coalitionalgame {ClaimsProblems}R Documentation

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 E0E\ge 0 be the endowment to be divided and dRnd\in \mathcal{R}^n the vector of claims with d0d\ge 0 and such that i=1ndiE,  \sum_{i=1}^{n} d_i\ge E,\; the sum of claims exceeds the endowment.

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

Given a claims problem (E,d)(E,d), its associated pessimistic coalitional game is the game vpes:2NRv_{pes}:2^N\rightarrow \mathcal{R} assigning to each coalition S2NS\in 2^N the real number:

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

Given a claims problem (E,d)(E,d), its associated optimistic coalitional game is the game vopt:2NRv_{opt}:2^N\rightarrow \mathcal{R} assigning to each coalition S2NS\in 2^N the real number:

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

The optimistic and the pessimistic coalitional games are dual games, that is, for all S2NS\in 2^N:

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

An efficient way to represent a nonempty coalition S2NS\in 2^N is by identifying it with the binary sequence anan1a1a_{n}a_{n-1}\dots a_{1} where ai=1a_i=1 if iSi\in S and ai=0a_i=0 otherwise. Therefore, each coalition SS is represented by the number associated with its binary representation: iT2i1\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)(E,d), its associated coalitional game vv can be represented by the vector whose coordinates are the values assigned by vv to all the nonempty coalitions. For instance. if n=3n=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})]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})]v = [v(\{1\}),v(\{2\}),v(\{3\}),v(\{1,2\}),v(\{1,3\}),v(\{2,3\}),v(\{1,2,3\})]

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

[Package ClaimsProblems version 0.2.1 Index]