Preferences over sets
Shapley extensions
Properties of Shapley extensions
Power set extensions of ordinal preferences using
marginal contributions
Stefano MORETTI
LAMSADE-CNRS, Paris Dauphine
Game Theory at the universities of Milano II, 12,13-05-2011
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Summary
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Preferences over sets
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Shapley extensions
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Properties of Shapley extensions
S. Moretti
Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Sets are important
- Standard approach in ‘decision theory’ has focused on pairwise
preferences or rankings over individual alternatives.
S. Moretti
Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Sets are important
- Standard approach in ‘decision theory’ has focused on pairwise
preferences or rankings over individual alternatives.
- many decision problems require learning preferences over sets of
alternatives.
S. Moretti
Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Sets are important
- Standard approach in ‘decision theory’ has focused on pairwise
preferences or rankings over individual alternatives.
- many decision problems require learning preferences over sets of
alternatives.
- For example, when selecting what to take on a backpacking trip,
a bottle of water can be most essential (thus ranked highest), but
two bottles of water may be less preferred than a bottle of water
and an apple.
S. Moretti
Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Sets are important
- Standard approach in ‘decision theory’ has focused on pairwise
preferences or rankings over individual alternatives.
- many decision problems require learning preferences over sets of
alternatives.
- For example, when selecting what to take on a backpacking trip,
a bottle of water can be most essential (thus ranked highest), but
two bottles of water may be less preferred than a bottle of water
and an apple.
- or picking the top k items ranked by google does not always yield
the optimal subset for building a music playlist.
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Ranking genes...
- statistical testing for gene selection in computational biology
aims at finding genes which are ’strongly’ differentially expressed
between two conditions (e.g., case-control studies),
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Ranking genes...
- statistical testing for gene selection in computational biology
aims at finding genes which are ’strongly’ differentially expressed
between two conditions (e.g., case-control studies),
- Following this approach, genes are usually ranked according to
their p-values, being genes with the smallest p-values the most
differentially expressed.
S. Moretti
Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Ranking genes...
- statistical testing for gene selection in computational biology
aims at finding genes which are ’strongly’ differentially expressed
between two conditions (e.g., case-control studies),
- Following this approach, genes are usually ranked according to
their p-values, being genes with the smallest p-values the most
differentially expressed.
- genes in a set can interact in ways that increase (via
complementarity), or decrease (via redundancy), the overall
valuation of a set.
S. Moretti
Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Ranking genes...
- statistical testing for gene selection in computational biology
aims at finding genes which are ’strongly’ differentially expressed
between two conditions (e.g., case-control studies),
- Following this approach, genes are usually ranked according to
their p-values, being genes with the smallest p-values the most
differentially expressed.
- genes in a set can interact in ways that increase (via
complementarity), or decrease (via redundancy), the overall
valuation of a set.
- a subset of genes identified as being individually differentially
expressed between two conditions can be less efficient in a disease
characterization than a subset of genes which show different levels
of interaction between the two conditions.
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Axiomatic approach
- many other problems in individual and collective decision making
involve the comparison of sets of alternatives (e.g., comparing the
outcomes of social choice correspondences; non probabilistic
models of choice under uncertainty; coalition formations etc...)
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Axiomatic approach
- many other problems in individual and collective decision making
involve the comparison of sets of alternatives (e.g., comparing the
outcomes of social choice correspondences; non probabilistic
models of choice under uncertainty; coalition formations etc...)
- Relevant number of papers focused on the problem of deriving a
preference relation on the power set of X from a preference
relation over single objects in X . Most of them provide an
axiomatic approach (Kannai and Peleg (1984), Barbera et al
(2004), Bossert (1995), Fishburn (1992), Roth (1985) etc.)
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Literature
- Complete uncertainty. In this kind of problems, set are considered
as formed by alternatives that are mutually exclusive (only one
object will be selected at the end); players may compare sets, but
cannot influence the selection of an alternative from a certain sets
(Kannai and Peleg (1981), Barberá et al. (2004) for a survey).
S. Moretti
Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Literature
- Complete uncertainty. In this kind of problems, set are considered
as formed by alternatives that are mutually exclusive (only one
object will be selected at the end); players may compare sets, but
cannot influence the selection of an alternative from a certain sets
(Kannai and Peleg (1981), Barberá et al. (2004) for a survey).
- Opportunity sets. Set contain again mutually exclusive
alternatives, but the agents can select a final outcome from a set.
Typical examples of opportunity sets or menus from which an
agent can make a choice (Kreps (1979), Pattanaik and Xu (1998),
Sen (1988)).
S. Moretti
Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Literature
- Complete uncertainty. In this kind of problems, set are considered
as formed by alternatives that are mutually exclusive (only one
object will be selected at the end); players may compare sets, but
cannot influence the selection of an alternative from a certain sets
(Kannai and Peleg (1981), Barberá et al. (2004) for a survey).
- Opportunity sets. Set contain again mutually exclusive
alternatives, but the agents can select a final outcome from a set.
Typical examples of opportunity sets or menus from which an
agent can make a choice (Kreps (1979), Pattanaik and Xu (1998),
Sen (1988)).
- Sets as final outcomes. In this setting sets contain objects that
are assumed to materialize simultaneously (i.e. the, the agent will
receive all of them simultaneously (Roth (1985), Fishburn (1992),
Bossert (1995)).
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example
-Let X = {x, y } be a universal set of two alternatives. Suppose
that an agent prefer x over y .
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example
-Let X = {x, y } be a universal set of two alternatives. Suppose
that an agent prefer x over y .
-Under the uncertainty interpretation, it is reasonable to expect
that the agent will prefer set {x} to {x, y }, since under the last
state the possibility that alternative y materializes does exist.
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example
-Let X = {x, y } be a universal set of two alternatives. Suppose
that an agent prefer x over y .
-Under the uncertainty interpretation, it is reasonable to expect
that the agent will prefer set {x} to {x, y }, since under the last
state the possibility that alternative y materializes does exist.
-Under the interpretation of opportunity sets, the two sets {x} and
{x, y } could be simply considered indifferent (or the agent could
prefer set {x, y } because it provides more freedom of choice...).
S. Moretti
Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example
-Let X = {x, y } be a universal set of two alternatives. Suppose
that an agent prefer x over y .
-Under the uncertainty interpretation, it is reasonable to expect
that the agent will prefer set {x} to {x, y }, since under the last
state the possibility that alternative y materializes does exist.
-Under the interpretation of opportunity sets, the two sets {x} and
{x, y } could be simply considered indifferent (or the agent could
prefer set {x, y } because it provides more freedom of choice...).
- Under the interpretation of sets as final outcomes, if objects are
goods, one could guess that to have {x, y } is better, because the
agent will receive both y and x. But the judgement depends on
the nature of x and y and possible effects of their interaction...
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Standard Assumption
A binary relation < on X (<⊆ X × X ):
reflexivity : for each x ∈ X , x < x;
transitivity : for each x, y , z ∈ X , x < y and y < z ⇒ x < z;
completeness: for each x, y ∈ X , x 6= y ⇒ x < y or y < x;
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Standard Assumption
A binary relation < on X (<⊆ X × X ):
reflexivity : for each x ∈ X , x < x;
transitivity : for each x, y , z ∈ X , x < y and y < z ⇒ x < z;
completeness: for each x, y ∈ X , x 6= y ⇒ x < y or y < x;
An total preorder is a reflexive, transitive and complete binary
relation. A reflexive, transitive, complete and antisymetric binary
relation is called total order or linear order.
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Standard Assumption
A binary relation < on X (<⊆ X × X ):
reflexivity : for each x ∈ X , x < x;
transitivity : for each x, y , z ∈ X , x < y and y < z ⇒ x < z;
completeness: for each x, y ∈ X , x 6= y ⇒ x < y or y < x;
An total preorder is a reflexive, transitive and complete binary
relation. A reflexive, transitive, complete and antisymetric binary
relation is called total order or linear order.
The strict preference relation and the indifference relation ∼ are
defined by letting, for all x, y ∈ X , x y if and only if x < y and
not y < x; x ∼ y if and only if x < y and y < x.
S. Moretti
Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Standard Assumption
A binary relation < on X (<⊆ X × X ):
reflexivity : for each x ∈ X , x < x;
transitivity : for each x, y , z ∈ X , x < y and y < z ⇒ x < z;
completeness: for each x, y ∈ X , x 6= y ⇒ x < y or y < x;
An total preorder is a reflexive, transitive and complete binary
relation. A reflexive, transitive, complete and antisymetric binary
relation is called total order or linear order.
The strict preference relation and the indifference relation ∼ are
defined by letting, for all x, y ∈ X , x y if and only if x < y and
not y < x; x ∼ y if and only if x < y and y < x.
A vector x ∈ RX is a numerical representation of the preference
relation < on X if for every i, j ∈ X , xi ≥ xj ⇔ i < j.
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Preferences extension
Preferences over sets
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Properties of Shapley extensions
Extension axiom
Given a binary relation < on X , we say that a binary relation w on
2X is an extension of < if and only if for each x, y ∈ X ,
{x} w {y } ⇔ x < y
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example: maxi-max and maxi-min
Perhaps the simplest extensions are the maxi-max wmax and the
maxi-min wmin extensions.
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example: maxi-max and maxi-min
Perhaps the simplest extensions are the maxi-max wmax and the
maxi-min wmin extensions.
- For instance, let X = {1, 2, 3} and 1 2 3.
According to the maxi-max criterion, for each S, T ∈ 2X , we have
(S wmax T ) ⇔ (max(S) <max max(T )).
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example: maxi-max and maxi-min
Perhaps the simplest extensions are the maxi-max wmax and the
maxi-min wmin extensions.
- For instance, let X = {1, 2, 3} and 1 2 3.
According to the maxi-max criterion, for each S, T ∈ 2X , we have
(S wmax T ) ⇔ (max(S) <max max(T )).
So The extension wmax of < is:
{1, 2, 3} wmax {1, 3} wmax {1, 2} wmax {1} Amax {2} wmax {2, 3} Amax {3}
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example: maxi-max and maxi-min
Perhaps the simplest extensions are the maxi-max wmax and the
maxi-min wmin extensions.
- For instance, let X = {1, 2, 3} and 1 2 3.
According to the maxi-max criterion, for each S, T ∈ 2X , we have
(S wmax T ) ⇔ (max(S) <max max(T )).
So The extension wmax of < is:
{1, 2, 3} wmax {1, 3} wmax {1, 2} wmax {1} Amax {2} wmax {2, 3} Amax {3}
- According to the maxi-min criterion, for each S, T ∈ 2X , we have
(S wmin T ) ⇔ (min(S) <max min(T )).
So The extension wmin of < is:
{1} Amin {1, 2} wmin {2} Amin {2, 3} wmin {1, 3} wmin {3} wmin {1, 2, 3}
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example: lexi-max and lex-min
Other extensions are the lexi-min and the lexi-max extensions,
which are obtained, respectively, as the lexicographical
generalizations of the maxi-max and maxi-min extensions.
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example: lexi-max and lex-min
Other extensions are the lexi-min and the lexi-max extensions,
which are obtained, respectively, as the lexicographical
generalizations of the maxi-max and maxi-min extensions.
- For instance, let X = {1, 2, 3} and 1 2 3.
According to the lexi-max criterion, we have
{1, 2, 3} AL max {1, 2} AL max {1, 3} AL max {1} AL max
{2, 3} AL max {2} AL max {3}.
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Preferences over sets
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Properties of Shapley extensions
Basic-Basic on coalitional games
A coalitional game (many names...) is a pair (N, v ), where N
denotes the finite set of players and v : 2N → R is the
characteristic function, with v (∅) = 0.
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Preferences over sets
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Properties of Shapley extensions
Basic-Basic on coalitional games
A coalitional game (many names...) is a pair (N, v ), where N
denotes the finite set of players and v : 2N → R is the
characteristic function, with v (∅) = 0.
A one-point solution (or simply a solution) for a class C N of
coalitional games is a function ψ that assigns a payoff vector ψ(v )
to every coalitional game in the class, that is ψ : C N → RN .
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Basic-Basic on coalitional games
A coalitional game (many names...) is a pair (N, v ), where N
denotes the finite set of players and v : 2N → R is the
characteristic function, with v (∅) = 0.
A one-point solution (or simply a solution) for a class C N of
coalitional games is a function ψ that assigns a payoff vector ψ(v )
to every coalitional game in the class, that is ψ : C N → RN .
The most widely used solution in the theory of coalitional games is
the Shapley value, introduced by Shapley in 1953. This solution
can be described in several ways. One possibility is to introduce
the Shapley value φ applied to game (N, v ) ∈ G N by the general
formula
X (s − 1)!(n − s)!
φi (v ) =
v (S) − v (S \ {i})
(1)
n!
S⊆N:i∈S
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Properties of Shapley extensions
Complementarity effects
- The attitude to interact among elements of X cannot be
excluded a priori, and the only information from which it can be
derived is the original ranking <.
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Properties of Shapley extensions
Complementarity effects
- The attitude to interact among elements of X cannot be
excluded a priori, and the only information from which it can be
derived is the original ranking <.
- objective: to characterize those extensions which are able to
represent complementarity effects that are ”compatible” with the
information provided by the original <.
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Preferences extension
Preferences over sets
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Properties of Shapley extensions
Example of complementarity effects we want to consider
- Let X = {x, y , z} and suppose that an agent’s preference is such
that x < y , x < z and y < z.
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Preferences over sets
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Properties of Shapley extensions
Example of complementarity effects we want to consider
- Let X = {x, y , z} and suppose that an agent’s preference is such
that x < y , x < z and y < z.
Trying to extend < to 2X , one could guess that set {x, y } is better
than {y , z}, because the agent will receive both y and x instead of
y and z (and x is preferred to z).
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example of complementarity effects we want to consider
- Let X = {x, y , z} and suppose that an agent’s preference is such
that x < y , x < z and y < z.
Trying to extend < to 2X , one could guess that set {x, y } is better
than {y , z}, because the agent will receive both y and x instead of
y and z (and x is preferred to z).
However, in case of incompatibility among x and y (determining,
for instance, a negative externality) the preference between the two
sets could be reversed.
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Preferences over sets
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Properties of Shapley extensions
Shapley extension
Let < be a total preorder on X . An extension w of < is a Shapley
extension of < iff the Shapley value φ(v ) is a numerical
representation of < for every numerical representation v of w.
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Properties of Shapley extensions
Example of Shapley extension
Let X = {1, 2, 3} and let wa be such that {1, 2, 3} Aa {3} Aa
{2} Aa {1, 3} Aa {2, 3} Aa {1} Aa {1, 2} Aa ∅.
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Properties of Shapley extensions
Example of Shapley extension
Let X = {1, 2, 3} and let wa be such that {1, 2, 3} Aa {3} Aa
{2} Aa {1, 3} Aa {2, 3} Aa {1} Aa {1, 2} Aa ∅.
Note that wa is an extension of 3 2 1.
X
For every v ∈ R2 representing wa
φ2 (v ) − φ1 (v ) =
1
1
v (2) − v (1) + v (2, 3) − v (1, 3) > 0
2
2
On the other hand
φ3 (v ) − φ2 (v ) =
1
1
v (3) − v (2) + v (1, 3) − v (1, 2) > 0.
2
2
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Properties of Shapley extensions
Simple observation
Let (X , v ) be a coalitional game. Let i, j ∈ X be such that for
each S ∈ 2X with S ∩ {i, j} = ∅ it holds that
v (S ∪ {i}) ≥ v (S ∪ {j}). Then φi (v ) ≥ φj (v ).
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Properties of Shapley extensions
Simple observation
Let (X , v ) be a coalitional game. Let i, j ∈ X be such that for
each S ∈ 2X with S ∩ {i, j} = ∅ it holds that
v (S ∪ {i}) ≥ v (S ∪ {j}). Then φi (v ) ≥ φj (v ).
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Preferences over sets
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Properties of Shapley extensions
A consequence of the simple observation
The following criteria are Shapley extensions
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Preferences extension
Preferences over sets
Shapley extensions
Properties of Shapley extensions
A consequence of the simple observation
The following criteria are Shapley extensions
- maxi-max and maxi-min
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Preferences over sets
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Properties of Shapley extensions
A consequence of the simple observation
The following criteria are Shapley extensions
- maxi-max and maxi-min
- lexi-min and lexi-max
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Preferences over sets
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Properties of Shapley extensions
A consequence of the simple observation
The following criteria are Shapley extensions
- maxi-max and maxi-min
- lexi-min and lexi-max
- median-based extensions (Nitzan and Pattanaik, 1984)
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Preferences over sets
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Properties of Shapley extensions
A consequence of the simple observation
The following criteria are Shapley extensions
- maxi-max and maxi-min
- lexi-min and lexi-max
- median-based extensions (Nitzan and Pattanaik, 1984)
- all the criteria above (and many others): if i < j than
S ∪ {i} w S ∪ {j} for each S which does not contain neither i nor j.
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Properties of Shapley extensions
Responsiveness (Roth (1985))
- a set S is preferred to a set T whenever a set S is obtained from
T by replacing some object t ∈ T with another n ∈ X not in T
which is preferred to t.
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Properties of Shapley extensions
Responsiveness (Roth (1985))
- a set S is preferred to a set T whenever a set S is obtained from
T by replacing some object t ∈ T with another n ∈ X not in T
which is preferred to t.
- introduced by Roth (1985) studying colleges’ preferences for the
“college admission problem” (colleges need to rank sets of
students based on their ranking of individual applicants, see also
Gale and Shapley (1962)).
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Properties of Shapley extensions
Responsiveness (Roth (1985))
- a set S is preferred to a set T whenever a set S is obtained from
T by replacing some object t ∈ T with another n ∈ X not in T
which is preferred to t.
- introduced by Roth (1985) studying colleges’ preferences for the
“college admission problem” (colleges need to rank sets of
students based on their ranking of individual applicants, see also
Gale and Shapley (1962)).
- Bossert (1995) used the same property for ranking sets of
alternatives with a fixed cardinality.
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Preferences extension
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Properties of Shapley extensions
Responsiveness (Roth (1985))
- a set S is preferred to a set T whenever a set S is obtained from
T by replacing some object t ∈ T with another n ∈ X not in T
which is preferred to t.
- introduced by Roth (1985) studying colleges’ preferences for the
“college admission problem” (colleges need to rank sets of
students based on their ranking of individual applicants, see also
Gale and Shapley (1962)).
- Bossert (1995) used the same property for ranking sets of
alternatives with a fixed cardinality.
- Together with a neutrality property (the labelling of the
alternatives is irrelevant in establishing the ranking), Bossert
(1995) characterized the class of rank-ordered lexicographic
extensions.
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Ax. Responsiveness, RESP
Let < be a binary relation on X . A binary relation w on 2X
satisfies the responsiveness property on 2X (and with respect to <)
iff for all A ∈ 2X , for all x ∈ A and for all y ∈ X \ A we have that
[A w (A \ {x}) ∪ {y } ⇔ x < y ] and [(A \ {x}) ∪ {y } w A ⇔ y < x].
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Equivalent formulation of Responsiveness
Let < be a binary relation on X . A binary relation w on 2X
satisfies the responsiveness property on 2X (and with respect to
<) iff for all i, j ∈ X and all S ∈ 2X , S ∩ {i, j} = ∅ we have that
S ∪ {i} w S ∪ {j} ⇔ i < j.
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Prop. 1
Let < be a total preorder on X and let w be an extension of <. If
w satisfies RESP property, than w is a Shapley extension.
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RESP is not a necessary conditions (a)
Note that the linear order wa introduced in the previous Example
does not satisfies RESP property. In fact,
{1, 2, 3} Aa {3} Aa {2} Aa {1, 3} Aa {2, 3} Aa {1} Aa {1, 2} Aa ∅.
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RESP is not a necessary conditions (a)
Note that the linear order wa introduced in the previous Example
does not satisfies RESP property. In fact,
{1, 2, 3} Aa {3} Aa {2} Aa {1, 3} Aa {2, 3} Aa {1} Aa {1, 2} Aa ∅.
- We have that 2 < 1 but A = {1, 3} Aa {2, 3} = A \ {1} ∪ {2}.
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Ax. Monotonicity, MON
A binary relation w on 2X satisfies the monotonicity property iff
for each S, T ∈ 2X such that i ∈ S we have that S ⊇ T ⇒ S w T .
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Examples
- The extension wa introduced in the previous example does not
satisfies MON.
{1, 2, 3} Aa {3} Aa {2} Aa {1, 3} Aa {2, 3} Aa {1} Aa {1, 2} Aa
∅.
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Properties of Shapley extensions
Examples
- The extension wa introduced in the previous example does not
satisfies MON.
{1, 2, 3} Aa {3} Aa {2} Aa {1, 3} Aa {2, 3} Aa {1} Aa {1, 2} Aa
∅.
- maxi-min criterion: for instance, if 2 1 we have that
{2} Aa {1, 2}. Maxi min is a Shapley extension, it satisfies RESP,
but it does not satisfy MON.
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Properties of Shapley extensions
Examples
- The extension wa introduced in the previous example does not
satisfies MON.
{1, 2, 3} Aa {3} Aa {2} Aa {1, 3} Aa {2, 3} Aa {1} Aa {1, 2} Aa
∅.
- maxi-min criterion: for instance, if 2 1 we have that
{2} Aa {1, 2}. Maxi min is a Shapley extension, it satisfies RESP,
but it does not satisfy MON.
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Preferences over sets
Shapley extensions
Properties of Shapley extensions
Sub-extensions
Let < be a total pre-order on X and let w be an extension of <.
For each S ∈ 2X \ {∅}, a sub-extension wS is a relation on 2S such
that for each U, V ∈ 2S ,
U w V ⇔ U wS V .
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Preferences over sets
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Properties of Shapley extensions
Sub-extensions
Let < be a total pre-order on X and let w be an extension of <.
For each S ∈ 2X \ {∅}, a sub-extension wS is a relation on 2S such
that for each U, V ∈ 2S ,
U w V ⇔ U wS V .
Note that wS is an extension of <S , where <S is the restriction of
< such that i <S j ⇔ i < j for each i, j ∈ S.
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Preferences over sets
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Properties of Shapley extensions
Ax. Sub-extendibility, SE
Let < be a total pre-order on X and let w be a Shapley extension
of < on 2X . Relation w satisfies the sub-extendibility property iff
for each S ∈ 2X \ {∅} we have that wS is a Shapley extension of
<S .
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Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example
Let X = {1, 2, 3, 4} and let wb be a total preorder such that
{1, 2, 3, 4} :Ab {2, 3, 4} Ab {1, 3, 4} Ab {1, 2, 4} Ab {3, 4} Ab
{1, 2, 3} Ab {1, 3} wb {2, 4} Ab {2, 3} wb {1, 4} Ab {4} Ab
{3} Ab {1, 2} Ab {2} Ab {1} Ab ∅. Note that wb is an extension
of 4 3 2 1.
It is easy to check that for every game v numerical representation
of wb the following relations hold.
φ2 (v ) − φ1 (v ) =
1
v (2) − v (1) + 31 v (2, 3, 4) − v (1, 3, 4)+
3
1
1
v (2, 3) − v (1, 4) + 6 v (2, 4) − v (1, 3) > 0
6
(2)
φ3 (v ) − φ2 (v ) =
1
v (3) − v (2) + 31 v (1, 3, 4) − v (1, 2, 4)+
3
1
v (1, 3) − v (2, 4) + 61 v (3, 4) − v (1, 2) > 0
6
(3)
φ4 (v ) − φ3 (v ) =
1
v (4) − v (3) + 31 v (1, 2, 4) − v (1, 2, 3)+
3
1
v (1, 4) − v (2, 3) + 61 v (2, 4) − v (1, 3) > 0
6
(4)
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Preferences over sets
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Properties of Shapley extensions
Example (follows)
- wb does not satisfies the RESP property.
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Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example (follows)
- wb does not satisfies the RESP property.
- Relation wb does not satisfies SE property.
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Preferences over sets
Shapley extensions
Properties of Shapley extensions
Example (follows)
- wb does not satisfies the RESP property.
- Relation wb does not satisfies SE property.
Consider the sub-extension wb{1,2,3} . A game that represents
Ab{1,2,3} is:
v ({1, 2, 3}) = 10, v ({1, 3}) = 9, v ({2, 3}) = 5, v ({3}) = 4,
v ({1, 2}) = 3, v ({2}) = 2, v ({1}) = 1, v (∅) = 0
and the corresponding Shapley value is φ1 (v ) = 3, φ2 (v ) = 1.5,
φ3 (v ) = 5.5.
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Properties of Shapley extensions
Prop. 2
Let < be a total pre-order on X and let w be a Shapley extension
of <. If w satisfies both MON and SE properties, then it also
satisfies RESP property.
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Preferences over sets
Shapley extensions
Properties of Shapley extensions
References
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Handbook of Utility Theory, Volume 2, Kluwer Academic Publishers.
-Bossert W. (1995) Preference extension rules for ranking sets of alternatives
with a fixed cardinality, Theory and Decision, 39, 301-317.
-Fishburn P.C. (1992) Signed orders and power set extensions, Journal of
Economic Theory, 56, 1-19.
-Gale D., Shapley L.S. (1962) College admissions and the stability of marriage,
American Mathematical Monthly, 69, 9-15.
-Kannai Y., Peleg B. (1981) A note on the extension of an order on a set to
the power set, Journal of Economic Theory, 32, 172-175.
-Kreps D.M. (1979) A representation theorem for ‘preference for flexibility’,
Econometrica, 47, 565-577.
-Pattanaik P.K., Xu Y. (1998) On preference and freedom. Theory and
Decision, 44, 173-198.
-Roth A.E. (1985) The college admissions problem is not equivalent to the
marriage problem, Journal of Economic Theory, 36, 277-288.
-Sen A. (1988) Freedom of choice: concept and content. European Economic
Review, 32, 269-294.
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