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 be the endowment to be divided and
the vector of claims
with
and such that
the sum of claims exceeds the endowment.
For each subset of the set of claimants
, let
be the sum of claims of the members of
and let
be the complementary coalition of
.
Given a claims problem , its associated pessimistic coalitional game is the game
assigning to each coalition
the real number:
Given a claims problem , its associated optimistic coalitional game is the game
assigning to each coalition
the real number:
The optimistic and the pessimistic coalitional games are dual games, that is, for all :
An efficient way to represent a nonempty coalition is by identifying it with the binary sequence
where
if
and
otherwise.
Therefore, each coalition
is represented by the number associated with its binary representation:
.
Then coalitions can be ordered by their associated numbers.
Alternatively, coalitions can be ordered lexicographically.
Given a claims problem , its associated coalitional game
can be represented by the vector whose coordinates are the values assigned by
to all the nonempty coalitions.
For instance. if
, the associated coalitional game can be represented by the vector of the values of all the 7 nonempty coalitions, ordered using the binary representation:
Alternatively, the coordinates can be ordered lexicographically:
When , the associated coalitional game can be represented by the vector of the values of all the 15 nonempty coalitions, ordered using the binary representation:
Alternatively, the coordinates can be ordered lexicographically:
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
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]