Definition Existence CG vs Potential Games. Congestion Games. Algorithmic Game Theory

Transcription

1 Algorithmic Game Theory

2 Definitions and Preliminaries Existence of Pure Nash Equilibria vs. Potential Games

3 (Rosenthal 1973) A congestion game is a tuple Γ = (N,R,(Σ i ) i N,(d r) r R) with N = {1,...,n}, set of players R = {1,...,m}, set of resources Σ i 2 R, strategy space of player i d r : {1,...,n} Z, delay function of resource r For any state S = (S 1,...,S n) Σ 1 Σ n, n r = number of players with r S i d r(n r) = delay of resource r δ i (S) = r S i d r(n r) = delay of player i The cost of player i in state S is c i (S) = δ i (S), that is, players aim at minimizing their delays.

4 Example: Network Given a directed graph G = (V,E) with delay functions d e : {1,...,n} Z, e E. Player i wants to allocate a path of minimal delay between a source s i and a target t i. 1,2,9 7,8,9 s 1,9,9 t 4,5,6 1,2,3 In this example, N = {1,2,3}, R = E, Σ i = set of s-t paths. This game is symmetric, i.e., all players have the same set of strategies.

5 Example: Network A sequence of (best reply) improvement steps: First step... 0,99 s 1,1 3,3 1,1 0,0 t 0,3 6,6 1,1 s 0,99 1,1 3,3 1,1 0,0 t 0,3 6,6 1,1

6 Example: Network... second step... 0,99 s 1,1 3,3 1,1 0,0 t 0,3 6,6 1,1 s 0,99 1,1 3,3 1,1 0,0 t 0,3 6,6 1,1

7 Example: Network... third step... 0,99 s 1,1 3,3 1,1 0,0 t 0,3 6,6 1,1 s 0,99 1,1 3,3 1,1 0,0 t 0,3 6,6 1,1 Pure Nash Equilibrium stop!

8 Transition Graph The transition graph of a congestion game Γ contains a node for every state S and a directed edge (S,S ) if S can be reached from S by an improvement step of a single player. The best response transiton graph contains only edges for best response improvement steps. The sinks of the (best response) transition graph are the pure Nash equilibria of Γ.

9 Questions Does every congestion game posses a pure Nash equilibrium? Is every sequence of improvement steps finite? How many steps are needed to reach a (pure) Nash equilibrium? What is the complexity of computing (pure) Nash equilibria in congestion games?

10 Definitions and Preliminaries Existence of Pure Nash Equilibria vs. Potential Games

11 Finite Improvement Property Theorem (Rosenthal 1973) For every congestion game, every sequence of improvement steps is finite. This result immediately implies Corollary Every congestion game has at least one pure Nash equilibrium. Rosenthal s analysis is based on a potential function argument. For every state S, let Φ(S) = n r(s) d r(k). r R k=1 This function is called Rosenthal s potential function.

12 Proof of Rosenthal s theorem Lemma: Let S be any state. Suppose we go from S to a state S by an improvement step of player i decreasing his delay by > 0. Then Φ(S ) = Φ(S). d r(k) d r (k) In the picture, the value of the potential is the shaded area. If a player changes from r to r, his delay changes exactly as the potential value.

13 Proof of Rosenthal s theorem Lemma: Let S be any state. Suppose we go from S to a state S by an improvement step of player i decreasing his delay by > 0. Then Φ(S ) = Φ(S). Proof: The potential φ(s) can be calculated by inserting the agents one after the other in any order, and summing the delays of the players at the point of time at their insertion. W.l.o.g., agent i is the last player that we insert when calculating Φ(S). Then the potential accounted for agent i corresponds to the delay of player i in state S. When going from S to S, the delay of i decreases by, and, hence, Φ decreases by as well. (Lemma)

14 Proof of Rosenthal s theorem The lemma shows that Φ is a so-called exact potential, i.e., if a single player decreases its latency by a value of > 0, then Φ decreases by exactly the same amount. Further observe that i) the delay values are integers so that, for every improvement step, 1, ii) for every state S, Φ(S) r R n i=1 dr(i), iii) for every state S, Φ(S) r R n i=1 dr(i). Consequently, the number of improvements is upper-bounded by 2 r R n i=1 dr(i) and hence finite. (Theorem)

15 Definitions and Preliminaries Existence of Pure Nash Equilibria vs. Potential Games

16 Recall: Strategic Games and Pure Nash Equilibrium Notation For a strategic game, we here use Γ = (N,(Σ i ) i N,(c i ) i N ) where N is the finite set of n players Σ i is the finite set of (pure) strategies of player i Σ = Π i N Σ i the set of states of the game c i : Σ R is the cost function of player i Denote by Σ i = Π i N\{i} Σ i S = (S i ) i N and (S i,s i ) states c i (S) the cost of player i in state S Definition Let Γ = (N,(Σ i ) i N,(c i ) i N ) be a strategic game. We call a state S (pure) Nash equilibrium if for every player i N and every strategy S i Σ i c i (S) c i (S i,s i ).

17 Potential Games Definition (Potential Game) We call a strategic game Γ = (N,(Σ i ) i N,(c i ) i N ) potential game if there exists a function Φ: Σ R such that for every i N, for every S i Σ i, and every S i,s i Σ i : c i (S i,s i ) c i (S i,s i ) = Φ(S i,s i ) Φ(S i,s i ). Observation Let Γ = (N,(Σ i ) i N,(c i ) i N ) be a potential game. Then Γ has the finite improvement property and, hence, there exists a state that is a (pure) Nash equilibrium.

18 Congestion versus Potential Games It follows from Rosenthal s potential function that Corollary Every congestion game is a potential game. In some sense, the reverse is true as well. Theorem (Monderer and Shapley, 1996) Every potential game is isomorphic to a congestion game.

19 Coordination and Dummy Games We present a proof for the theorem of Monderer and Shapley based on a simplified analysis by Voorneveld, Boom, van Megen, Tijs, Facchini (1999) using the notion of coordination and dummy games. Definition (Coordination and Dummy Games) A strategic game Γ = (N,(Σ i ) i N,(c i ) i N ) is a coordination game if there exists a function c: Σ R such that c = c i for every player i N. dummy game if for all every player i N, every S i Σ i, and every S i,s i Σ i ci(si,s i) = ci(s i,s i ).

20 Examples of Coordination and Dummy Games A coordination game, a dummy game and the sum of these games

21 Potential, Coordination, and Dummy Games Theorem Let Γ = (N,(Σ i ) i N,(c i ) i N ) be a strategic game. Γ is a potential game if and only if there exist cost functions (c c i ) i N and (c d i ) i N such that for every player i N, c i = c c i +c d i, (N,(Σ i ) i N,(c c i ) i N ) is a coordination game, and (N,(Σ i ) i N,(c d i ) i N ) is a dummy game.

22 Proof when player i moves, c i changes by the same amount as c c i hence, c = c c i is a potential function of Γ let Φ be a potential function of Γ let c c i = Φ and c d i = c i Φ so that c i = c c i +c d i (N,(Σ i ) i N,(c c i ) i N ) is a coordination game because c c i = Φ does not depend on i (N,(Σ i ) i N,(c d i ) i N ) is a dummy game because, for every i N, S i Σ i and S i,s i Σ i, c i (S i,s i ) c i (S i,s i ) = Φ(S i,s i ) Φ(S i,s i ) c i (S i,s i ) Φ(S i,s i ) = c i (S i,s i ) Φ(S i,s i ), which gives c d i (S i,s i ) = c d i (S i,s i ).

23 Isomorphisms between Games Definition (Isomorphic Games) Let Γ = (N,(Σ i ) i N,(c i ) i N ) and Γ = (N,(Σ i) i N,(c i) i N ) be two strategic games with identical player set N. Γ and Γ are isomorphic if for every i N there exists a bijection ϕ i : Σ i Σ i such that for every (S 1,...,S n) Σ c i (S 1,...,S n) = c i(ϕ 1(S 1),...,ϕ n(s n)). Informally, Γ and Γ are called isomorphic if they are identical apart from reordering (or renaming) the strategies of the players.

24 Isomorphisms between Games Lemma For every coordination game Γ = (N,(Σ i ) i N,(c c i ) i N ), there is a congestion game Γ = (N,R,(Σ i) i N,(d r) r R) that is isomorphic to Γ. Lemma For every dummy game Γ = (N,(Σ i ) i N,(c d i ) i N ), there is a congestion game Γ = (N,R,(Σ i) i N,(d r) r R) that is isomorphic to Γ.

25 Proof for Coordination Games... We construct Γ = (N,R,(Σ i) i N,(d r) r R) as follows. For every state S of Γ, let R contain a resource r(s). For every strategy S i in Σ i, let Σ i contain a corresponding strategy R Si (i.e., a set of resources) defined by R Si = {r(s i,s i ) S i Σ i }. For Γ, we define the delay function of resource r(s) by d r(s) (k) = { c(s) if k = n 0 otherwise where c(s) is the cost of the players in state S of Γ.

26 Proof for Coordination Games Correctness: Consider a state S = (S 1,...,S n) of Γ. In S every player i has the same cost ci c (S) = c(s). In the corresponding state of Γ, only resource r(s) is used by all n players. Thus, resource r(s) has delay c(s). All other resources have delay 0. Consequently, the delay of any player in this state is c(s). Hence, Γ and Γ have the same utility for all players in all states so that they are isomorphic.

27 Illustration of the construction for coordination games Suppose the coordination game is a 2-player game with 3 rows and 3 columns corresponding to the following states (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3) Then the set of resources in the corresponding congestion game Γ is R = {r(1,1),r(1,2),...,r(3,2),r(3,3)}

28 Illustration of the construction for coordination games The set of strategies for player 1 is Σ 1 = { {r(1,1),r(1,2),r(1,3)}, {r(2,1),r(2,2),r(2,3)}, {r(3,1),r(3,2),r(3,3)} }. The set of strategies for player 2 is Σ 2 = { {r(1,1),r(2,1),r(3,1)}, {r(1,2),r(2,2),r(3,2)}, {r(1,3),r(2,3),r(3,3)} }. Exercise: Describe resources and strategy sets of a 3-player coordination game with strategies.

29 Proof for Dummy Games We construct Γ = (N,R,(Σ i) i N,(d r) r R ) as follows. For every i N and S i Σ i, define a resource r(s i ). For every i N and S i Σ i, introduce a strategy R Si = {r(s i ) S i Σ i } {r(s j ) S j Σ j with (S j ) i S i }. j N\{i} Define the delay function of resource r(s i ) by d r(s i )(k) = { c d i (S i ) if k = 1 0 otherwise where c d i (S i ) denotes the cost of player i in Γ when the other players choose S i.

30 Proof for Dummy Games Correctness: Consider a state S = (S 1,...,S n) of Γ. In this state, player i has cost c d i (S) = c d i (S i ). In the corresponding state of Γ, player i, for i N, is the only user of resource r(s i ). Thus, for every i N, resource r(s i ) has delay c d i (S i ), and all other resources have delay 0. Consequently, player i has delay c d i (S i ) in this state. Hence, Γ and Γ have the same utility for all players in all states so that they are isomorphic.

31 Illustration of the construction for dummy games Consider a dummy game is a 2-player game with 3 rows and 3 columns, that is, Σ = {(1,1),(1,2),(1,3),...,(3,2),(3,3)} and Σ 1 = {(,1),(,2),(,3)}, Σ 2 = {(1, ),(2, ),(3, )}. Then the set of resources in the corresponding congestion game Γ is R = {r(,1),r(,2),r(,3)} {r(1, ),r(2, ),r(3, )}.

32 Illustration of the construction for dummy games The set of strategies for player 1 is Σ 1 = { {r(,1),r(,2),r(,3),r(2, ),r(3, )}, {r(,1),r(,2),r(,3),r(1, ),r(3, )}, {r(,1),r(,2),r(,3),r(1, ),r(2, )} }. The set of strategies for player 2 is Σ 2 = { {r(1, ),r(2, ),r(3, ),r(,2),r(,3)}, {r(1, ),r(2, ),r(3, ),r(,1),r(,3)}, {r(1, ),r(2, ),r(3, ),r(,1),r(,2)} }. Exercise: Describe resources and strategy sets of a 3-player dummy game with strategies.

33 Finishing the Proof Now we are ready to prove that every potential game Γ is isomorphic to a congestion game. Split Γ into a coordination and a dummy game. Construct the isomorphic congestion games for these two games as described before. Combine these two congestion games by taking the union of the resource sets. Let each strategy contain the union of resource sets that it contains in coordination and dummy games. The resulting congestion game is isomorphic to the potential game Γ.