Title: Extension of the Core and the Nucleolus to Games in Partition Function Form, Not Necessarily Superadditive
1Extension of the Core and the Nucleolus to Games
in Partition Function Form, Not Necessarily
Superadditive
The paper is available at http//faculty.lebow.dre
xel.edu/mccainr/top/eco/PFpap.pdf
2Objective
- A model of rational interdependent decisions that
resembles the world we observe in that - We live our lives in coalitions. -- Maskin
- Most incomes are side coalitional side payments
- Coalitions often compete against one another
- Decentralization often seems advantageous
- Free riders and inefficiencies are persistent
- This presentation is based on a newer paper at
http//faculty.lebow.drexel.edu/mccainr/top/eco/PF
pap.pdf
3Simplifying Assumptions 1
- Shapley and Shubik 1969, a very distinguished and
important paper, nevertheless illustrates a range
of simplifying assumptions that frustrate these
objectives. The assumptions are highly coherent,
and thus must be reconsidered as a whole. - Coalition function representation
- Assurance principle
- Perfect Recall
- Superadditivity
- Nontransferable Utility
4Simplifying Assumptions 2
- Coalition function representation
- While SS model externalities in the underlying
game, they are not visible in the coalition
function. - Assurance principle
- In effect rules out free riders.
- Perfect Recall
- Rules out information problems in the context of
a coalitional game. Real-world coalitions in
which we live our lives do not seem to escape
information problems, though.
5Simplifying Assumptions 3
- Superadditivity
- While superadditivity follows from perfect
recall, it rules out a case in which
decentralization is efficient. Again, this may be
modeled in the underlying game, but is not
visible in the coalition value function. - Nontransferable Utility
- The advantage of TU games is that side payments
are modeled directly in the coalitional game in
NTU games they are left for the underlying game
in extensive form (and almost never explicitly
modeled.) This study assumes TU.
6Partition Functions
- Representing the game in partition function form
enables us to relax assumptions 1-4. - This is argued extensively in my forthcoming
book, Game Theory and Public Policy. - The Partition Function Form was proposed by
Thrall and Lucas in 1963. - A pair P, Ci with coalition Ci ÎP a partition
is called an embedded coalition. - A coalition value function v(P, Ci) assigns a
real number (TU) value to coalition Ci in the
context of the partition P.
7Some Terminolgy
- A game in partition function form is proper if
v(P, S)v(Q ,S) ?P, Q , S ? PÎ??? Q Î????
P?Q , , SÎP, SÎQ . - If P is a partition and QB1, , Bs, Æ is a
partition and ??i 1, , s, ??k Î??????????r
??Ck?Æ???, then Q is said to be a refinement of
P. - For any PÎ?N and SÏP,
- PSC ???Î?P ??CB\SÈ?S?, PS will be
called the residual partition of P with respect
to S.
8More Terminology
- For PÎ????SÎP, a partition Q is said to be
granular with respect to S, P, iff ??BÎQ ,
either BS or ?CÎP??C?S, ?? - For PÎ????SÎP, a refinement Q is said to be
particulate with respect to S, P, iff ??BÎQ ,
either or ?CÎP??C?S, ??BC.
9Superadditivity 1
- The first section of the paper returns to some
quite old papers for a critique of
superadditivity. - However, the concept of superadditivity is not
transparent in the case of partition function
games. - Some of that terminology will help sort this out.
10Superadditivity 2
- The argument for superadditivity is essentially
that any vector of strategies available to the
two coalitions separately is also available to
the merged coalition, so that they can do no
worse than to adopt the strategies adopted by the
two coalitions separately.
11Superadditivity 3
- This does not violate superadditivity because
partition 1 is not particulate with respect to 3.
There is no reason to think the strategies
adopted by A,B separately will yield the same
payoff after the merger of C, D. - Thus, superadditivity has to be defined relative
to particulate refinements. (Section iii).
12Superadditivity in the Paper
- This will illustrate the importance of explicit
modeling in terms of partition function games. - The relation among partition functions,
externalities, and superadditivity proves to be
complex. - Even as superadditivity is redefined here,
Section i, a remark expressed in intuitive terms,
argues that the assertion of superadditivity,
however logically consistent, is not suited for
our purposes.
13Imputation
- This study follows Aumann and Dreze, 1974 (and
departs from much subsequent literature) in
constraining an imputation for partition P so
that - xCv(P,C) for all CÎ P
- Then x is admissible for P
- A candidate solution is a partition P with an
admissible imputation x.
14The Core
- The problem is not really that it is difficult to
extend the core to PF games. - Rather, it can be extended in a number of
different ways. - Consider the NIMBY Game
15A Deviation
- Consider partition 2, and suppose a deviates.
- PS , the immediate result, is line 5.
- But the residual, b,c, can benefit by
reorganizing as b,c. - They are playing the residual game. (Koczy)
16Successor Function
- Thinking along those lines, postulate a successor
function - Let coalition C deviate from partition P.
- After the residual has taken any steps of
reorganization that improve their payoffs
relative to PS, we arrive at partition Q . - Then Q R(P)
- For some games, though, this will not be unique.
17Core
- For the moment, suppose R is unique.
- Consider a deviation C from a candidate solution
P,x. - For a deviation C from P, compute the excess
(Schmeidler) as - e(P,C,x)v(Q ,C)- xC, where Q R(P,C)
- The core comprises P,x for which, for any S? P,
the excess is nonpositive.
18Optimism and Pessimism
- If the successor function is nonunique, we can
set limits as follows - The optimistic successor R-(P,C) yields the
greatest value v(P,C) among all rational
successors. - The pessimistic successor R(P,C) yields the
smallest value v(P,C) among all rational
successors. - Denoting the optimistic core as X, the
pessimistic core as X-, and a a core computed
from an arbitrary rational successor function as
X, we have
19Ambiguity
- For many games, the optimistic and pessimistic
successors coincide, and so the ambiguity
disappears. - In other cases, we can do no better than set
limits via optimistic and pessimistic cores. - Similarly, an extension of the nucleolus can be
constructed for any successor function that
arises from rational play in the residual game. - When the nucleolus is computed from the
corresponding successor function, and if the core
is not null, the nucleolus is an element of the
coalition structure core for that partition.
20Concluding
Slides marked by are not from the paper in the
program but
http//faculty.lebow.drexel.edu/mccainr/top/eco/PF
pap.pdf
Extension of concepts such as the core to
partition function games requires some
information in addition to that supplied in the
game, such as a successor function but that
information should itself be consistent with
whatever assumptions of rationality we make
generally and that requirement at least sets
some limits on what the extensions may be.