In this thesis, the atomic congestion games we consider are assumed to be splittable. In these games, we have a finite set of players, each player having a non-negligible demand to transfer from an origin to a destination. In contrary to unsplittable games, players can divide their demand among the different routes. In practice, these games can model companies having a stock to ship for one point to an other, and paradoxes similar as in Section0.1.2can occur, see for example Catoni and Pallottino(1991).
Another way to see atomic splittable games is to consider nonatomic games with coalitions. In this case, a subset of nonatomic users can decide to act in a centralized way, as a coalition.
1.3.1 Definition
In atomic splittable games, we are given a digraph D = (V, A) and K players. The players are identified with the integers 1, . . . , K and the set of all players is denoted by [K]. Each player k ∈ [K] has to send dk units of flow, his
demand, from an origin sk ∈ V to a destination tk ∈ V in the digraph.
For a given digraph and set of players with their demand and origin- destination pair, a strategy profile is a multiflow ~x = (x1, . . . , xK) ∈ RA×K+ such that for each player k, xk ∈ RA
flow for player k is an element of Fk = ( y ∈ RA+ : X a∈δ+(sk) ya− X a∈δ−(sk) ya = dk and X a∈δ+(v) ya= X a∈δ−(v) ya, ∀v ∈ V \ {sk, tk} )
and is referred as a feasible flow for player k. A strategy profile is then a feasible multiflow, i.e. an element of F1× · · · × FK.
Each player k has his own vector of cost functions ck= (ck
a)a∈A where for
each arc a the cost function cka(·) is a R+ → R+ function. We assume given
a set of allowable cost functions C in which ~c = (c1, . . . , cK) is taken. Since each player has his own vector of cost functions, the game is said to be with player-specific cost functions.
Remark 1.7. In the literature, this game is sometimes called a multiclass game. On the contrary, if c1 = · · · = cK for every ~c ∈ C, one speaks of a
single-class game. However, in order to avoid confusion between nonatomic and atomic games, we will use the terminology class only for nonatomic games.
We denote the total flow on an arc a by xa = Pk∈[K]xka. The cost
experienced by a player k is
Qk(~x) =X
a∈A
xkacka(xa).
The goal of this player consists in sending his dk units of flows while mini- mizing this cost.
The game we are interested in is defined by the digraph D, the set of players [K] with their demand, origin-destination pair, and set of cost func- tions.
A strategy profile ~x = (x1, . . . , xK) is a Nash equilibrium if for each player
k, we have Qk(~x) = min y∈FkQ k(y, ~x−k ), (1.2) where (y, ~x−k) = (x1, . . . , xk−1, y, xk+1, . . . , xK). The social cost of a multiflow is defined as
Q(~x) = X k∈[K] Qk(~x) = X k∈[K] X a∈A xkacka(xa).
1.3.2 Existence of a Nash equilibrium
The existence of a Nash equilibrium is ensured when the game is a convex game (Rosen, 1965). For an arc a and a player k, the cost function ck
a is a
per-unit cost. The contribution of this arc to the total cost of the player is xkacka(xa). To have a convex game, it is sufficient that the functions xk 7→
xkck
a(x) are convex. However we make the stronger assumption that the cost
functions ck
a are nondecreasing.
In this case, the result of Rosen (1965) holds, using the Kakutani fixed point theorem. The formulation of the proof in the framework of congestion games can be found inOrda et al. (1993).
Theorem 1.8 (Rosen(1965),Orda et al. (1993)). Consider an atomic split- table congestion game. Suppose that the cost functions ck
a are continuous
and nondecreasing for every arc a and player k. Then there exists a Nash equilibrium.
The definition of a Nash equilibrium can be reformulated, giving then new characterizations.
1.3.3 Characterizations of a Nash equilibrium
Throughout the thesis, the components of x ∈ RK
+ are denoted xk. Given
a K-tuple of continuously differentiable cost functions c = (c1, . . . , cK), we define for every k the marginal costeck: RK
+ → R+ by e ck(x) = ∂ ∂xk x kck(x) where x = X `∈[K] x`.
For the ease of reading, we drop the parenthesis over the derivatives and note ck 0 instead of (ck)0. To avoid any confusion, no index with a prime k0 will be
used in the thesis. We have then e
ck(x) = xkck 0(x) + ck(x).
The characterizations of a Nash equilibrium in nonatomic games still hold for atomic splittable games where the cost functions ck are replaced
by the marginal costeck. In particular, Proposition 1.6 can be reformulated. However, the following proposition is standard in this context and can be directly obtained by writing the optimality conditions of Equation (1.2). This characterization has been used in particular in Haurie and Marcotte
Proposition 1.9. Suppose that the cost functions ck
a are continuously dif-
ferentiable and nondecreasing. The multiflow ~x is a Nash equilibrium if and only if, for all k, it satisfies
X
a∈A
e
cka(xa)(yka− x k
a) ≥ 0, for any feasible flow y k
for player k, (1.3) where xa = (x1a, . . . , xKa) ∈ RK+.
Furthermore, Proposition 1.5 can also be transposed for atomic games. We reformulate it in a more convenient way that will be used in the following. Proposition 1.10. Suppose that the cost functions ck
a are continuously dif-
ferentiable and nondecreasing. The multiflow ~x is a Nash equilibrium flow if and only if, for all k, xk is a feasible flow for player k such that there exists πk ∈ RV with
˜
cka(xa) ≥ πvk− πuk for all a = (u, v) ∈ A,
˜
cka(xa) = πvk− π k
u for all a = (u, v) ∈ A such that x k a> 0.
1.3.4 Uniqueness of equilibrium
Uniqueness of the Nash equilibrium is not guaranteed in general. In their seminal paper, Orda et al. (1993) showed that the equilibrium is unique for symmetric players i.e. when all players have the same demand, origin- destination pair and cost functions. They gave a counterexample for more general games. The conditions for uniqueness can be divided into two types: restrictions on the set of allowable cost functions, and restrictions on the graph topology.
Applying the general result of Rosen (1965), uniqueness is guaranteed when we assume conditions on the cost functions, related to the notion of diagonal strict convexity. More recently, Altman et al. (2002) proved that when players have the same monomial cost functions of degree at most three, the equilibrium flow is unique.
Topological restrictions on the graph ensure uniqueness as well. Richman and Shimkin (2007) extended the result of Milchtaich (2005) holding for nonatomic graphs. They proved that when all players have the same origin- destination pair, the uniqueness property holds if and only if the graph is nearly parallel, see Figure 1.1. Bhaskar et al. (2009) extended the result to generalized parallel graphs, when all players have the same cost functions. The generalization to graphs with several origin-destination pairs is still an open problem. In particular, whether the results of Chapter2for nonatomic games can be extended to atomic games deserves future work.
1.3.5 Computation of an equilibrium
Since the equilibria for both nonatomic multiclass and atomic games are solu- tions of a variational inequality, the methods introduced in Section1.2.5hold for atomic games, when replacing the cost functions ck
a by the marginal costs
˜
cka. In particular, the algorithms given in Chapters 3 and 4 can be adapted to give an equilibrium flow for atomic games with affine cost functions.
More specific algorithms exist for games with a finite number of players, beginning with Rosen (1965). The algorithm introduced by Rosen (1965) needs restrictive conditions, but the dynamic is shown to converge to an equilibrium. Algorithms have been designed in other specific cases, using supermodularity or monotonicity, see Altman et al. (2006) for a review. In particular,Altman et al.(2001) andBoulogne et al.(2002) proved that some basic algorithms converge to an equilibrium in simple graphs with linear cost functions.