1. Congestion games Computational game theory Fall 2010
by Rotem Arnon & Eytan Kidron

2. Congestion Games
In a Congestion Game, each agent chooses a set of resources.
The cost function for each resource depends only on the number of agents who chose the resource.
Agents: A,B,C
Resources: the Edges
Cost function: (for ex.)

3. Example 1: Multicast Routing
In the Multicast Routing problem, we are given a graph G=(V,E).
Each agent i must buy edges connecting si∊V to ti∊V.
The cost of an edge e∊E is distributed equally between the agents that bought it ⇒ The more agents buy e, the cheaper it is for each agent.
The cost of path P:siti is ∑e∊P(Ce / Ne) where Ce is the cost of e and Ne is the number of agents that bought e.
The goal is to minimize the cost.

4. Example 1: Multicast Routing
agent 1 cost: 1 2
1+4/2+3=6 7 4 7 6
agent 2 cost: 3 1 t1 t2
2+4/2+1=5
2+4+1=7 6

5. Example 2: Traffic
In the Traffic problem, we are given a graph G=(V,E).
Each agent i must choose a path connecting si∊V to ti∊V.
The delay on an edge e∊E is proportional to the number of agents using this edge ⇒ The more agents use e, the higher the delay for each agent.
The cost for using path P:siti is ∑e∊P(Ce * Ne) where Ce is the cost of e and Ne is the number of agents that use e.
The goal is to minimize the cost (delay).

6. Example 2: Traffic 7 2+2+1=5 6 2+2*2+1=7 agent 1 cost 1+2*2+3=8

7. Congestion Games – Formal Definition
Let R be the set of all resources.
Let Si⊂2R be the set of strategies from which agent i can choose. Each strategy si∊Si is a set of resources.
For each resource r∊R, let Nr denote the number of agents that choose resource r. Let cr be a cost function for resource r. cr=f(Nr), meaning, cr depends only on Nr and not on the agent i.
The cost function for agent i is ∑r∊s(i)cr(Nr)

8. Congestion Games vs. Potential Games
Theorem 1: Every congestion game
is an (exact) potential game.
Theorem 2: Every finite (exact) potential game is isomorphic to a congestion game.

9. Congestion Game ⇒ Potential Game
Given: a congestion game with a set of resources R
and cost functions cr.
Goal: define an exact potential function which
describes the game.
The potential function is f(S) = Σr Σ1 ≤ j ≤ Nr(S) cr(j)
Where:
S=(s1,…, sn) is the combined strategy of all agents
Nr(S) is the number of agents using resource r in S.
One interpretation: the sum of the costs that the agents would have received if each agent were unaffected by all later agents.

10. Congestion Game ⇒ Potential Game
f(S) = Σr Σ1 ≤ j ≤ Nr(S) cr(j)
Why is this a correct potential function?
Suppose: agent i changes it’s strategy from si to si’.
⇒ the combined strategy changes from
S =(si,s-i) to S’=(si’,s-i)
Let R+ = si’-si be the new resources the agent added.
Let R- = si-si’ be the resources the agent removed.
The increase in the agent’s cost equals to: ΣrR+ cr(Nr(S) + 1) - ΣrR- cr(Nr(S))
This is exactly the change in the potential function above!
Conclusion: congestion games are exact potential games

11. Congestion Games vs. Potential Games
Theorem 1: Every congestion game
is an (exact) potential game.
Theorem 2: Every finite (exact) potential game is isomorphic to a congestion game.

12. Potential Game ⇒ Congestion Game
Given: a potential game with a potential function f(S).
Goal: define the game as a congestion game.
What does it mean to define the game as a
congestion game?
We need to define:
The resources R
The cost functions cr for every r∊R such that:
∀combined strategy S and ∀ player i, ui(S) will be the same as in the potential game.

13. Define The Resources Let:
n be the number of agents in the potential game
k be the number of possible strategies (for a single agent)
Define the resources R as all the sequences of nk bits, R={0,1}nk.
Which subset of resources will agent i choose?
If an agent i chooses strategy si in the potential game,
he will choose all resources which have a 1 bit in the (i*k+si)-th bit in the congestion game.
⇒ In total, each agent will choose 2nk-1 resources.

14. Type A Resources
Type A resources are resources with the following format: ∀ agent, in the k bits associated with that agent, there is exactly one set bit (1 bit).
Example: for n=3 and k=4, the string:
“ ” is a type A resource
“ ” is not a type A resource
“ ” is not a type A resource
For a resource r of type A, let si denote the index of the set bit of the i-th agent.
Example: “ ”
s0=1
s1=2
s2=0

15. Cost Function: Type A Resources
Reminder:
Let n be the number of agents in the potential game.
For a resource r of type A, let si denote the index of the set bit of the i-th agent.
Now we can define the cost function for type A resources:
The cost function of r is:
cr(n)=f(s0,…, sn-1)
cr(x)=0 for x≠n
Note: there is exactly one type A resource for which each agent receives cost, and it is the same resource for all agents.

16. Type B Resources
Type B resources are resources with the following format: ∃agent i, such that the k bits associated with agent i are set. For all other agents, k-1 out of the k bits are set.
Example: for n=3 and k=4, the string:
“ ” is a type B resource
“ ” is not a type B resource
“ ” is not a type B resource
For a resource r of type B, let i denote the agent whose bits are all set and for ∀j≠i let sj denote the index of the unset bit of the j-th agent.
Example: “ ” i=1
s0=3
s2=2

17. Cost Function: Type B Resources
Reminder: For a resource r of type B, let i denote the agent whose bits are all set and for ∀j≠i let sj denote the index of the unset bit of the j-th agent.
Now we can define the cost function for type B resources:
The cost function of r is:
cr(1)=ui(s0,…, sn-1) - f(s0,…, sn-1)
cr(x)=0 for x≠1
Note: for any combined strategy S=(s0,…, sn-1), each agent gets a non-zero cost for exactly one type B resource, a resource in which i is the agents index and the sj values are the other agents’ strategies.
But what is si? si is undefined...

18. Cost Function: Type B Resources
Interestingly, the value of ui(s0,…, sn-1) - f(s0,…, sn-1)
does not depend on si!
Why?
f is a potential function ⇒ for every si and si’:
ui(si,s-i) - ui(si’,s-i) = f(si,s-i) - f(si’,s-i)
⇒ ui(si,s-i) - f(si,s-i) = ui(si’,s-i) - f(si’,s-i)
Hence ui(s0,…, sn-1) - f(s0,…, sn-1) does not depend on si.

19. Potential Game ⇒ Congestion Game
∀resource r which is not a type A or type B resource the cost function is 0, cr(x)=0.
What cost does agent i receive for the combined strategy S=(s0,…, sn-1)?
In the potential game, he receives ui(s0,…, sn-1).
In the congestion game, he receives:
1 type A resource  f(s0,…, sn-1)
1 type B resource  ui(s0,…, sn-1) - f(s0,…, sn-1)
⇒ In total he receives ui(s0,…, sn-1), exactly the same utility as in the potential game.
Conclusion: exact potential games are congestion games

20. Congestion Games vs. Potential Games
Theorem 1: Every congestion game
is an (exact) potential game.
Theorem 2: Every finite (exact) potential game is isomorphic to a congestion game.

21. Pure Nash equilibrium
Theorem: Every finite congestion game has a pure Nash equilibrium.
Why?
Congestion game ⇒
Exact potential game ⇒
Every exact potiential game has a pure Nash equilibrium

22. Intrinsic Robustness of the Price of Anarchy
Tim Roughgarden
Stanford University

23. Inefficiency of Nash Flows
Note: Nash flows do not minimize the cost
- observed informally by [Pigou 1920]
Cost of Nash flow = 1•1 + 0•1 = 1
Cost of optimal (min-cost) flow = ½•½ +½•1 = ¾
Price of anarchy := Nash/OPT ratio = 4/3 x 1 s t 1

24. Unbounded POA Example: Nash flow has cost 1 Min cost  0xd 1
1-Є Є
Nash flow has cost 1
Min cost  0
⇒ Nash flow can cost arbitrarily more than the optimal (min-cost) flow
even if cost functions are polynomials

25. Linear Cost Functions cost of Nash flow cost of opt flow
Definition: linear cost fn is of form ce(x)=aex+be
Theorem: [Roughgarden/Tardos 00] for every network with linear cost fns:
≤ 4/3 ×
i.e., price of anarchy ≤ 4/3 in the linear case.
cost of Nash flow
cost of opt flow

26. Sources of Inefficiency
Corollary of previous Theorem:
For linear cost fns, worst Nash/OPT ratio is realized in a two-link network!
simple explanation for worst inefficiency
confronted w/two routes, selfish users overcongest one of them s t x 1
Cost of Nash = 1
Cost of OPT = ¾

27. Weaker Equilibrium Concepts
no regret
correlated eq
mixed Nash
pure
Nash
best-
response
dynamics

28. Main Result (Informal)
Informal Theorem: [Roughgarden 09] under “surprisingly general” conditions, a bound on the price of anarchy (for pure Nash) extends automatically to all 5 bigger sets.
Example Application: selfish routing games (nonatomic or atomic) with cost functions in an arbitrary fixed set.

29. The Setup n players, each picks a strategy si
player i incurs a cost Ci(s)
Important Assumption: objective function is cost(s) := i Ci(s)
Next: generic template for upper bounding price of anarchy of pure Nash equilibria.
notation: s = a Nash eq; s* = an optimal

30. An Upper Bound Template
Suppose we have:
cost(s) = i Ci(s) [defn of cost]
≤ i Ci(s*i,s-i) [s a Nash eq]
≤ λ●cost(s*) + μ●cost(s) [(*)]
Then: POA (of pure Nash eq) ≤ λ/(1-μ).
Definition: A game is (λ,μ)-smooth if (*) holds for every pair s,s* outcomes.
not only when s is a pure Nash eq!

31. Main Result #1 Examples: selfish routing, linear cost fns.
every nonatomic game is (1,1/4)-smooth
every atomic game is (5/3,1/3)-smooth
Theorem 1: in a (λ,μ)-smooth game, expected cost of each outcomes in the 5 sets above is at most λ/(1-μ).
such a POA bound “automatically” far more general 31

32. Illustration
So: in every (λ,μ)-smooth game with a sum objective, inefficiency of outcomes in the 5 sets looks like:
worst
pure
Nash
worst
mixed
Nash
worst
correlated
equilibium
worst
no regret
sequence
optimal
outcome 1
λ/(1-μ) 32

33. Main Result #2
Theorem 2 (informal): in sufficiently rich classes of games, smoothness arguments suffice for a tight worst-case bound (even for pure Nash equilibria).
pure
Nash
correlated
equilibium
optimal
outcome
no regret
sequence
mixed
Nash 1
λ/(1-μ)
for tightest
choice of λ,μ 33