This function returns the pessimistic and optimistic coalitional games associated with a claims problem.
coalitionalgame(E, d, opt = FALSE, lex = FALSE)The pessimistic (and optimistic) associated coalitional game(s).
The endowment.
The vector of claims.
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.
Logical parameter. If lex = TRUE, coalitions of claimants are ordered lexicographically. By default, lex = FALSE, and coalitions are ordered using their binary representations.
Let \(N=\{1,\ldots,n\}\) be the set of claimants, \(E\ge 0\) the endowment to be divided and \(d\in \mathbb{R}_+^N\) the vector of claims such that \(\sum_{i \in N} d_i\ge E\). For each coalition \(S\in 2^N\), let \(d(S)=\sum_{j\in S}d_j\) and \(N\backslash S\) be the complementary coalition of \(S\).
Given a claims problem \((E,d)\), its associated pessimistic coalitional game is the game \(v_{pes}:2^N\rightarrow \mathbb{R}\) assigning to each coalition \(S\in 2^N\), $$v_{pes}(S)=\max\{0,E-d(N\backslash S)\}.$$
Given a claims problem \((E,d)\), its associated optimistic coalitional game is the game \(v_{opt}:2^N\rightarrow \mathbb{R}\) assigning to each coalition \(S\in 2^N\), $$v_{opt}(S)=\min\{E,d(S)\}.$$
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: \(\sum_{i\in S}2^{i-1}\). 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:
\(v = [v(\{1\}),v(\{2\}),v(\{1,2\}),v(\{3\}),v(\{1,3\}),v(\{2,3\}),v(\{1,2,3\}),v(\{4\}),\dots\)
\(\dots,v(\{1,4\}),v(\{2,4\}),v(\{1,2,4\}),v(\{3,4\}),v(\{1,3,4\}),v(\{2,3,4\}),v(\{1,2,3,4\})].\)
Alternatively, the coordinates can be ordered lexicographically:
\(v=[v(\{1\}),v(\{2\}),v(\{3\}),v(\{4\}),v(\{1,2\}),v(\{1,3\}),v(\{1,4\}),v(\{2,3\}),\dots\)
\(\dots v(\{2,4\}),v(\{3,4\}),v(\{1,2,3\}),v(\{1,2,4\}),v(\{1,3,4\}),v(\{2,3,4\}),v(\{1,2,3,4\})].\)
O’Neill, B. (1982) A problem of rights arbitration from the Talmud. Mathematical Social Sciences 2, 345–371.
setofawards.
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]
Run the code above in your browser using DataLab