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Convergence to Nash Michal Feldman and Amos Fiat Congestion Games – Approx Nash • Convergence to Approximate Nash in Congestion Games – Steve Chien and Alistair Sinclair, SODA 2007 • ε-Nash – no player can improve her cost by more than a factor of ε • A congestion game satisfies the α≥1 bounded jump condition if de(t+1) ≤ α de(t) – delay on edge/resource e Bounds the increase of one additional player using the resource Symmetric Congestion Games, α-bounded resources/edges • The ε-Nash dynamics converges from any initial state in n log( nC ) steps (if the player with the largest relative gain gets to play first) C is an upper bound on the cost of any player Symmetric Congestion Games, α-bounded resources/edges • The ε-Nash dynamics converges from any initial state in n( 1) (1 ) log(nC ) T steps (if every player gets to play at least once within every T time steps) C is an upper bound on the cost of any player Symmetric Congestion Games, α-bounded resources/edges Exact Potential Function fs (e) For Congestion Game ( s ) d e (t ) eE t 1 Player pi with current cost ci ( s) ( s) / Makes a move, the cost must Decrease by a factor of ε Which means that the potential drops by ( s) / n log( nC ) Symmetric Congestion Games, α-bounded resources/edges fs (e) ( s ) d e (t ) eE t 1 Player pi with current cost ci ( s) ( s ) / makes a move, the cost must decrease by a factor of ε Which means that the potential drops by ( s) / after log / steps we have max reached an ε-Nash. In any case, (s) i ci (s) So, # steps is ≤ log(nC ) Symmetric Congestion Games, α-bounded resources/edges fs (e) ( s ) d e (t ) If this player has largest cost, then we get the result needed eE t 1 Player pi with current cost ci ( s) ( s ) / makes a move, the cost must decrease by a factor of ε Which means that the potential drops by ( s) / after log / steps we have max reached an ε-Nash. In any case, (s) i ci (s) So, # steps is ≤ log(nC ) Symmetric Congestion Games, α-bounded resources/edges • So, if we always let the player with the largest relative gain play, and every edge has α-bounded jumps, and if the next step is made by player i, then the cost for i is at least 1/α the cost for j Symmetric Congestion Games, α-bounded resources/edges • The ε-Nash dynamics converges from any initial state in n( 1) (1 ) log(nC ) T steps (if every player gets to play at least once within every T time steps) C is an upper bound on the cost of any player Why care about Games? Users with a multitude of diverse economic interests sharing a Network (Internet) • browsers • routers • servers Model Resulting Issues as Selfishness: Parties deviate from their protocol if it is Games on Networks in their interest A simple game: load balancing Each job wants to be on a lightly loaded machine. With coordination we 1 2 can arrange them to minimize load 3 2 Example: load of 4 machine 1 machine 2 A simple game: load balancing Each job wants to be on a lightly loaded machine. • Without coordination? 2 • Stable arrangement: 1 No job has incentive to switch 3 2 • Example: some have load of 5 Games: setup • A set of players (in example: jobs) • for each player, a set of strategies (which machine to choose) Game: each player picks a strategy For each strategy profile (a strategy for each player) a payoff to each player (load on selected machine) Nash Equilibrium: stable strategy profile: where no player can improve payoff by changing strategy Games: setup Deterministic (pure) or randomized (mixed) strategies? Pure: each player selects a strategy. simple, natural, but stable solution may not exists Mixed: each player chooses a probability distribution of strategies. • equilibrium exists (Nash), • but pure strategies often make more sense Pure versus Mixed strategies in load balancing • Pure strategy: load of 1 1 1 • A mixed equilibrium 1 1 Expected load of 3/2 50% 50% 50% for both jobs 50% Machine 1 Machine 2 Quality of Outcome: Goal’s of the Game Personal objective for player i: min load Li or expected load E(Li) Overall objective? • Social Welfare: i Li or expected value E(i Li ) • Makespan: maxi Li or max expected value maxi E(Li) or expected makespan E(maxi Li ) Example: simple load balancing n identical jobs and n machines 1 1 1 1 1 All pure equilibria: load of 1 (also optimum) A mixed equilibrium: prob 1/n each machine 1 expected load: E(Li)= 1+(n-1) 1 n <2 for each i E(maxi Li ): balls and bins: log n/log log n Results on load balancing: Theorem for E(maxi Li ): • w/uniform speeds, p.o.a ≤ log m/log log m • w/general speeds, worst-case p.o.a. is Θ(log m/log log log m) Proof idea: balls and bins is worst case?? Requence of results by [Koutsoupias/Papadimitriou 99], [Mavronicolas/Spirakis 01], [Koutsoupias/Mavronicolas/Spirakis 02], [Czumaj/Vöcking 02] Today: focus on pure equilibria Does a pure equilibria exists? Does a high quality equilibria exists? Are all equilibria high quality? some of the results extend to sum/max of E(Li) load balancing and routing Load balancing: Delay as a function ℓe(x) = x of load: jobs x unit of load causes delay ℓe(x) machines Routing network: Allow more complex networks ℓe(x) = x x 1 s t s 0 t 1 1 x Atomic vs. Non-atomic Game Non-atomic game: 80% r=1 x 1 • Users control an infinitesimally s t 0 small amount of flow 1 x • equilibrium: all flow path 20% carrying flow are minimum total delay r=1 x 1 Atomic Game: s 0 t • Each user controls a unit of flow, and 1 x • selects a single path or machine Both congestion games: cost on edge e depends on the congestion (number of users) Example of nonatomic flow on two links • One unit of flow sent from s to t x Flow = .5 Traffic on lower s t edge is envious. 1 Flow = .5 x Flow = 1 An envy free solution: s t 1 No-one is Flow = 0 better off Infinite number of players • will make analysis cleaner by continuous math Braess’s Paradox Original Network x .5 .5 1 s t .5 .5 Cost of Nash flow 1 x = 1.5 Added edge: x 1 .5 .5 s .5 0 .5 t 1 x Effect? Braess’s Paradox Original Network x .5 .5 1 s t .5 .5 Cost of Nash flow 1 x = 1.5 Added edge: x 1 1 s 1 0 1 t 1 x Cost of Nash flow = 2 All the flow has increased delay! Model of Routing Game • A directed graph G = (V,E) • source–sink pairs si,ti for x .5 1 i=1,..,k s .5 t r1 =1 • rate ri 0 of traffic 1 .5 .5 x between si and ti for each i=1,..,k • Load-balancing jobs wanted min load • Here want minimum delay: delay adds along path edge-delay is a function ℓe(•) of the load on the edge e Delay Functions r1 =1 Assume ℓe(x) continuous and x .5 .5 1 s t monotone increasing in load 1 .5 .5 x x on edge No capacity of edges for now Example to model capacity u: ℓe(x)= a/(u-x) ℓe(x) x u Goal’s of the Game Personal objective: minimize ℓP(f) = sum of latencies of edges along P (wrt. flow f) No need for mixed strategies Overall objective: C(f) = total latency of a flow f: = P fP•ℓP(f) =social welfare Routing Game?? Flow represents x 1 • cars on highways s t • packets on the Internet x 1 individual packets or small continuous model User goal: Find a path selfishly minimizing user delay true for cars, packets?: users do not choose paths on the Internet: routers do! With delay as primary metric router protocols choose shortest path! Connecting Nash and Opt • Min-latency flow • for one s-t pair for simplicity • minimize C(f) = e fe• ℓe(fe) • subject to: f is an s-t flow • carrying r units • By summing over edges rather than paths where fe = amount of flow on edge e Characterizing the Optimal Flow • Optimality condition: all flow travels along minimum-gradient paths x .5 1 gradient is: s 0 t 1 x (x ℓ(x))’ .5 = ℓ(x)+x ℓ’(x) Characterizing the Optimal Flow • Optimality condition: all flow travels along minimum-gradient paths x .5 1 gradient is: s 0 t 1 x (x ℓ(x))’ .5 = ℓ(x)+x ℓ’(x) Recall: flow f is at Nash equilibrium iff all flow travels along minimum-latency paths Nash Min-Cost Corolary 1: min cost is “Nash” with delay ℓ(x)+x ℓ’(x) Corollary 2: Nash is ‘’min cost’’ with cost fe Ф(f) = e 0 ℓe(x) dx Why? gradient of: fe (0 ℓe(x) dx )’ = ℓ(x) Using function Ф • Nash is the solution minimizing Ф Theorem (Beckmann’56) • In a network latency functions ℓe(x) that are monotone increasing and continuous, • a deterministic Nash equilibrium exists, and is essentially unique Using function Ф (con’t) • Nash is the solution minimizing value of Ф • Hence, Ф(Nash) < Ф(OPT). Suppose that we also know for any solution Ф ≤ cost ≤ A Ф cost(Nash) ≤ A Ф(Nash) ≤ A Ф(OPT) ≤ A cost(OPT). There exists a good Nash! Example: Ф ≤ cost ≤ A Ф Example: ℓe(x) =x then – total delay is x·ℓe(x)=x2 – potential is ℓe() d = x2/2 More generally: linear delay ℓe(x) =aex+be – delay on edge x·ℓe(x) = aex2+be x – potential on edge: ℓe() d = aex2/2+be x – ratio at most 2 Degree d polynomials: – ratio at most d+1 Sharper results for non-atomic games Theorem 1 (Roughgarden-Tardos’00) • In a network with linear latency functions – i.e., of the form ℓe(x)=aex+be • the cost of a Nash flow is at most 4/3 times that of the minimum-latency flow Sharper results for non-atomic games Theorem 1 (Roughgarden-Tardos’00) • In a network with linear latency functions – i.e., of the form ℓe(x)=aex+be • the cost of a Nash flow is at most 4/3 times that of the minimum-latency flow x r=1 x 1 Flow = .5 s t s t 0 1 1 x Flow = .5 Nash cost 1 optimum 3/4 Nash cost 2 optimum 1.5 Braess paradox in springs (aside) Cutting middle string makes the weight rise and decreases power flow along springs Flow=power; delay=distance Bounds for spring paradox Theorem 1’ (Roughgarden-Tardos’00) In a network with springs and strings cutting some strings can increase the height by at most a factor of 4/3. Cutting middle string General Latency Functions • Question: what about more general edge latency functions? • Bad Example: (r = 1, d large) A Nash flow can 1 xd 1- cost arbitrarily s t more than the 0 1 optimal (min-cost) flow Sharper results for non-atomic games Theorem 2 (Roughgarden’02): • In any network with any class of convex continuous latency functions • the worst price of anarchy is always on two edge network 1- x x Corollary: 1 price of anarchy for s t s t degree d polynomials is 1 01 O(d/log d). Another Proof idea Modify the network Nash: ℓe(x) fe ℓe(x) fe ℓ(x)= • Add a new fixed delay parallel edge – fixed cost set = ℓe(fe) • Nash not effected • Optimum can only improve Modified Network Nash: e ℓe(x) fe ℓe(x) fe-e ℓ(x)= – fixed cost set = ℓe(fe) • Optimum on modified network splits flow so that marginal costs are equalized • and common marginal cost is = ℓe(fe) Proof of better bound Nash: e ℓe(x) fe ℓe(x) fe-e ℓ(x)= • Theorem 2: the worst price of anarchy is always two edge network • Proof: Prize of anarchy on G is median of ratios for the edges More results for non-atomic games Theorem 3 (Roughgarden-Tardos’00): • In any network with continuous, nondecreasing latency functions cost of Nash with cost of opt with rates ri for all i rates 2ri for all i Proof … Proof of bicriteria bound Nash: e ℓe(x) fe ℓe(x) fe-e ℓ(x)= common marginal cost on two edges in opt is = ℓ e ( fe ) • Proof: Opt may cost very little, but marginal cost is as high as latency in Nash • Augmenting to double rate costs at least as much as Nash More results for non-atomic games Theorem 3 (Roughgarden-Tardos’00): • In any network with continuous, nondecreasing latency functions cost of Nash with cost of opt with rates ri for all i rates 2ri for all i Morale for the Internet: build for double flow rate Morale for IP versus ATM? Corollary: with M/M/1 delay fns: ℓ(x)=1/(u-x), where u=capacity Nash w/cap. 2u opt w/cap. u Doubling capacity is more effective than optimized routing (IP versus ATM) Part II • Discrete potential games: • network design • price of anarchy stability Continuous Potential Games Continuous potential game: there is a function (f) so that Nash equilibria are exactly the local minima of also known as Walrasian equilibrium convex then Nash equilibrium are the minima. For example fe Ф(f) = e 0 ℓe(x) dx Discrete Analog Atomic Game t • Each user controls s one unit of flow, and t s • selects a single path Theorem Change in potential is same as function change perceived by one user [Rosenthal’73, Monderer Shapley’96,] (f) = e ( ℓe(1)+…+ ℓe(fe)) = e e Even though moving player ignores all other users Potential: Tracking Happiness Theorem Change in potential is same as function change perceived by one user [Rosenthal’73, Monderer Shapley’96,] (f) = e ( ℓe(1)+…+ ℓe(fe)) = e e Potential before move: Reason? ℓe(1)+… ℓe(fe -1) + ℓe(fe) e + ℓe’(1)+…+ ℓe’(fe’) e’ Potential: Tracking Happiness Theorem Change in potential is same as function change perceived by one user [Rosenthal’73, Monderer Shapley’96,] (f) = e ( ℓe(1)+…+ ℓe(fe)) = e e Potential after move: Reason? ℓe(1)+… ℓe(fe -1) + ℓe(fe) e + ℓe’(1)+…+ ℓe’(fe’) + ℓe’(fe’+1) e’ Change in is -ℓe(fe) + ℓe’(fe’+1) same as change for player What are Potential Games Discrete potential game: there is a function (f) so that change in potential is same as function change perceived by one user Theorem [Monderer Shapley’96] Discrete potential games if and only if congestion game (cost of using an element depends on the number of users). Proof of “if” direction (f) = e ( ℓe(1)+…+ ℓe(fe)) Corollary: Nash equilibria are local min. of (f) Why care about Best Nash/Opt ratio? Papadimitriou-Koutsoupias ’99 Nash = outcome of selfish behavior worst Nash/Opt ratio: Price of Anarchy Non-atomic game: Nash is unique… Atomic Nash not unique! Best Nash is good quality… cost of best selfish outcome Price of Stability= “socially optimum” cost cost of worst selfish outcome Price of Anarchy= “socially optimum” cost Potential argument Low price of stability But do we care? Atomic Game: Routing with Delay Theorems 1&2 true for the Nash minimizing the potential function, assuming all players carry the same amount of flow Worst case on 2 edge network Atomic Game: Price of Anarchy? Theorem: Can be bounded for some classes of delay functions e.g., polynomials of degree at most d at most exponential in d. Suri-Toth-Zhou SPAA’04 + Awerbuch-Azar-Epstein STOC’05+ Christodoulou-Koutsoupias STOC’05 Network Design as Potential Game Given: G = (V,E), costs ce (x) for all e є E, k terminal sets (colors) Have a player for each color. Network Design as Potential Game Given: G = (V,E), costs ce (x) for all e є E, k terminal sets (colors) Have a player for each color. Each player wants to build a network in which his nodes are connected. Player strategy: select a tree connecting his set. Costs in Connection Game Players pay for their trees, want to minimize payments. What is the cost of the edges? ce (x) is cost of edge e for x users. Assume economy of scale for costs: ce (x) x Costs in Connection Game Players pay for their trees, want to minimize payments. What is the cost of the edges? ce (x) is cost of edge e for x users. Assume economy of scale for costs: ce (x) How do players share the cost of an edge? x A Connection Game How do players share the cost of an edge? Natural choice is fair sharing, or Shapley cost sharing: A Connection Game How do players share the cost of an edge? Natural choice is fair sharing, or Shapley cost sharing: Players using e pay for it evenly: ci(P) = Σ ce (ke ) /ke where ke number of users on edge e [Herzog, Shenker, Estrin’97] A Connection Game How do players share the cost of an edge? Natural choice is fair sharing, or Shapley cost sharing: Players using e pay for it evenly: ci(P) = Σ ce (ke ) /ke where ke number of users on edge e [Herzog, Shenker, Estrin’97] This is congestion game: ℓe(x) =ce(x)/x with decreasing “latency” A Simple Example t 1, t 2 , … t k t 1 k s s1, s2, … sk A Simple Example t 1, t 2 , … t k t t 1 k 1 k s s s1, s2, … sk One NE: each player pays 1/k A Simple Example t 1, t 2 , … t k t t t 1 k 1 k 1 k s s s s1, s2, … sk One NE: Another NE: each player each player pays 1/k pays 1 Maybe Best Nash is good? We know price of anarchy is bad. Game is a potential game so maybe Price of Stability is better. cost of best selfish outcome Price of Stability= “socially optimum” cost Do we care? Nash as Stable Design Need to Find a Nash equilibrium – Stable design: as no user finds it in their interest to deviate Need to find a “good” Nash – Best Nash/Opt ratio? = Price of Stability [ADKTWR 2004] Design with a constraint for stability Results for Network Design Theorem [Anshelevich, Dasgupta, Kleinberg, Tardos, Wexler, Roughgarden FOCS’04] Price of Stability is at most O(log k) for k players proof: • edge cost ce with ke > 0 users • edge potential with ke > 0 users e =ce·(1+1/2+1/3+…+1/k) Ratio at most Hk=O(log k) Example: Bound is Tight t 1 1 2 1 3 1 1 k k-1 1+ 1 2 3 ... k-1 k 0 0 0 0 0 Example: Bound is Tight t cost(OPT) = 1+ε 1 1 2 1 3 1 1 k k-1 1+ 1 2 3 ... k-1 k 0 0 0 0 0 Example: Bound is Tight t cost(OPT) = 1+ε …but not a NE: 1 1 2 1 3 1 1 k player k k-1 1+ ... pays (1+ε)/k, 1 2 3 k-1 k could pay 1/k 0 0 0 0 0 Example: Bound is Tight t 1 1 2 1 3 1 1 k so player k k-1 1+ ... would deviate 1 2 3 k-1 k 0 0 0 0 0 Example: Bound is Tight t 1 1 2 1 3 1 1 k now player k-1 k-1 1+ ... pays (1+ε)/(k-1), 1 2 3 k-1 k could pay 1/(k-1) 0 0 0 0 0 Example: Bound is Tight t 1 1 2 1 3 1 1 k so player k-1 k-1 1+ ... deviates too 1 2 3 k-1 k 0 0 0 0 0 Example: Bound is Tight t Continuing this process, all 1 1 1 1 players defect. 2 3 1 k k-1 1+ 1 2 3 ... k-1 k 0 0 0 0 0 This is a NE! (the only Nash) 1 1 cost = 1 + 2 + … + k Price of Stability is Hk = Θ(log k)! Congestion games Routing with delay: xd 1 • cost increasing with s t 0 congestion 1 xd e.g., ce(x)= xℓe(x) =xd+1 Network Design Game: • cost decreasing with congestion e.g., ℓe(x)= c(x)e/x Contrast with Routing Games Routing games Design with Fair Sharing • ce(x) increasing • ce(x) decreasing • Traffic maybe non- • Choice atomic atomic OK? to split traffic need to select single path • Nash is unique • Many equilibria • Price of Stability grows • Price of Stability with steepness of c: bounded by log n – worst case on 2 links – bicriteria bound x Flow = .5 s t 1 Flow = .5 Part III Is Nash a reasonable concept? Is the price of anarchy always small? and what can be do when its too big (mechanism design) Examples: • Network design and • Resource allocation Why stable solutions? Plan: analyze the quality of Nash equilibrium. But will players find an equilibrium? • Can a stable solution be found in poly. time? • Does natural game play lead to an equilibrium? • We are assuming non-cooperative players, what if there is cooperation? Answer 1: A clean solution concept and exists ([Nash 1952] if game finite) Does life lead to clan solutions? Why stable solutions? • Finding an equilibrium? Nonatomic games: we’ll see that equilibrium can be found via convex optimization [Beckmann’56] Atomic game: finding an equilibrium is polynomial local search (PLS) complete [Fabrikant, Papadimitriou, Talwar STOC’04] Why stable solutions? • Does natural game play lead to equilibrium? we’ll see that natural “best response play” leads to equilibrium if players change one at-a-time See also: Fischer\Räcke\Vöcking’06, Blum\Even-Dar\Ligett’06 also if players simultaneously play natural learning strategies Why stable solutions? • We are assuming non-cooperative players Cooperation? No great models, see some partial results on Thursday. How to Design “Nice” Games? (Mechanism Design) Traditional Mechanism Design (VCG): • use payments to induce all players to tell us his utility for connection • Select a network to maximize social welfare (minimize cost) How to Design “Nice” Games? (Mechanism Design) Traditional Mechanism Design (VCG): • use payments to induce all players to tell us his utility for connection • Select a network to maximize social welfare (minimize cost) Cost lot of money; lots of information to share How to Design “Nice” Games? (Mechanism Design) Traditional Mechanism Here: Design (VCG): • design a simple/natural • use payments to induce Nash game where users all players to tell us his select their own graphs utility for connection and • Select a network to maximize social welfare (minimize cost) • analyze the Prize of Anarchy Cost lot of money; lots of information to share Network Design Mechanism How should multiple players on a single edge split costs? We used fair sharing … [Herzog, Shenker, Estrin’97] ci(P) = Σ ce (ke ) /ke where ke number of users on edge e which makes network design a potential game Network Design Game Revisited How should multiple players on a single edge split costs? We used fair sharing … [Herzog, Shenker, Estrin’97] Another approach: Why not free market? players can also agree on shares? ...any division of cost agreed upon by players is OK. Near-Optimal Network Design with Selfish Agents STOC ‘03 Anshelevich, Dasgupta, Tardos, Wexler. Network Design without Fairness Results [Anshelevich, Dasgupta, Tardos, Wexler STOC’03] Good news: Price of Stability 1 when all users want to connect to a common source (as compared to log n for fair sharing) But: with different source-sink pairs • Nash may not exists (free riding problem) • and may be VERY bad when it exists Partial good news: low cost Approximate Nash No Deterministic Nash: Free Riding problem Network Design s1 t2 [ADTW STOC’03] 1 1 1 1 Users bid contribution on s2 t1 individual edges. • Single source game: ? Price of Anarchy = 1 s1 t2 1 1 1 • Multi source: no Nash 1 s2 t1 Mechanism Design Example: Network design. Results can be used to answer question: Should one promote “fair sharing” or “free market”? Another Example: Bandwidth Allocation Many Users with diverse utilities for bandwidth. How should we share a given B bandwidth? Bandwidth Sharing Game Assumption: Users have a utility function Ui(x) for receiving x bandwidth. Ui(x) Assume elastic users (concave utility functions) xi x A Mechanism: Kelly: proportional sharing • Players offer money wi for bandwidth. • Bandwidth allocated proportional to payments: Many Users with diverse – effective price p= (i wi )/B utilities for bandwidth. – player allocation xi = wi /p How should we share a given B bandwidth? A Mechanism: Kelly: proportional sharing • Players offer money wi for bandwidth. • allocation proportional: – unit price p= (i wi )/B Many Users with diverse utilities for – player i gets xi = wi /p bandwidth. Thm: If players are price-takers (do not anticipate the effect How should we share a of their bid on the price) given B bandwidth? Selfish play results in optimal allocation Price Taking Users Given price p: how much bandwidth does user i want? Ui(x) slope p Answer: keeps asking for more until marginal increase in happiness is xi at least p: x Assume elastic users Ui’(x)=p (concave utility functions) Price Taking Users: Kelly Mechanism Optimal Equilibrium at price p: slope p each user i wants xi such Ui(x) that Ui’(xi)=p Total bandwidth used up at price p result optimal division of bandwidth xi x Assume elastic users (concave utility functions) Price taking users standard assumption if many players Kelly Proportional Sharing: Johari-Tsitsikis, 2004: what if players do anticipate their effect on the price? Players offer money wi for bandwidth. Theorem: Price of Anarchy Bandwidth allocated at most ¾ on any proportional to networks, and any payments number of users Kelly Proportional Sharing: Theorem [Johari-Tsitsikis, 2004] Price of Anarchy at most ¾ on any networks, and any number of users Why not optimal? big users “shade” their price. User Players offer money wi for bandwidth. choice Bandwidth allocated Ui’(xi)(1-xi)=p proportional to assuming total bandwidth is 1 payments Worst case: one large user and many small users Summary We talked about many issues Price of Anarchy/Stability/Coalitions in the context of some Network Games: – routing, load balancing, network design, bandwidth sharing • Designing games (mechanism design) – network design Algorithmic Game Theory • The main ingredients: – Lack of central control like distributed computing – Selfish participants game theory • Common in many settings e.g., Internet Most results so far: – Price of anarchy/stability in many games, including many I did not mention – e.g. Facility location (another potential game) [Vetta FOCS’02] and [Devanur-Garg- Khandekar-Pandit-Saberi’04]: Some Open Directions: • Other natural network games with low lost of anarchy • Design games with low cost of anarchy • Better understand dynamics of natural game play • Dynamics of forming coalitions

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