Oligopolistic Competitive Packet Routing Games B. Peis V. Timmermans L. Vargas Koch B. Tauer 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 s i v t i directed graph G Finite set of n players, each owning a finite set of k i packets. Player i desires to route her packets from source s i to sink t i . Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17
Oligopolistic Packet Routing Games - Intuition 0 , 2 1 , 1 time = 0 s i v t i directed graph G Finite set of n players, each owning a finite set of k i packets. Player i desires to route her packets from source s i to sink t i . Edges are equipped with integer transit times τ e and capacities u e . All s i - t i 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 0 , 2 1 , 1 time = 0 s i v t i directed graph G Finite set of n players, each owning a finite set of k i packets. Player i desires to route her packets from source s i to sink t i . Edges are equipped with integer transit times τ e and capacities u e . All s i - t i 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 0 , 2 1 , 1 time = 0 s i v t i directed graph G Finite set of n players, each owning a finite set of k i packets. Player i desires to route her packets from source s i to sink t i . Edges are equipped with integer transit times τ e and capacities u e . All s i - t i 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 0 , 2 1 , 1 time = 0 s i v t i directed graph G Finite set of n players, each owning a finite set of k i packets. Player i desires to route her packets from source s i to sink t i . Edges are equipped with integer transit times τ e and capacities u e . All s i - t i 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 0 , 2 1 , 1 time = 0 s i v t i Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 0 Red 2 0 Blue 1 0 Blue 2 0 Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17
Oligopolistic Packet Routing Games - Intuition 0 , 2 1 , 1 time = 0 s i v t i Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 0 Red 2 0 Blue 1 0 Blue 2 0 Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17
Oligopolistic Packet Routing Games - Intuition 0 , 2 1 , 1 time = 1 s i v t i Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 0 1 Red 2 0 Blue 1 0 Blue 2 0 Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17
Oligopolistic Packet Routing Games - Intuition 0 , 2 1 , 1 time = 2 s i v t i Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 0 1 Red 2 0 2 Blue 1 0 Blue 2 0 Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17
Oligopolistic Packet Routing Games - Intuition 0 , 2 1 , 1 time = 3 s i v t i Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 0 1 Red 2 0 2 Blue 1 0 3 Blue 2 0 Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17
Oligopolistic Packet Routing Games - Intuition 0 , 2 1 , 1 time = 4 s i v t i Example: 2 players owning 2 packets each. Priority list: (Red Player > Blue player) Packet strategy arrival time Red 1 0 1 Red 2 0 2 Blue 1 0 3 Blue 2 0 4 Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 2 / 17
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 Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 3 / 17
Release times as part of the strategy 0 , 2 time = 0 s v t 1 , 1 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 3 , 1 0 , 2 time = 0 s v t 1 , 1 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 3 , 1 0 , 2 time = 0 s v t 1 , 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 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 3 , 1 0 , 2 time = 0 s v t 1 , 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 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 3 , 1 0 , 2 time = 1 s v t 1 , 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 3 , 1 0 , 2 time = 2 s v t 1 , 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 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 3 , 1 0 , 2 time = 3 s v t 1 , 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 2 Blue 1 lower,0 3 lower,2 3 Blue 2 lower,0 4 upper,0 3 Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 4 / 17
Our Work We study the computation and efficiency of pure Nash equilibria in oligopolistic 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: Cost in worst NE PoA := Cost of social optimum , Cost in best NE PoS := Cost of social optimum . Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 5 / 17
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. Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 6 / 17
Existence and Computation of Nash equilibria Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 7 / 17
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] Peis et al. (RWTH Aachen) Oligopolistic Packet Routing Games Aussious 2019 8 / 17
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