A Heterogeneous Routing Game - Walid Krichene
A Heterogeneous Routing Game Farhad Farokhi
Walid Krichene
Alexandre M. Bayen
Karl H. Johansson
KTH Royal Institute of Technology, Stockholm, Sweden University of California at Berkeley, CA, USA
October 2, 2013
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
1 / 27
Outline
1
The non-atomic routing game
2
An equivalent finite-player game
3
Heterogeneous routing games and potential games
4
Tolling
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
2 / 27
Outline
1
The non-atomic routing game
2
An equivalent finite-player game
3
Heterogeneous routing games and potential games
4
Tolling
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
3 / 27
Introduction
Most studies of routing games assume that drivers or vehicles are of the same type. Motivated by I
I
Transportation networks: drivers only care about the travel time (Wardrop, 1952); Packet routing in communication networks (Altman et al., 2006; Banner & Orda, 2007; Czumaj, 2004).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
4 / 27
Introduction
We would like to relax this assumption, because of I I I
Fuel consumption (Farokhi & Johansson, 2013; Alam et al., 2010); Sensitivity to Latency (Stern, 1999; Stern & Richardson, 2005); Sensitivity to Tolls (Inregia, 2001; Engelson & Lindberg, 2006).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
5 / 27
Introduction
We would like to relax this assumption, because of I I I
Fuel consumption (Farokhi & Johansson, 2013; Alam et al., 2010); Sensitivity to Latency (Stern, 1999; Stern & Richardson, 2005); Sensitivity to Tolls (Inregia, 2001; Engelson & Lindberg, 2006).
Heterogeneous routing games have been considered (Baldacci et al., 2008; Engevall et al., 2004; Fleischer et al., 2004; Fotakis et al., 2010; Karakostas & Kolliopoulos, 2004; Marcotte & Zhu, 2009). However, they adjust the sensitivity of the agents either to the observed latencies or the tolls through a multiplicative weight.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
5 / 27
Introduction
We would like to relax this assumption, because of I I I
Fuel consumption (Farokhi & Johansson, 2013; Alam et al., 2010); Sensitivity to Latency (Stern, 1999; Stern & Richardson, 2005); Sensitivity to Tolls (Inregia, 2001; Engelson & Lindberg, 2006).
Heterogeneous routing games have been considered (Baldacci et al., 2008; Engevall et al., 2004; Fleischer et al., 2004; Fotakis et al., 2010; Karakostas & Kolliopoulos, 2004; Marcotte & Zhu, 2009). However, they adjust the sensitivity of the agents either to the observed latencies or the tolls through a multiplicative weight. More general classes of latency functions?
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
5 / 27
Heterogeneous Routing Game
0
1
5 4 6
2
3
7
8
A directed graph G = (V, E) models the transportation network. A set of commodities {(sk , tk )}K k=0 .
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
6 / 27
Heterogeneous Routing Game
0
1
5 4 6
2
3
7
8
A directed graph G = (V, E) models the transportation network. A set of commodities {(sk , tk )}K k=0 . Pk : set of all admissible paths over the graph G that connect sk ∈ V to tk ∈ V. Denote P = ∪K k=1 Pk .
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
6 / 27
Heterogeneous Routing Game 0
1
fpθ
5
fθp1 + fθp2 + fθp3 = Fθ1
4 6
2 7
3 8
The type of a player is determined by θ ∈ Θ where Θ is a finite set. fpθ ∈ R≥0 denotes the flow of players of type θ ∈ Θ that use path p ∈ P.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
7 / 27
Heterogeneous Routing Game 0
1
5
fpθ1 + fpθ2 + fpθ3 = Fθ1
4 6
2 7
3 8
The type of a player is determined by θ ∈ Θ where Θ is a finite set. fpθ ∈ R≥0 denotes the flow of players of type θ ∈ Θ that use path p ∈ P. |Θ|
Commodity k ∈ JK K = {1, . . . , K } transfers a flow equal to (Fkθ )θ∈Θ ∈ R≥0 .
Feasibility A flow vector f = (fpθ )p∈P,θ∈Θ ∈ R|P|·|Θ| is feasible if k ∈ JK K and θ ∈ Θ.
Farokhi et al.
A Heterogeneous Routing Game
θ p∈Pk fp
P
= Fkθ for all
October 2, 2013
7 / 27
Heterogeneous Routing Game 0
1
φθ(4,5)
5
φθ(4,5) = fpθ1 + fpθ10 + fp100θ
4 6
2 7
3 8
The type of a player is determined by θ ∈ Θ where Θ is a finite set. fpθ ∈ R≥0 denotes the flow of players of type θ ∈ Θ that use path p ∈ P. |Θ|
Commodity k ∈ JK K = {1, . . . , K } transfers a flow equal to (Fkθ )θ∈Θ ∈ R≥0 .
Feasibility P A flow vector f = (fpθ )p∈P,θ∈Θ ∈ R|P|·|Θ| is feasible if p∈Pk fpθ = Fkθ for all k ∈ JK K and θ ∈ Θ. P φθe = p∈P:e∈p fpθ denotes the flow of drivers of type θ on edge e ∈ E. Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
7 / 27
Heterogeneous Routing Game
0
1
5
0 `˜θe ((φθ(4,5) )θ0 ∈Θ )
4 6
2 7
3 8
Driver of type θ ∈ Θ traveling along e ∈ E: experiences edge latency 0 `˜θe ((φθe )θ0 ∈Θ ).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
8 / 27
Heterogeneous Routing Game
0
1
θ0 )θ0 ∈Θ ) `θp (f ) = `˜θe ((φ(0,4) θ0 )θ0 ∈Θ ) +`˜θe ((φ(4,5) θ θ0 ˜ +`e ((φ(5,1) )θ0 ∈Θ )
5 4 6
2 7
3 8
Driver of type θ ∈ Θ traveling along e ∈ E: experiences edge latency 0 `˜θe ((φθe )θ0 ∈Θ ). P 0 Total latency on path p ∈ Pk : `θp (f ) = e∈p `˜θe ((φθe )θ0 ∈Θ ).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
8 / 27
Heterogeneous Routing Game
0
1
θ0 )θ0 ∈Θ ) `θp (f ) = `˜θe ((φ(0,4) θ0 )θ0 ∈Θ ) +`˜θe ((φ(4,5) θ θ0 ˜ +`e ((φ(5,1) )θ0 ∈Θ )
5 4 6
2 7
3 8
Driver of type θ ∈ Θ traveling along e ∈ E: experiences edge latency 0 `˜θe ((φθe )θ0 ∈Θ ). P 0 Total latency on path p ∈ Pk : `θp (f ) = e∈p `˜θe ((φθe )θ0 ∈Θ ). A driver ⇔ infinitesimal amount of flow, strategically tries to minimize its own latency min `θp (f ) p∈Pk
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
8 / 27
Nash Equilibrium in Heterogeneous Routing Game
Nash Equilibrium∗ 0
A flow vector f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium if for all k ∈ JK K and θ ∈ Θ, fpθ > 0 for a path p ∈ Pk implies that `θp (f ) ≤ `θp0 (f ) for all p 0 ∈ Pk .
∗ Also called Wardrop equilibrium due to pioneering work of Wardrop (1952), and the fact that pure strategy Nash equilibrium was primarily defined in the context of games with finitely many players. See (Chau & Sim, 2003; Haurie & Marcotte, 1985; Roughgarden & Tardos, 2002). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
9 / 27
Nash Equilibrium in Heterogeneous Routing Game
Nash Equilibrium∗ 0
A flow vector f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium if for all k ∈ JK K and θ ∈ Θ, fpθ > 0 for a path p ∈ Pk implies that `θp (f ) ≤ `θp0 (f ) for all p 0 ∈ Pk .
For a commodity k ∈ JK K and type θ ∈ Θ
paths with nonzero flow for drivers of type θ have equal costs the rest have larger than or equal costs.
∗ Also called Wardrop equilibrium due to pioneering work of Wardrop (1952), and the fact that pure strategy Nash equilibrium was primarily defined in the context of games with finitely many players. See (Chau & Sim, 2003; Haurie & Marcotte, 1985; Roughgarden & Tardos, 2002). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
9 / 27
Illustrative Example: Platooning Incentives Let Θ = {c, t} where t denotes trucks and c denotes cars. Edge cost functions `˜ce (φce , φte ) = ξe (φce + φte ), `˜te (φce , φte ) = ξe (φce + φte ) + ζe (φce + φte )γe (φte ), where
For an experimental study of improvements in the fuel efficiency caused by platooning in heavy-duty vehicles, see (Alam et al., 2010). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
10 / 27
Illustrative Example: Platooning Incentives Let Θ = {c, t} where t denotes trucks and c denotes cars. Edge cost functions `˜ce (φce , φte ) = ξe (φce + φte ), `˜te (φce , φte ) = ξe (φce + φte ) + ζe (φce + φte )γe (φte ), where I
ξe : R≥0 → R≥0 : latency for using edge e ∈ E, function of the total edge flow;
For an experimental study of improvements in the fuel efficiency caused by platooning in heavy-duty vehicles, see (Alam et al., 2010). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
10 / 27
Illustrative Example: Platooning Incentives Let Θ = {c, t} where t denotes trucks and c denotes cars. Edge cost functions `˜ce (φce , φte ) = ξe (φce + φte ), `˜te (φce , φte ) = ξe (φce + φte ) + ζe (φce + φte )γe (φte ), where I I
ξe : R≥0 → R≥0 : latency for using edge e ∈ E, function of the total edge flow; ζe : R≥0 → R≥0 : fuel consumption of trucks, function of the total flow;
For an experimental study of improvements in the fuel efficiency caused by platooning in heavy-duty vehicles, see (Alam et al., 2010). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
10 / 27
Illustrative Example: Platooning Incentives Let Θ = {c, t} where t denotes trucks and c denotes cars. Edge cost functions `˜ce (φce , φte ) = ξe (φce + φte ), `˜te (φce , φte ) = ξe (φce + φte ) + ζe (φce + φte )γe (φte ), where I I I
ξe : R≥0 → R≥0 : latency for using edge e ∈ E, function of the total edge flow; ζe : R≥0 → R≥0 : fuel consumption of trucks, function of the total flow; γe : R≥0 → R≥0 : inverse of fuel efficiency of the trucks, function of the flow of trucks: platooning reduces air drag
For an experimental study of improvements in the fuel efficiency caused by platooning in heavy-duty vehicles, see (Alam et al., 2010). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
10 / 27
Standing Assumptions
Assumption For all θ ∈ Θ and e ∈ E, `˜θe satisfies: (i) `˜θe ∈ C 1 ; (ii) `˜θe is positive; R φθ 0 (iii) 0 e `˜θe (u, (φθe )θ0 ∈Θ\{θ} )du is a convex function in φθe .
0 Assumption (iii) can be replaced with the assumption that `˜θe ((φθe )θ0 ∈Θ ) is an θ non-decreasing function in φe .
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
11 / 27
Outline
1
The non-atomic routing game
2
An equivalent finite-player game
3
Heterogeneous routing games and potential games
4
Tolling
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
12 / 27
An equivalent finite-player game Let Θ = {θ1 , . . . , θN }. Consider an abstract game with N players in which player i ∈ JNK corresponds to type θi ∈ Θ.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
13 / 27
An equivalent finite-player game Let Θ = {θ1 , . . . , θN }. Consider an abstract game with N players in which player i ∈ JNK corresponds to type θi ∈ Θ. The action of player i is denoted by ai = (fpθ0i )p0 ∈P , in the action set Ai =
(fpθ0i )p0 ∈P
∈
|P| R≥0
X θi θi fp0 = Fk . p 0 ∈Pk
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
13 / 27
An equivalent finite-player game Let Θ = {θ1 , . . . , θN }. Consider an abstract game with N players in which player i ∈ JNK corresponds to type θi ∈ Θ. The action of player i is denoted by ai = (fpθ0i )p0 ∈P , in the action set Ai =
(fpθ0i )p0 ∈P
∈
|P| R≥0
X θi θi fp0 = Fk . p 0 ∈Pk
Utility of player i: θ
Ui (ai , a−i ) =
XZ e∈E
φe i
θ `˜θe i (u, (φe j )θj ∈Θ\{θi } )du,
0
An action profile a ∈ ×N j=1 Aj is a pure strategy Nash equilibrium if for all i ∈ JNK, Ui (ai , a−i ) ≥ Ui (¯ai , a−i ), ∀¯ai ∈ Ai . Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
13 / 27
An equivalent finite-player game
Lemma 0
A flow vector (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium of the heterogeneous routing game if and only if ((fpθ01 )p0 ∈P , . . . , (fpθ0N )p0 ∈P ) is a pure strategy Nash equilibrium of the abstract game. proof: write KKT conditions for the problem minai Ui (ai , a−i ).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
14 / 27
Existence of a Nash Equilibrium
Theorem The heterogeneous routing game admits at least one Nash equilibrium.
Proof. Equivalent to showing that the game with |Θ| players admits a pure strategy Nash equilibrium. The rest is an application of seminal results of (Arrow & Debreu, 1954; Debreu, 1952).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
15 / 27
Outline
1
The non-atomic routing game
2
An equivalent finite-player game
3
Heterogeneous routing games and potential games
4
Tolling
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
16 / 27
Potential Game The abstract game is a potential game and admits a potential function V : ×N i=1 Ai → R if for all i ∈ JNK, V (ai , a−i ) − V (¯ai ,a−i ) = Ui (ai , a−i ) − Ui (¯ai , a−i ), ∀ai , ¯ai ∈ Ai and a−i ∈ ×j∈JNK\{i} Aj .
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
17 / 27
Potential Game The abstract game is a potential game and admits a potential function V : ×N i=1 Ai → R if for all i ∈ JNK, V (ai , a−i ) − V (¯ai ,a−i ) = Ui (ai , a−i ) − Ui (¯ai , a−i ), ∀ai , ¯ai ∈ Ai and a−i ∈ ×j∈JNK\{i} Aj . In general, finding a Nash equilibrium is difficult (Fabrikant et al., 2004; Papadimitriou, 2007; Roughgarden, 2010).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
17 / 27
Potential Game The abstract game is a potential game and admits a potential function V : ×N i=1 Ai → R if for all i ∈ JNK, V (ai , a−i ) − V (¯ai ,a−i ) = Ui (ai , a−i ) − Ui (¯ai , a−i ), ∀ai , ¯ai ∈ Ai and a−i ∈ ×j∈JNK\{i} Aj . In general, finding a Nash equilibrium is difficult (Fabrikant et al., 2004; Papadimitriou, 2007; Roughgarden, 2010). A minimizer of the potential function is a pure strategy Nash equilibrium of a potential game (Monderer & Shapley, 1996).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
17 / 27
Potential Game The abstract game is a potential game and admits a potential function V : ×N i=1 Ai → R if for all i ∈ JNK, V (ai , a−i ) − V (¯ai ,a−i ) = Ui (ai , a−i ) − Ui (¯ai , a−i ), ∀ai , ¯ai ∈ Ai and a−i ∈ ×j∈JNK\{i} Aj . In general, finding a Nash equilibrium is difficult (Fabrikant et al., 2004; Papadimitriou, 2007; Roughgarden, 2010). A minimizer of the potential function is a pure strategy Nash equilibrium of a potential game (Monderer & Shapley, 1996). Convergence results in many learning algorithms (e.g., fictitious play, myopic learning, multiplicative updates) rely heavily on potential functions (Drighès et al., 2013; Fudenberg 1998; Marden et al., 2009; Monderer & Shapley, 1996)
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
17 / 27
Existence of a Potential Function
Lemma (Necessary Condition) If the abstract game admits a potential function V ∈ C 2 , then " # X ∂ ˜θi ∂ ˜θj θ0 θ0 ` ((φe )θ0 ∈Θ ) − ` ((φe )θ0 ∈Θ ) = 0, θi e θ e ∂φe j e∈p1 ∩p2 ∂φe for all i, j ∈ JNK and p1 , p2 ∈ P.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
18 / 27
Existence of a Potential Function Lemma (Sufficient Condition) Assume that |Θ| = 2. If X e∈p1 ∩p2
∂ ˜θ2 θ1 θ2 ∂ ˜θ1 θ1 θ2 ` ` (φ (φ , φ , φ ) − ) = 0, e e e e e e ∂φθe 1 ∂φθe 2
for all p1 , p2 ∈ P, then V ((fpθ01 )p0 ∈P , (fpθ02 )p0 ∈P )
=
XZ e∈E
φθe 1
`˜θe 1 (u1 , φθe 2 )du1 +
0
Z
φθe 2
`˜θe 2 (φθe 1 , u2 )du2
0
Z
φθe 2Z φθe 1
− 0
0
∂ ˜θ1 ` (t, u)dtdu ∂u e
is a potential function for the abstract game.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
19 / 27
Finding a Nash Equilibrium Theorem (Sufficient condition for Nash equilibria) Assume that |Θ| = 2, and X ∂ ˜θe 2 (φθe 1 , φθe 2 ) − ∂ `˜θe 1 (φθe 1 , φθe 2 ) = 0, ` θ1 ∂φθe 2 e∈p1 ∩p2 ∂φe 0
for all p1 , p2 ∈ P. If f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a solution of the optimization problem min V ((fpθ01 )p0 ∈P , (fpθ02 )p0 ∈P ), X X s.t. fpθ1 = Fkθ1 and fpθ2 = Fkθ2 , ∀k ∈ JK K, p∈Pk
fpθ1 , fpθ2
p∈Pk
∈ R≥0 , ∀p ∈ P,
0
then f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium of the heterogeneous routing game.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
20 / 27
Characterizing All Nash Equilibria
Theorem Furthermore, assume that potential function V is a convex function. Then 0 f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium of the heterogeneous routing game if and only if it is a solution of the convex optimization problem
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
21 / 27
Outline
1
The non-atomic routing game
2
An equivalent finite-player game
3
Heterogeneous routing games and potential games
4
Tolling
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
22 / 27
Imposing Tolls If finding a Nash equilibrium in the heterogeneous routing game is numerically intractable, it might be unlikely for the drivers to find it.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
23 / 27
Imposing Tolls If finding a Nash equilibrium in the heterogeneous routing game is numerically intractable, it might be unlikely for the drivers to find it. 0
Driver of type θ ∈ Θ must pay a toll τ˜eθ ((φθe )θ0 ∈Θ ) for using an edge e ∈ E. P 0 For using path p ∈ Pk , she must pay τpθ (f ) = e∈p τ˜eθ ((φθe )θ0 ∈Θ ).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
23 / 27
Imposing Tolls If finding a Nash equilibrium in the heterogeneous routing game is numerically intractable, it might be unlikely for the drivers to find it. 0
Driver of type θ ∈ Θ must pay a toll τ˜eθ ((φθe )θ0 ∈Θ ) for using an edge e ∈ E. P 0 For using path p ∈ Pk , she must pay τpθ (f ) = e∈p τ˜eθ ((φθe )θ0 ∈Θ ).
Nash Equilibrium 0
A flow vector f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium for the routing game with tolls if, for all k ∈ JK K and θ ∈ Θ, whenever fpθ > 0 for some path p ∈ Pk , then `θp (f ) + τpθ (f ) ≤ `θp0 (f ) + τpθ0 (f ) for all p 0 ∈ Pk .
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
23 / 27
Imposing Tolls If finding a Nash equilibrium in the heterogeneous routing game is numerically intractable, it might be unlikely for the drivers to find it. 0
Driver of type θ ∈ Θ must pay a toll τ˜eθ ((φθe )θ0 ∈Θ ) for using an edge e ∈ E. P 0 For using path p ∈ Pk , she must pay τpθ (f ) = e∈p τ˜eθ ((φθe )θ0 ∈Θ ).
Nash Equilibrium 0
A flow vector f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium for the routing game with tolls if, for all k ∈ JK K and θ ∈ Θ, whenever fpθ > 0 for some path p ∈ Pk , then `θp (f ) + τpθ (f ) ≤ `θp0 (f ) + τpθ0 (f ) for all p 0 ∈ Pk . Consider the case that the tolls are type-independent (i.e., τ˜eθ1 (φθe 1 , φθe 2 ) = τ˜eθ2 (φθe 1 , φθe 2 ) = τ˜e (φθe 1 , φθe 2 )) which is a harder case.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
23 / 27
Imposing Tolls Proposition Assume that |Θ| = 2. The abstract game admits the potential function V ((fpθ01 )p0 ∈P , (fpθ02 )p0 ∈P ) =
φθe 1
XZ
(`˜θe 1 (u1 , φθe 2 ) + τ˜e (u1 , φθe 2 ))du1
0
e∈E
Z +
φθe 2
(`˜θe 2 (φθe 1 , u2 ) + τ˜e (φθe 1 , u2 ))du2
0
Z
φθe 2Z φθe 1
− 0
0
∂ ˜θ1 (` (t, u) + τ˜e (t, u))dtdu ∂u e
if ∂ τ˜e (φθe 1 , φθe 2 ) ∂ τ˜e (φθe 1 , φθe 2 ) ∂ `˜θ2 (φθ1 , φθ2 ) ∂ `˜θ1 (φθ1 , φθ2 ) − = e eθ1 e − e eθ2 e , θ2 θ1 ∂φe ∂φe ∂φe ∂φe for all e ∈ E. Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
24 / 27
Imposing Tolls
Corollary Assume that |Θ| = 2. The abstract game admits a potential function V ∈ C 2 if the imposed tolls are of the following from τ˜e (φθe 1 , φθe 2 ) = ce + ψe (φθe 1 + φθe 2 ) # Z φθe 2 " ˜θ2 ∂ `e (y , x) ∂ `˜θe 1 (y , x) − + ∂y ∂x 0
dq, θ
θ
x=q,y =φe 1 +φe 2 −q
where c ∈ R≥0 and ψe ∈ C 1 are arbitrarily chosen for all e ∈ E.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
25 / 27
Conclusions and Future Work
Conclusions Proved the existence of a Nash equilibrium in heterogeneous routing game. Characterized necessary and sufficient conditions for the existence of a potential function. Calculated tolls to ensure the existence of a potential function.
Future Work Study how heterogeneous populations can learn Nash equilibria; Extend the results to |Θ| > 2, or a continuum of types.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
26 / 27
Thank you.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
27 / 27
Walid Krichene
Alexandre M. Bayen
Karl H. Johansson
KTH Royal Institute of Technology, Stockholm, Sweden University of California at Berkeley, CA, USA
October 2, 2013
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
1 / 27
Outline
1
The non-atomic routing game
2
An equivalent finite-player game
3
Heterogeneous routing games and potential games
4
Tolling
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
2 / 27
Outline
1
The non-atomic routing game
2
An equivalent finite-player game
3
Heterogeneous routing games and potential games
4
Tolling
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
3 / 27
Introduction
Most studies of routing games assume that drivers or vehicles are of the same type. Motivated by I
I
Transportation networks: drivers only care about the travel time (Wardrop, 1952); Packet routing in communication networks (Altman et al., 2006; Banner & Orda, 2007; Czumaj, 2004).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
4 / 27
Introduction
We would like to relax this assumption, because of I I I
Fuel consumption (Farokhi & Johansson, 2013; Alam et al., 2010); Sensitivity to Latency (Stern, 1999; Stern & Richardson, 2005); Sensitivity to Tolls (Inregia, 2001; Engelson & Lindberg, 2006).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
5 / 27
Introduction
We would like to relax this assumption, because of I I I
Fuel consumption (Farokhi & Johansson, 2013; Alam et al., 2010); Sensitivity to Latency (Stern, 1999; Stern & Richardson, 2005); Sensitivity to Tolls (Inregia, 2001; Engelson & Lindberg, 2006).
Heterogeneous routing games have been considered (Baldacci et al., 2008; Engevall et al., 2004; Fleischer et al., 2004; Fotakis et al., 2010; Karakostas & Kolliopoulos, 2004; Marcotte & Zhu, 2009). However, they adjust the sensitivity of the agents either to the observed latencies or the tolls through a multiplicative weight.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
5 / 27
Introduction
We would like to relax this assumption, because of I I I
Fuel consumption (Farokhi & Johansson, 2013; Alam et al., 2010); Sensitivity to Latency (Stern, 1999; Stern & Richardson, 2005); Sensitivity to Tolls (Inregia, 2001; Engelson & Lindberg, 2006).
Heterogeneous routing games have been considered (Baldacci et al., 2008; Engevall et al., 2004; Fleischer et al., 2004; Fotakis et al., 2010; Karakostas & Kolliopoulos, 2004; Marcotte & Zhu, 2009). However, they adjust the sensitivity of the agents either to the observed latencies or the tolls through a multiplicative weight. More general classes of latency functions?
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
5 / 27
Heterogeneous Routing Game
0
1
5 4 6
2
3
7
8
A directed graph G = (V, E) models the transportation network. A set of commodities {(sk , tk )}K k=0 .
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
6 / 27
Heterogeneous Routing Game
0
1
5 4 6
2
3
7
8
A directed graph G = (V, E) models the transportation network. A set of commodities {(sk , tk )}K k=0 . Pk : set of all admissible paths over the graph G that connect sk ∈ V to tk ∈ V. Denote P = ∪K k=1 Pk .
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
6 / 27
Heterogeneous Routing Game 0
1
fpθ
5
fθp1 + fθp2 + fθp3 = Fθ1
4 6
2 7
3 8
The type of a player is determined by θ ∈ Θ where Θ is a finite set. fpθ ∈ R≥0 denotes the flow of players of type θ ∈ Θ that use path p ∈ P.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
7 / 27
Heterogeneous Routing Game 0
1
5
fpθ1 + fpθ2 + fpθ3 = Fθ1
4 6
2 7
3 8
The type of a player is determined by θ ∈ Θ where Θ is a finite set. fpθ ∈ R≥0 denotes the flow of players of type θ ∈ Θ that use path p ∈ P. |Θ|
Commodity k ∈ JK K = {1, . . . , K } transfers a flow equal to (Fkθ )θ∈Θ ∈ R≥0 .
Feasibility A flow vector f = (fpθ )p∈P,θ∈Θ ∈ R|P|·|Θ| is feasible if k ∈ JK K and θ ∈ Θ.
Farokhi et al.
A Heterogeneous Routing Game
θ p∈Pk fp
P
= Fkθ for all
October 2, 2013
7 / 27
Heterogeneous Routing Game 0
1
φθ(4,5)
5
φθ(4,5) = fpθ1 + fpθ10 + fp100θ
4 6
2 7
3 8
The type of a player is determined by θ ∈ Θ where Θ is a finite set. fpθ ∈ R≥0 denotes the flow of players of type θ ∈ Θ that use path p ∈ P. |Θ|
Commodity k ∈ JK K = {1, . . . , K } transfers a flow equal to (Fkθ )θ∈Θ ∈ R≥0 .
Feasibility P A flow vector f = (fpθ )p∈P,θ∈Θ ∈ R|P|·|Θ| is feasible if p∈Pk fpθ = Fkθ for all k ∈ JK K and θ ∈ Θ. P φθe = p∈P:e∈p fpθ denotes the flow of drivers of type θ on edge e ∈ E. Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
7 / 27
Heterogeneous Routing Game
0
1
5
0 `˜θe ((φθ(4,5) )θ0 ∈Θ )
4 6
2 7
3 8
Driver of type θ ∈ Θ traveling along e ∈ E: experiences edge latency 0 `˜θe ((φθe )θ0 ∈Θ ).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
8 / 27
Heterogeneous Routing Game
0
1
θ0 )θ0 ∈Θ ) `θp (f ) = `˜θe ((φ(0,4) θ0 )θ0 ∈Θ ) +`˜θe ((φ(4,5) θ θ0 ˜ +`e ((φ(5,1) )θ0 ∈Θ )
5 4 6
2 7
3 8
Driver of type θ ∈ Θ traveling along e ∈ E: experiences edge latency 0 `˜θe ((φθe )θ0 ∈Θ ). P 0 Total latency on path p ∈ Pk : `θp (f ) = e∈p `˜θe ((φθe )θ0 ∈Θ ).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
8 / 27
Heterogeneous Routing Game
0
1
θ0 )θ0 ∈Θ ) `θp (f ) = `˜θe ((φ(0,4) θ0 )θ0 ∈Θ ) +`˜θe ((φ(4,5) θ θ0 ˜ +`e ((φ(5,1) )θ0 ∈Θ )
5 4 6
2 7
3 8
Driver of type θ ∈ Θ traveling along e ∈ E: experiences edge latency 0 `˜θe ((φθe )θ0 ∈Θ ). P 0 Total latency on path p ∈ Pk : `θp (f ) = e∈p `˜θe ((φθe )θ0 ∈Θ ). A driver ⇔ infinitesimal amount of flow, strategically tries to minimize its own latency min `θp (f ) p∈Pk
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
8 / 27
Nash Equilibrium in Heterogeneous Routing Game
Nash Equilibrium∗ 0
A flow vector f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium if for all k ∈ JK K and θ ∈ Θ, fpθ > 0 for a path p ∈ Pk implies that `θp (f ) ≤ `θp0 (f ) for all p 0 ∈ Pk .
∗ Also called Wardrop equilibrium due to pioneering work of Wardrop (1952), and the fact that pure strategy Nash equilibrium was primarily defined in the context of games with finitely many players. See (Chau & Sim, 2003; Haurie & Marcotte, 1985; Roughgarden & Tardos, 2002). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
9 / 27
Nash Equilibrium in Heterogeneous Routing Game
Nash Equilibrium∗ 0
A flow vector f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium if for all k ∈ JK K and θ ∈ Θ, fpθ > 0 for a path p ∈ Pk implies that `θp (f ) ≤ `θp0 (f ) for all p 0 ∈ Pk .
For a commodity k ∈ JK K and type θ ∈ Θ
paths with nonzero flow for drivers of type θ have equal costs the rest have larger than or equal costs.
∗ Also called Wardrop equilibrium due to pioneering work of Wardrop (1952), and the fact that pure strategy Nash equilibrium was primarily defined in the context of games with finitely many players. See (Chau & Sim, 2003; Haurie & Marcotte, 1985; Roughgarden & Tardos, 2002). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
9 / 27
Illustrative Example: Platooning Incentives Let Θ = {c, t} where t denotes trucks and c denotes cars. Edge cost functions `˜ce (φce , φte ) = ξe (φce + φte ), `˜te (φce , φte ) = ξe (φce + φte ) + ζe (φce + φte )γe (φte ), where
For an experimental study of improvements in the fuel efficiency caused by platooning in heavy-duty vehicles, see (Alam et al., 2010). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
10 / 27
Illustrative Example: Platooning Incentives Let Θ = {c, t} where t denotes trucks and c denotes cars. Edge cost functions `˜ce (φce , φte ) = ξe (φce + φte ), `˜te (φce , φte ) = ξe (φce + φte ) + ζe (φce + φte )γe (φte ), where I
ξe : R≥0 → R≥0 : latency for using edge e ∈ E, function of the total edge flow;
For an experimental study of improvements in the fuel efficiency caused by platooning in heavy-duty vehicles, see (Alam et al., 2010). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
10 / 27
Illustrative Example: Platooning Incentives Let Θ = {c, t} where t denotes trucks and c denotes cars. Edge cost functions `˜ce (φce , φte ) = ξe (φce + φte ), `˜te (φce , φte ) = ξe (φce + φte ) + ζe (φce + φte )γe (φte ), where I I
ξe : R≥0 → R≥0 : latency for using edge e ∈ E, function of the total edge flow; ζe : R≥0 → R≥0 : fuel consumption of trucks, function of the total flow;
For an experimental study of improvements in the fuel efficiency caused by platooning in heavy-duty vehicles, see (Alam et al., 2010). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
10 / 27
Illustrative Example: Platooning Incentives Let Θ = {c, t} where t denotes trucks and c denotes cars. Edge cost functions `˜ce (φce , φte ) = ξe (φce + φte ), `˜te (φce , φte ) = ξe (φce + φte ) + ζe (φce + φte )γe (φte ), where I I I
ξe : R≥0 → R≥0 : latency for using edge e ∈ E, function of the total edge flow; ζe : R≥0 → R≥0 : fuel consumption of trucks, function of the total flow; γe : R≥0 → R≥0 : inverse of fuel efficiency of the trucks, function of the flow of trucks: platooning reduces air drag
For an experimental study of improvements in the fuel efficiency caused by platooning in heavy-duty vehicles, see (Alam et al., 2010). Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
10 / 27
Standing Assumptions
Assumption For all θ ∈ Θ and e ∈ E, `˜θe satisfies: (i) `˜θe ∈ C 1 ; (ii) `˜θe is positive; R φθ 0 (iii) 0 e `˜θe (u, (φθe )θ0 ∈Θ\{θ} )du is a convex function in φθe .
0 Assumption (iii) can be replaced with the assumption that `˜θe ((φθe )θ0 ∈Θ ) is an θ non-decreasing function in φe .
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
11 / 27
Outline
1
The non-atomic routing game
2
An equivalent finite-player game
3
Heterogeneous routing games and potential games
4
Tolling
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
12 / 27
An equivalent finite-player game Let Θ = {θ1 , . . . , θN }. Consider an abstract game with N players in which player i ∈ JNK corresponds to type θi ∈ Θ.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
13 / 27
An equivalent finite-player game Let Θ = {θ1 , . . . , θN }. Consider an abstract game with N players in which player i ∈ JNK corresponds to type θi ∈ Θ. The action of player i is denoted by ai = (fpθ0i )p0 ∈P , in the action set Ai =
(fpθ0i )p0 ∈P
∈
|P| R≥0
X θi θi fp0 = Fk . p 0 ∈Pk
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
13 / 27
An equivalent finite-player game Let Θ = {θ1 , . . . , θN }. Consider an abstract game with N players in which player i ∈ JNK corresponds to type θi ∈ Θ. The action of player i is denoted by ai = (fpθ0i )p0 ∈P , in the action set Ai =
(fpθ0i )p0 ∈P
∈
|P| R≥0
X θi θi fp0 = Fk . p 0 ∈Pk
Utility of player i: θ
Ui (ai , a−i ) =
XZ e∈E
φe i
θ `˜θe i (u, (φe j )θj ∈Θ\{θi } )du,
0
An action profile a ∈ ×N j=1 Aj is a pure strategy Nash equilibrium if for all i ∈ JNK, Ui (ai , a−i ) ≥ Ui (¯ai , a−i ), ∀¯ai ∈ Ai . Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
13 / 27
An equivalent finite-player game
Lemma 0
A flow vector (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium of the heterogeneous routing game if and only if ((fpθ01 )p0 ∈P , . . . , (fpθ0N )p0 ∈P ) is a pure strategy Nash equilibrium of the abstract game. proof: write KKT conditions for the problem minai Ui (ai , a−i ).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
14 / 27
Existence of a Nash Equilibrium
Theorem The heterogeneous routing game admits at least one Nash equilibrium.
Proof. Equivalent to showing that the game with |Θ| players admits a pure strategy Nash equilibrium. The rest is an application of seminal results of (Arrow & Debreu, 1954; Debreu, 1952).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
15 / 27
Outline
1
The non-atomic routing game
2
An equivalent finite-player game
3
Heterogeneous routing games and potential games
4
Tolling
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
16 / 27
Potential Game The abstract game is a potential game and admits a potential function V : ×N i=1 Ai → R if for all i ∈ JNK, V (ai , a−i ) − V (¯ai ,a−i ) = Ui (ai , a−i ) − Ui (¯ai , a−i ), ∀ai , ¯ai ∈ Ai and a−i ∈ ×j∈JNK\{i} Aj .
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
17 / 27
Potential Game The abstract game is a potential game and admits a potential function V : ×N i=1 Ai → R if for all i ∈ JNK, V (ai , a−i ) − V (¯ai ,a−i ) = Ui (ai , a−i ) − Ui (¯ai , a−i ), ∀ai , ¯ai ∈ Ai and a−i ∈ ×j∈JNK\{i} Aj . In general, finding a Nash equilibrium is difficult (Fabrikant et al., 2004; Papadimitriou, 2007; Roughgarden, 2010).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
17 / 27
Potential Game The abstract game is a potential game and admits a potential function V : ×N i=1 Ai → R if for all i ∈ JNK, V (ai , a−i ) − V (¯ai ,a−i ) = Ui (ai , a−i ) − Ui (¯ai , a−i ), ∀ai , ¯ai ∈ Ai and a−i ∈ ×j∈JNK\{i} Aj . In general, finding a Nash equilibrium is difficult (Fabrikant et al., 2004; Papadimitriou, 2007; Roughgarden, 2010). A minimizer of the potential function is a pure strategy Nash equilibrium of a potential game (Monderer & Shapley, 1996).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
17 / 27
Potential Game The abstract game is a potential game and admits a potential function V : ×N i=1 Ai → R if for all i ∈ JNK, V (ai , a−i ) − V (¯ai ,a−i ) = Ui (ai , a−i ) − Ui (¯ai , a−i ), ∀ai , ¯ai ∈ Ai and a−i ∈ ×j∈JNK\{i} Aj . In general, finding a Nash equilibrium is difficult (Fabrikant et al., 2004; Papadimitriou, 2007; Roughgarden, 2010). A minimizer of the potential function is a pure strategy Nash equilibrium of a potential game (Monderer & Shapley, 1996). Convergence results in many learning algorithms (e.g., fictitious play, myopic learning, multiplicative updates) rely heavily on potential functions (Drighès et al., 2013; Fudenberg 1998; Marden et al., 2009; Monderer & Shapley, 1996)
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
17 / 27
Existence of a Potential Function
Lemma (Necessary Condition) If the abstract game admits a potential function V ∈ C 2 , then " # X ∂ ˜θi ∂ ˜θj θ0 θ0 ` ((φe )θ0 ∈Θ ) − ` ((φe )θ0 ∈Θ ) = 0, θi e θ e ∂φe j e∈p1 ∩p2 ∂φe for all i, j ∈ JNK and p1 , p2 ∈ P.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
18 / 27
Existence of a Potential Function Lemma (Sufficient Condition) Assume that |Θ| = 2. If X e∈p1 ∩p2
∂ ˜θ2 θ1 θ2 ∂ ˜θ1 θ1 θ2 ` ` (φ (φ , φ , φ ) − ) = 0, e e e e e e ∂φθe 1 ∂φθe 2
for all p1 , p2 ∈ P, then V ((fpθ01 )p0 ∈P , (fpθ02 )p0 ∈P )
=
XZ e∈E
φθe 1
`˜θe 1 (u1 , φθe 2 )du1 +
0
Z
φθe 2
`˜θe 2 (φθe 1 , u2 )du2
0
Z
φθe 2Z φθe 1
− 0
0
∂ ˜θ1 ` (t, u)dtdu ∂u e
is a potential function for the abstract game.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
19 / 27
Finding a Nash Equilibrium Theorem (Sufficient condition for Nash equilibria) Assume that |Θ| = 2, and X ∂ ˜θe 2 (φθe 1 , φθe 2 ) − ∂ `˜θe 1 (φθe 1 , φθe 2 ) = 0, ` θ1 ∂φθe 2 e∈p1 ∩p2 ∂φe 0
for all p1 , p2 ∈ P. If f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a solution of the optimization problem min V ((fpθ01 )p0 ∈P , (fpθ02 )p0 ∈P ), X X s.t. fpθ1 = Fkθ1 and fpθ2 = Fkθ2 , ∀k ∈ JK K, p∈Pk
fpθ1 , fpθ2
p∈Pk
∈ R≥0 , ∀p ∈ P,
0
then f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium of the heterogeneous routing game.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
20 / 27
Characterizing All Nash Equilibria
Theorem Furthermore, assume that potential function V is a convex function. Then 0 f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium of the heterogeneous routing game if and only if it is a solution of the convex optimization problem
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
21 / 27
Outline
1
The non-atomic routing game
2
An equivalent finite-player game
3
Heterogeneous routing games and potential games
4
Tolling
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
22 / 27
Imposing Tolls If finding a Nash equilibrium in the heterogeneous routing game is numerically intractable, it might be unlikely for the drivers to find it.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
23 / 27
Imposing Tolls If finding a Nash equilibrium in the heterogeneous routing game is numerically intractable, it might be unlikely for the drivers to find it. 0
Driver of type θ ∈ Θ must pay a toll τ˜eθ ((φθe )θ0 ∈Θ ) for using an edge e ∈ E. P 0 For using path p ∈ Pk , she must pay τpθ (f ) = e∈p τ˜eθ ((φθe )θ0 ∈Θ ).
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
23 / 27
Imposing Tolls If finding a Nash equilibrium in the heterogeneous routing game is numerically intractable, it might be unlikely for the drivers to find it. 0
Driver of type θ ∈ Θ must pay a toll τ˜eθ ((φθe )θ0 ∈Θ ) for using an edge e ∈ E. P 0 For using path p ∈ Pk , she must pay τpθ (f ) = e∈p τ˜eθ ((φθe )θ0 ∈Θ ).
Nash Equilibrium 0
A flow vector f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium for the routing game with tolls if, for all k ∈ JK K and θ ∈ Θ, whenever fpθ > 0 for some path p ∈ Pk , then `θp (f ) + τpθ (f ) ≤ `θp0 (f ) + τpθ0 (f ) for all p 0 ∈ Pk .
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
23 / 27
Imposing Tolls If finding a Nash equilibrium in the heterogeneous routing game is numerically intractable, it might be unlikely for the drivers to find it. 0
Driver of type θ ∈ Θ must pay a toll τ˜eθ ((φθe )θ0 ∈Θ ) for using an edge e ∈ E. P 0 For using path p ∈ Pk , she must pay τpθ (f ) = e∈p τ˜eθ ((φθe )θ0 ∈Θ ).
Nash Equilibrium 0
A flow vector f = (fpθ0 )p0 ∈P,θ0 ∈Θ is a Nash equilibrium for the routing game with tolls if, for all k ∈ JK K and θ ∈ Θ, whenever fpθ > 0 for some path p ∈ Pk , then `θp (f ) + τpθ (f ) ≤ `θp0 (f ) + τpθ0 (f ) for all p 0 ∈ Pk . Consider the case that the tolls are type-independent (i.e., τ˜eθ1 (φθe 1 , φθe 2 ) = τ˜eθ2 (φθe 1 , φθe 2 ) = τ˜e (φθe 1 , φθe 2 )) which is a harder case.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
23 / 27
Imposing Tolls Proposition Assume that |Θ| = 2. The abstract game admits the potential function V ((fpθ01 )p0 ∈P , (fpθ02 )p0 ∈P ) =
φθe 1
XZ
(`˜θe 1 (u1 , φθe 2 ) + τ˜e (u1 , φθe 2 ))du1
0
e∈E
Z +
φθe 2
(`˜θe 2 (φθe 1 , u2 ) + τ˜e (φθe 1 , u2 ))du2
0
Z
φθe 2Z φθe 1
− 0
0
∂ ˜θ1 (` (t, u) + τ˜e (t, u))dtdu ∂u e
if ∂ τ˜e (φθe 1 , φθe 2 ) ∂ τ˜e (φθe 1 , φθe 2 ) ∂ `˜θ2 (φθ1 , φθ2 ) ∂ `˜θ1 (φθ1 , φθ2 ) − = e eθ1 e − e eθ2 e , θ2 θ1 ∂φe ∂φe ∂φe ∂φe for all e ∈ E. Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
24 / 27
Imposing Tolls
Corollary Assume that |Θ| = 2. The abstract game admits a potential function V ∈ C 2 if the imposed tolls are of the following from τ˜e (φθe 1 , φθe 2 ) = ce + ψe (φθe 1 + φθe 2 ) # Z φθe 2 " ˜θ2 ∂ `e (y , x) ∂ `˜θe 1 (y , x) − + ∂y ∂x 0
dq, θ
θ
x=q,y =φe 1 +φe 2 −q
where c ∈ R≥0 and ψe ∈ C 1 are arbitrarily chosen for all e ∈ E.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
25 / 27
Conclusions and Future Work
Conclusions Proved the existence of a Nash equilibrium in heterogeneous routing game. Characterized necessary and sufficient conditions for the existence of a potential function. Calculated tolls to ensure the existence of a potential function.
Future Work Study how heterogeneous populations can learn Nash equilibria; Extend the results to |Θ| > 2, or a continuum of types.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
26 / 27
Thank you.
Farokhi et al.
A Heterogeneous Routing Game
October 2, 2013
27 / 27
