Dynamic Atomic Congestion Games with Seasonal Flows
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1 Dynamic Atomic Congestion Games with Seasonal Flows Marc Schröder Marco Scarsini, Tristan Tomala Maastricht University Department of Quantitative Economics Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 1 / 9
2 Model Dynamic congestion games Most models of congestion games are static. The static game represents the steady state of a dynamic model with constant flow over time. Even if the flow of travellers is constant, how is the steady state reached? In real life traffic flows are rarely constant, although often (nearly) periodic. How does this affect the steady state? Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 2 / 9
3 Model Edge dynamics Each edge had a travel time and a capacity. For example, τ e = 2 and γ e = 2. v w Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games / 9
4 Model Edge dynamics Each edge had a travel time and a capacity. For example, τ e = 2 and γ e = 2. t = 0 v 1,2, w Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games / 9
5 Model Edge dynamics Each edge had a travel time and a capacity. For example, τ e = 2 and γ e = 2. t = 1 v 1,2, w Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games / 9
6 Model Edge dynamics Each edge had a travel time and a capacity. For example, τ e = 2 and γ e = 2. t = 2 v w Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games / 9
7 Model Edge dynamics Each edge had a travel time and a capacity. For example, τ e = 2 and γ e = 2. t = v w Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games / 9
8 Model Edge dynamics Each edge had a travel time and a capacity. For example, τ e = 2 and γ e = t = 0 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 4 / 9
9 Model Edge dynamics Each edge had a travel time and a capacity. For example, τ e = 2 and γ e = t = t = 1 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 4 / 9
10 Model Edge dynamics Each edge had a travel time and a capacity. For example, τ e = 2 and γ e = t = t = 1 t = 2 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 4 / 9
11 Model Edge dynamics Each edge had a travel time and a capacity. For example, τ e = 2 and γ e = t = t = 1 t = 2 t = Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 4 / 9
12 Model Related literature Continuous time and flows Koch and Skutella (2011) provide a characterization of Nash flows over time via a sequence of thin flows with resetting. Cominetti, Correa and Larré (2011) prove existence and uniqueness of Nash flows over time. Macko, Larson and Steskal (201) analyse Braess s paradox for flows over time. Discrete time and flows Werth, Holzhauser and Krumke (2014). Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 5 / 9
13 Model Model A directed network N = (V, E, (τ e ) e E, (γ e ) e E ) with a single source and sink, where - τ e N is the travel time, - γ e N is the capacity. Time is discrete and players are atomic. Inflow is deterministic, but is allowed to be periodic. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 6 / 9
14 Model Model At each stage t, a generation G t of δ t players departs from the source. Players are ordered by priority. At time t, player [it] observes the choices of players [js] [it] and chooses an edge e = (s, v) E. Player [it] arrives at time t + τ e at the exit of e. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 7 / 9
15 Model Model At this exit a queue might have formed by 1 players who entered e before [it], 2 players who entered e at the same time as [it], but have higher priority. Recall at most γ e players can exit e simultaneously. When exiting edge e = (s, v), player [it] chooses an outgoing edge e = (v, v ). This is repeated until player [it] arrives at the destination. This defines a game with perfect information Γ(N, K, δ). Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 8 / 9
16 Model Latencies c it (σ) = e r it (σ) τ e is the travel time of player [it], w it (σ) is the waiting time of player [it], l it (σ) is the total latency suffered by player [it]: l it (σ) = c it (σ) + w it (σ). l t (σ) = l it (σ) is the total cost of generation G t. [it] G t Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 9 / 9
17 Model Solution concepts Equilibrium. Each player minimizes her own total latency given the queues in the system. - Exists: multiple equilibria - Subgame perfect Markov equilibrium Optimum. A social planner controls all players and seeks to minimize the long-run total costs, averaged over a period. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 10 / 9
18 Parallel networks Overview 1 Model 2 Parallel networks Uniform departures Periodic departures Extensions Chain-of-parallel networks Braess s networks Series-parallel networks 4 Conclusion Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 11 / 9
19 Parallel networks Uniform inflow In a parallel network each route is made of a single edge. The capacity of the network is γ = e γ e. e 1 s e 2 e d We assume that δ t = γ for all t N. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 12 / 9
20 Parallel networks Uniform departures Example Inflow=(,,,...). What happens in the equilibrium? τ 1 = 1 s τ 2 = τ = 5 d Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 1 / 9
21 Parallel networks Uniform departures Equilibrium 2 1 t = 1 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 14 / 9
22 Parallel networks Uniform departures Equilibrium 2 1 t = t = 2 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 14 / 9
23 Parallel networks Uniform departures Equilibrium 2 1 t = t = t = Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 14 / 9
24 Parallel networks Uniform departures Equilibrium 2 1 t = t = t = 1 2 t = 4 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 14 / 9
25 Parallel networks Uniform departures Equilibrium 2 1 t = t = t = 1 2 t = t = 5 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 14 / 9
26 Parallel networks Uniform departures Equilibrium 2 1 t = t = t = 1 2 t = t = t = 6 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 14 / 9
27 Parallel networks Uniform departures Equilibrium 2 1 t = t = t = 1 2 t = t = t = t = 7 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 14 / 9
28 Parallel networks Uniform departures Equilibrium 2 1 t = t = t = 1 2 t = t = t = t = t = 8... Total costs= 5=15. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 14 / 9
29 Parallel networks Uniform departures Optimum Can we do better in the long-run than 15 per generation? Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 15 / 9
30 Parallel networks Uniform departures Optimum Can we do better in the long-run than 15 per generation? 2 1 t = 1 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 15 / 9
31 Parallel networks Uniform departures Optimum Can we do better in the long-run than 15 per generation? 2 1 t = t = 2 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 15 / 9
32 Parallel networks Uniform departures Optimum Can we do better in the long-run than 15 per generation? 2 1 t = t = t =... Total costs=1++5=9. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 15 / 9
33 Parallel networks Uniform departures Steady state Proposition Let N be a parallel network. Then WEq(N, γ) = γ max e E τ e, Opt(N, γ) = e E γ e τ e. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 16 / 9
34 Parallel networks Uniform departures Steady state Proposition Let N be a parallel network. Then WEq(N, γ) = γ max e E τ e, Opt(N, γ) = e E γ e τ e. Equilibrium flows eventually coincide with optimal flows, but equilibrium costs are higher. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 16 / 9
35 Parallel networks Uniform departures Price of anarchy Let N be a parallel network. Then PoA(N, γ) = WEq(N, γ) Opt(N, γ) max e τ e min e τ e. The price of anarchy is unbounded over the class of parallel networks. Example Bad network: τ 1 = 1, γ 1 = N, τ 2 = N, γ 2 = 1. PoA(N, γ) = (N + 1) N. 2N Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 17 / 9
36 Parallel networks Periodic departures Periodic departures Inflow is a K-periodic vector: δ = (δ 1,..., δ K ) N K such that K k=1 δ k = K γ. We denote N K (γ) the set of such sequences. When δ is not-uniform, queues have to be created when there is a surge of players. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 18 / 9
37 Parallel networks Periodic departures Example Equilibrium for inflow (4,2,). Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 19 / 9
38 Parallel networks Periodic departures Example Equilibrium for inflow (4,2,) t = 1 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 19 / 9
39 Parallel networks Periodic departures Example Equilibrium for inflow (4,2,) t = 1 2 t = 2 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 19 / 9
40 Parallel networks Periodic departures Example Equilibrium for inflow (4,2,) t = 1 2 t = t = Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 19 / 9
41 Parallel networks Periodic departures Example Equilibrium for inflow (4,2,) t = 1 2 t = t = t = 4 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 19 / 9
42 Parallel networks Periodic departures Example Equilibrium for inflow (4,2,) t = 1 2 t = t = t = 4 2 t = 5 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 19 / 9
43 Parallel networks Periodic departures Example Equilibrium for inflow (4,2,) t = t = 6 2 t = t = t = 4 2 t = 5 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 19 / 9
44 Parallel networks Periodic departures Example Equilibrium for inflow (4,2,) t = t = 6 2 t = t = t = t = 4 2 t = 5 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 19 / 9
45 Parallel networks Periodic departures Example Equilibrium for inflow (4,2,) t = t = 6 2 t = t = t = 2 t = t = 4 2 t = 5 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 19 / 9
46 Parallel networks Periodic departures Example Equilibrium for inflow (4,2,) t = t = 6 2 t = t = t = 2 t = t = t = 9 2 t = 5 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 19 / 9
47 Parallel networks Periodic departures Example Optimum for inflow (4,2,). Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 20 / 9
48 Parallel networks Periodic departures Example Optimum for inflow (4,2,) t = 1 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 20 / 9
49 Parallel networks Periodic departures Example Optimum for inflow (4,2,) t = 1 2 t = 2 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 20 / 9
50 Parallel networks Periodic departures Example Optimum for inflow (4,2,) t = 1 2 t = t = Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 20 / 9
51 Parallel networks Periodic departures Example Optimum for inflow (4,2,) t = 1 2 t = t = Both in the equilibrium as in the optimum, the fourth player behaves as if he was postponed by one stage. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 20 / 9
52 Parallel networks Periodic departures Measure of periodicity Definition For any two elements δ, δ N K (γ), we say that δ is obtained from δ by an elementary operation if there exist an i with δ i > γ such that δ i = δ i 1, δ i+1 = δ i + 1. Let D(δ) be the minimal number of elementary operations one has to perform to transform δ into γ K. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 21 / 9
53 Parallel networks Periodic departures Measure of periodicity Figure: 1 operation needed to transform (, 1, 2) into (2, 2, 2). Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 22 / 9
54 Parallel networks Periodic departures Measure of periodicity Figure: 1 operation needed to transform (, 1, 2) into (2, 2, 2) Figure: 2 operations needed to transform (, 2, 1) into (2, 2, 2). Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 22 / 9
55 Parallel networks Periodic departures Steady state Theorem Let N be a parallel network and δ N K (γ). Then WEq(N, K, δ) = K γ max τ e + D(δ), e E Opt(N, K, δ) = K γ e τ e + D(δ). e E Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 2 / 9
56 Parallel networks Periodic departures Steady state Theorem Let N be a parallel network and δ N K (γ). Then WEq(N, K, δ) = K γ max τ e + D(δ), e E Opt(N, K, δ) = K γ e τ e + D(δ). e E Equilibrium flows eventually coincide with optimal flows. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 2 / 9
57 Extensions Overview 1 Model 2 Parallel networks Uniform departures Periodic departures Extensions Chain-of-parallel networks Braess s networks Series-parallel networks 4 Conclusion Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 24 / 9
58 Extensions Chain-of-parallel networks Parallel network below capacity τ 1 = 1 s τ 2 = 2 τ = τ 4 = d Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 25 / 9
59 Extensions Chain-of-parallel networks Parallel network below capacity τ 1 = 1 s τ 2 = 2 τ = τ 4 = d Equilibrium. If δ =, then WEq(N, 1, δ) = 9. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 25 / 9
60 Extensions Chain-of-parallel networks Parallel network below capacity τ 1 = 1 s τ 2 = 2 τ = τ 4 = d Equilibrium. If δ =, then WEq(N, 1, δ) = 9. If δ = (6, 0), then WEq(N, 2, δ) = 16 < 18. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 25 / 9
61 Extensions Chain-of-parallel networks Steady state below capacity Proposition Let N be a parallel network with capacity γ and let δ N K (γ ), where γ γ. Then WEq(N, K, δ) K γ max e E τ e + D(δ). Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 26 / 9
62 Extensions Chain-of-parallel networks Chain-of-parallel network τ 1 = 1 τ 2 = 2 s v d τ = τ 4 = τ 5 = 1 γ 5 = Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 27 / 9
63 Extensions Chain-of-parallel networks Chain-of-parallel network τ 1 = 1 τ 2 = 2 s v d τ = τ 4 = τ 5 = 1 γ 5 = Equilibrium. If δ = (6, 0), then t = Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 27 / 9
64 Extensions Chain-of-parallel networks Chain-of-parallel network τ 1 = 1 τ 2 = 2 s v d τ = τ 4 = τ 5 = 1 γ 5 = Equilibrium. If δ = (6, 0), then t = Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 27 / 9
65 Extensions Chain-of-parallel networks Chain-of-parallel network τ 1 = 1 τ 2 = 2 s v d τ = τ 4 = τ 5 = 1 γ 5 = Equilibrium. If δ = (6, 0), then t = Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 27 / 9
66 Extensions Chain-of-parallel networks Chain-of-parallel network τ 1 = 1 τ 2 = 2 s v d τ = τ 4 = τ 5 = 1 γ 5 = Equilibrium. If δ = (6, 0), then t = Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 27 / 9
67 Extensions Chain-of-parallel networks Chain-of-parallel network τ 1 = 1 τ 2 = 2 s v d τ = τ 4 = τ 5 = 1 γ 5 = Equilibrium. If δ = (6, 0), then t = t = 4 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 27 / 9
68 Extensions Chain-of-parallel networks Chain-of-parallel network τ 1 = 1 τ 2 = 2 s v d τ = τ 4 = τ 5 = 1 γ 5 = Equilibrium. If δ = (6, 0), then t = t = t = 5 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 27 / 9
69 Extensions Chain-of-parallel networks Chain-of-parallel network τ 1 = 1 τ 2 = 2 s v d τ = τ 4 = τ 5 = 1 γ 5 = Equilibrium. If δ = (6, 0), then t = t = t = 5 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 27 / 9
70 Extensions Chain-of-parallel networks Chain-of-parallel network τ 1 = 1 τ 2 = 2 s v d τ = τ 4 = τ 5 = 1 γ 5 = Equilibrium. If δ = (6, 0), then t = t = t = 5 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 27 / 9
71 Extensions Chain-of-parallel networks Chain-of-parallel network τ 1 = 1 τ 2 = 2 s v d τ = τ 4 = τ 5 = 1 γ 5 = Equilibrium. If δ = (6, 0), then WEq(N, 2, δ) = 22 (earliest-arrival property). WEq (N, 2, δ) = 25 (no overtaking). WEq (N, 2, δ) = 27 (allow overtaking). Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 28 / 9
72 Extensions Chain-of-parallel networks Optimum Let F be the (static) min-cost flow. Define M r p(σ) = Theorem (p+1)k t=pk+1 N r p(σ). Let δ N K (γ). Then there exists an optimal strategy profile σ such that Mp(σ) r = K Fr for each route r and each period p, and Opt(N, K, δ) = Opt(N, K, γ) + D(δ). Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 29 / 9
73 Extensions Braess s networks Braess s network s τ 1 = 0 τ 2 = 1 v τ = 0 w τ 4 = 1 τ 5 = 0 d Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 0 / 9
74 Extensions Braess s networks Braess s network Worst equilibrium. Player [11] and [21] choose e 1 e e 5. Player [12] chooses e 1 e e 5 and [22] chooses e 2 e 5. Player [1] chooses e 1 e e 5 and [2] chooses e 1 e 4. Player [14] chooses e 2 e 5 and [24] chooses e 1 e e 5. For t 5, player [1t] chooses e 1 e 4 and [2t] chooses e 2 e 5. Total costs=+=6. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 1 / 9
75 Extensions Braess s networks Braess s network Best equilibrium. Player [11] chooses e 1 e e 5 and [21] chooses e 2 e 5. For t 2, player [1t] chooses e 1 e 4 and [2t] chooses e 2 e 5. Total costs=1+1=2. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 2 / 9
76 Extensions Braess s networks Braess s network Proposition For every even integer n, there exists a network N in which V = n such that PoA(N, γ) = WEq(N, γ) BEq(N, γ) = BR(N, γ) = n 1. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games / 9
77 Extensions Series-parallel networks Series-parallel network τ 1 = 1 τ 2 = 0 s v d γ 2 = 2 τ = 0 τ 4 = 1 Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 4 / 9
78 Extensions Series-parallel networks Series-parallel network τ 1 = 1 τ 2 = 0 s v d γ 2 = 2 τ = 0 τ 4 = 1 Equilibrium. Player [11] chooses e 2 e, [21] chooses e 2 e 4, [1] chooses e 2 e. For t 2, [1t] chooses e 1, [2t] chooses e 2 e, [t] chooses e 2 e 4. Total costs=1+1+2=4. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 4 / 9
79 Extensions Series-parallel networks Series-parallel network τ 1 = 1 τ 2 = 0 s v d γ 2 = 2 τ = 1 τ 4 = 1 Suppose e contains a queue, then total costs decrease to Another view on Braess s paradox: initial queues can improve total costs. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 5 / 9
80 Conclusion Overview 1 Model 2 Parallel networks Uniform departures Periodic departures Extensions Chain-of-parallel networks Braess s networks Series-parallel networks 4 Conclusion Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 6 / 9
81 Conclusion Summary Two main contributions: We propose a measure of periodicity that characterizes the additional delay due to periodicity. We illustrate a new form of Braess s paradox: the presence of initial queues in a network may decrease the long-run costs in equilibrium. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 7 / 9
82 Conclusion Open problems General networks Multiple sources and destinations Connection with continuous time and flows Stochastic inflow Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 8 / 9
83 Conclusion Apologies for congesting your brain. Scarsini, Schröder, Tomala Dynamic Atomic Congestion Games 9 / 9
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