The term cooperative game theory is a phenomenon where it is assumed that utility or level of satisfaction is transferable between players, implying that a single unit is equivalent to the utility for a player. It is worth to note that this phenomenon can be explored in the bargaining problem. For the purpose of this essay, I will discuss the solution concepts as it relates to the terms core and sharply valley. Basically, a solution refers to the expected outcome of an event.
The core comprises of a set of utility vectors that are not susceptible to improvement on any given coalition. It implies that it cannot be improved on by any coalition. For instance if we are given a set of values such or members of a set say x such that x = (xa, xb, xc)) where xa ≥ v(a), xb ≥ v(b), xc ≥ v(c), and given that v represents some value of a known vector scalar. The set members of the whole x, when crossed and added, the constant scalar v ought to be maintained as in the commutative property. For instance, xa+xb≥ v(ab), xa+xc≥ v(ac), xb+xc≥ v(bc), shows that when crossed, the scalar v is maintained throughout (Moulin 19).
This introduces a new concept such that when the constants of the scalar v are equated, the result is unity implying that the core is empty. Practically it holds in this respect, v(a)=v(b)=v(c)=0, v(ab)=v(ac)=v(bc)=v(abc)=1. But when the values in the scalar constant are unity, then it holds that the linear relation provided above would make the core be unity. If v(a)=v(b)=v(c)= v(bc)= 0, v(ab)=v(ac)=1=v(abc)=1, then it follows that the core is single valued in this respect. A numerical example, given that three people (“Tom, Mary and Paul”) are given real numerical values of v(a)= 6, v(b)=3, v(c)=0, the cross cumutative property v(ab)=18, v(ac)=6 and v(bc)=o. conclusively, it would imply that v(abc)=18. From the definition, this means that the core solution has many values (Peleg 200).
On the other hand, the term Shapley value can be pegged on some criteria, namely in terms of efficiency, where the solution is a representation of the payoff of the whole coalition called the grand coalition. Secondly, is the property of anonymity, implying that the solution would depend solely on the anonymous game. A null player has a null or zero outcome or result, and lastly the additive property, where if a solution is got from joining two games, then the outcome represents a sum of the two individual games. With these properties in mind, it would be wise to note that Shapley value assigns a value of an arithmetic mean over the others. For instance if a basis of the game is assumed to be vR,c R is a member of the set of N, implicitly there is a coalition of r subset members. Therefore, all the other null players would be assigned a zero value. Conclusively, it implies that there can be only one Shapley value solution for all games that uphold the properties of efficiency, anonymity, nullity and additivity. From the abstract values stated above herein, an equal number of games can be obtained from the characteristic functions by either subtracting or adding an equal share of the number of basis. Numerically, if v(a)=v(b)=v(c)=0, v(ab)=v(ac)=v(bc)=v(abc)=1, where v is a scalar vector, then from adding the scalars, the Shapley solution will be given by SVa = SVb = SVc = 1/3.
Moulin, Hervés. Cooperative microeconomics: a game-theoretic introduction. Princeton University Press, 1995.
Peleg, Bezalel. "An axiomatization of the core of cooperative games without side payments." Journal of Mathematical Economics 14.2 (1985): 203-214.