We provide several properties for union values, based on the properties defined in van den Brink and Dietz (2014). Some properties correspond to properties discussed before, but are adapted to fit union values. Let (N, v, P)∈ GP be a game with coalition structure andγ a union value. The first property is discussed before and implies that all costs are recovered.
EF Efficiency: the cost of the grand coalition is exactly allocated amongst the unions inP.
For allk∈M we have thatP
k∈MγPk(N, v, P) =v(N), for all (N, v, P)∈ GP. In section 3.2 we presented the definition of a null agent. The following property considers the effect of deleting a null agent from a union for the cost allocation. The null agent out property was introduced by Derks and Haller (1999) for TU games.6
NAO Null Agent Out: deleting a null agent from any union has no effect on the cost shares of the unions.
If agenti∈Pk is a null agent and|Pk| ≥2, thenγ(N, v, P) =
γ(N\ {i}, cN\{i},(P \ {Pk})∪ {Pk\ {i}}), for all (N, v, P)∈ GP.
Note that the unionPkcannot be a singleton, since then the union would not exist once
agentileaves, therefore the condition|Pk| ≥2 is required. Due to this condition, agents
can be excluded from the game, but not entire unions. Next we consider symmetric unions. Unions k, l ∈ M are symmetric if |Pk| = |Pl| and there exist a permutation
πk = (π1, ..., π|Pk|) on Pk and a permutation π
l = (π
1, ..., π|Pl|) on Pl such that v(S∪
{πik}) = v(S ∪ {πil}) for all i ∈ 1, ...,|Pk| and S ⊆ N \ {πik, πjl}. So two unions are
symmetric if the agents of the unions can be ordered such that there is a one to one symmetry correspondence between the agents in one union and the other union in game (N, v, P)∈ GP.
SYM Symmetry: symmetric unions are allocated the same cost share.
If unionsk, l ∈Mare symmetric, thenγPk(N, v, P) =γPl(N, v, P), for all (N, v, P)∈
GP.
Strong monotonicity is adapted from the definition of Young (1985) and states that if the marginal contributions of all agents of a union in a gamev are at least as high as in a gamew, then this union gets a cost share in game vthat is at least as high as in game
w. Marginality is a weaker version of strong monotonicity.
6
SM Strong Monotonicity: if the marginal contributions for all agents in union Pk
in gamevare at least as high as the marginal contributions for all agents in union
Pk in gamew, then the cost shares for the union in gamev should be at least as
high as in gamew .
For alli∈Pk,k∈M andS ⊆N\ {i}such that v(S∪i)−v(S)≥w(S∪i)−w(S)
it holds thatγPk(N, v, P)≥γPk(N, w, P), for all (N, v, P),(N, w, P)∈ GP. MR Marginality: if the marginal costs for all agents in union Pk in game v are
equivalent to the marginal costs for all agents in union Pk in game w, then they
should obtain equal cost shares.
For alli∈Pk,k∈M andS ⊆N\ {i}such that v(S∪i)−v(S) =w(S∪i)−w(S)
it holds thatγPk(N, v, P) =γPk(N, w, P), for all (N, v, P),(N, w, P)∈ GP. For games with a priori unions some extra properties with respect to collusion are impor- tant. In van den Brink and Dietz (2014) two types of collusion properties with respect to union values are considered, collusion between agents and collusion between unions. The first concept is based on two agents colluding defined by the proxy agree- ment described in Haller (1994). If agenti∈Nacts as a proxy agent for agentj∈N\{i}
then we define the characteristic function (N, vij)∈ G instead of v, as follows
vij =
v(S\ {j}) ifi /∈S
v(S∪ {j}) ifi∈S. (3.14)
So the meaning of agent i being a proxy agent for an agent j is that an agent j only incurs its cost in a coalition when agentiis also in that coalition. As long as agent j is in a coalition without agenti, agentj can be seen as a null agent. This brings us to the agent collusion neutrality axiom.
ACN Agent Collusion Neutrality: collusion of two agentsi, j belonging to the same union, does not change the cost share of this union.
For all i, j ∈Pk and k∈ M it holds that γPk(N, v, P) =γPk(N, vij, P), for every (N, v, P)∈ GP.
Now we consider collusion between unions instead of between agents. Collusion between unionsPk andPlis described by the union of the these two unions, such that we obtain
the partitionPkl= (P\ {Pk, Pl} ∪ {Pk∪Pl}). Without loss of generality we may assume
thatk < l, such that we may reorder the partitionPklwithPkkl=Pk∪Pl andPhkl=Ph
for all h∈M\ {k, l}. Note that unions Pk,Pl do not have to be consecutive.
UCN Union Collusion Neutrality: collusion of two unionsPk, Pldoes not change the
For all k ∈ M it holds that γPk(N, v, P) +γPl(N, v, P) = γPk(N, v, P
kl), for all
(N, v, P)∈ GP.
Note that collusion of agents or unions changes the cardinality of the agent set or the union set, respectively.