Congestion Games Karousatou Christina Algor. Game Theory June 2, 2011 Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 1 / 25
Congestion Games with Player-Specific Payoff Functions 1 Congestion Games with Player-Specific Payoff Functions The model The Existence of a Pure-Strategy Nash Equilibrium 2 Congestion Games with Player-Specific Constants Congestion Games on Parallel Links Arbitrary Congestion Games Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 2 / 25
Congestion Games with Player-Specific Payoff Functions The model 1 Congestion Games with Player-Specific Payoff Functions The model The Existence of a Pure-Strategy Nash Equilibrium 2 Congestion Games with Player-Specific Constants Congestion Games on Parallel Links Arbitrary Congestion Games Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 3 / 25
Congestion Games with Player-Specific Payoff Functions The model (Unweighted) Congestion Games The n players share a common set of r strategies. The payoff the i th player receives for playing the j th strategy S ij is a monotonically nonincreasing function of the total number of players playing the j th strategy. We denote the strategy played by the i th player by σ i . Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 4 / 25
Congestion Games with Player-Specific Payoff Functions The model The strategy-tuple σ = ( σ , σ , . . . , σ n ) is a Nash equilibrium iff each σ i is a best-reply strategy: S iσ i ( n σ i ) ≥ S ij ( n j + 1) for all i and j . Here n j = # { 1 ≤ i ≤ n | σ i = j } . Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 5 / 25
Congestion Games with Player-Specific Payoff Functions The model The strategy-tuple σ = ( σ , σ , . . . , σ n ) is a Nash equilibrium iff each σ i is a best-reply strategy: S iσ i ( n σ i ) ≥ S ij ( n j + 1) for all i and j . Here n j = # { 1 ≤ i ≤ n | σ i = j } . Theorem Congestion games involving only two strategies possess the Finite Improvement Property. Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 5 / 25
Congestion Games with Player-Specific Payoff Functions The Existence of a Pure-Strategy Nash Equilibrium 1 Congestion Games with Player-Specific Payoff Functions The model The Existence of a Pure-Strategy Nash Equilibrium 2 Congestion Games with Player-Specific Constants Congestion Games on Parallel Links Arbitrary Congestion Games Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 6 / 25
Congestion Games with Player-Specific Payoff Functions The Existence of a Pure-Strategy Nash Equilibrium Theorem Every (unweighted) congestion game possesses a Nash equilibrium in pure strategies. Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 7 / 25
Congestion Games with Player-Specific Payoff Functions The Existence of a Pure-Strategy Nash Equilibrium Theorem Every (unweighted) congestion game possesses a Nash equilibrium in pure strategies. Lemma (a) If j (0) , j (1) , . . . , j ( M ) is a sequence of strategies, σ (0) , σ (1) , . . . , σ ( M ) is a best-reply improvement path, and σ ( k ) results from the deviation of one player from j ( k − 1) to j ( k ) ( k = 1 , 2 , . . . , M ), then M ≤ n . (b) Similarly, if the deviation in the k th step is from j ( k ) to j ( k − 1) ( k = 1 , 2 , . . . , M ), then M ≤ n · ( r − 1). Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 7 / 25
Congestion Games with Player-Specific Payoff Functions The Existence of a Pure-Strategy Nash Equilibrium Proof of Theorem By induction on the number n of players. n = 1 trivial. Assume that the theorem holds for all ( n − 1)-player congestion games. We prove it for n -player games. We reduce an n -player congestion game Γ into an ( n − 1)-player game ¯ Γ by ”deleting” the last player. Γ is also a congestion game. The payoff functions ¯ ¯ S ij are defined by ¯ S ij (¯ n j ) = S ij (¯ n j ) for 1 ≤ i ≤ n − 1 and all j , ¯ n j = # { 1 ≤ i ≤ n − 1 | σ i = j } . Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 8 / 25
Congestion Games with Player-Specific Payoff Functions The Existence of a Pure-Strategy Nash Equilibrium Proof contd. By induction hypothesis, there exists a pure-strategy Nash σ = ( σ 1 (0) , σ 2 (0) , . . . , σ n − 1 (0)) for ¯ equilibrium ¯ Γ . Let σ n (0) be a best reply of player n against ¯ σ . Starting with j (0) = σ n (0), we can find a sequence j (0) , j (1) , . . . , j ( M ) of strategies and a best-reply improvement path σ (0) , σ (1) , . . . , σ ( M ), as in part (a) of the lemma, such that M is maximal. Claim: σ ( M ) = ( σ 1 ( M ) , σ 2 ( M ) , . . . , σ n ( M )) is an equilibrium. Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 9 / 25
Congestion Games with Player-Specific Payoff Functions The Existence of a Pure-Strategy Nash Equilibrium Proof contd. Case σ i (0) � = σ i ( M ). Strategy σ i ( M ) is a best-reply against σ ( M ), by the proof of the lemma. Case σ i (0) = σ i ( M ). If σ i ( M ) = j ( M ), then j ( M ) is a best reply against σ ( M ), otherwise there is contradiction to the maximality of M . If σ i ( M ) � = j ( M ), then the number of players playing σ i ( M ) = σ i (0) is the same in σ ( M ) and ¯ σ . Note that S iσ i (0) (¯ n σ i (0) ) ≥ S ij (¯ n j + 1) for all i and j . Also, n j ( M ) ≥ ¯ n j for all j . Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 10 / 25
Congestion Games with Player-Specific Payoff Functions The Existence of a Pure-Strategy Nash Equilibrium Proof contd. Case σ i (0) � = σ i ( M ). Strategy σ i ( M ) is a best-reply against σ ( M ), by the proof of the lemma. Case σ i (0) = σ i ( M ). If σ i ( M ) = j ( M ), then j ( M ) is a best reply against σ ( M ), otherwise there is contradiction to the maximality of M . If σ i ( M ) � = j ( M ), then the number of players playing σ i ( M ) = σ i (0) is the same in σ ( M ) and ¯ σ . Note that S iσ i (0) (¯ n σ i (0) ) ≥ S ij (¯ n j + 1) for all i and j . Also, n j ( M ) ≥ ¯ n j for all j . We conclude that S iσ i ( M ) ( n σ i ( M ) ( M )) ≥ S ij ( n j ( M ) + 1) for all j , and thus σ i ( M ) is a best reply for i against σ ( M ). � Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 10 / 25
Congestion Games with Player-Specific Payoff Functions The Existence of a Pure-Strategy Nash Equilibrium As a result of the proof of the theorem and the second part of the previous lemma we get the next theorem. Theorem Given an arbitrary strategy tuple σ (0) in a congestion game Γ , there exists a best-reply improvement path σ (0) , σ (1) , . . . , σ ( L ) such that � n +1 � σ ( L ) is an equilibrium and L ≤ r · . 2 Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 11 / 25
Congestion Games with Player-Specific Constants 1 Congestion Games with Player-Specific Payoff Functions The model The Existence of a Pure-Strategy Nash Equilibrium 2 Congestion Games with Player-Specific Constants Congestion Games on Parallel Links Arbitrary Congestion Games Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 12 / 25
Congestion Games with Player-Specific Constants Some Definitions A weighted congestion game with player specific constants is a weighted congestion game Γ = ( n, E, ( w i ) i ∈ [ n ] , ( S i ) i ∈ [ n ] , ( f ie ) i ∈ [ n ] ,e ∈ E ) with player-specific latency functions such that ( i ) for each resource e ∈ E , there is a non-decreasing delay function g e : R > 0 → R > 0 , and ( ii ) for each player i ∈ [ n ] and a resource e ∈ E , there is a player-specific constant c ie > 0, so that for each player i ∈ [ n ] and a resource e ∈ E , f ie = c ie · g e . Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 13 / 25
Congestion Games with Player-Specific Constants A profile is a tuple s = ( s 1 , . . . , s n ) ∈ S 1 × . . . × S n . The load δ e ( s ) for the profile s, on resource e ∈ E is given by δ e ( s ) = � i ∈ [ n ] | s i ∋ e w i . The Individual Cost of a player i ∈ [ n ], for the profile s, is given by IC i ( s ) = � e ∈ s i f ie ( δ e ( s )) = � e ∈ s i c ie · g e ( δ e ( s )). ։ In the unweighted case, w i = 1 for all players i ∈ [ n ]. Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 14 / 25
Congestion Games with Player-Specific Constants Congestion Games on Parallel Links 1 Congestion Games with Player-Specific Payoff Functions The model The Existence of a Pure-Strategy Nash Equilibrium 2 Congestion Games with Player-Specific Constants Congestion Games on Parallel Links Arbitrary Congestion Games Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 15 / 25
Congestion Games with Player-Specific Constants Congestion Games on Parallel Links Theorem Every unweighted congestion game with player-specific constants on parallel links has an ordinal potential. Karousatou Christina (Algor. Game Theory) Congestion Games June 2, 2011 16 / 25
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