[Solved]Consider Following Two Player Congestion Game Ai R R2 Player Select Either Resource R1 R2 Q37245032

(a) Consider the following two player congestion game: ·Ai {rı, r2): (a player can select either resource r1 or r2 but not both) Write down the payoff matrix for this two player routing game. Prove that a pure Nash equilibrium must exist irrespective of the congestion functions for route rı and r2 (b) Consider any arbitrary routing game with with n players: r1, …, Tm N-1.,n Consider the following two potential functions rER Let a and a, = (a, a-i) be any two routing profiles that differ by a unilateral deviation. Prove or disprove the following statement: (c) Prove or disprove the following statement: (d) Prove that a pure Nash equilibrium must exist in any congestion game. (e) (Bonus) Derive an anonymous tolling scheme, i.e., t 0, 1, 2,…^ ->R for each resource r E R such that the resulting game where player’s cost functions are now rEai is a potential game with potential function φιώ Show transcribed image text (a) Consider the following two player congestion game: ·Ai {rı, r2): (a player can select either resource r1 or r2 but not both) Write down the payoff matrix for this two player routing game. Prove that a pure Nash equilibrium must exist irrespective of the congestion functions for route rı and r2 (b) Consider any arbitrary routing game with with n players: r1, …, Tm N-1.,n Consider the following two potential functions rER Let a and a, = (a, a-i) be any two routing profiles that differ by a unilateral deviation. Prove or disprove the following statement: (c) Prove or disprove the following statement: (d) Prove that a pure Nash equilibrium must exist in any congestion game. (e) (Bonus) Derive an anonymous tolling scheme, i.e., t 0, 1, 2,…^ ->R for each resource r E R such that the resulting game where player’s cost functions are now rEai is a potential game with potential function φιώ

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Answer to (a) Consider the following two player congestion game: ·Ai {rı, r2): (a player can select either resource r1 or r2 but… . . .