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Algorithmic Game Theory September 22, 2008
Lecture 3: Network Design
Lecturer: Sergei Vassilvitskii Scribe:Etienne Vouga
1 Overview
We will examine the network design problem, which features
- A directed graph G = (V, E).
- A number of players. Each player i wants to go from some source vertex si to a sink vertex ti.
- A cost ce for each edge e, for which the edge can be bought. Several players can split the cost of an edge. Once an edge is bought, all players can use it.
A valid network is then any set of purchases that allows every player to travel from his source to his sink by travelling only along bought edges. If only one player wishes to cross a certain edge, naturally that player pays the full cost of the edge. If several players wish to cross the same edge, however, such as the middle edge in Figure (1), we must specify how they are to split the cost. For now we impose a fair cost sharing scheme, where all players who cross an edge split its cost evenly. Later we will look at an alternative scheme.
2 Optimal Solution
Given a graph, sources and sinks, and edge costs as described above, one can ask for a subset T ⊆ E of edges such that
- Every source si is connected (using the edges in T ) to its sink ti, and
- T minimizes the total cost C(T ) =
e∈T ce.
Such a T is called the optimal solution. Finding T is equivalent to the Steiner tree problem, which is known to be N P -complete.
Figure 1: A sample two-player network. The optimal and Nash solutions (green) are iden- tical.
Figure 2: Left: An example network where the optimal solution is a Nash solution (green), but a second Nash solution (blue) exists as well. Right: An example where the Nash solutions are optimal.
3 Selfish Solution
Alternatively, we could let each player individually choose a path pi from si to ti. For each edge e, we then let ke be the number of people using e, so that the total cost charged to the i-th player for his path is Ci(pi) =
e∈pi
ce ke
A solution Tˆ ⊂ E =
i pi^ is then^ selfish^ or^ Nash^ if, for any alternative path ˜pi^ for player^ i, Ci(˜pi) ≥ Ci(pi). As illustrated in Figure (2), the optimal solution may or may not be Nash, and a Nash solution is not necessary unique.
Figure 3: A network for k players where the price of stability is a big improvement over the price of anarchy. The optimal solution (which is also a Nash solution) has cost 1 + e, for any small e > 0. (green). The worst Nash solution has cost k (blue).
Edges of type 1 or 4 have no effect on∑ Ci. For each edge of type 2, player i receives a refund
e∈ 2
ce ke , and he pays a fee^
e∈ 3
ce ke+1 for each edge of type 3. The total change in cost is thus Ci(p′ i) − Ci(pi) =
e∈ 3
ce ke + 1
e∈ 2
ce ke
The change in potential is just the sum of the changes in the individual edge potentials,
Φ(S′) − Φ(S) =
e
φe(S′) − φe(S)
The potentials of edges of type 1 or 4 do not change, so they contribute nothing to the above sum. For the other two types, ∑
e∈ 2
φe(S′) − φe(S)
e∈ 2
(ceHke− 1 − ceHke ) = −
e∈ 2
ce ke ∑
e∈ 3
φe(S′) − φe(S)
e∈ 3
(ceHke+1 − ceHke ) =
e∈ 3
ce ke + 1
the same two terms as in (1), so
Ci(p′ i) − Ci(pi) = Φ(S′) − Φ(S). 2
Corollary 4.2 Let Sn^ be a set of paths that minimize Φ. Then Sn^ is a Nash solution.
Proof. Suppose, for contradiction, that it’s not. Then there is some player i who can deviate from path pi to p′ i to cheapen his cost, Ci(p′ i) − Ci(pi) < 0. But then
Φ(S′) − Φ(Sn) < 0
Φ(S′) < Φ(Sn)
and Sn^ is not a minimum, a contradiction. 2
Now for a network with k players let Sn^ be the minimizer of Φ, S the best Nash solution, and S∗^ the optimum solution. Then
- C(S) ≤ C(Sn) by the corollary.
- C(Sn) ≤ Φ(Sn) since for each edge, ce ≤ ce
1 + 12 +... + (^) k^1 e
= φe.
- Φ(Sn) ≤ Φ(S∗) since Sn^ minimizes Φ.
- Φ(S∗) ≤ C(S∗)Hk since for each edge, φe = ceHke ≤ ceHk.
Chaining together inequalities, C(S) ≤ C(S∗)Hk
Price of Stability =
C(S)
C(S∗)
≤ Hk ≈ log k.
In fact this bound is tight: Figure (4) shows a worst-case network where the optimal solution has cost 1 + for → 0, and the only Nash solution has cost 1 + 12 +... + (^) k^1 = Hk.
Figure 4: A worst-case network for k players. In the Nash solution (blue), the i-th player pays (^1) i , whereas the total cost of the optimal solution (yellow) is just 1 + e, for e → 0.
5 Potential Games
The method of potential functions used here can also be used to analyze other games. The key is the existence of a magical potential Φ that directly relates the overall potential of the system to the benefit to individual players, as described in the theorem. Games with such a Φ are called potential games. As with network design, the minimizer of a potential games’s Φ is a Nash solution. Moreover, suppose there is some lower bound > 0 such that, if player i can deviate from his current strategy to decreases his cost, his cost does so by at least (this condition is easy to show for finite games.) Then we have an algorithm for finding a Nash solution: start with any solution S. If no player can deviate to a cheaper strategy, S is Nash. Otherwise, let a player deviate, and check again. Each deviation decreases Φ by at least , and Φ is bounded below since, at best, every player pays 0 cost, so this algorithm is guaranteed to terminate and find a Nash solution. Unfortunately, does so may take a while. Consider again the specific case of the network design problem, and suppose all edge costs are integral with C = maxe ce be the cost of the most expensive edge (So ce ∈ { 0 , 1 , 2 ,... , C}.). Then we know, for any starting solution S, 0 ≤ Φ(S) ≤ C |E| Hk. Fur- thermore it’s easy to lower bound the change in potential at every step. In the worst case, at number of players sharing one edge will drop from k to k −1, thereby reducing φe by at least 1 k. Therefore after^ O(C|E|k^ log^ k) best responses we must have found a Nash Equilibrium. (Otherwise the potential Φ would have become negative.)
6 Open Research Question
What if G is undirected? When we bounded the price of stability above by Hk, we never used that G was directed; however, the example in Figure (4) used to show tightness of this bound no longer works. What is the new tight bound? For simplicity, you can assume that all players share the same sink t. Even in this setting, no example networks have been found with price of stability greater than 127.
7 Other Cost Shares
In all of the above, we charged all players equally for the edges they shared. We now look at alternative ways to split the cost.
