routing games
TRANSCRIPT
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Introduction2 Routing models
◦Nonatomic selfish routing
Reasons of explain◦Simple but general◦Understandability of POA◦Different techniques needed for
analyze
◦Atomic selfish routing
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Multicommodity Flow Network
Directed graph G = (V, E), may have parallel edgesCommodities:
set of source/sink pairs: (s1, t1),…,(sk , tk )Assign each player to a\some commoditynonnegative, continuous, non-decreasing cost
function ce
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Multicommodity Flow Network (cnt)
Paths
Routes◦ Traffic described with flow f
◦ fP: amount of traffic of commodity i that chooses path to travel from si to ti
◦ Prescribed amount of traffic for commodity i : ri
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Cost DefinitionCost of a path P with respect to a flow
f
fe amount of traffic using paths that contain the edge e.
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Nonatomic EquilibriumA flow (f) is equilibrium if:
no commodity can increase its gain by changing its traffic
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Existence & Uniqueness of Equilibrium
There exists at least one equilibrium
All equilibriums are equivalent (of equal cost)
Using Potential Function
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Potential FunctionsLet ce(x) be the cost of transporting x
over some edge eThe total amount spent on that edge
is x.ce(x)
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Marginal Cost and Optimality(cnt)
How to minimize the total marginal cost?
Create a new network with the marginal cost as cost function, and find an equilibrium.
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Back to Finding EquilibriumRemaining questions?
◦ Can we find equilibrium by finding optimal flows?
◦ What function should we minimize?
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Potential FunctionWe want to minimize the functions
he(x) of all edges. The sum of those is called the potential function and should be minimized.
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ConsequencesThe potential function is convex:
◦ there exists a minimum, and therefore an equilibrium
◦ and all equilibriums form a set◦ with the same cost
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Atomic Selfish RoutingAlmost the same as the nonatomic
instance.Differences:
◦ Each commodity represents ONE player instead of a large number of players.
◦ Player i must route traffic amount ri on a
SINGLE path.Result: Each player must route a
significant amount of traffic instead of a negligible amount.
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Feasibility of a flowA flow f is feasible if:
◦it routes all traffic◦it uses one path per player i to route
ri units
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Atomic equilibrium flow
Also note:Different equilibrium flows can have different
costs
no uniqueness
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Nonexistence of equilibria
Traffic amount: r1 = 1; r2 = 2 P1 s -> t P2 s -> v -> t P3 s -> w -> t P4 s -> v -> w -> t
1- If player 2 takes P1 or P2, player 1 takes P4.
2- If player 1 takes P4, player 2 takes P3. 3- If player 2 takes P3 or P4, player 1
takes P1. 4- If player 1 takes P1, player 2 takes P2. This does never end. . .
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Difference of Atomic and NonDifferent EQ of an atomic
instance can have different costsNonatomic EQ have same costPOA in atomic instance can be
larger than nonatomic
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Overcome nonexistence of equilibrium
To guarantee an equilibrium, additional restrictions are placed on atomic instances. For instance:
Give all players an equal amount of traffic which has to be routed.
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ProofIn atomic instances we can
discretize the potential function that was used for nonatomic instances:
#Possible flows is finite. One flow f is a global minimum of