Extension of the Core and the Nucleolus to Games in Partition Function Form, Not Necessarily Superadditive

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Extension of the Core and the Nucleolus to Games
in Partition Function Form, Not Necessarily
Superadditive

• By Roger A. McCain

The paper is available at http//faculty.lebow.dre
xel.edu/mccainr/top/eco/PFpap.pdf
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Objective

• 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
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Simplifying 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

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Simplifying 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.

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Simplifying 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.

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Partition 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.

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Some 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.

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More 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.

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Superadditivity 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.

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Superadditivity 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.

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Superadditivity 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).

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Superadditivity 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.

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Imputation

• 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.

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The 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

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A 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)

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Successor 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.

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Core

• 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.

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Optimism 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

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Ambiguity

• 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.

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Concluding
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.