Equilibria of Atomic Flow Games are not Unique

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

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

4. 5 10 s t 20 15 20 Atomic Splittable Flow (Game Version: More Players) Here the red player has a total delay: 5 ×(10 + 5 + 1) (top edge) + 15 × (0.5 ×(20 + 15)) (bottom edge) = 262.5 Objective of the red player minimize 30

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 delay 1-d New blue player delay = (1-d)*(2*(1-d)+6) + (4+d)(4+d+2) = (8 – 10d + 2d2) + (24 + 10d + d2) s t 4+d 2 1 Blue player delay 1*(2*1+6) + 4*(4+2) = 32; Red player delay 2*(4+2) = 12; s t 4 2

8. Nash Equilibrium (cont’d) Marginal Delay: 1 s 4 t 2 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 differentdelay functions on each edge Richman and Shimkin, 2007

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

13. Circulations and Agreeing Cycles (cont’d) This is a blueagreeing cycle Blue Circulation Red Circulation 1 7 6 1 + = 7 1 5 2 2 1 1 And this is NOT a red agreeing cycle 3 2 This is a redagreeing 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. Intuition: Suppose there is a blue agreeing cycle • 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.

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. 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 1 4 Wrong direction 3 5 4 1 5 4 8 4 5 5 5 4 4 Wrong direction Wrong direction Lemma 2 also gives a novel characterization of Generalized Series-Parallel Graphs:

17. 4 5 4 8 5 “Convex” function 5 5 4 4 Delay “Elbow” function Flow • 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. • 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. 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 For instance… 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:

20. Questions?

21. 2 3 3 s 6 t 2 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. 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 Coalition A Coalition B Nonatomic flow + Collusion = atomic splittable flow game Hayrapetyan, Tardos and Wexler, STOC 06 s t