U00 100 100 100 100j

While the Shapley value can address the allocation of initial endowments between world regions, it cannot address the chain of allocations from the world regional levels down to country and subcountry levels given the allocations, each level of which has its own players with varying levels of bargaining power. Methodology is developed here to capture the bargaining activity through several administrative or other political power levels. To account for negotiations over conservation of PGRFA being held at several levels, we assume that fictitious delegates are elected to represent each world region, each country, and one subcountry level, which can be states, provinces, lobbyist groups, or even firms. The set of players at the lowest level is denoted as/V = {l,..i,...,«}. The world regions and the countries they belong to are denoted by the level structure B = {v51,S2}, where BX = {RX,... ,Rm} is the set of all world regions and

B2 = |Ci,...,Cg| the set of all countries that describe iV'sa priori coalition structure. Given this coalition structure framework, if we assume that in each world region j, Rj eB^ , a delegate is selected to represent the coalition, the bargaining situation is a problem of how to divide the conservation funds and can be formally represented by the quotient (M,V) = (N,U) IBl , where (M,V) is a game with a set of players (world region representatives) M = {i,..., j,...,m} and its characteristic function is given by:

Considering the world region b, its marginal contribution is positive for three coalitions: (b,e), (a,b,c), and (b,c,d). Then applying formula given in (3), we have

(2-l)!(S-2)! (3-l)!(5-3)! (3-l)l(5-3)! Ill 7 , Vb 5! 5! 5! 20 30 30 60

Applying the formula for all world regions, the Shapley value of this game is thus i. 30 ' 60 ' 5 ' 5 ' 20 J

Thus, a reasonable expectation for the jth world region is the amount <Pi(M,V), which is the element in the Shapley value corresponding to player j in game (M ,V), and would be the value normally expected by this world region.

Given the allocations of the permits among world regions, subregional allocations take place. Obviously, the payoff a player (i.e., a firm) receives depends crucially on the definition of the bargaining relationships between countries and the lower level players, which we can denote as "firms" for convenience. To address this topic, we propose in the following subsections three bargaining and payoff principle scenarios that are based on the same bargaining relationship among world regions, but on different bargaining relationships among countries and among firms. The scenarios depend on the capacities of threat (bargaining power) of some countries over other countries within the same world region or on the capacity of threat of a firm toward other firms within the same country. Section 2.1 presents a payoff function XF| for a base scenario in which we consider only the amount each lowest-level player can obtain on its own. Section 2.2 presents a payoff function that captures the "lobbying game" between countries as played out in the IU. Section 2.3 defines a payoff function motivated by the subsidiarity principle. In these three scenarios, to determine the final distribution of initial endowments in a manner that takes into account interest firms of various types, various countries, and various world regions, we assume a three-step process. In the first step, the world regions bargain with each other to determine the division of the surplus. In the second step, countries belonging to the same world region bargain with each other over the allocation received by their world region. Finally, the firms of a given country divide among themselves the share the country receives. However, because the Shapley value formula in Eq. (3) does not take into account the fact that the lowest-level players are organized a priori into level structure B, the model must be extended for determining the sharing of conservation funds.

2.1 Null threat and egalitarian principle

Under the "null threat," a player has no power to negotiate with other players in the same coalition. Hence, in this minimum information game, we need only account for the amount a player can obtain by himself (that is, his value in the characteristic function), and the amount the coalition to which he belongs to can obtain. According to the egalitarian principle, no player has a greater opportunity than another player to form a minimal winning coalition. Hence, every player has the same power, and the symmetry axiom of the Shapley value implies that all players will obtain the same share. For example, if the United States is not a winning coalition (i.e., receives no payoff) at the international level, the value of the characteristic function for the U.S. regions is always the same, and the United States receives an egalitarian division of the PGRFA funds.6 In sum, the allocation rule proposed in this section supposes that each country [and each firm] passively accepts the funds proposed by the IU.

Formally, we model this bargaining procedure within world regions by only considering the amount U(k) that every coalition of countries K

(such as K cz Rj, K eB ) can obtain on its own. For this game, we can define the sub-game (Nr.,Wr.) on the world region R., where NR

represents the set of countries in world region i? .. The characteristic function of this sub-game is given by :

Eq. (4) says that the empty set is worth nothing and (6) says that the amount a coalition K can itself obtain in the sub-game, defined on the world region that K belongs to, is the same amount K can obtain in the initial game. Given these equations, a reasonable expectation for the country C/, c Rj is the amount (Ph[^R .,WR . j, which is the Shapley value of player h in the sub-game (Nr.,Wr.).

We can use the same reasoning to divide the amount received by the country C/, c Rj among the lowest-level players—we can called these

"firms" for convenience—considered in the model. Thus, each firm ieC/j c Rj can obtain the amount <Pi{Nch^Ch \ which is the Shapley value of firm / in the sub-game defined on the country Ch c Rj.

6 In our simple game in Eq. (2), a losing coalition is one that has less than ¡3% of the world population. Hence, every subcoalition of this coalition is also a losing coalition. However, this situation does not imply that a loosing coalition will obtain nothing. Instead, the outcome depends on the definition of the game. For instance, two losing coalitions may become winners if they act together.

For this bargaining game, we can define a payoff function Y1, in which the share a firm ieQc R.will receive is (see Chantreuil, 2000):

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