To give a demonstration of the influence of characteristics of graph-based games, we had to select a promising sample type of a coalitional game, which offers various different characteristics that can be analysed in a complexity-theoretic context. For a start, shortest path games are a promising candidate, because there are two different types of shortest path games in the literature, which have quite different properties (e.g. expressive power). As indicated in the introduction, we introduce shortest path games also for another reason: They are similar to other graph-based games in the literature, which have been analysed from a computational perspective.
Shortest path games are a class of coalitional games that has not been considered, to our knowledge, in the context of computational complexity yet. There are two different game-theoretic approaches to the problem of shortest path problems, which have been introduced by Fragnelli, Garc´ıa-Jurado, and M´endez-Naya [33] and Voorneveld and Grahn [58].
We first introduce a strongly restricted variant of a shortest path game to give a flavour of the type of game representation that we are going to consider. Based on this basic pattern of how shortest path games are modeled, we present both variants from the literature and furthermore introduce a wider range of different variants including threshold versions of shortest path games. So, we introduce a framework of shortest path games, where each game possesses specific characteristics, which we are going to present and analyse in this section.
3.5.1
Basic Concepts of Shortest Path Games
Before we introduce the different variants of shortest path games, we want to give a detailed account of the basic construction ideas of this type of coalitional game. We start by giving an overview of a shortest path problem and the corresponding coalitional game.
The shortest path problems considered here are limited to a finite set of players, where each player owns arcs or vertices in a finite network. There are costs associated to the use of each arc or vertex and there are rewards involved with the transport of an item from the source to the sink of the network. The variant of a shortest path game that we are going to introduce now, borrows its basic outline from the original definition of shortest path games by Fragnelli et al. [33] and Voorneveld et al. [58], whereas we decided to restrict the model considerably.
Definition 3.5.1. Ashortest path pre-problemΣis a tuplehV, A, So, Sii, where
• (V, A)is a directed graph with two special elements, namely thesource(So) and thesink(Si)
• We have a setA⊆V ×V of directed arcs in the network.
Definition 3.5.2. ApathP (So → Si) in a directed graph is a sequence of vertices such that from each of its vertices there is an arc to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertexSo, called its source, and a last vertexSi, called its sink.
Now we present a minimal class of shortest path games, calledVSPG(standing for value shortest path game). LetΣbe a shortest path pre-problem, where the arcs of the graphhV, Aiare owned by a finite set of playersN according to a total bijective mapo:A→N, such thato(a) = imeans that playeri
is the owner of edgea. Hence, every arc is assigned to exactly one player. We have a cost mapcthat assigns to every arca∈Aa non-negative real numberc(a)(c:A→R+
0).
Given the simple structure ofVSPG, a path owned by players of coalitionS is simply a sequence of vertices(v1, v2, ..., vm)such thatv1 =So,vm =Siand for eachk ∈ {1, ..., m−1}the arc(vk, vk+1)
is owned by playerik ∈S. LetP(S)denote the collection of all paths owned by coalitionS. For any
pathP we denote byo(P)the set of owners of the arcs inP.
Suppose that the transportation of a certain good from the source to the sink ofΣproduces an incomer
and a cost given by the length of the path that was used. In particular, the costs associated to a pathp= (v1, v2, ..., vm) ∈ P(S)are defined as the sum of the costs of its arcs: cost(p) = Pm
−1
k=1 c(vk, vk+1).
Note that a coalitionS ⊆ N can transport the good only through paths owned by it and a pathP is basically owned by a coalitionS ifo(P)⊆S.
Obviously, if a coalitionShas to find a path from source to sink, it will choose among its alternatives inP(S)the path with minimal costs. Define for eachS∈2N \ {∅}:
c(S) =
minp∈P(S)cost(p) ifP(S)6=∅
∞ otherwise
We overloadedcat this point, but it will be obvious from the context which function is meant. Note that the shortest path can be computed in polynomial time using for instance the well-known algorithm of Dijkstra [25].
We now put all the information necessary to describe a shortest path game together, and introduce the notion of ashortest path game environmentσ, which is any such tuplehN,Σ, o, c, ri.
Remember that a coalitional game with transferable utility can be represented as a pair(N, v), where
N is a finite set of players andv : 2N \ {∅} →
R is a function that assigns to each coalition S ∈
2N \ {∅} its value v(S) ∈ R. We can associate withσ the T U-game hN, vσi whose characteristic
functionvσis given by:
vσ(S) =max{r−c(S),0}=
r−c(S) ifSowns a path inΣandc(S)< r
0 otherwise for everyS ⊂N. Hence, the environment σ and the definition of vσ is our game representationGN. The coalitional
game associated withGN has the following underlying intuition: if a coalitionS ∈2N\{∅}transports
its goods from source to sink, it will receive a total reward r and incur costs c(S), the costs of the shortest path owned byS. Ifr−c(S) > 0, coalition Smakes a profit. Ifr−c(S) ≤ 0, coalitionS
can generate profit zero by simply doing nothing. Therefore, coalitionS’s profit ismax{r−c(S),0}. Let’s sum it up: A shortest path game of type VSPGis any such game hN, vigenerated by a game representationGN as defined above.
3.5.2
Notions and Properties
In this subsection we give some basic notions and definitions, necessary to reason about properties of shortest path games.
Definition 3.5.3. Given a sourceSoand a sinkSi, a pathP (So → Si) is ashortest pathif there is no other pathP0 such thatc(P0)< c(P), wherecis the cost function.
Definition 3.5.4. A pathP inΣ is said to be aprofitable path in σ if r > cost(P), wherer is the reward.
Definition 3.5.5. A shortest-veto (briefly s-veto) player of a shortest path game(N, v)is a player in
N who owns at least one edge (vertex) of every shortest path (inΣ).
To avoid trivial case, it is often assumed that0≤ c(N) < r(N), where r(N)is the maximal reward with respect toN. This assumption implies that
• we have a strictly positive reward;
• the valuev(N)of the grand coalition is positive;
This assumption also avoids the zero game. In some papers it is even assume that0< c(N)< r(N)
to makes sure that there are indeed costs to divide over the players. We do not apply this assumption, becausec(N) = 0offers a lot of interesting scenarios for shortest path games, as will be shown later.