View source: R/coalitionalgame.R

coalitionalgame | R Documentation |

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

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

`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. |

Let *E≥ 0* be the endowment to be divided and *d* the vector of claims
with *d≥ 0* and such that the sum of claims exceeds the endowment.

For each subset *S* of the set of claimants *N*, let *d(S)* be the sum of claims of the members of *S*
and let *N-S* be the complementary coalition of *S*.

Given a claims problem *(E,d)*, its associated pessimistic coalitional game is the game *vp* assigning to each coalition *S*
the real number:

*vp(S)=max {0,E-d(N-S)}.*

Given a claims problem *(E,d)*, its associated optimistic coalitional game is the game *vo*
assigning to each coalition *S*
the real number:

*vo(S)=min {E,d(S)}.*

The optimistic and the pessimistic coalitional games are dual games, that is, for all *S*:

*vo(S)=E-vp(N-S).*

An efficient way to represent a nonempty coalition *S* is by identifying it with the binary sequence
*a(n)a(n-1) … a(1)* where *a(i)=1* if *i belongs to S*
and *a(i)=0* otherwise.
Therefore, each coalition *S* is represented by the number associated with its binary representation: *∑ a(i)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}), 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})]*

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

O’Neill B (1982) A problem of rights arbitration from the Talmud. Math Soc Sci 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]

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