coalitionalgame {ClaimsProblems}

Coalitional game associated with a claims problem

Description

This function returns the pessimistic and optimistic coalitional games associated with a claims problem.

Usage

coalitionalgame(E, d, opt = FALSE, lex = FALSE)

Arguments

E

The endowment.

d

The vector of claims.

opt

Logical parameter. If opt = TRUE, both the pessimist and optimistic associated coalitional games are given. By default, opt = FALSE, and only the associated pessimistic coalitional game is computed.

lex

Logical parameter. If lex = TRUE, coalitions of claimants are ordered lexicographically. By default, lex = FALSE, and coalitions are ordered using their binary representations.

Details

Let E\ge 0 be the endowment to be divided and d\in \mathcal{R}^n the vector of claims with d\ge 0 and such that \sum_{i=1}^{n} d_i\ge E,\; the sum of claims exceeds the endowment.

For each subset S of the set of claimants N, let d(S)=\sum_{j\in S}d_j be the sum of claims of the members of S and let N\backslash S be the complementary coalition of S.

Given a claims problem (E,d), its associated pessimistic coalitional game is the game v_{pes}:2^N\rightarrow \mathcal{R} assigning to each coalition S\in 2^N the real number:

v_{pes}(S)=\max\{0,E-d(N\backslash S)\}.

Given a claims problem (E,d), its associated optimistic coalitional game is the game v_{opt}:2^N\rightarrow \mathcal{R} assigning to each coalition S\in 2^N the real number:

v_{opt}(S)=\min\{E,d(S)\}.

The optimistic and the pessimistic coalitional games are dual games, that is, for all S\in 2^N:

v_{opt}(S)=E-v_{pes}(N\backslash S).

An efficient way to represent a nonempty coalition S\in 2^N is by identifying it with the binary sequence a_{n}a_{n-1}\dots a_{1} where a_i=1 if i\in S and a_i=0 otherwise. Therefore, each coalition S is represented by the number associated with its binary representation: \sum_{i\in T}2^{i-1}. Then coalitions can be ordered by their associated numbers.

Alternatively, coalitions can be ordered lexicographically.

Given a claims problem (E,d), its associated coalitional game v can be represented by the vector whose coordinates are the values assigned by v to all the nonempty coalitions. For instance. if n=3, the associated coalitional game can be represented by the vector of the values of all the 7 nonempty coalitions, ordered using the binary representation:

v = [v(\{1\}),v(\{2\}),v(\{1,2\}),v(\{3\}),v(\{1,3\}),v(\{2,3\}),v(\{1,2,3\})]

Alternatively, the coordinates can be ordered lexicographically:

v = [v(\{1\}),v(\{2\}),v(\{3\}),v(\{1,2\}),v(\{1,3\}),v(\{2,3\}),v(\{1,2,3\})]

When n=4, the associated coalitional game can be represented by the vector of the values of all the 15 nonempty coalitions, ordered using the binary representation:

v = [v(\{1\}),v(\{2\}),v(\{1,2\}),v(\{3\}),v(\{1,3\}),v(\{2,3\}),v(\{1,2,3\}),v(\{4\}),

v(\{1,4\}),v(\{2,4\}),v(\{1,2,4\}),v(\{3,4\}),v(\{1,3,4\}),v(\{2,3,4\}),v(\{1,2,3,4\})]

Alternatively, the coordinates can be ordered lexicographically:

v=[v(\{1\}),v(\{2\}),v(\{3\}),v(\{4\}),v(\{1,2\}),v(\{1,3\}),v(\{1,4\}),v(\{2,3\}),\dots

\dots v(\{2,4\}),v(\{3,4\}),v(\{1,2,3\}),v(\{1,2,4\}),v(\{1,3,4\}),v(\{2,3,4\}),v(\{1,2,3,4\})]

Value

The pessimistic (and optimistic) associated coalitional game(s).

References

O’Neill B (1982) A problem of rights arbitration from the Talmud. Math Soc Sci 2:345–371.

See Also

setofawards

Examples

E=10
d=c(2,4,7,8)
v=coalitionalgame(E,d,opt=TRUE,lex=TRUE)
#The pessimistic and optimistic coalitional games are dual games
v_pes=v$v_pessimistic_lex
v_opt=v$v_optimistic_lex
v_opt[1:14]==10-v_pes[14:1]