# We denition system wardrop equilibrium the ow proles x

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Unformatted text preview: tisﬁes the L.P. W.E. conditions and α(1) ≥ β (1) ¯ H where α(1) = Tmin + C ¯ ∗ = 0, x ∗H satisﬁes the H.P. W.E. conditions and α(0) ≤ β (0) ¯ ¯ p ¯ ¯(0) = α(0) + C L where β Vijay Kamble (IIT Kharagpur, INDIA) Hierarchical Routing Games MURI meeting, Sept 9 2010 12 / 21 System Wardrop Equilibrium S.W.E. Deﬁnition (System Wardrop equilibrium) ¯ ¯ The ﬂow proﬁles x ∗H , x ∗L and p∗ constitute a System Wardrop equilibrium if either of the three conditions are satisﬁed: ¯ ¯ p∗ ∈ (0, 1), x ∗H and x ∗L satisfy the multilevel W.E. conditions and ¯ α (p ∗ ) = β (p ∗ ) ¯ ¯ ¯ p∗ = 1, x ∗L satisﬁes the L.P. W.E. conditions and α(1) ≥ β (1) ¯ H where α(1) = Tmin + C ¯ ∗ = 0, x ∗H satisﬁes the H.P. W.E. conditions and α(0) ≤ β (0) ¯ ¯ p ¯ ¯(0) = α(0) + C L where β Related to N.E : At the S.W.E., any packet has no incentive to change its route or its priority level Vijay Kamble (IIT Kharagpur, INDIA) Hierarchical Routing Games MURI meeting, Sept 9 2010 12 / 21 System Wardrop Equilibrium S.W.E. Deﬁnition (System Wardrop equilibrium) ¯ ¯ The ﬂow proﬁles x ∗H , x ∗L and p∗ constitute a System Wardrop equilibrium if either of the three conditions are satisﬁed: ¯ ¯ p∗ ∈ (0, 1), x ∗H and x ∗L satisfy the multilevel W.E. conditions and ¯ α (p ∗ ) = β (p ∗ ) ¯ ¯ ¯ p∗ = 1, x ∗L satisﬁes the L.P. W.E. conditions and α(1) ≥ β (1) ¯ H where α(1) = Tmin + C ¯ ∗ = 0, x ∗H satisﬁes the H.P. W.E. conditions and α(0) ≤ β (0) ¯ ¯ p ¯ ¯(0) = α(0) + C L where β Related to N.E : At the S.W.E., any packet has no incentive to change its route or its priority level Not a potential game!! (Externality symmetry condition not satisﬁed) Vijay Kamble (IIT Kharagpur, INDIA) Hierarchical Routing Games MURI meeting, Sept 9 2010 12 / 21 Existence, Uniqueness Outline 1 Overview System Wardr...
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