CSCE475/875 Multiagent
Systems
Handout 13: Coalitional
Game Payoffs and The Shapley Value
November 3, 2011
(Based on Shoham
and Leyton-Brown 2011)
First, let
denote
a mapping from a coalitional game (that is, a set of agents
and a value function
)
to a vector of
real values, and let
denote the
th
such
real value. Denote such a vector of
real values as
.
Each
denotes the share of the grand
coalition’s payoff that agent
receives. When the coalitional game
is understood from context, we
write
as a shorthand for
.
Definition
12.2.1 (Feasible payoff) Given a coalitional game (N,
v),
the
feasible payoff set is defined
as
In other words, the feasible
payoff set contains all payoff vectors that do not distribute more than the
worth of the grand coalition. We can view this as requiring the payoffs to be weakly budget balanced.
Definition
12.2.2 (Pre-imputation) Given a coalitional game
,
the pre-imputation set, denoted
,
is defined as
We can view the pre-imputation
set as the set of feasible payoffs that are efficient, that is, they distribute the entire worth of the grand
coalition. Looked at another way, the pre-imputation set is the set of
feasible payoffs that are strictly
budget balanced. (In this setting these two concepts are
equivalent; do you see why?)
Definition
12.2.3 (Imputation) Given a coalitional game
,
the imputation set,
,
is defined as
Imputations
are payoff vectors that are not only efficient but individually rational. That is, each agent is guaranteed a payoff
of at least the amount that he could achieve by forming a singleton coalition.
Perhaps
the most straightforward answer to the question of how payoffs should be
divided is that the division should be fair.
First, say that agents
and
are interchangeable
if they always contribute the same amount to every coalition of the other
agents. That is, for all
that contains neither
nor
,
The symmetry axiom states that such agents
should receive the same payments.
Axiom 12.2.4
(Symmetry) For
any
,
if
and
are interchangeable
then
Second, say that an agent
is a dummy player if the amount
that
contributes to any coalition is exactly the
amount that
is able to achieve alone. That is, for
all
such that
The dummy player axiom
states that dummy players should receive a payment equal to exactly the amount
that they achieve on their own.
Axiom 12.2.5
(Dummy player) For
any
,
if
is a dummy player then
.
Finally, consider two different
coalitional game theory problems, defined by two different characteristic
functions
and
,
involving the same set of agents. The additivity
axiom states that if we re-model the setting as a single game in which each
coalition
achieves a payoff of
,
the agents’ payments in each coalition should be the sum of the payments they
would have achieved for that coalition under the two separate games.
Axiom 12.2.6 (Additivity) For any two
and
,
we have for any player
that
,
where the game
is defined by
for every coalition
.
Theorem 12.2.7 Given a coalitional
game
,
there is a unique pre-imputation
that satisfies the Symmetry,
Dummy player, Additivity axioms.
Definition
12.2.8 (Shapley value) Given a coalitional
game
,
the Shapley
value of player
is given by
This expression can be viewed as
capturing the “average marginal
contribution” of agent
,
where we average over all the different sequences according to which the grand coalition
could be built up from the empty coalition. More specifically, imagine that the
coalition is assembled by starting with the empty set and adding one agent at a
time, with the agent to be added chosen uniformly at random. Within any such
sequence of additions, look at agent
′s marginal
contribution at the time it is added to the set
,
its contribution is
Now
multiply this quantity by the
different
ways the set
could have been formed prior to agent
’s
addition and by
different ways the remaining
agents could be added afterward. Finally, sum over all possible sets
and obtain an average by dividing by
the number of possible orderings of all the
agents.
Example of the Shapley value in
action …
Consider the voting game. Recall that the four political parties
and
have 45, 25, 15, and 15 representatives,
respectively, and a simple majority (51 votes) is required to pass the $100 million
spending bill.
If
we want to analyze how much money it is fair for each party to demand, we can
calculate the Shapley values of the game.
Note that every coalition with 51
or more members has a value of $100 million, and others have $0. In this game,
therefore, the parties
,
and
are interchangeable, since they add the
same value to any coalition. (They add $100 million to the coalitions
that do not include them already and to
;
they add $0 to all other coalitions.)
The Shapley value of
is given by:
The Shapley value for
(and, by symmetry, also for
and
)
is given by:
Thus the Shapley values are
,
which add up to the entire $100 million.