**Augusto Rufasto**

**Strategic Intelligence**

**Shapley Value**

Shapley value is a Game
Theory concept proposed by Lloyd Shapley in 1953 aiming to propose the fairest
allocation of collectively gained profits between the several collaborative
agents. The basic criterion is to find the relative importance of every agent
regarding the cooperative activities.

Let *N* be a set of *n* agents. Let us define the characteristic function *v(S)* as follows:

The whole set of
characteristic functions *V(N)* is:

_{}

The *v* function has the following properties:

_{} (1)

_{} (2)

Here *S* and *T* are disjoint subsets from *N*.
Function *v* should be interpreted as
follows: If *S* is a coalition of *s* agents all of which agree to cooperate,
then *v(S)* is the total expected
revenue from this *S* cooperation,
regardless of the actions of any agent external to *S* may do. As stated in the condition of super-additivity,
collaboration from agents new to the coalition should increase the revenue to
the coalition, but never decrease it.

Let us define the
allocation function Φ as:

_{}

_{}

Given *v(N)*, Φ Hill specify the weight
Φ* _{i}* for each agent

Let’s now consider
that there exists one original coalition *S*
and that a new agent called *i* studies
the possibility of joining it. That will imply that *i* can create value added after joining the group. His contribution
should be assessed by the method of comparing the value of the coalition *S* including *i* minus the value of the sole coalition *S*. Afterwards, it will be necessary to build the fair distribution
criterion. To do that, we will use the propositions of Shapley.

So let’s now compare
the value of the new coalition minus the value of the original one. We will
assess *i*’s contribution by
calculating:

_{}

The total
sub-coalitions of *s* agents (*s* does not include newcomer agent *i*) that can be built between the *n* members of the *N* set is:

_{}

We are interested in
formalizing the event of i associating himself with the s members of a group.
Consider then that the total possible coalitions with size *s* that necessarily do not include *i* is exactly the number of the total possible coalitions size *s+1* that do necessarily include i.

When the new agent *i* is willing to participate in an (*n+1)*-size coalition, then we may say
that *s* equals *n*, and therefore the total number of possible coalitions with n+1
members including agent *i* will be:

_{}

Let us build weight
factor *q* as:

_{}

Now, the contribution
of new agent *i* to one specific
coalition will be divided by the total number of possible coalitions, that is,
his contribution will be multiplied by the following factor:

_{}

The coalition of agent
*i* with subset *S* will therefore get this distributed revenue. But inside a
specific subset *S* there are *s* agents. The final coalition has
therefore *(s+1)* agents. We need now
to distribute the contribution share already allocated to the coalition between
*(s+1)* agents. And in this way we get
weight factor *q*. We may think of q as
the area share of a small rectangle which width equals the total number of
coalitions that do necessarily include agent I, and which height equals the *(s+1)* number of agents in such
coalition.

Let us consider:

_{}

The sum includes all
subsets size *s* that necessarily do
not include agent *i*, that is, the
total subsets size *(s+1)* that do
necessarily include agent *i*.

Consider agent *i* analyzes his future collaboration with
group *S*. Let’s call the shares
corresponding to *i* and to each member
of group *S* as r(α_{i}) y
r(α_{s}). May *j* be the
general notation for agent *i* and for
each member of *S*. Then, each share rj
will be calculated as follows:

_{}

Consider the
possibility of a new firm to be built upon the coalition of one entrepreneur
α_{0} and *m* workers
α* _{i}*. Entrepreneur
α

Now suppose that all m
workers joined that group. Then the size of the coalition will be m+1, and
Shapley allocation factor according to the rule are:

_{} for entrepreneur
α_{0}

_{} for any worker
α_{i}

Let us build the
shares that each agent will receive of the coalition revenues, let these shares
be called r(α_{0}) y r(α_{i}). Each share will be
calculated as follows:

_{}

So that calculated
shares are:

_{}

_{}

This means that the
entrepreneur gets 50% of the coalition revenue and that the remaining 50% is to
be distributed between the *m* workers.

The revenue for the
coalition is *mB*. Entrepreneur earns:

_{}

Each worker earns:

_{}

This corresponds to
half his marginal contribution to the coalition.

This states a rule for
defining salary in a firm. As before, we suppose that the presence of the
entrepreneur is necessary for the firm to operate. Let’s say the firm is has
now the manager and *m* employees
producing *B* each one and is also
willing to hire a new employee. If, by staying out of the firm the future
employee produces zero but entering in the firm he produces C, then the values
for the allocation rule are:

So that calculated
share factor are:

_{} for
entrepreneur α_{0}

_{} for
any worker α_{i}

_{} for
the new employee_{}

The revenue for the
coalition is *mB+C*. Entrepreneur
earns:

_{}

Each old worker earns:

_{}

The new employee
earns:

_{}

Any agent which joins
any coalition under the same conditions described here will deserve to retrieve
an income equal to half his marginal contribution:

According to that, if
a new employee joins a firm which without his participation cannot generate new
incomes, then the new employee’s fair salary will be half his marginal
contribution to the firm. Let’s put it this way: if the new employee is capable
to generate 200,000 dollars a year within the firm and zero when out of it,
then his annual fair salary within the company will be 100,000 dollars. Also
note that if this new employee is capable to make money out of the firm, then
his fair salary will automatically rise, because the firm’s marginal
contribution to his earnings will be smaller.

Having already stated
the salary rule, one may think about the differences that can be found in the
real world, regarding salaries. The one main reason for these differences is the
bargaining power. If a firm has great bargaining power, then new employees will
be pushed to accept salaries at any point below the fair level. On the side of
the firm, there will appear extraordinary earnings. On the side of the new
employee, there will be a relative loss when accepting the salary offered by
the firm. The negotiation is performed in a conventional zero-sum-game
scenario, so extra earnings in one side are covered by losses on the opposite
side.

The following are the
desired properties for a coalition (it has been stated that Shapley value is
the unique criteria set that has properties 2, 3 and 5):

For each *i* in *N*:

_{}

It means that each
agent gets from the coalition at least the value he can generate by himself
alone.

Total revenues are
fully allocated between the agents:

_{}

Let *w* be value earned by a coalition which
includes an agent called j, and let w be equivalent to *v*, the value earned by a coalition which includes agent i. Then:

_{}

The names of the
agents do not affect the allocation of revenues.

For any pair of agents
*i* and *j*, if, for every S that belongs to *2 ^{n}*, neither

_{}

then:

_{}

If we add the two
coalition sets described by earning functions *v* and *w*, the earnings’
distribution will be:

_{}

This will be valid for
every agent *i* in *N*.

Wikipedia: http://en.wikipedia.org/wiki/Shapley_value,
which refers to: Lloyd S. Shapley’s ‘A
Value for n-person Games’ in ‘Contributions
to the Theory of Games, volume II’, H.W. Kuhn and A.W. Tucker, editors, Annals of Mathematical Studies v. 28,
pp. 307-317, Princeton University Press

**Augusto Rufasto – Strategic Intelligence**