Oligopolistic Competitive Packet Routing Games B. Peis V. - - PowerPoint PPT Presentation

Oligopolistic Competitive Packet Routing Games

• B. Peis
• B. Tauer
• V. Timmermans
• L. Vargas Koch

RWTH Aachen

Aussious 2019

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 1 / 17

Oligopolistic Packet Routing Games - Intuition

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17

Oligopolistic Packet Routing Games - Intuition

si v ti directed graph G Finite set of n players, each owning a finite set of ki packets. Player i desires to route her packets from source si to sink ti.

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Oligopolistic Packet Routing Games - Intuition

si v ti time = 0 0, 2 1, 1 directed graph G Finite set of n players, each owning a finite set of ki packets. Player i desires to route her packets from source si to sink ti. Edges are equipped with integer transit times τe and capacities ue. All si-ti paths have a length at least 1.

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17

Oligopolistic Packet Routing Games - Intuition

si v ti time = 0 0, 2 1, 1 directed graph G Finite set of n players, each owning a finite set of ki packets. Player i desires to route her packets from source si to sink ti. Edges are equipped with integer transit times τe and capacities ue. All si-ti paths have a length at least 1. Forwarding policy is a priority list on the players. (1 > 2 > ... > n).

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17

Oligopolistic Packet Routing Games - Intuition

si v ti time = 0 0, 2 1, 1 directed graph G Finite set of n players, each owning a finite set of ki packets. Player i desires to route her packets from source si to sink ti. Edges are equipped with integer transit times τe and capacities ue. All si-ti paths have a length at least 1. Forwarding policy is a priority list on the players. (1 > 2 > ... > n). Each player chooses a path and a start time for each of her packets.

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17

Oligopolistic Packet Routing Games - Intuition

si v ti time = 0 0, 2 1, 1 directed graph G Finite set of n players, each owning a finite set of ki packets. Player i desires to route her packets from source si to sink ti. Edges are equipped with integer transit times τe and capacities ue. All si-ti paths have a length at least 1. Forwarding policy is a priority list on the players. (1 > 2 > ... > n). Each player chooses a path and a start time for each of her packets. Goal: minimize the sum of arrival times.

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17

Oligopolistic Packet Routing Games - Intuition

si v ti time = 0 0, 2 1, 1 Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 Red 2 Blue 1 Blue 2

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17

Oligopolistic Packet Routing Games - Intuition

si v ti time = 0 0, 2 1, 1 Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 Red 2 Blue 1 Blue 2

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17

Oligopolistic Packet Routing Games - Intuition

si v ti time = 1 0, 2 1, 1 Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 1 Red 2 Blue 1 Blue 2

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Oligopolistic Packet Routing Games - Intuition

si v ti time = 2 0, 2 1, 1 Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 1 Red 2 2 Blue 1 Blue 2

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Oligopolistic Packet Routing Games - Intuition

si v ti time = 3 0, 2 1, 1 Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 1 Red 2 2 Blue 1 3 Blue 2

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Oligopolistic Packet Routing Games - Intuition

si v ti time = 4 0, 2 1, 1 Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 1 Red 2 2 Blue 1 3 Blue 2 4

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Motivation: Self driving vehicles - Road Priorities

Different priority classes: higher paying customers, or special vehicles, have priority over other vehicles Vehicles that have equal priority are controlled by the same operator, that aims to minimize average travel time for all cars in that class. Choosing your start time may affect congestion

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Release times as part of the strategy

s v t 0, 2 1, 1 time = 0 Example: 2 players owning 2 packets each. No start times Packet strategy arrival time Red 1 lower,0 1 Red 2 lower,0 2 Blue 1 lower,0 3 Blue 2 lower,0 4

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Release times as part of the strategy

s v t 0, 2 3, 1 1, 1 time = 0 Example: 2 players owning 2 packets each. No start times Packet strategy arrival time Red 1 lower,0 1 Red 2 lower,0 2 Blue 1 lower,0 3 Blue 2 lower,0 4

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 4 / 17

Release times as part of the strategy

s v t 0, 2 3, 1 1, 1 time = 0 Example: 2 players owning 2 packets each. No start times Start times Packet strategy arrival time strategy arrival time Red 1 lower,0 1 lower,0 Red 2 lower,0 2 lower,1 Blue 1 lower,0 3 lower,2 Blue 2 lower,0 4 upper,0

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 4 / 17

Release times as part of the strategy

s v t 0, 2 3, 1 1, 1 time = 0 Example: 2 players owning 2 packets each. No start times Start times Packet strategy arrival time strategy arrival time Red 1 lower,0 1 lower,0 Red 2 lower,0 2 lower,1 Blue 1 lower,0 3 lower,2 Blue 2 lower,0 4 upper,0

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 4 / 17

Release times as part of the strategy

s v t 0, 2 3, 1 1, 1 time = 1 Example: 2 players owning 2 packets each. No start times Start times Packet strategy arrival time strategy arrival time Red 1 lower,0 1 lower,0 1 Red 2 lower,0 2 lower,1 Blue 1 lower,0 3 lower,2 Blue 2 lower,0 4 upper,0

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 4 / 17

Release times as part of the strategy

s v t 0, 2 3, 1 1, 1 time = 2 Example: 2 players owning 2 packets each. No start times Start times Packet strategy arrival time strategy arrival time Red 1 lower,0 1 lower,0 1 Red 2 lower,0 2 lower,1 2 Blue 1 lower,0 3 lower,2 Blue 2 lower,0 4 upper,0

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 4 / 17

Release times as part of the strategy

s v t 0, 2 3, 1 1, 1 time = 3 Example: 2 players owning 2 packets each. No start times Start times Packet strategy arrival time strategy arrival time Red 1 lower,0 1 lower,0 1 Red 2 lower,0 2 lower,1 2 Blue 1 lower,0 3 lower,2 3 Blue 2 lower,0 4 upper,0 3

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Our Work

We study the computation and efficiency of pure Nash equilibria in

• ligopolistic competitive packet routing games.

In a Nash equilibrium, no player can unilaterally deviate to decrease her

• cost. A social optimum minimizes the total cost.

We measure the efficiency of equilibria: PoA := Cost in worst NE Cost of social optimum, PoS := Cost in best NE Cost of social optimum.

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Our Results

In multi commodity games, Nash equilibria exist and can be computed within pseudo-polynomial time. In single commodity games, we can compute a social optimum. Furthermore: PoS = 1 and PoA = n. In the special case of equal demands PoA = 1

2(n + 1).

In single source games, PoS ≥ 2, and given a number of players and their packets, we can create a game that maximizes the PoA.

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Existence and Computation

• f Nash equilibria

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Earliest Arrival Flows

Definition

An integer flow over time has the earliest arrival property if it maximizes the total outflow at any point in time.

Lemma

One can compute an earliest arrival flow for single commodity games with varying capacities. [A. Tjandra, 2003]

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Computation of a Nash Equilibrium

Lemma

Given strategies of players 1, . . . , i − 1, an earliest arrival flow is a best response for player i.

Theorem

For multi commodity oligopolistic competitive packet routing games, one can compute a Nash equilibrium by sequentially computing an earliest arrival flow for each player.

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Price of Anarchy in Symmetric Games

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Upperbound PoA - Equal demands

Theorem

In a symmetric oligopolistic competitive packet routing game, PoA ≤ 1

2(n + 1).

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 11 / 17

Upperbound PoA - Equal demands

Theorem

In a symmetric oligopolistic competitive packet routing game, PoA ≤ 1

2(n + 1).

Proof: Assume we know the optimal arrival pattern of the first player. Example: Arrival pattern example: 3,4,4,5

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 11 / 17

Upperbound PoA - Equal demands

Theorem

In a symmetric oligopolistic competitive packet routing game, PoA ≤ 1

2(n + 1).

Proof: Assume we know the optimal arrival pattern of the first player. Example: Arrival pattern example: 3,4,4,5 We call q the arrival spread. (q = 3)

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 11 / 17

Upperbound PoA - Equal demands

Theorem

In a symmetric oligopolistic competitive packet routing game, PoA ≤ 1

2(n + 1).

Proof: Assume we know the optimal arrival pattern of the first player. Example: Arrival pattern example: 3,4,4,5 We call q the arrival spread. (q = 3) In any NE, player i can use the same strategy as player 1, where all start times are increased by (i − 1)q. 3,4,4,5 → 6,7,7,8 → 9,10,10,11 → . . .

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 11 / 17

Upperbound PoA - Equal demands

Theorem

In a symmetric oligopolistic competitive packet routing game, PoA ≤ 1

2(n + 1).

Proof: Assume we know the optimal arrival pattern of the first player. Example: Arrival pattern example: 3,4,4,5 We call q the arrival spread. (q = 3) In any NE, player i can use the same strategy as player 1, where all start times are increased by (i − 1)q. 3,4,4,5 → 6,7,7,8 → 9,10,10,11 → . . .

Lemma

In any NE, player i can use the same strategy as player 1, where all start times are increased by (i − 1)q, without being delayed by any other player.

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 11 / 17

Let (ap) denote the arrival pattern of player 1, then we know that: C(NE) ≤

n

• i=1

q

• p=1

ap(S + p − 1 + (i − 1)q).

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 12 / 17

Let (ap) denote the arrival pattern of player 1, then we know that: C(NE) ≤

n

• i=1

q

• p=1

ap(S + p − 1 + (i − 1)q). Continue example: Player 1 Player 2 Player 3 . . . NE 3 4 4 5 6 7 7 8 9 10 10 11 . . .

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 12 / 17

Let (ap) denote the arrival pattern of player 1, then we know that: C(NE) ≤

n

• i=1

q

• p=1

ap(S + p − 1 + (i − 1)q). Continue example: Player 1 Player 2 Player 3 . . . NE 3 4 4 5 6 7 7 8 9 10 10 11 . . . OPT 3 4 4 5 5 5 5 5 5 5 5 5 . . .

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 12 / 17

Let (ap) denote the arrival pattern of player 1, then we know that: C(NE) ≤

n

• i=1

q

• p=1

ap(S + p − 1 + (i − 1)q). Continue example: Player 1 Player 2 Player 3 . . . NE 3 4 4 5 6 7 7 8 9 10 10 11 . . . OPT 3 4 4 5 5 5 5 5 5 5 5 5 . . . Optimizing over S, q and the arrival pattern (ap)1≤p≤q results in q = 1, S = 1 and PoA ≤ 1

2(n + 1).

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 12 / 17

Let (ap) denote the arrival pattern of player 1, then we know that: C(NE) ≤

n

• i=1

q

• p=1

ap(S + p − 1 + (i − 1)q). Continue example: Player 1 Player 2 Player 3 . . . NE 3 4 4 5 6 7 7 8 9 10 10 11 . . . OPT 3 4 4 5 5 5 5 5 5 5 5 5 . . . Optimizing over S, q and the arrival pattern (ap)1≤p≤q results in q = 1, S = 1 and PoA ≤ 1

2(n + 1).

Player 1 Player 2 Player 3 . . . NE 1 1 1 1 2 2 2 2 3 3 3 3 . . . OPT 1 1 1 1 1 1 1 1 1 1 1 1 . . .

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Example with maximal PoA

Assume there are n players with k packets each. ue = k for all e ∈ E s t . . . . . . . . . 1 1 1 1 1

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Example with maximal PoA

Assume there are n players with k packets each. ue = k for all e ∈ E s t . . . . . . . . . C(OPT) = kn n disjoint parallel paths 1 1 1 1 1

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Example with maximal PoA

Assume there are n players with k packets each. ue = k for all e ∈ E s t . . . . . . . . . C(OPT) = kn n disjoint parallel paths C(NE) = 1

2kn(n + 1)

1 zig-zag paths 1 1 1 1 1

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Price of Stability in Single Source Games

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PoS > 1

s t1 v t2 2 1 1 Two players, k1 = 1 and k2 = 2 and ue = 1

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PoS > 1

s t1 v t2 2 1 1 2 1 1 Two players, k1 = 1 and k2 = 2 and ue = 1 C(NE) = 1 + 2 + 3 = 6

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PoS > 1

s t1 v t2 2 1 1 2 1 1 Two players, k1 = 1 and k2 = 2 and ue = 1 C(NE) = 1 + 2 + 3 = 6 > 5 = 2 + 1 + 2 = C(OPT)

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Theorem

In single source oligopolistic competitive packet routing games, PoS ≥ 2. Assume n players. k1 = · · · = kn−1 = 1, kn = n. s t1 t2 tn−2 tn−1 tn . . . 1 2 3 n − 1 n 1 1 1 1

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Theorem

In single source oligopolistic competitive packet routing games, PoS ≥ 2. Assume n players. k1 = · · · = kn−1 = 1, kn = n. s t1 t2 tn−2 tn−1 tn . . . 1 2 3 n − 1 n 1 1 1 1

• P. 1
• P. 2

. . .

• P. n − 1
• P. n

NE 1 2 . . . n − 1 n . . . 2n − 1 OPT 2 3 . . . n 1 . . . n

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 16 / 17

Overview and Open Problems

Overview In multi commodity games, Nash equilibria exist and can be computed within pseudo-polynomial time. In single commodity games, we can compute a social optimum. Furthermore: PoS = 1 and PoA = n. In the special case of equal demands PoA = 1

2(n + 1).

In single source games, PoS ≥ 2, and given a number of players and their packets, we can create a game that maximizes the PoA. Open Problems How efficient are equilibria in multi commodity games? How can we adapt the model to get constant bounds for PoA?

Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 17 / 17

Overview and Open Problems

Overview In multi commodity games, Nash equilibria exist and can be computed within pseudo-polynomial time. In single commodity games, we can compute a social optimum. Furthermore: PoS = 1 and PoA = n. In the special case of equal demands PoA = 1

2(n + 1).

In single source games, PoS ≥ 2, and given a number of players and their packets, we can create a game that maximizes the PoA. Open Problems How efficient are equilibria in multi commodity games? How can we adapt the model to get constant bounds for PoA? Thank you!

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