1. Equilibria of Atomic Flow Games are not Unique
Umang Bhaskar Dartmouth
Lisa Fleischer Dartmouth
Darrell Hoy Bridgewater Associates
Chien-Chung Huang Max-Planck-Institut

2. Atomic Splittable Flow
Our job: send a certain amount of flow from s to t.
Every edge has a delay function 20 10 30 t s
Total delay
Goal: Make it small

3. Atomic Splittable Flow (cont’d)20 10 s t 30
Here the total delay is:
10 (10 + 1) (top edge) +
20 (0.5×20) (bottom edge)
= 310
We consider the following class of delay functions
Non-negative
Non-decreasing
is convex
For instance…

4. Atomic Splittable Flow (Game Version: More Players)
Here the red player has a total delay:
5 ×( ) (top edge) +
15 × (0.5 ×( )) (bottom edge) = 262.5
Objective of the red player
minimize 15 5 s t 20 10 30 20

5. Motivation I (shipping companies)
Blue Company s t
Red Company

6. Motivation II (ISPs)
Blue ISP
Red ISP

7. Nash Equilibrium A flow pattern where no player can change its flow and reduce its total delay1
Blue player delay 1*(2*1+6) + 4*(4+2) = 32;
Red player delay 2*(4+2) = 12; s t 4 2 s t
1-d
4+d 2
New blue player delay
= (1-d)*(2*(1-d)+6) + (4+d)(4+d+2)
= (8 – 10d + 2d2) + ( d + d2)

8. Nash Equilibrium (cont’d)1 s 4 t 2
Marginal Delay:
Blue player’s marginal delay:
Top edge: (2*1+6) + 1*2 = 10
Bottom edge: (4+2) + 4*1 = 10
Theorem: At equilibrium, each player uses paths with minimum marginal delay.

9. Do equilibria always exist in atomic splittable flow games ?
Yes. They are convex games. Rosen, 1965
For a given instance, is the equilibrium unique?
Yes, if…
(0) all players control infinitesimal amounts of flow Beckman et al., 1956
(1) all players have the same amount of flow, and the same source and destination Orda et al., 1993
(2) delay functions are polynomials of degree at most 3
Altman et al., 2002
(3) the network is a two-terminal nearly-parallel graph Richman and Shimkin, 2007
No? (open question)
However, there are multiple Nash equilibrium flows IF the players see different delay functions on each edge Richman and Shimkin, 2007

10. Our Main Results
A complete characterization of graph topologies that have a unique equilibrium
(1) For two players, there is a unique equilibrium if and only if the network is a generalized series-parallel graph.
(1.1) For two types of players, there is a unique equilibrium if and only if the network is a series-parallel graph. s t 2 6 3
Two players are of the same type if they have the same amount of flow
Red and pink are of the same type (2 units); blue and green are of the same type (6 units)
(2) For more than two types of players, there is a unique equilibrium if and only if the network is a generalized nearly-parallel graph.
Bonus: We derive new characterizations for these two classes of graphs based on properties of circulations

11. Series-Parallel Graphs
A graph is (generalized) series-parallel if it is a single edge , or is constructed from a sequence of operations applied on the single edge
Copy an edge
Split an edge
Add an edge (only for generalized series-parallel graphs)
A graph is generalized series-parallel if and only if it does not contain K4 as a minor.

12. - = - = Our Techniques Circulations and Agreeing Cycles:
What if there are really two Nash equilibrium flows, say f and g? 3 4 8 1 6 = - 1 2 7 5 4 5 6 8 1 5 7 4 1 - 2 = 8 6 3

13. Circulations and Agreeing Cycles (cont’d)
Blue Circulation
Red Circulation 1 7 6 1 + = 7 1 5 2 2 1 1
This is a blue agreeing cycle
And this is NOT a red agreeing cycle 3 2
This is a red agreeing cycle 3
Definition: Let f be the sum of k circulations, f1, f2,…fk. A cycle is an i-agreeing cycle if it is a directed cycle in fi and it runs in the same direction as f on every edge of the cycle.

14. Theorem: For two players,
in a generalized series-parallel graph, there is a unique Nash equilibrium.
If the graph is not generalized series-parallel, there may exist multiple Nash equilibria.
Uniqueness follows from:
Lemma 1: If the difference of flows f and g has an agreeing cycle, then both cannot be equilibrium flows.
Intuition: Suppose there is a blue agreeing cycle

15. Theorem: For two players,
in a generalized series-parallel graph, there is a unique Nash equilibrium.
If the graph is not generalized series-parallel, there may exist multiple Nash equilibria.
Uniqueness follows from:
Lemma 1: If the difference of flows f and g has an agreeing cycle, then both cannot be equilibrium flows.
Lemma 2: A graph is generalized series-parallel iff given any two circulations, there is always an agreeing cycle with their sum.
Proof of Theorem: By Lemma 2 for two players the difference of two flows always has agreeing cycle.
Hence by Lemma 1 both cannot be equilibrium flows.

16. Lemma 2 also gives a novel characterization of Generalized Series-Parallel Graphs:
A graph is generalized series-parallel given any two circulations, there exists an agreeing cycle with their sum.
( ) by induction
( ) we show that if a graph is not generalized series-parallel, it is possible to have two circulations without an agreeing cycle 4 8 5 4 3 1 5
Wrong direction
Wrong direction
Wrong direction

17. Designing the counterexample
Theorem: Suppose there are two players.
In a generalized series-parallel graph, there is a unique Nash equilibrium.
If the graph is not generalized series-parallel, there may exist multiple Nash equilibria. 4 8 5
Designing the counterexample
Flow pattern: we need to avoid agreeing cycles. (Lemma 1)
Delay functions: recall that if the delay functions are polynomials of degree at most 3, there is a unique Nash equilibrium.
“Elbow” function
Flow
Delay
“Convex” function
Marginal Cost:

18. Summary
Our results give a complete characterization of graph topologies that have a unique equilibrium
For two players, equilibrium is unique iff graph is generalized series-parallel
For two types of players equilibrium is unique iff graph is series-parallel
For more than two types of players equilibrium is unique iff graph is generalized nearly-parallel
We introduce the concept of agreeing cycles to show uniqueness results
We give a new characterization of generalized series-parallel and generalized nearly-parallel graphs in terms of agreeing cycles

19. Concluding Remarks
Our uniqueness results hold for the following extensions…
If each player has its own source and destination.
Indeed, even if each player has multiple sources and multiple destinations.
Even if each player has its own delay function on each edge.
And even if each player’s objective is not measured by the “product” of player’s own flow and the delay:
For instance…

20. Questions?

21. Our Main Results
A complete characterization of graph topologies that have a unique equilibrium
(1) For two players, there is a unique equilibrium if and only if the network is a generalized series-parallel graph.
(1.1) For two types of players, there is a unique equilibrium if and only if the network is a series-parallel graph. s t 2 6 3
Red and pink are of the same type (2 units); blue and green are of the same type (6 units)
Two players are of the same type if they have the same amount of flow
(2) For more than two types of players, there is a unique equilibrium if and only if the network is a generalized nearly-parallel graph.

22. Proof of Lemma 1
Lemma 1: If the difference of flows f and g has an agreeing cycle, then both cannot be equilibrium flows.

23. Motivation III
Nonatomic flow + Collusion = atomic splittable flow game Hayrapetyan, Tardos and Wexler, STOC 06
Coalition A
Coalition B s t