Presentation is loading. Please wait.

Presentation is loading. Please wait.

Inefficiency of equilibria, and potential games Computational game theory Spring 2008 Michal Feldman TexPoint fonts used in EMF. Read the TexPoint manual.

Similar presentations


Presentation on theme: "Inefficiency of equilibria, and potential games Computational game theory Spring 2008 Michal Feldman TexPoint fonts used in EMF. Read the TexPoint manual."— Presentation transcript:

1 Inefficiency of equilibria, and potential games Computational game theory Spring 2008 Michal Feldman TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

2 Inefficiency of equilibria Outcome of rational behavior might be inefficient How to measure inefficiency? – E.g., prisoners dilemma Define an objective function – Social welfare (= sum of players payoffs): utilitarian – Maximize min i u i (egalitarian) – … 0,53,3 1,15,0

3 Inefficiency of equilibria To measure inefficiency we need to specify: – Objective function – Definition of approximately optimal – Definition of an equilibrium – If multiple equilibria exist, which one do we consider?

4 Common measures Price of anarchy (poa)=cost of worst NE / cost of OPT Price of stability (pos)=cost of best NE / cost of OPT – Note: poa, pos 1 (by definition) Approximation ratio: Measures price of limited computational resources Competitive ratio: Measures price of not knowing future Price of anarchy: Measures price of lack of coordination

5 Price of anarchy Example: in prisoners dilemma, poa = pos = 3 – But can be as large as desired Wish to find games in which pos or poa are bounded – NE approximates OPT – Might explains Internet efficiency. Suppose we define poa and pos w.r.t. NE in pure strategies – we first need to prove existence of pure NE 0,53,3 1,15,0 Prisoners dilemma

6 Max-cut game Given undirected graph G = (V,E) players are nodes v in V An edge (u,v) means u hates v (and vice versa) Strategy of node i: s i {Black,White} Utility of node i: # neighbors of different color Lemma: for every graph G, corresponding game has a pure NE

7 Proof 1 Claim: OPT of max-cut defines a NE Proof: – Define strategies of players by cut (i.e., one side is Black, other side is White) – Suppose a player i wishes to switch strategies: is benefit from switching = improvement in value of the cut – Contradicting optimality of cut u i =1 u i =2

8 Proof 2 Algorithm greedy-find-cut (GFC): – Start with arbitrary partition of nodes into two sets – If exists node with more neighbors in other side, move it to other side (repeat until no such node exists) Claim 1: GFC provides 2-approx. to max-cut, and runs in polynomial time Proof: – Poly time: GFC terminates within at most |E| steps (since every step improves the value of the solution in at least 1, and |E| is a trivial upper bound to solution) – 2-approx.: Each node ends up with more neighbors in other side than in own side, so at least |E|/2 edges are in cut (since #edges in cut > #edges not in cut)

9 Proof 2 (contd) Claim 2: cut obtained by GFC defines a NE Proof: obvious, as each player stops only if his strategy is the best response to the other players strategies Conclusion: max-cut game admits a NE in pure strategies

10 Extensions What would happen if the edges were weighted? Say, +5 – hate a lot, -5 love a lot? What would happen if love/hate were not symmetric? Home work assignment – I dont know: - Find NE? Complexity?

11 Potential games Definition: a game is an ordinal potential game if there exists :S1×…×Sn R, s.t. i,s i,s -i,s i, c i (s i,s -i ) > c i (s i,s -i ) IFF (s i,s -i ) > (s i,s -i ) Note: G is an exact potential game if c i (s i,s -i ) - c i (s i,s -i ) = (s i,s -i ) - (s i,s -i ) Example: max-cut is an exact potential game, where is the cut size – Unfortunately, is not always so natural

12 Potential games Lemma: a game is a potential game IFF local improvements always terminate proof: – Define a directed graph with a node for each possible pure strategy profile – Directed edge (u,v) means v (which differs from u only in the strategy of a single player, i) is a (strictly) better action for i, given the strategies of the other players – A potential function exists IFF graph does not contains cycles If cycle exists, no potential function; e.g., (a,b,c,a) means f(a) { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/7/1634897/slides/slide_12.jpg", "name": "Potential games Lemma: a game is a potential game IFF local improvements always terminate proof: – Define a directed graph with a node for each possible pure strategy profile – Directed edge (u,v) means v (which differs from u only in the strategy of a single player, i) is a (strictly) better action for i, given the strategies of the other players – A potential function exists IFF graph does not contains cycles If cycle exists, no potential function; e.g., (a,b,c,a) means f(a)

13 Examples direction of local improvement 1,-1-1,1 1,-1 0,53,3 1,15,0 0,02,2 3,30,0 Matching pennies Prisoners dilemma Coordination game C D DC col row Which are potential games? Exact potential games? Are the potential functions unique? 0,02,1 1,20,0 Battle of the sexes

14 Properties of potential games Admit a pure strategy Nash equilibrium Best-response dynamics converge to NE Price of stability is bounded

15 Existence of a pure NE Theorem: every potential game admits a pure NE Proof: we show that the profile minimizing is a NE – Let s be pure profile minimizing – Suppose it is not a NE, so i can improve by deviating to a new profile s – (s) - (s) = c i (s) – c i (s) < 0 – Thus (s) < (s), contradicting s minimizes More generally, the set of pure-strategy Nash equilibria is exactly the set of local minima of the potential function –Local minimum = no player can improve the potential function by herself

16 Best-response dynamics converge to a NE Best-response dynamics: –Start with any strategy profile –If a player is not best-responding, switch that players strategy to a better response (must decrease potential) –Terminate when no player can improve (thus a NE) –Alas, no guarantee on the convergence rate

17 17 Multicast (and non-multicast) Routing Multicast routing: Given a directed graph G = (V, E) with edge costs c e 0, a source node s, and k agents located at terminal nodes t 1, …, t k. Agent j must construct a path P j from node s to its terminal t j. Routing: Given a directed graph G = (V, E) with edge costs c e 0, and k agents seeking to connect s j,t j pairs, Agent j must construct a path P j from node sj to its terminal t j. Fair share: If x agents use edge e, they each pay c e / x. Slides on cost sharing based on slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved.

18 18 Multicast Routing : Shapley price sharing (fair cost sharing) outer 2 middle 4 1 pays 5 + 1 5/2 + 1 middle4 1 outer middle outer 8 2 pays 8 5/2 + 1 5 + 1 s t1t1 v t2t2 4 8 11 5

19 19 Nash Equilibrium Example: – Two agents start with outer paths. – Agent 1 has no incentive to switch paths (since 4 5 + 1). – Once this happens, agent 1 prefers middle path (since 4 > 5/2 + 1). – Both agents using middle path is a Nash equilibrium. s t1t1 v t2t2 4 8 11 5

20 Recall price of anarchy and stability Price of anarchy (poa)=cost of worst NE / cost of OPT Price of stability (pos)=cost of best NE / cost of OPT

21 Socially Optimum Social optimum: Minimizes total costs of all agents. Observation: In general, there can be many Nash equilibria. Even when it is unique, it does not necessarily equal the social optimum. s t1t1 v t2t2 355 1 1 Social optimum = 7 Unique Nash equilibrium = 8 s t k 1 + Social optimum = 1 + Nash equilibrium A = 1 + Nash equilibrium B = k k agents pos=1, poa=kpos=poa=8/7

22 Price of anarchy Claim: poa k Proof: – Let N be the worst NE – Suppose by contradiction c(N) > k OPT – Then, there exists a player i s.t. c i (N) > OPT – But i can deviate to OPT (by paying OPT alone), contradicting that N is a NE Note: bound is tight (lower bound in prev. slide)

23 23 Price of Stability What is price of stability in multicast routing? Lower bound of log k: s t2t2 t3t3 tktk t1t1... 11/2 1/31/k 0 000 1 + 1 + 1/2 + … + 1/k Social optimum: Everyone Takes bottom paths. Unique Nash equilibrium: Everyone takes top paths. Price of stability: H(k) / (1 + ). upper bound will follow..

24 24 Finding a potential function Attempt 1: Let (s) = j=1 k cost(t i ) be the potential function. A problem: The potential might increase when some agent improve. Example: When all 3 agents use the right path, each pays 4/3 and the potential (total cost) is 4. After one agent moves to the left path the potential increases to 5. s t 4 1 3 agents

25 25 Finding a potential function Consider a set of paths P 1, …, P k. – Let x e denote the number of paths that use edge e. – Let (P 1, …, P k ) = e E c e · H(x e ) be a potential function. – Consider agent j switching from path P j to path P j '. – Change in agent js cost: H(0) = 0,

26 26 Potential function – increases by – decreases by – Thus, net change in is identical to net change in player js cost

27 27 Bounding the Price of Stability Claim: Let C(P 1, …, P k ) denote the total cost of selecting paths P 1, …, P k. For any set of paths P 1, …, P k, we have Proof: Let x e denote the number of paths containing edge e. – Let E + denote set of edges that belong to at least one of the paths.

28 28 Bounding the Price of Stability Theorem: There is a Nash equilibrium for which the total cost to all agents exceeds that of the social optimum by at most a factor of H(k) (i.e., price of stability H(k)). Proof: – Let (P 1 *, …, P k * ) denote set of socially optimal paths. – Run best-response dyn algorithm starting from P *. – Since is monotone decreasing (P 1, …, P k ) (P 1 *, …, P k * ). previous claim applied to P previous claim applied to P*

29 Local search and PLS (polynomial local search) Local optimization problem: find a local optimum (i.e., no improvement in neighborhood Local optimization problem is in PLS if exists an oracle that for every instance and solution s decides if s is a local optimum; if not returns a better solution s in neighborhood of s Finding NE in potential games is in PLS – Define neighborhood of a profile s to be profiles obtained by deviation of a single player – s is local optimum for c(s) = (s) iff s is a NE

30 Congestion games [Rosenthal 73] There is a set of resources R Agent is set of actions (pure strategies) A i is a subset of 2 R, representing which subsets of resources would meet her needs –Note: different agents may need different resources There exist cost functions c r : {1, 2, 3, …} such that agent is cost for a = (a i, a -i ) is Σ r a i c r (n r (a)) –n r (a) is the number of agents that chose r as one of their resources in the profile a

31 Example: multicast routing Resources = edges Each resource r has a cost c r Player 1s action set: {{A}, {C,D}} Player 2s action set: {{B}, {C,E}} For all resources r, c r (n r (a)) = c r / n r (a) s t1t1 v t2t2 E 8 11 5 A 4 C D B

32 Every congestion game is an exact potential game Use potential (a) = Σ r Σ 1 i nr(a) c r (i) –One interpretation: the sum of the costs that the agents would have received if each agent were unaffected by all later agents Why is this a correct potential function? Suppose an agent changes action: stop using some resources (R-), start using others (R+) increase in the agents cost equals Σ r R+ c r (n r (a) + 1) - Σ r R- c r (n r (a)) This is exactly the change in the potential function above –Conclusion: congestion games are exact potential games TexPoint Display

33 Computational Game Theory: Network Creation Game Arbitrary Payments (Not a congestion game) Credit to Slides To Eva Tardos Modified/Corrupted/Added to by Michal Feldman and Amos Fiat

34 Network Creation Game – Arbitrary Cost partition G = (V,E) is an undirected graph with edge costs c(e). There are k players. Each player i has a source s i and a sink t i he wants to have connected. s1s1 t3t3 t1t1 t2t2 s2s2 s3s3

35 Model (cont) Player i picks payment p i (e) for each edge e. e is bought if total payments c(e). Note: any player can use bought edges s1s1 t3t3 t1t1 t2t2 s2s2 s3s3

36 The Game Each player i has only 2 concerns: 1) Must be a bought path from s i to t i s1s1 t3t3 t1t1 t2t2 s2s2 s3s3 bought edges

37 The Game Each player i has only 2 concerns: 1) Must be a bought path from s i to t i 2) Given this requirement, i wants to pays as little as possible. s1s1 t3t3 t1t1 t2t2 s2s2 s3s3

38 Nash Equilibrium A Nash Equilibium (NE) is set of payments for players such that no player wants to deviate. Note: player i doesnt care whether other players connect. s1s1 t3t3 t1t1 t2t2 s2s2 s3s3

39 An Example One NE: Each player pays 1/k to top edge. Another NE: Each player pays 1 to bottom edge. Note: No notion of fairness; many NE that pay unevenly for the cheap edge. s 1 …s k t 1 …t k c(e) = 1 c(e) = k

40 Three Observations 1) The bought edges in a NE form a forest. 2) Players only contribute to edges on their s i -t i path in this forest. 3) The total payment for any edge e is either c(e) or 0.

41 Example 2: No Nash s1s1 t1t1 t2t2 s2s2 all edges cost 1 a b c d

42 Example 2: No Nash s1s1 t1t1 t2t2 s2s2 We know that any NE must be a tree: WLOG assume the tree is a,b,c. all edges cost 1 a b c d

43 Example 2: No Pure Nash s1s1 t1t1 t2t2 s2s2 We know that any NE must be a tree: WLOG assume the tree is a,b,c. Only player 1 can contribute to a. all edges cost 1 a b c d

44 Example 2: No Pure Nash s1s1 t1t1 t2t2 s2s2 We know that any NE must be a tree: WLOG assume the tree is a,b,c. Only player 1 can contribute to a. Only player 2 can contribute to c. all edges cost 1 a b c d

45 Example 2: No Pure Nash s1s1 t1t1 t2t2 s2s2 We know that any NE must be a tree: WLOG assume the tree is a,b,c. Only player 1 can contribute to a. Only player 2 can contribute to c. Neither player can contribute to b, since d is tempting deviation. all edges cost 1 a b c d


Download ppt "Inefficiency of equilibria, and potential games Computational game theory Spring 2008 Michal Feldman TexPoint fonts used in EMF. Read the TexPoint manual."

Similar presentations


Ads by Google