Congestion Games with Strategic Departure Time Thomas Rivera 1 Marco - - PowerPoint PPT Presentation

Model Equilibria Efficiency Correlation

Congestion Games with Strategic Departure Time

Thomas Rivera1 Marco Scarsini2 Tristan Tomala1

1HEC, Paris 1LUISS, 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 Si “ Z.
• Set of pure strategy profiles S :“ ˆiPNSi.
• Set of mixed strategy profiles Σ :“ ˆiPN∆pSiq.

Model Equilibria Efficiency Correlation

´n ´n ` 1 ´n ` 2 ´2 ´1

Model Equilibria Efficiency Correlation

The model, continued

• Each player needs to arrive at destination by time 0,
• therwise she incurs a penalty cost C, assumed very large.
• ai is the arrival time of player i.
• The cost (negative utility) of player i departing at time ´t is

Riptq “ riptq ` 1pai ą 0q ¨ C where rip¨q is a strictly increasing function.

• For any mixed strategy profile σ P Σ the social cost is

C pσq “ ´ ÿ

iPI

ÿ

siPZ

Ripsiqσipsiq.

• In particular, for a pure strategy profile s P S

C psq “ ´ ÿ

iPI

Ripsiq

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 iff C pσ˚q “ max

σPE C pσq.

• Call B the set of best Nash equilibria, i.e.,

σ˚ P B iff C pσ˚q “ min

σPE C pσq.

• Call O the set of social optima, i.e.,

σ˝ P O iff C pσ˝q “ min

σPΣ C pσq.

Model Equilibria Efficiency Correlation

Nash Equilibrium

• Assume riptq “ t.
• The game has no pure equilibria.

Model Equilibria Efficiency Correlation

Nash Equilibrium, continued

Proposition

For C ą n, if σ˚ P W , then supppσ˚

i q “ t´n, ´pn ´ 1qu for all

i P N.

Proposition

For C ą n2, if σ˚ P B, then, for some j P N, σ˚

j “ ´n and

supppσ˚

i q “ t´pn ´ 1q, ´pn ´ 2qu for all i P Nztju.

Proposition

For C ą n, if σ˝ P O, then one player leaves at each time ´n, p´n ´ 1q, . . . , ´1.

Model Equilibria Efficiency Correlation

Worse equilibrium ε “ p n

C q

1 n´1

´n ´n ` 1 ´n ` 2 ´2 ´1 1 ´ ε ε Best equilibrium η “ pn´1

C q

1 n´2

´n ´n ` 1 ´n ` 2 ´2 ´1 1 pl 1 ´ η η

Model Equilibria Efficiency Correlation

Price of anarchy

Definition

The Price of anarchy is PoA “ C pσ˚q C pσ˝q with σ˚ P W and σ˝ P O.

Corollary

For C ą n PoA “ n2

npn`1q 2

“ 2 ´ 2 n ` 1

Model Equilibria Efficiency Correlation

Price of stability

Definition

The Price of stability is PoS “ C pσ˚q C pσ˝q with σ˚ P B and σ˝ P O.

Corollary

For C ą n PoS “ 2 ` 2 npn ` 1q ´ 4 n ` 1.

Model Equilibria Efficiency Correlation

Correlated equilibria

Definition

A planner draws a profile of strategies according to Q P ∆pSq 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

sPS

Qps|si “ ´kqRipk, s´iq ď ÿ

sPS

Qps|si “ ´kqRipk1, s´iq. for all k1 “ 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 xk the outcome

xk

t “

$ ’ & ’ % k ´ 1 for t “ ´pk ´ 1q, 1 for t P t´n, . . . , ´ku,

• therwise.

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 x2, . . . , xn. This correlated equilibrium has a cost that is close to the

• ptimal cost.

Model Equilibria Efficiency Correlation