) fo each set of playes S N and playe i S, the distibution ule satisfies f i S W S W S\i (iii) fo each set of playes S N, the distibution ule satisfies i S f i S W S, then if an equilibium exists, the pice of anachy is geate than o equal to 1/2. To povide a basis fo compaison, computing the optimal assignment fo a geneal distibuted welfae game with submodula welfae functions is NP-complete (Muphey 1999). Futhemoe, the best-known appoximation algoithms guaantee only to povide a solution that is within 1 1/e of the optimal (Feige and Vondak 2006, Ageev and Sviidenko 2004, Ahuja et al. 2004). Thus, the 1/2 pice of anachy in this scenaio is compaable to the best centalized solution. It is impotant to note that these best-known centalized appoximations ae in polynomial time, wheeas finding a Nash equilibium is geneally not polynomial time. Howeve, ecent esults suggest that fo this class of poblems thee ae dynamics that get close to this 1/2 pice of anachy guaantee in polynomial time (Roughgaden 2009). While the geneality of Poposition 4 is useful, the applicability is limited because it does not guaantee the existence of an equilibium. Hence, its applicability depends on the esults we have poven in 3. Coollay 1. If G is a distibuted welfae game whee fo each esouce R: (i) the distibution ule f coesponds to the maginal contibution as in (5), o (ii) the distibution ule f coesponds to the (weighted) Shapley value as in (6) o (9), then an equilibium exists and the pice of anachy is geate than o equal to 1/2 The fou distibution ules depicted in Coollay 1 all guaantee the existence of an equilibium. Note that the wondeful life utility design satisfies Condition (ii) with equality in addition to Condition (iii) because the welfae function is submodula. Additionally, the Shapley and weighted Shapley values satisfies Condition (iii) with equality and can easily be seen to satisfy Condition (ii) when the welfae function is submodula. The following examples highlights a specific distibuted welfae game meeting the conditions set foth in Coollay 1, which yields a pice of anachy of 1/2; hence, the only way to attain a pice of anachy > 1/2 is to impose additional stuctue on the game envionment, which we will exploe in the ensuing section. Example 1 (Tightness of Pice of Anachy). Conside a distibuted welfae game with playe set N = 1 n, esouces R = 1 n, actions set i = R fo all playes i N, and anonymous esouce specific welfae functions of the fom W i S = c i fo any playe set S. Let c 1 = 1 and c 2 = = c n = 1/n. If the distibution ule is of the fom (4), than an equilibium is all chaacteized by all playes choosing 1. The optimal allocation is all playes choosing diffeent esouces. The efficiency of this situation is n/2n 1, which goes to 1/2 fo lage n. This example demonstates that the equal shae utility design and Shapley value utility design have a pice of anachy (and pice of stability) of 1/2. Altenative examples can be constucted to show that the wondeful life utility and the weighted Shapley value utility also have a tight pice of anachy of 1/2. To this point we have focused exclusively on bounding the pice of anachy. Inteestingly, when we focus on the pice of stability thee is a distinction between ules that ae budget balanced and those that ae not, as highlighted in Table 2. When consideing the class of distibuted welfae games with submodula welfae functions, the pice of anachy and the pice of stability is 1/2 fo any budget balanced distibution ule. If budget balance is not a equiement, then it is possible to obtain a pice of anachy of 1/2 and a pice of stability is 1.

8 Maden and Wieman: Distibuted Welfae Games 162 Opeations Reseach 61(1), pp , 2013 INFORMS Table 2. PoA and PoS compaison ove all distibuted welfae games with submodula welfae functions. Budget Pice of Pice of Distibution ule balanced anachy stability Equally shaed (anonymous) Yes 1/2 1/2 WLU No 1/2 1 Shapley value Yes 1/2 1/ Single Selection Distibuted Welfae Games To povide a tighte chaacteization of the pice of anachy we shift attention to single-selection distibuted welfae games. In this case, we can stengthen the esults of Poposition 4 utilizing Conditions , which guaantee the existence of an equilibium. Poposition 5. If G is a distibuted welfae game whee: (i) fo each esouce R the welfae function W is submodula, (ii) fo all playes i N the action sets satisfy i = R, and (iii) the distibution ules f ae budget balanced and satisfy Conditions , then an equilibium exists and the pice of anachy is 1/2. Poof. The poof elies on showing that Conditions combine to ensue that Condition (ii) of Poposition 4 is satisfied, i.e., fo any esouce R, set of playes S N, and playe i S, we have f i S W S W S\i Let i j N be any two playes whee i is stonge than j. Let S N \i j be any playe set. Condition 4.3 gives us that fo any esouce R, f j S j f i S i f i S i j f j S i j Because playe i is stonge than playe j, fom Condition 4.2 we know that f i S i f i S i j, which gives us f j S j f j S i j (13) Theefoe, Condition 4.2 holds fo any playes i j N and set of playes S N \i j. Using the fact that the distibution ule is budget balanced and satisfies (13), we have f i S + W S\i W S = f i S + f j S\i f j S j S\i j S = f j S\i f j S 0 j S\i Theefoe, we have f i S W S W S\i, which completes the poof Anonymous Distibuted Welfae Games Ou bounds on the pice of anachy to this point have been independent of the numbe of playes. In this section, we investigate the elationship between the pice of anachy and the numbe of playes, albeit in the limited case whee playes ae anonymous with egad to thei impact on the global welfae. Futhemoe, we analyze the pice of anachy when the numbe of playes at the equilibium and optimal allocations diffes. Specifically, let W a ne n + be the total welfae ganeed by an equilibium consisting of n + playes, +. Likewise, let W a opt n be the total welfae ganeed by an optimal allocation consisting of n playes. While we allow vaiations in the numbe of playes, the esouces R and thei espective welfae functions W emain fixed. Theoem 2. If G is a distibuted welfae game whee (i) fo each esouce R the welfae function W is anonymous and submodula, (ii) the action set of any two playes i j N ae identical, i.e., i = j 2 R, (iii) fo any set of playes S N and playe i N, the distibution ule f satifies f i S W S W S\i then if an equilibium exists, the elative pice of anachy satisfies W a ne n + W a opt n n + 2n + 1 Poof. We pove the esult by bounding W a opt n in tems of W a ne n +. Rathe than poving this theoem in tems of the distibution ules, we use the utility functions, which ae of the fom U i a i a i = a i f i a. Rewiting condition (iii) in tems of utility functions, letting a 0 i = we have that U i a i a i W a W a 0 i a i (14) Fist, notice that an uppe bound on the W a opt n is if one playe in the optimal allocation can attain the entie welfae ganeed at the equilibium, W a ne n +, and all othe playes attain min i N U i a ne n +, whee U i a ne n + epesents the utility playe i eceives fo the allocation a ne consisting of n + playes. To see that this uppe bound holds, note fist that (14) guaantees that each playe s utility is an uppe bound on the playe s contibution to the global welfae. Futhemoe, by combining the definition of an equilibium with the fact that the welfae function is submodula, we see that no additional playe can attain a utility highe than min i N U i a ne n + once W a ne n + is coveed. Thus, we have W a opt n W a ne n + + n 1 min i N U ia ne n +

9 Maden and Wieman: Distibuted Welfae Games Opeations Reseach 61(1), pp , 2013 INFORMS 163 Now, noting that min i N U ia ne n + W ane n + n + gives W a ne n + + n 1 min U ia ne n + i N ( W a ne n n 1 ) n + which easily gives the bound in the theoem W a ne n + W a opt n n 1/n + = n + 2n + 1 Notice that Theoem 2 shows that the wost-case pice of anachy is inceasing as the numbe of playes inceases and that as n the pice of anachy appoaches 1/2, which matches Poposition 4. Example 1 illustates that this bound is tight by slightly modifying the coefficients to c 2 = = c n+ = 1/n +. Futhemoe, note that all the utility design methods peviously studied, i.e., equally shaed, wondeful life, and (weighted) Shapley value utility, satisfy the thee conditions of Theoem 2. Hence, if the welfae function is submodula, then an equilibium is guaanteed to exist and the bound on the elative pice of anachy holds. Lastly, note that the pice of anachy, = 0, is bounded by W a ne n W a opt n n 2n 1 6. Illustative Examples To illustate the elevance of the esults discussed to this point, we now focus on two boad classes of esouce allocation poblems: coveage poblems and coloing poblems. Ou fist goal in this section is to highlight that utility design fo distibuted esouce allocation does not need to be ad-hoc and application specific, it can often follow immediately fom the geneal famewok pesented hee. To illustate this, we focus on the esults in Panagopoulou and Spiakis (2008), whee the authos popose and analyze utility functions fo the poblem of gaph coloing. We illustate that the poposed design is equivalent to Shapley value utility design, which highlights that utilizing the Shapley value utility design would have eliminated the need fo having to pove existence of an equilibium o a potential game stuctue. Ou second goal in this section is to highlight that many of the methodologies discussed in this pape povide stong efficiency guaantees in boad settings. Fo example, in coveage poblems, all the methodologies discussed in this pape guaantee that all equilibia ae at least 50% efficient since the welfae functions ae submodula. Futhemoe, we demonstate that the sufficient conditions established in 4 lead to the constuction of equally desiable utility functions that ae less demanding than eithe the maginal contibution o Shapley value utility functions. Lastly, we demonstate how the stuctue of the specific welfae functions can be exploited to tighten the efficiency guaantees Gaph Coloing Poblems A gaph coloing poblem is defined as follows. Thee is a finite set of colos (o esouces) denoted by and a gaph epesented by the tuple N E, whee N is a finite numbe of nodes (o playes) and E 2 N N is a set of diected edges on the gaph G. Each node is allowed to choose any colo, i.e., i = fo all nodes i N. A colo assignment is a tuple a = a 1 a n that associates a colo with each node. We call a colo assignment valid if c i c j fo all nodes i j such that e i e j E. The goal of the gaph coloing poblem is to find a valid coloing assignment using the least numbe of possible coloings. Gaph coloing poblems play a pominent ole in seveal class of esouce allocation poblems anging fom distibuted caching (Chun et al. 2004) to spectum allocation in cognitive adio netwoks (Mosciboda and Wattenhofe 2005, Schneide and Wattenhofe 2009), and the game theoetic techniques have ecently been suggested at a useful appoach fo developing distibuted potocols fo such poblems (Panagopoulou and Spiakis 2008, Chatzigiannakis et al. 2010). To fomally descibe the gaph coloing poblem in the context of distibuted welfae games, we associate with each colo c a welfae function W c 2 N whee fo any subset of nonconflicted playes S N, i.e., if i j S, then e i e j E, we have W c S = { 0 S = 1 S If S contains conflicted playes, i.e., thee exists playes i j S such that e i e j E, then we adopt the convention that W c S =. The goal of the gaph coloing poblem is to find a coloing assignment a to maximize W a = c W c a c. In Panagopoulou and Spiakis (2008), the authos model the gaph coloing poblem as a noncoopeative game whee each node is assigned a utility function of the fom U i a i a i = c a i a c whee a is any valid coloing assignment. Because this utility design was constucted specifically fo the gaph coloing poblem, the authos needed to pove esults petaining to existence and efficiency of equilibium and the undelying potential game stuctue. It tuns out that the poposed design is equivalent to assigning each playe a utility in accodance with the Shapley value U SV i a i a i = c a i f SV c a c = c a i 1 a c

10 Maden and Wieman: Distibuted Welfae Games 164 Opeations Reseach 61(1), pp , 2013 INFORMS whee a is any valid coloing assignment. By equivalent, we mean that fo any assignment a i, playe i N, and altenative choice a i i we have U i a i a i U i a i a i > 0 U SV i a i a i U SV i a i a i > 0 This equivalence implies that many of the esults petaining to equilibium existence and the esulting potential game stuctue would be obtained fo fee because utilizing these methodologies eliminates the guess and check potocols commonly used in utility design. Lastly, because the Shapley value utility design povides guaantees iespective of the stuctue of the playes action sets i, many of the esults hold immediately fo a moe geneal setting Coveage Poblems In a coveage poblem thee is a finite set of esouces denoted by, and each location/esouce t has a elative value v t 0. Thee ae a finite numbe of agents denoted by N. The set of possible assignments fo agent i is i 2 and epesents the set of joint assignments. Lastly, each agent i N is paameteized with a success pobability denoted by p i t a i 0 1, which indicates the pobability that agent i will successfully cove esouce t given the assignment a i. We assume that the success pobabilities satisfy t a i p i t a i > 0 The goal of a coveage poblem is to find a joint assignment that maximizes the global welfae function W a = ( v t 1 ) 1 p i t a i (15) i a t t a t whee 1 i a t 1 p i t a i epesents the expected value that esouce t is coveed by the assignment a. Note that computing the optimal assignment fo this class of poblems is an NP-had combinatoial optimization poblem (Muphey 1999) and, esultantly, eseach has taditionally centeed aound developing heuistic methods to quickly obtain nea optimal assignment, whee the degee of suboptimality is dependent on the stuctue of the global objective, e.g., Ahuja et al. (2004). To view coveage poblems in the context of distibuted welfae games, we simply note that they ae esouce allocation poblems with a sepaable welfae function, whee the welfae function fo any location/esouce t and any set of agents S N is ( W t S = v t 1 ) p i t a i (16) i S1 Theefoe, we can appeal to the utility design methodologies developed fo distibuted welfae game to constuct local utility fo the agents such that the esulting game has a host of desiable popeties including existence and efficiency of equilibium. One possible design choice is the maginal contibution distibution ule, which takes on the fom U i a i a i = f MC t i a t t a i = t a i v t (p i t a i j a t \i ) 1 p j t a j (17) An altenative design choice is the weighted Shapley value distibution ule, whee fo a given set of playe weights w R n + the utility takes on the fom U i a i a i = f WSV t i a t t a i = ( ( w v i T R t 1 t a i S a t i S j S w j R T ( 1 j R1 ))) p j t a j (18) whee the Shapley value can be attained with w i = 1 fo all i N. Both (17) and (18) guaantee the existence of an equilibium. Futhemoe, because the welfae function is submodula, both ules yield a pice of anachy of 1/2. Futhemoe, the pice of stability of the wondeful life design is 1 and the pice of stability of the weighted Shapley design is 1/ Othe Distibution Rules. We have just seen that cost shaing methodologies yield useful distibution ules fo coveage poblems. Howeve, a question that emains is whethe we ae bound to using the (weighted) Shapley value if the goal is to design utility functions that ae budget balanced and guaantee the existence of an equilibium. The following theoem identifies one altenative distibution ule, albeit fo the case of singleton stategies. Theoem 3. If G = N i W t f t is a distibuted welfae game, whee: (i) the welfae function fo each location t takes on the fom in (16), (ii) the action set of each agent is i =, (iii) the distibution ule fo each location t fo any set of agents S N and playe i S is f t i S = p i j S p j W t S (19) then the utility functions ae budget balanced, an equilibium exists, and the pice of anachy is at least 1/2. Poof. We pove this esult by veifying Conditions Fist, Condition 4.1 is satisfied tivially

11 Maden and Wieman: Distibuted Welfae Games Opeations Reseach 61(1), pp , 2013 INFORMS 165 because fo any location t, set of agents S N and agents i j S we have f t i S f t j S = p i p j Veifying Condition 4.2 equies showing that fo any location t, set of agents S N and agents i S and j S whee p i > p j, we have p i W p t S p i W S p j + p t S j S whee p S = k S p k. Using the fact that W t S j = W t S + p j v W t S and eaanging the above expession, we need to show that p S v t W t S W t S Woking with the left-hand side of the above expession, we have p S v t W t S= p i v t W t S p i v t W t S\i i S i S i S p i p S W t S=W t S (20) whee the fist and step follow fom submodulaity of W t. Veifying Condition 4.3 equies showing that fo any location t, set of agents S N, and agents i j S whee p i > p j, we have p i + p S W t S j p j + p S W t S i Using the fact that W t S i = W t S + p i v t W t S fo any playe i S this is equivalent to showing that p S v t W t S W t S, which is tue fom the pevious analysis in (20). A few notes ae in ode as to the meaning of the esulting utility design in Theoem 3. Fist, note that the utility design set foth in Theoem 3 is budget balanced and guaantees the existence of an equilibium egadless of the game setup, i.e., the numbe of agents, thei espective coveage pobabilities, o the numbe of locations, povided that the action sets satisfy i =. Futhemoe, this utility has an infomation dependency simila to that of equal shae utility design in (4). This utility design has seveal inteesting popeties. Fist of all, one can view the distibution ule in (19) as a weighted shae distibution ule of the geneal fom f i S = s i j S s j W S whee s i is the stength (o weight) of playe i. While setting s i = p i is the obvious choice, it tuns out that letting the stength of each agent i N be defined by the solutions to the equation s i p i = 1 ks i + k 1 + s i whee 1 k 0 also povides the same guaantees as in Theoem 3 with a simila poof that we omit fo bevity. The impotance of this is that thee ae family-of-stength coefficients that guaantee equilibium existence; hence, this weighted shae distibution ule is not a azo-edge phenomenon. Undestanding when weighted shaed distibution ules ae possible is fundamentally impotant to undestanding utility design. Fom a design pespective, having a complete undestanding of this space of admissible utility function is impotant as it allows the designe to optimize ove this class. Lastly, this esult in some sense contadicts the esults in Chen et al. (2008) that suggest that the weighted Shapley value is the only distibution ule that guaantees the existence of an equilibium and is budget balanced. It is staightfowad to show that the utility design in Theoem 3 does not coespond to a weighted Shapley value fo any fixed weights w i R +. It is fundamentally impotant to undestand whethe the oot of the discepancy aises fom (i) cost minimization vesus welfae maximization, (ii) stuctue of the action set, i.e., i = R vesus i 2 R, o (iii) stuctue of the welfae functions, i.e., anonymous vesus nonanonymous Efficiency Impovement. So fa, we have appoached coveage poblems using only the geneal esults povided ealie in the pape. Howeve, it is impotant to note that the pice of anachy guaantees povided ealie can fequently be stengthened by exploiting the stuctue (o cuvatue) of the welfae function fo the specific application of inteest. Hee, we illustate such a tightening in the context of coveage poblems. To simplify ou analysis, we focus puely on the anonymous single selection case whee each agent has the same success pobability p. In this setting, we can diectly appeal to Theoem 2 to show that the elative pice of anachy is W a ne n + W a opt n n + 2n + 1 We can stengthen these bounds futhe to attain the following bound, which illustates the impact of the coveage pobability on the pice of anachy. Theoem 4. If G = N i W t f t is a distibuted welfae game whee (i) the welfae function fo each location t is anonymous and takes on the fom in (16), (ii) the action set of each agent is i =, (iii) each agent i N has the same detection pobability p i = p whee p 0 1,