Equilibria of Atomic Flow Games are not Unique

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Equilibria of Atomic Flow Games are not Unique

• Umang Bhaskar Dartmouth
• Lisa Fleischer Dartmouth
• Darrell Hoy Bridgewater Associates
• Chien-Chung Huang Max-Planck-Institut

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Atomic Splittable Flow
Our job send a certain amount of flow from s to
t.
30
t
s
3
Atomic Splittable Flow (contd)
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Here the total delay is 10 (10 1) (top edge)
20 (0.520) (bottom edge) 310
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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
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Motivation I (shipping companies)
Blue Company
s
t
Red Company
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Motivation II (ISPs)
Blue ISP
Red ISP
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Nash Equilibrium A flow pattern where no player
can change its flow and reduce its total delay
1
Blue player delay 1(216) 4(42) 32 Red
player delay 2(42) 12
s
t
4
2
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Nash Equilibrium (contd)
1
s
t
4
2
Blue players marginal delay
Top edge (216) 12 10
Bottom edge (42) 41 10
Theorem At equilibrium, each player uses paths
with minimum marginal delay.
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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
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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.
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
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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.

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Our Techniques
Circulations and Agreeing Cycles What if there
are really two Nash equilibrium flows, say f and
g?
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Circulations and Agreeing Cycles (contd)
Blue Circulation
Red Circulation
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1
7
1


7
2
1
5
2
1
1
And this is NOT a red agreeing cycle
3
2
This is a red agreeing cycle
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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.
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• 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.
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• 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.
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Lemma 2 also gives a novel characterization of
Generalized Series-Parallel Graphs
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• 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
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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

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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 players objective is not
  measured by the product of players own flow
  and the delay

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Questions?
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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.
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Proof of Lemma 1
Lemma 1 If the difference of flows f and g has
an agreeing cycle, then both cannot be
equilibrium flows.
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Motivation III
Nonatomic flow Collusion atomic splittable
flow game Hayrapetyan, Tardos and Wexler, STOC 06
s
t