# The Core

For this chapter, a coalition is *feasible* if its members are linked in appropriate ways. McCain (2013, p. 193) suggests that such a group will comprise a central agent, along with agents directly linked to that central agent. More complex definitions of “appropriate linkage” might be suggested (McCain, 2013, also briefly mentions a k-degrees-of-separation criterion as an alternative) but, for simplicity, we will adopt the former, single-link criterion. A feasible coalition structure (FCS) is a set of appropriately linked coalitions that may meet some other criteria as well. For this chapter, the coalition structure will not be feasible if certain kinds of members, including employees, proprietors, and the central agent to whom all are linked, are duplicated in more than one coalition.

For this chapter we consider an FCS game in which considerable solutions are determined as in Chapter 11. A candidate solution is a (feasible) coalition structure *Q = {S _{1},* S

_{2}, . . .} and a schedule of net payoffs

*{y}*from the coalitions to their members, with

*g*

_{г}-

_{e}

*s.y*(Q

_{i}= V*,Sj),*where V(Q,

*Sj)*is the value created by coalition

*Sj*if the coalition structure

*Q*is formed. In the context of considerable solutions, the value of

*C*will depend on the coalition structure

*Q*in which

*C*is imbedded via the Nash equilibrium among the separate coalitions. Suppose then that a deviation

*D*is proposed. That is,

*D*is a coalition among agents who are appropriately linked but

*D*0 Q. Let

*s(Q, D)*be the rational successor function

^{1}for the FCS game formed by the existing information-sharing links. If

*R = s(Q,*D),

*D [ R*and

*V(R*,D)

*> gi*[

*Dy,*then

*Q*is

*dominated*by

*R*via D. That is, a candidate solution is dominated if a group of agents can form a new coalition

*D*, forming a new coalition structure, which realizes more value among them than they derive from the existing coalition structure. The set of undominated candidate solutions comprises the

*core*of the game. If for

*Q*there is a payoff schedule

*{y}*such that the candidate solution {Q,{yJ} is undominated, then

*Q*is

*stable.*

Consider the following example. There are seven agents with links as

*Figure 12.1 Links among seven agents*

*Table 12.1 Some feasible coalition structures*

I |
1. {a,b,c,d}(16) 2. {e,f,g}(9) |

II |
1. {a,b,c,d,f}(25) 2. {e}(1) 3. {g}(1) |

III |
1. {a}(1) 2. {b}(1) 3. {d}(1) 4. {c,e,f,g}(16) |

IV |
1. {a}(1) 2. {b}(1) 3. {c,d}(4) 4. {e,f,g}(9) |

shown by the lines connecting nodes in Figure 12.1. For this example, coalitions can be formed only among agents linked to a common central agent, and no agent can be a member of more than one coalition. (These are links of the employee type, not of the customer type.) Assume further that the value of a coalition is the square of the number of agents in the coalition.

Table 12.1 shows four of a larger number of coalition structures feasible for this example, with coalitions indicated by brackets, {}, and the values of the coalitions following in parentheses.

The first three will be of primary interest, since they cannot be consolidated. By contrast, structure IV can be consolidated by merging {a}, {b}, and {c, d} producing structure I. Moreover, clearly, structure I and payouts of 4,4,4,4 for coalition I.1 dominates structure IV with any payoff schedule feasible for IV. In general, any structure comprising proper subsets of coalitions I, II, or III will be dominated (in this example) and thus not stable. Thus, we limit attention from this point to I, II, and III.

We will find that II is the only stable coalition structure. For II to be stable against I we require

and for stability against III,

For feasibility

Consider a payoff schedule

Inequalities (12.1)-(12.4) are satisfied, so the candidate solution comprising II with payoff schedule (12.5) is undominated and II is a stable coalition.

Similar reasoning will establish that I and III are not stable; and we have already noted that there are no other stable coalition structures in the example. Thus, for this example, recontracting specifies a unique stable coalition structure. (That will not be true in general without some powerful additional assumptions.) However, the payoff schedule is not uniquely specified. Consider the payoff schedule

This payoff schedule is also undominated. In this example, as we assumed in the previous chapter, the payoffs will be determined in part by bargaining power.