Model Equilibria Efficiency Correlation Congestion Games with Strategic Departure Time Thomas Rivera 1 Marco Scarsini 2 Tristan Tomala 1 1 HEC, Paris 1 LUISS, Rome Dynamic Traffic Models in Transportation Science Dagstuhl, October 2015
Model Equilibria Efficiency Correlation Congestion games • In a static congestion game players use resources and the cost of using a resource is an increasing function of the number of players that use it. • Time is not a variable in the description of the game. • In a dynamic congestion game time is a relevant parameter but the inflow of players in the system is exogenously given. • We consider a game where the time a player enters the system is strategically chosen.
Model Equilibria Efficiency Correlation The model • Single path from source to destination. • Capacity of the path γ “ 1. • Length of the path β “ 1. • Discrete time. • Atomic game with n players. The set of players is N . • A player chooses a time when to enter the path at the source. • The set of pure strategies of player i is S i “ Z . • Set of pure strategy profiles S : “ ˆ i P N S i . • Set of mixed strategy profiles Σ : “ ˆ i P N ∆ p S i q .
Model Equilibria Efficiency Correlation ´ n ´ n ` 2 ´ 2 ´ 1 0 ´ n ` 1
Model Equilibria Efficiency Correlation The model, continued • Each player needs to arrive at destination by time 0, otherwise she incurs a penalty cost C , assumed very large. • a i is the arrival time of player i . • The cost (negative utility) of player i departing at time ´ t is R i p t q “ r i p t q ` 1 p a i ą 0 q ¨ C where r i p¨q is a strictly increasing function. • For any mixed strategy profile σ P Σ the social cost is ÿ ÿ C p σ q “ ´ R i p s i q σ i p s i q . i P I s i P Z • In particular, for a pure strategy profile s P S ÿ C p s q “ ´ R i p s i q i P I
Model Equilibria Efficiency Correlation Equilibria and optima • Call E the set of Nash equilibria. • Call W the set of worst Nash equilibria, i.e., σ ˚ P W C p σ ˚ q “ max iff σ P E C p σ q . • Call B the set of best Nash equilibria, i.e., σ ˚ P B C p σ ˚ q “ min iff σ P E C p σ q . • Call O the set of social optima, i.e., σ ˝ P O C p σ ˝ q “ min iff σ P Σ C p σ q .
Model Equilibria Efficiency Correlation Nash Equilibrium • Assume r i p t q “ t . • The game has no pure equilibria.
Model Equilibria Efficiency Correlation Nash Equilibrium, continued Proposition For C ą n, if σ ˚ P W , then supp p σ ˚ i q “ t´ n , ´p n ´ 1 qu for all i P N. Proposition For C ą n 2 , if σ ˚ P B , then, for some j P N, σ ˚ j “ ´ n and supp p σ ˚ i q “ t´p n ´ 1 q , ´p n ´ 2 qu for all i P N zt j u . Proposition For C ą n, if σ ˝ P O , then one player leaves at each time ´ n , p´ n ´ 1 q , . . . , ´ 1 .
Model Equilibria Efficiency Correlation Worse equilibrium 1 ε “ p n C q n ´ 1 1 ´ ε ε ´ n ´ n ` 2 ´ 2 ´ 1 0 ´ n ` 1 Best equilibrium 1 η “ p n ´ 1 C q n ´ 2 1 pl 1 ´ η η ´ n ´ n ` 2 ´ 2 ´ 1 0 ´ n ` 1
Model Equilibria Efficiency Correlation Price of anarchy Definition The Price of anarchy is PoA “ C p σ ˚ q C p σ ˝ q with σ ˚ P W and σ ˝ P O . Corollary For C ą n n 2 2 PoA “ “ 2 ´ n p n ` 1 q n ` 1 2
Model Equilibria Efficiency Correlation Price of stability Definition The Price of stability is PoS “ C p σ ˚ q C p σ ˝ q with σ ˚ P B and σ ˝ P O . Corollary For C ą n 2 4 PoS “ 2 ` n p n ` 1 q ´ n ` 1 .
Model Equilibria Efficiency Correlation Correlated equilibria Definition A planner draws a profile of strategies according to Q P ∆ p S q and makes the drawn recommendation to the players. The distribution Q is a correlated equilibrium if whenever player i is told to depart at time ´ k , then this is optimal for her to accept the recommendation, i.e., for all i P N ÿ ÿ Q p s | s i “ ´ k q R i p k 1 , s ´ i q . Q p s | s i “ ´ k q R i p k , s ´ i q ď s P S s P S for all k 1 “ 1 , . . . , n .
Model Equilibria Efficiency Correlation • The outcome x induced by the profile s is the vector that indicates the number of departures at any given time. • Call X the set of outcomes and Y the set of outcomes where no player is late. • Any distribution on S induces a distribution on X . • Call x k the outcome $ ’ k ´ 1 for t “ ´p k ´ 1 q , & x k t “ 1 for t P t´ n , . . . , ´ k u , ’ % 0 otherwise.
Model Equilibria Efficiency Correlation Efficiency and correlation Proposition There exists C such that for all C ą C we can construct the best correlated equilibrium Q. This equilibrium induces a distribution r Q on X that is supported on x 2 , . . . , x n . This correlated equilibrium has a cost that is close to the optimal cost.
Model Equilibria Efficiency Correlation
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