Download The position value is the Myerson value, in a sense

The position value is the Myerson value, in a sense
accepted for publication in the International Journal of Game Theory
as of March 15, 2007
André Casajus†
(May 2006, this version: March 17, 2007, 16:00)
Abstract
We characterize the position value for TU games with a cooperation
structure in terms of the Myerson value of some natural modification of
the original game–the link agent form. This construction is extended to
TU games with a conference structure.
Journal of Economic Literature Classification Number: C71.
Key Words: TU game, cooperation structure, conference structure,
graph, hypergraph, link agent form.
†
Universität Leipzig, Wirtschaftswissenschaftliche Fakultät, Professur für Mikroökonomik, PF
100920, D-04009 Leipzig, Germany. e-mail: [email protected]
I thank the editor of this journal, two anonymous referees, and an associate editor for their
helpful comments on this note. Of course, the usual disclaimer applies.
1
2
1. Introduction
Generalizing the Shapley (1953) value for TU games and the Aumann and Drèze
(1974) value for TU games with a coalition structure (partition of the player set),
Myerson (1977) introduced a value, now called the Myerson value, for TU games with
a cooperation structure (graph on the player set) (henceforth CO-games and COvalue). As an alternative, Meessen (1988) suggests the position value for CO-games.
This value was popularized by Borm, Owen and Tijs (1992). Yet another CO-value
was introduced by Hamiache (1999). It was discussed by Bilbao, Jiménez and López
(2006). Van den Nouweland, Borms, and Tijs (1992) extended the Myerson value
and the position value to TU games with a conference structure (hypergraph on the
player set) (henceforth CF-games and CF-value).
Besides the elegant Myerson (1977) axioms, there are alternative axiomatizations
of this value for CO-games (Myerson, 1980; Borm et al., 1992; Slikker and van
den Nouweland, 2001), both for general games and for the class of cycle-free COgames. The position value for CO-games was axiomatized by Borm et al. (1992)
for the class of cycle-free CO-games. Only recently, Slikker (2005) gave a general
characterization. While van den Nouweland, Borm and Tijs (1992) give some general
characterizations for the Myerson value for CF-games, their characterization of the
position value for CF-games is restricted to cycle-free CF-games. Algaba, Bilbao,
Borm and López (2000) characterize the position value for a class of union stable
systems. So it seems to be an open problem to find an axiomatization of the position
value for the class of all CF-games.
Our main result is a new characterization of the position value for CO-games. In
particular, we express the position value in terms of the Myerson value. In contrast
to the Myerson value, which emphasizes the role of the players, the position value
focuses on the links. Therefore, one may be tempted to split the players into separate
“agents”, one for each link, and then to “connect” a player’s agents. Based on this
idea, we introduce the link agent form (LAF) of a CO-game. It turns out that the
Myerson payoffs of a player’s agents in the LAF sum up to the position value payoff
of that player in the original CO-game.
In contrast to the links of a cooperation structure, which all connect exactly
two players, the hyperlinks of a conference structure may connect any number of
players. This impedes the most straightforward extension of the LAF to CF-games.
Instead, we introduce the hyperlink agent form (HAF) of a CO-game, which is
a generalization of the LAF. As by the LAF, players are split into agents, but not
3
necessarily into only one for each hyperlink. Again, the Myerson payoffs of a player’s
agents in the HAF sum up to the position value payoff of that player in the original
CF-game.
Now, one might be interested to know whether the construction of the LAF/HAF
can be reversed in such a way that the Myerson value can be expressed in terms of
the position value. Yet, there seems to be no obvious way to do so. At least, our
own attempts failed.
The plan of this note is as follows: Basic definitions and notation are given in
second section. The third section introduces the link agent form of a CO-game
and presents our characterization of the position value. In the fourth section, these
results are extended to CF-games.
2. Basic definitions and notation
A TU game is a pair (N, v) consisting of a non-empty and finite set of players N
and the coalition function v : 2N → R, v (∅) = 0. Subsets of N are called coalitions,
and v (K) is called the worth of coalition K. For ∅ 6= T ⊆ N, the game (N, uT ),
uT (K) = 1 if T ⊆ K and uT (K) = 0 otherwise, is called a unanimity game. The
restriction of v to N 0 ⊆ N is denoted v|N 0 . A value is an operator ϕ that assigns
payoff vectors to all games, ϕ (N, v) ∈ RN . An order of a set N is a bijection σ :
N → {1, . . . , |N|} with the interpretation that i is the σ (i)th player in σ. The set
of these orders is denoted by Σ (N) . The set of players not after i in σ is denoted
by Ki (σ) = {j : σ (j) ≤ σ (i)} . The marginal contribution of i in σ is defined as
MCiv (σ) := v (Ki (σ)) − v (Ki (σ) \ {i}) . The Shapley value Sh (Shapley, 1953) is
defined by
X
(1)
MCiv (σ) , i ∈ N.
Shi (N, v) := |Σ (N)|−1
σ∈Σ(N)
For K ⊆ N, we denote by ϕK (N, v, ·) the sum
P
i∈K
ϕi (N, v, ·) .
Since conference structures generalize cooperation structures and since the CFvalues under consideration extend the respective CO-values and for notational parsimony, we first provide the CF-formalism explicitly and then indicate the relation
to the CO-formalism.
A conference structure for (N, v) is a hypergraph (N, H) where H is a system
of non-singleton subsets of N, H ⊆ H N := {h ⊆ N| |h| > 1} . A typical element,
hyperlink, of H is written as h. The set of player i’s hyperlinks is denoted by Hi :=
4
{h ∈ H|i ∈ h} . Given any hypergraph (N, H) , N splits into (maximal connected)
components which constitute the partition C (N, H) of N; Ci (N, H) ∈ C (N, H)
denotes the component containing i ∈ N. The restriction of H to K ⊆ N is denoted
by H|K ,
(2)
H|K := {h ∈ H|h ⊆ K} .
A cooperation structure on N is a graph (N, L), L ⊆ LN := {h ⊆ N| |h| = 2}
⊆ H N , i.e., a hypergraph where all hyperlinks connect exactly two players; these
hyperlinks are called links, ij := {i, j} , a typical link is denoted λ. A CF-game
(N, v, H) is a game (N, v) together with a conference structure (N, H); a CF-value
is an operator ϕ that assigns payoff vectors to all CF-games, ϕ (N, v, H) ∈ RN . A
CO-game (N, v, L) is a game (N, v) together with a cooperation structure (N, L). A
CO-value is an operator ϕ that assigns payoff vectors to all CO-games, ϕ (N, v, H)
∈ RN .
The Myerson value μ (Myerson, 1977; van den Nouweland et al., 1992) is defined
by
X
¡
¢
(3) μ (N, v, H) := Sh N, v H ,
v (S) ,
K ⊆ N.
v H (K) :=
S∈C(K,H|K )
¡
¢
For any CF/CO-game (N, v, H) consider the hyperlink/link game H, vN where
X
(4)
v (S) ,
F ⊆ H.
v N (F ) =
S∈C(N,F )
Since vN (∅) may not vanish and for convenience, we restrict attention to 0-normalized TU games, i.e. v ({i}) = 0 for all i ∈ N. The position value (Meessen, 1988;
van den Nouweland et al., 1992) is given by
X −1
¡
¢
|h| Shh H, vN .
π i (N, v, H) =
(5)
h∈Hi
Since μ and π are component efficient, i.e. μS (N, v, H) = π S (N, v, H) = v (S) for
all S ∈ C (N, H) , we assume that (N, H) does not contain isolated players, i.e.
|Hi | > 0 for all i ∈ N.
3. A characterization of the position value for CO-games
Even though the construction and the result of this section are special cases of
those in Section 4, we feel that they should be presented separately because they
are much more natural for CO-games than for CF-games, in general. We express
the position value for a CO-game in terms of the Myerson value of the link agent
5
4.3
......
............................•
.
...................................
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.. .... ...
.•
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.
..... 4.2
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4.1 .... ..... ..... .... .•......
3.1 ... .....
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2.1 ...... .... ..... ........
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2.2
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•
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1.1
5
G
•
•
5.1
LAF(G)
Figure 1. The graph of a link agent form
form (LAF) of the original game. While the position value emphasizes the role of
the links, the Myerson value focuses on the players. Therefore, one could think of
splitting the players into separate agents which represent/control exactly one of a
player’s links. This is what the LAF does.
Definition 1. For any CO-game G = (N, v, L), its link agent form LAF (G) =
¢
¡
N̄, v̄, L̄ is defined as follows:
[
(6)
N̄ (i) ,
N̄ (i) := {(i, λ) |λ ∈ Li }
N̄ =
i∈N
(7)
(8)
L̄ = L̄o ∪
[
i∈N
LN̄(i) ,
¡ ¡ ¢¢
¡ ¢
v̄ K̄ = v N K̄ ,
L̄o := {ı̄j̄|ij ∈ L} ,
ı̄j̄ := {(i, ij) , (j, ij)}
©
ª
¡ ¢
N K̄ := i ∈ N |N̄ (i) ∩ K̄ 6= ∅ ,
K̄ ⊆ N̄
The player set N̄ comprises the link agents (i, λ) for all players i ∈ N and all
links λ ∈ Li . Since we assume that there are no isolated players, all link agent sets
N̄ (i) are non-empty. The cooperation structure L̄ contains the original links ij as
the links ı̄j̄ in the link set L̄o . Further, L̄ completely connects the set of link agents
N̄ (i) of any original player i ∈ N via the link set LN̄(i) . By (8), any of a player’s
agents is as productive as the original player, but any one of them suffices to do the
¢
¡
job. The construction of N̄, L̄ is visualized with the following example.
6
Example 1. Figure 1 shows the graph of some CO-game G and the graph of its
link agent form LAF (G) . In LAF (G) , the links which correspond to the original
links in G are drawn as solid lines while the links which connect a player’s agents are
represented by dashed ones. For example, the link {3, 4} in G corresponds to the
link {3.2, 4.3} in LAF (G) . Player 1 in G has just one link. Hence in LAF (G) , he is
represented by the single agent 1.1. Player 2, for example, has four links in G that
are represented by the agents 2.1 to 2.4 in LAF (G) who are completely connected
with each other.
Now, we can provide our characterization of the position value for CO-games. Its
proof is very similar to the proof of Theorem 2 (see Remark 1 below).
Theorem 1. For any 0-normalized CO-game without isolated players G = (N, v, L) ,
we have π i (G) = μN̄(i) (LAF (G)) for all i ∈ N.
Based on the LAF, one could think of the axiom PSI below and then wonder
whether there are interesting CO-values which are component efficient and invariant
to player splitting. It is clear that neither the Myerson value nor the position value
is.
Player splitting invariance, PSI. For all 0-normalized CO-games without isolated players G = (N, v, L), we have
¡
¢
ϕi (N, v, L) = ϕN̄(i) N̄, v̄, L̄
¡
¢
for all i ∈ N where N̄, v̄, L̄ = LAF (G) .
The following example reveals that the AD-value AD (Aumann and Drèze, 1974)
(applied to C (N, L)) and the Hamiache value Ha (Hamiache, 1999) also do not
meet PSI. Consider the CO-game (N, uN , L) , N = {1, 2, 3} , L = {12, 23} and its
¡
¢
LAF N̄, v̄, L̄ , N̄ = {1, 20 , 200 , 3} , L̄ = {120 , 20 200 , 200 3} , v̄ = uN̄\{200 } + uN̄\{20 } − uN̄ .
Straightforward calculations yield
¡
¡
¢¢
1
1
AD2 (N, v, C (N, L)) = 13 6= 12
+ 12
= AD{20 ,200 } N̄, v̄, C N̄, L̄ .
Since v̄ L̄ = uN̄ , Hamiache (1999, Thm. 2 and Tab. A.I) imply
¢
¡
Ha2 (N, v, L) = 12 6= 38 + 38 = Ha{20 ,200 } N̄, v̄, L̄ .
7
4. A characterization of the position value for CF-games
As we will see later on, the validity of Theorem 1 crucially depends on the fact
that the links in a CO-game all connect the same number of players, namely, exactly two players. Since hyperlinks may contain different numbers of players, the
obvious extension of the LAF to CF-games does not work. We adjust to this fact by
introducing the hyperlink agent form (HAF) below which, however, lacks some of
the appeal of the LAF. Nevertheless, for CO-games the HAF and the LAF coincide.
Definition 2. For any CF-game G = (N, v, H) without isolated players, its hyper¢
¡
link agent form HAF (G) = N̄, v̄, H̄ is defined as follows: Let χ (H) denote the
least common multiple of the numbers in {|h| |h ∈ H} . Then set
ª
©
(9)
N̄ (i, h) = (i, h, k) |k = 1, 2, . . . , χ (H) · |h|−1 ,
[
[
(10)
N̄ (i) , N̄ (i) :=
N̄ (i, h) ,
N̄ =
i∈N
(11)
(12)
H̄ = H̄ o ∪
[
i∈N
h∈Hi
LN̄(i) ,
¡ ¢
¡ ¡ ¢¢
v̄ K̄ = v N K̄ ,
©
ª
H̄ o := h̄|h ∈ H ,
h̄ :=
[
N̄ (i, h) ,
i∈h
¡ ¢
©
ª
N K̄ := i ∈ N|N̄ (i) ∩ K̄ 6= ∅ ,
K̄ ⊆ N̄.
The player set N̄ (i, h) contains χ (H) · |h|−1 hyperlink agents (i, h, k) who control
player i’s hyperlink h. In contrast to the LAF where a single agent (i, λ) controls the
link λ ∈ Li , by (9), N̄ (i, h) may contain more than one player if not all hyperlinks
in H connect the same number of players. This deviation from the LAF is necessary
in order to ensure that all representatives h̄ ∈ H̄ o of original hyperlinks h have the
¯ ¯
same cardinality. By (9) and (11), we have ¯h̄¯ = χ (H) for all h̄ ∈ H̄ o . Since we
assume that there are no isolated players, all hyperlink agent sets N̄ (i) are nonempty. Further, H̄ completely connects the hyperlink agents of an original player
i ∈ N via the link set LN̄ (i) , i.e., a player’s hyperlink agents are pairwise connected
via bilateral hyperlinks. Hence, whenever two agents of the same player are members
of some coalition then they are connected within this coalition. This reflects the fact
that they actually “are” the same player. The motivation for v̄ is as for the LAF.
¡
¢
The construction of N̄, L̄ is visualized with the following example.
Example 2. Figure 2 shows the hypergraph of some CF-game G with four players
and two hyperlinks, h = {1, 2} and h0 = {2, 3, 4} , indicated by a solid line and
by a solid line encircling the players involved, respectively. The hypergraph of the
CF-game Ḡ seems to be the most obvious generalization of the LAF for G. Player
8
3.1
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3.1
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2.2
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• ...
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4.1
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2.1 . . . .
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2.1 ...........
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1
1.1
1.1
G
Ḡ
HAF(G)
Figure 2. The graph of a hyperlink agent form
2 who is the only player in G with more than one hyperlink is split into two agents,
2.1 and 2.2, which are (completely) connected by a link (dotted line); the other
players are represented by a single agent, respectively. The representatives of h and
h0 , {1.1, 2.1} and {2.2, 3.1, 4.1} , are drawn as in G. By this construction, however,
these representatives contain the same numbers of players as their representees, i.e.,
different ones. Therefore, Ḡ does not extend Theorem 1 to the CF-game G (see (*)
in the proof below).
In HAF (G) , 1.1, 2.1, 2.2, 3.1, and 4.1 stand for groups of agents. The representatives of the original hyperlinks are drawn as solid lines encircling the respective
agents, h̄ = 1.1 ∪ 2.1 and h̄0 = 2.2 ∪ 3.1 ∪ 4.1. Original player 1, for example, has the
group 1.1 of three agents who control his single original hyperlink h, i.e., h̄; original
player 2 has two groups of agents, 2.1 and 2.2, containing three or two agents who
control his original hyperlinks h and h0 , i.e., h̄ and h̄0 , respectively. Further, all of a
player’s agents are completely connected by links (dotted lines). Note that all agent
groups referring to the same original hyperlink have the same number of members,
but the group sizes are different for h̄ and h̄0 , namely 3 and 2. Thus, the representatives of h and h0 contain the same overall number of agents, namely 6. Yet, this
comes at a price: Since HAF (G) comprises much more agents than Ḡ, HAF (G) is
less natural than Ḡ.
9
¯
¯
Remark 1. If H is a graph then χ (H) = |h| = 2 and therefore ¯N̄ (i, h)¯ = 1
for all h ∈ H, i.e., for any player there is just a single agent for any of his links.
Since (11) and (12) then become (7) and (8), respectively, the HAF and the LAF
coincide for CO-games. And since the CF-versions of μ and π are extensions of their
CO-versions, respectively, the proof of Theorem 2 below also proves Theorem 1.
Now, we can express the position value for CF-games in terms of the Myerson
value for CF-games.
Theorem 2. For any 0-normalized CF-game without isolated players G = (N, v, H) ,
we have π i (G) = μN̄(i) (HAF (G)) for all i ∈ N.
Proof. Let G be as in the theorem. For any K̄ ⊆ N̄ define the set of original
hyperlinks which this player set establishes,
¡ ¢
©
ª
(13)
H K̄ := h ∈ H|∀i ∈ h : N̄ (i, h) ⊆ K̄ .
For (i, h, k) ∈ K̄ ⊆ N̄, this implies
¡ ¢
¡
¢
v̄ H̄ K̄ − v̄H̄ K̄\ {(i, h, k)} =
(3)
=
X
S̄∈C (K̄,H̄|K̄ )
(12)
=
X
S̄∈C (K̄,H̄|K̄ )
=
X
¡ ¢
v̄ S̄ −
0-norm.
=
X
S∈C (N,H (K̄ ))
(14)
(4)
=
S̄∈C (K̄\{(i,h,k)},H̄|K̄\{(i,h,k)} )
¡ ¡ ¢¢
v N S̄ −
S∈C (N (K̄ ),H (K̄ ))
X
v (S) −
v (S) −
X
¡ ¢
v̄ S̄
S̄∈C (K̄\{(i,h,k)},H̄|K̄\{(i,h,k)} )
X
¡ ¡ ¢¢
v N S̄
v (S)
S∈C (N (K̄\{(i,h,k)}),H (K̄\{(i,h,k)}))
X
v (S)
S∈C (N,H (K̄\{(i,h,k)}))
¡ ¡ ¢¢
¡ ¡
¢¢
vN H K̄ − v N H K̄\ {(i, h, k)} .
where the third equation holds for the following reasons: By (11), (i, h, k) and
(i, h0 , k0 ) are connected within K̄ whenever both are contained in K̄. Also, if (i, h, k)
¡
¢
and (j, h0 , k) are connected in K̄, H̄|K̄ then by (13) and (12), i and j are connected
¡ ¡ ¢
¡ ¢¢
¡ ¢
in N K̄ , H K̄ and vice versa. Any order ρ ∈ Σ N̄ induces a unique order
σ ∗ (ρ) ∈ Σ (H) such that
(15)
σ ∗ (ρ) (h) < σ ∗ (ρ) (h0 )
⇔
max ρ (ı̄) < max ρ (ı̄)
ı̄∈h̄
ı̄∈h̄0
10
H̄
v̄
for all h, h0 ∈ H. Using (12), (14), (13), and (15), we derive MC(i,h,k)
(ρ) = 0 if
H̄
N
v̄
(ρ) = MChv (σ ∗ (ρ)) if ρ (i, h, k) = maxı̄∈h̄ ρ (ı̄) .
ρ (i, h, k) < maxı̄∈h̄ ρ (ı̄) and MC(i,h,k)
¯ ¯
¡
¡
¢¢
By (15), we have σ ∗ Σ N̄ = Σ (H) . (*) Since ¯h̄¯ = χ (H) for all h ∈ H, all induced orders σ ∗ (ρ) are equally likely if the orders ρ are so. Further, it is also clear
that for all induced orders σ ∗ (ρ) , the probability that ρ (ı̄) = maxj̄∈h̄ ρ (j̄) , ı̄ ∈ h̄
¡ ¢
is χ (H)−1 . Hence, taking expectations over all sequences in Σ N̄ and Σ (H) ,
respectively, we obtain
¡
¢
μ(i,h,k) (HAF (G)) = χ (H)−1 Shh H, vN
by (1) and (3), and then
¡
¢
μN̄(i,h) (HAF (G)) = |h|−1 Shh H, vN
by (9). Summing up over N̄ (i) finally gives
μN̄(i) (HAF (G)) = π i (G)
by (5) and (10).
References
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