Problem Formulation

Consider a group of N travellers, T R = {tr1, tr2, ., tri, ., trN}, in which a traveler tri

is contemplating a trip from an initial location stri _{to a final destination f}tri_{. Traveler}

tri can make the trip by following path ptri ∈ Ptri. Ptri denotes the feasible paths

between stri _{to f}tri _{that are available to tr}

i. There is a certain cost value ctri associated

with each path ptri_{. c}tri _{is a function of several factors including transportation modals}

used to complete the trip (e.g, public vs private), environmental conditions along the path, as well as traffic conditions. The notion of cost in this work is multi-aspect, in the sense that it explicitly quantifies monetary costs, temporal costs, safety costs, and comfort costs, to the extent that a multi-criteria cost formulation is employed to facilitate the trip route optimization process. The concept of doctrine is introduced to capture travellers’ preferences, demands, and constraints. For each path ptri_{, we use}

γtri _{to denote traveler tr}

i’s doctrine for this path. Γtri denotes the set of all doctrines

associated with tri’s feasible paths Ptri. The traveler tri’s desirable route can then

be stated as

ptri

Opt = M in

∀ptri∈Ptri

where Υ represents a selfish selection process by which an optimum path is chosen. ptri

k represents a possible path, and P

tri _{represents the set of all paths. A selfish}

selection process is a process in which the drivers attempt to maximize their benefits regardless of the impact of their chosen actions on the system- negatively or positively. For N travellers, their interaction process and their travelling decisions must be formulated such that optimizing traveller tri’s plan does not negatively impact other

travellers’, tr−i, chosen routes. To deal with these kinds of interactions, we define a

team trip planning game. The team trip planning game, Σ, is a 4-tuple (G, Γtri_{, s}tri_,

ftri_{) such that:}

• G is a directed acyclic graph that includes all possible routes, Ptri_{. For each}

traveller tri, stri and ftri are defined.

• Γtri is a traveller selfish assessment process which assigns a non-negative value

to each road segment, lpk

j , denoting the cost of this segment.

A solution set of paths Ptri _{is defined for the game Σ such that each path, p}tri

k

connecting stri _{and f}tri_{, is composed of connected links l}p tri k j | l ptri_k j ∈ Lp tri k . The cost of these paths is ctri pk = Σ |Lptrik | j=1 l ptri_k

j ∗ξr, where ξris a weight value that reflects the degree of

preference that each traveller has for a certain path. For the mobility planning game Σ, traveller tri has an ordered set of strategies: Ptri : Ptri =ptr1i, p

tri 2 , , , , ptrni in which ptri k  p tri −k : p tri k = p tri

Opt for tri . Travellers own road segments, l

pk

j s, according

to a mapping function o : Lpk → T R such that o(lpk

j ) = tri (i.e., road segment lpjk is

chosen by traveler trias a possible path). For any path, pk, o(L

pk

S ) is a set of connected

segments satisfying certain travelling criteria and owned by a group of travellers. Suppose that for each traveller tri there is a travelling cost ctrpki and mobility plan-

ning reward Dtri

R_pk. Both cost and reward values are non-negative,

n ctri pk, D tri R_pk ∈ R+ o ,

and they belong to the same currency domain. For example, if the reward is monetary, the cost should be also monetary. For Dtri

R_pk > c tri

pk, traveller tri is able to generate

profit. Furthermore, assume that a group of travellers, i.e., coalition S ⊂ N , through certain agreements, can generate a profit using only paths PS owned by the coalition through certain agreements; for example, the set of paths P is owned by a coalition S if o(P ) ⊂ S. A situation in which there exists a successful team trip plan is called a team trip planning game (N, vσ). vσ is a characteristic function for this game, which

is computed as follows: vσ(S) = ζ(cS P, DRS) if S owns P S _{∈ Σ, and ζ(c}S P, DRS) > 0 0 otherwise (3.2)

Where ζ is the mapping function which associates every coalition S with a non- negative value according to cS

P and DRS.

The cost of each path, ctri

pk, for traveller tri, is not independent from its reward

value, Dtri

R_pk. Thus, it is necessary to define a positive function, for the coalition S,

that defines the relationship between cS_P and DP_RS. Furthermore, assume that the

ctri P 6= c tr(−i) P and D tri R_pk 6= D tr(−i)

R_pk . This game should be considered a game with Non-

Transferable Utility (N-TU). In this cooperative game, the solution, i.e., the core, is described as follows: core = ( x ∈ RN|X i∈N xi(N ) = v(N ) and X i∈S xi(S) ≥ v(S) ∀S ∈ 2N \ ∅ ) (3.3) whereP

i∈Nxi(N ) = v(N ) guarantees the efficiency of the outcome, and

P

i∈Sxi(S) ≥

v(S) ∀S ∈ 2N _{\ ∅ guarantees the stability of the game.}

Furthermore, each traveller, tri, has a set of strategies based on which the decision

of the best path, ptri

k , can be obtained. For each traveller, tri, there is a finite non-

u, such that u: Ptri → R. Paths in Ptri _{are associated with preference relation %}

such that u(ptri

i ) ≥ u(p tri j ) if p tri i % p tri

j . Cost functions and rewards can correspond to

utility values, and the ordering of Ptri _{is conducted through the knowledge of %}(tri)_.

Moreover, Ptri _{can be expanded to include a non-route related actions. The chosen}

departure and the expected arrival times for a trip can be considered as strategic actions, which might change the outcome of the game. For traveller, tri, Ptri can

be ordered according to %(tri) _{such that p}tri

k is better than p tri k+1 iff u(p tri k ) ≥ u(p tri k+1).

This game is described as (N, P, u) game.