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Definition Existence CG vs Potential Games

Congestion Games

Algorithmic Game Theory

Alexander Skopalik Algorithmic Game Theory 2012

Congestion Games


Definitions and Preliminaries

Existence of Pure Nash Equilibria

Congestion Games vs. Potential Games


Congestion Games


Congestion Games (Rosenthal 1973)

A congestion game is a tuple Γ = (N ,R, (Σi )i∈N , (dr )r∈R) with

◮ N = {1, . . . , n}, set of players

◮ R = {1, . . . ,m}, set of resources

◮ Σi ⊆ 2R, strategy space of player i

◮ dr : {1, . . . , n} → Z, delay function of resource r

For any state S = (S1, . . . , Sn) ∈ Σ1 × · · ·Σn,

◮ nr = number of players with r ∈ Si

◮ dr (nr ) = delay of resource r

◮ δi (S) =∑

r∈Sidr (nr ) = delay of player i

The cost of player i in state S is ci (S) = δi (S), that is, players aim atminimizing their delays.


Congestion Games


Example: Network Congestion Games

◮ Given a directed graph G = (V ,E ) with delay functionsde : {1, . . . , n} → Z, e ∈ E .

◮ Player i wants to allocate a path of minimal delay between a source si anda target ti .

1,2,9

4,5,6 1,2,3

1,9,9

7,8,9

s t

◮ In this example, N = {1, 2, 3}, R = E , Σi = set of s-t paths.

◮ This game is symmetric, i.e., all players have the same set of strategies.


Congestion Games



A sequence of (best reply) improvement steps: First step ...

1,1 3,3

0,99

0,01,1

6,6 1,1

0,3

s t1,1 3,3

0,99

0,01,1

6,6 1,1

0,3

s t


Congestion Games



... second step ...

1,1 3,3

0,99

0,01,1

6,6 1,1

0,3

s t1,1 3,3

0,99

0,01,1

6,6 1,1

0,3

s t


Congestion Games



... third step ...

1,1 3,3

0,99

0,01,1

6,6 1,1

0,3

s t1,1 3,3

0,99

0,01,1

6,6 1,1

0,3

s t

Pure Nash Equilibrium – stop!


Congestion Games


Transition Graph

◮ The transition graph of a congestion game Γ contains a node for everystate S and a directed edge (S , S ′) if S ′ can be reached from S by animprovement step of a single player.

◮ The best response transiton graph contains only edges for best responseimprovement steps.

The sinks of the (best response) transition graph are the pure Nash equilibriaof Γ.


Congestion Games


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 congestiongames?


Congestion Games






Congestion Games


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) =∑

r∈R

nr (S)∑

k=1

dr (k) .

This function is called Rosenthal’s potential function.


Congestion Games


Proof of Rosenthal’s theorem

Lemma: Let S be any state. Suppose we go from S to a state S ′ by animprovement step of player i decreasing his delay by ∆ > 0. ThenΦ(S ′) = Φ(S)−∆.

1 2 3 4 5 6

dr (k)

1 2 3 4 5 6

dr ′(k)

In the picture, the value of the potential is the shaded area. If a player changesfrom r ′ to r , his delay changes exactly as the potential value.


Congestion Games



Lemma: Let S be any state. Suppose we go from S to a state S ′ by animprovement step of player i decreasing his delay by ∆ > 0. ThenΦ(S ′) = Φ(S)−∆.

Proof:

◮ The potential φ(S) can be calculated by inserting the agents one after theother in any order, and summing the delays of the players at the point oftime 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 ofplayer i in state S .

◮ When going from S to S ′, the delay of i decreases by ∆, and, hence, Φdecreases by ∆ as well. (Lemma)


Congestion Games



The lemma shows that Φ is a so-called exact potential, i.e., if a single playerdecreases its latency by a value of ∆ > 0, then Φ decreases by exactly thesame 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 by2 ·

∑

r∈R

∑n

i=1 |dr (i)| and hence finite. (Theorem)


Congestion Games






Congestion Games


Recall: Strategic Games and Pure Nash Equilibrium

NotationFor a strategic game, we here use Γ = (N , (Σi )i∈N , (ci )i∈N ) where

◮ N is the finite set of n players

◮ Σi is the finite set of (pure) strategies of player i

◮ Σ = Πi∈NΣi the set of states of the game

◮ ci : Σ → R is the cost function of player i

Denote by

◮ Σ−i = Πi∈N\{i}Σi

◮ S = (Si )i∈N and (Si , S−i ) states

◮ ci (S) the cost of player i in state S

DefinitionLet Γ = (N , (Σi )i∈N , (ci )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 ∈ Σi

ci (S) ≤ ci (S′i , S−i ) .


Congestion Games


Potential Games

Definition (Potential Game)

We call a strategic game Γ = (N , (Σi )i∈N , (ci )i∈N ) potential game if thereexists a function Φ: Σ → R such that for every i ∈ N , for every S−i ∈ Σ−i ,and every Si , S

′i ∈ Σi :

ci (Si , S−i )− ci (S′i , S−i ) = Φ(Si , S−i )− Φ(S ′

i , S−i ) .

ObservationLet Γ = (N , (Σi )i∈N , (ci )i∈N ) be a potential game. Then Γ has the finiteimprovement property and, hence, there exists a state that is a (pure) Nashequilibrium.


Congestion Games


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.


Congestion Games


Coordination and Dummy Games

We present a proof for the theorem of Monderer and Shapley based on asimplified 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 , (Σi )i∈N , (ci )i∈N ) is a

◮ coordination game if there exists a function c : Σ → R such that c = ci forevery player i ∈ N .

◮ dummy game if for all every player i ∈ N , every S−i ∈ Σ−i , and everySi , S

′i ∈ Σi

ci (Si , S−i ) = ci (S′i , S−i ) .


Congestion Games


Examples of Coordination and Dummy Games

A coordination game,...

2 2 52 2 5

3 6 43 6 4

2 1 32 1 3

...a dummy game...

3 3 31 2 1

1 1 11 2 1

2 2 21 2 1

... and the sum of these games

5 5 83 4 6

4 7 54 8 5

4 3 53 3 4


Congestion Games


Potential, Coordination, and Dummy Games

TheoremLet Γ = (N , (Σi )i∈N , (ci )i∈N ) be a strategic game. Γ is a potential game ifand only if there exist cost functions (cci )i∈N and (cdi )i∈N such that

◮ for every player i ∈ N , ci = cci + cdi ,

◮ (N , (Σi )i∈N , (cci )i∈N ) is a coordination game, and

◮ (N , (Σi )i∈N , (cdi )i∈N ) is a dummy game.


Congestion Games


Proof

⇐

◮ when player i moves, ci changes by the same amount as cci◮ hence, c = cci is a potential function of Γ

⇒ let Φ be a potential function of Γ

◮ let cci = Φ and cdi = ci − Φ so that ci = cci + cdi◮ (N , (Σi )i∈N , (cci )i∈N ) is a coordination game because cci = Φ does not

depend on i

◮ (N , (Σi )i∈N , (cdi )i∈N ) is a dummy game because, for every i ∈ N ,S−i ∈ Σ−i and Si , S

′i ∈ Σi ,

ci (Si , S−i )− ci (S′i , S−i ) = Φ(Si , S−i )− Φ(S ′

i , S−i )⇔ ci (Si , S−i )− Φ(Si , S−i ) = ci (S

′i , S−i )− Φ(S ′

i , S−i ) ,

which gives cdi (Si , S−i ) = cdi (S′i , S−i ).


Congestion Games


Isomorphisms between Games

Definition (Isomorphic Games)

Let Γ = (N , (Σi )i∈N , (ci )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 ∈ Nthere exists a bijection ϕi : Σi → Σ′

i such that for every (S1, . . . , Sn) ∈ Σ

ci (S1, . . . , Sn) = c ′i (ϕ1(S1), . . . , ϕn(Sn)) .

Informally, Γ and Γ′ are called isomorphic if they are identical apart fromreordering (or renaming) the strategies of the players.


Congestion Games


Isomorphisms between Games

LemmaFor every coordination game Γ = (N , (Σi )i∈N , (cci )i∈N ), there is a congestiongame Γ′ = (N ,R, (Σ′

i )i∈N , (dr )r∈R) that is isomorphic to Γ.

LemmaFor every dummy game Γ = (N , (Σi )i∈N , (cdi )i∈N ), there is a congestion gameΓ′ = (N ,R, (Σ′

i )i∈N , (dr )r∈R) that is isomorphic to Γ.


Congestion Games


Proof for Coordination Games

... We construct Γ′ = (N ,R, (Σ′i )i∈N , (dr )r∈R) as follows.

◮ For every state S of Γ, let R contain a resource r(S).

◮ For every strategy Si in Σi , let Σ′i contain a corresponding strategy RSi

(i.e., a set of resources) defined by

RSi = {r(Si , S−i ) | S−i ∈ Σ−i} .

◮ For Γ′, we define the delay function of resource r(S) by

dr(S)(k) =

{

c(S) if k = n0 otherwise

where c(S) is the cost of the players in state S of Γ.


Congestion Games


Proof for Coordination Games

Correctness:

◮ Consider a state S = (S1, . . . , Sn) of Γ.

◮ In S every player i has the same cost cci (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 theyare isomorphic.


Congestion Games


Illustration of the construction for coordination games

◮ Suppose the coordination game is a 2-player game with 3 rows and 3columns 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)}


Congestion Games


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)} } .


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 2×2×2strategies.


Congestion Games


Proof for Dummy Games

We construct Γ′ = (N ,R, (Σ′i )i∈N , (dr )r∈R) as follows.

◮ For every i ∈ N and S−i ∈ Σ−i , define a resource r(S−i ).

◮ For every i ∈ N and Si ∈ Σi , introduce a strategy

RSi = {r(S−i ) | S−i ∈ Σ−i} ∪⋃

j∈N\{i}

{r(S−j) | S−j ∈ Σ−j with (S−j)i 6= Si} .

◮ Define the delay function of resource r(S−i ) by

dr(S−i )(k) =

{

cdi (S−i ) if k = 10 otherwise

where cdi (S−i ) denotes the cost of player i in Γ when the other playerschoose S−i .


Congestion Games


Proof for Dummy Games

Correctness:

◮ Consider a state S = (S1, . . . , Sn) of Γ.

◮ In this state, player i has cost cdi (S) = cdi (S−i ).

◮ In the corresponding state of Γ′, player i , for i ∈ N , is the only user ofresource r(S−i ).

◮ Thus, for every i ∈ N , resource r(S−i ) has delay cdi (S−i ), and all otherresources have delay 0.

◮ Consequently, player i has delay cdi (S−i ) in this state.

Hence, Γ and Γ′ have the same utility for all players in all states so that theyare isomorphic.


Congestion Games


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, ∗)} .


Congestion Games


Illustration of the construction for dummy games


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, ∗)} } .


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 2× 2× 2strategies.


Congestion Games


Finishing the Proof

Now we are ready to prove that every potential game Γ is isomorphic to acongestion game.

◮ Split Γ into a coordination and a dummy game.

◮ Construct the isomorphic congestion games for these two games asdescribed before.

◮ Combine these two congestion games by taking the union of the resourcesets. Let each strategy contain the union of resource sets that it containsin coordination and dummy games.

The resulting congestion game is isomorphic to the potential game Γ. ✷


Congestion Games

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