To Participate or Not in a Coalition in Adversarial Games

Cooperative Ressource Sharing With Adamant Player

“…γ represents the cost factor, λ i represents the influence factor of i th player, and, a i represents the action of i th player with a i ∈ (0, â) for some n/γ < â < ∞ which ensures the existence of a unique Nash Equilibrium ( [5]). The first component of equation ( 1) is the fraction of resource allocated to player i and the other component represents the cost.…”

“…This is the problem setting and our aim is to study the coalitions/partitions that emerge out successfully (at an appropriate equilibrium), when the C-players (henceforth referred as players) seek opportunities to come together in a noncooperative manner. There is a brief initial study of this problem in [5], for the special case when players only form grand coalition, i.e., when P = {{0}, {1, • • • , n}}. For this case, it has been shown that:…”

“…We say a partition P ′ is (strictly) better than partition P, represented by the symbol P ′ ≺ P, if every coalition of the latter is a subset of a coalition of the former (with at least one of them being a strict subset), i.e., if P ′ = P, and, for all S ∈ P ∃ S ′ ∈ P ′ such that S ⊂ S ′ . (5) Note that the size (number of coalitions) of the better partition is strictly smaller than that of the other; in other words, there exists at least two coalitions S 1 , S 2 ∈ P such that S 1 ∪ S 2 ⊂ S for some S ∈ P ′ .…”

“…In [5] authors illustrated that the sum utility of the C-players improve significantly, when all players come together to form a grand coalition (as n increases). However we will see in this paper that for n > 4, the only NE-partition is ALC (all alone).…”

“…Remarks: In [5], authors defined BoC (benefit of cooperation) as the normalized improvement in sum of utilities that the players achieve at GC in comparison with that achieved when they compete alone. They showed that BoC increases significantly as n increases ([5, Lemma 3]).…”