Next question is of how to allocate

the joint payoff along the cooperative trajectory.

So, now the institutions or

players made an agreement on strategies,

but how to allocate the joint payoff so that

this agreement would be beneficial for all of the participants,

because sometimes the cooperative agreement that assigns

such a behavior is not beneficial for one of the participants.

Then we need to reallocate

the cooperative payoff among the players

so that it would be individually rational and group rational.

So, in order to do that, firstly,

we need to define the characteristic function.

In one of the previous sections,

we also worked with characteristic function.

But there, it was a static game.

So, the characteristic function was predefined.

In here, we need to define

characteristic function using the multi-stage game.

So, we will use the following approach.

The characteristic function of coalition S will

be defined as a value of

the zero-sum game between coalitions S and N-S. So,

for a given coalition S,

we consider a zero-sum game between all players acting as one from

coalition S and all players acting as

one from college and N minus S. So,

the payoff of players from coalition S in

saddle point will be the value of characteristic function.

For this case, the value of

characteristic function of all players,

so the value of characteristic function for coalition N will be

equal to the maximum joined payoff in the whole game.

Also, the characteristic function

which is defined by the following approach,

and it is called a Maximin Approach,

satisfies the superadditivity property.

This property tells us

that a grand coalition

of all players is beneficial for all of them.

On the basis of characteristic function,

we will define a set of imputations in

the whole game or set of ways

for allocating joint payoff in the game.

Each imputation is a vector ksi from ksi_1 to ksi_n,

which satisfies

the group rationality and individual rationality properties.

So, the individual rationality property tells us

that the imputation or the payoff that player i receives

in the cooperation is more or equal to his payoff that

he obtains if he's not in the cooperation or not cooperating,

or the value of characteristic function

of coalition consisting of only player i.

The group rationality property tells us that we

allocate a maximum joint payoff of all players.

Why? Because the V(N) in

this particular approach actually

is the maximum joint payoff of players.

So, the payoff that the players

obtain along the cooperative trajectory.

Next thing we need to do is,

we need to choose a subset from the set of imputation or

the set of imputation that we would

use for our cooperative agreement.

The subset we will call a cooperative solution.

In this case, we will use a Shapley Values cooperative solution.

On the slide, you can see the explicit formula for Shapley Value.

Of course, we would need to define it using the set of axioms,

but we already did that in one of the previous sections,

and then we derive the explicit formula.

Now we will just use it for our particular game model.

Let's go back to our model of signing of package from documents.

On the slide, you can see values of characteristic function.

The value of characteristic function for coalition

1,2,3 or for all players is equal to six.

Why? Because the maximum joint payoff

of all players is equal to six.

Let's consider the way we can

calculate the value of characteristic function

for coalition consisting of only player one.

It is equal to one.

Why? On the first step,

player one makes a move.

He has two alternatives: to approve the package

of documents or to decline the package of documents.

Let's suppose that he chooses to approve a package of documents.

Then, according to the procedure

of calculating the characteristic function,

players two and three should minimize his payoff

or they should choose strategies so

that the payoff of player one would be minimum.

Then, according to this procedure,

player two would choose to approve a package of documents,

but the player three would choose to decline.

Then, as a result,

player one would receive payoff equal to one-third.

Of course, on the first step,

player one would choose an alternative B, so to decline.

Then his payoff would be equal to one

no matter what strategies will the players two and three chose.

So, the characteristic function of player one is equal to one.

In the same way, we can calculate the values of

characteristic function for other coalitions.

On the basis of values of characteristic function,

we can define a set of imputations in the following way.

This is set of vectors,

Xi one, Xi two, and Xi three,

where Xi one is more or equal to one,

Xi two is more or equal to one-half,

and Xi three is more or equal to one-third.

The sum of them is equal to six.

Then, in a set of imputations,

we can choose a specific cooperative solution or,

in this case, we will choose a Shapley Value.

The formula for Shapley Value is presented on slide.

On this slide, you can see a list of references,

where you can find more information of how to

define a cooperative multi-stage game,

how to define a characteristic function for this game

and what cooperative solutions can be

used for a multi-stage games.