**Cooperative Game theory (CGT) **is a model of Game theory wherein the participants (players or a set of players called coalitions) though in competition, will work with a cooperative behavior of an external force. It is also referred to as a **Coalitional game**. The coalition behavior of the participants is also tracked and monitored by some external agency, like a contract, set rules by the regulatory authority or such an association, trade organizations, etc.

## What is Game Theory?

A mother left her home for a chore. The house had only two children. After returning from the chore, their mother identified a few missing cookies. Irrespective of whether she should be upset or not, she still wanted to know, after all, who was the naughty kid, the culprit for missing cookies.

She separates the two children and places them in different rooms. She gives each one of them an option.

- If both admit, they will each be grounded for five weeks.
- Whoever admits first will be grounded for three weeks, and the silent partner will be grounded for nine weeks.
- Consequentially, if neither admits, each will be grounded for two weeks.

In this scenario, the mother has inevitably set up a **‘Game Theory.’**

### Game

She has set up a scenario or situation where the outcome will be dependent on the action taken by one of her kids.

### Players

The kids

### External Agency

The agency determining the rules is the Mother

The **payoff** is the benefit available for each kid.

Thus, the option available before each kid is the **Information Set**

- Silence or
- Admission

Each option discussed above has its own consequence

When either of the kids decides to take action, the action is based on

- The above statement made by his mother
- His perception of what the other kid would do
- And also what is best for him

it is Strategy

### Equilibrium

It is a point where the kids decide what to do and implement their strategy.

## Utility of Game Theory

The GT pioneers are J. von Neumann and O. Morgenstern. They proposed the Cooperative GT.

With the growth of Artificial Intelligence (AI), the Game Theory (GT) has gained a new perspective and is useful in almost all fields, including Business, Economics, Political Science, and biology. Wherever and whenever multiple parties are there, there arises a need for a decision. It becomes easier and more advantageous to decide by using Game theory.

## Branches

There are two main branches of game theory:

- Cooperative GT
- Non-cooperative GT

## Cooperative Game Theory (CGT)

- If players form a group and try to come to an agreement on pay-off with or without strategies, they are called a coalition.
- A Coalition is a subset of players. It is an elaborate theory, but in a nutshell, the players can make a pact or agreement then; it is called Cooperative Games Theory.
- Under a Cooperative Game Theory, players can share their payoff and coordinate their strategies.

### Cooperative Games with Transferable Utility

- When the advantages/utility can be
**transferred**by one player (or coalitions) to another without incurring any losses for themselves, then it is called Transferable Utility (TU)

### Practical Example of Cooperative Game Theory

- Assume a scenario where two entities, X Ltd and Y Ltd, are in the beverage industry and are launching a new fizzy fruit soda based on a flavor to be imported from another country. (
**Game**) - If the product clicks, it adds to their customer base and ends the profit line. They will be pioneers in the industry by introducing a flavor that was non existent (Payoff)
- What would be the quantum of fruit the companies should import?. (Strategy)
- Assuming, Government imposes restrictions on the quantity that can be imported in a period. (External forces influencing decision)
- The company’s now have many options, including (Agreement between
**players or a coalition**) X Ltd and Y Ltd can enter into an agreement where both the entities would be importing the same quantity but in different quarters to ensure the quota restriction is met. - This is a simple scenario. But real-time situations can get more messier with too many combinations.
- A coalition might not always be possible, and the assumption that the parties act rationally may not always be true.

### Core in Co-operative Game Theory

- The Core is a set of all feasible options/allocations with the Core Property.
- Therefore, a core property is an ultimate allocation; no other coalition can improve it.

### Shapely Value

- When a group of players together come into an agreement and share the payoffs, the individual contributions might not be the same.
- One of the players or a group could have contributed more compared to the others.
- As a result, the question arises, how should the payoff be shared among the group?
- Lloyd Stowell Shapley has come out with a solution concept in cooperative game theory to take care of this issue.
- A characteristic function specifies for every group of players the total payoff that the members can obtain by entering into an agreement (cooperation between themselves)

### Illustration

Suppose that there are two players and v({P1}) = 10, v({P2}) = 12 and v({P1,P2}) = 23

Therefore, only two outcomes could be considered.

P1 comes first, and then P2 comes, or P2 comes first, and then P1 arrives.

**Scenario 1**: P1 has arrived first.

Therefore, we consider the value of P1. i.e.) v({P1}) = 10

Now, the second player, P2 has arrived. The Grand Coalition is 23. So contribution of v({P2}) = 23 – 10 = 13.

**Scenario 2**: The next **Scenario** is that P2 has arrived first.

Here, we consider the value of P2 i.e.) v({P2}) = 12

Then P1 has arrived. The Grand Coalition is 23. So, contribution of v({P1}) = 23 – 12 = 11

**A summary of the above is**

Probability | Order of arrival | P1’s Marginal Contribution | P2’s Marginal Contribution |

0.5 | P1, P2 | 10 | 13 |

0.5 | P2, P1 | 11 | 12 |

=(10*0.5)+(11*0.5)=10.5 | =(13*0.5)+(12*0.5)=12.5 |

The **shapely value** is = P1 = 10.5 & P2 = 12.5

### Limitations of Co-operative Game Theory

- As the number of participants increases, the calculations become complex
- It is a general rule of logic and not the ultimate winning idea
- Uncertainties of reality are not considered
- The assumption that the parties cooperate and always act rationally is questionable in real-life scenarios.

## Final Words

As we discussed, there are game theories to determine the strategies and likely outcomes. And all that varies depending upon the type of games are being played- Cooperative, Non-cooperative, Hybrid, etc. The most important point to note in Cooperative Game Theory is that here the expectations, opportunities, and payoffs for all the players of the coalition are given. And then, an optimal strategy is worked upon to get the optimal payoff for the coalition or for all the players. So the individual payoffs are surrendered for the optimal payoff of the Group. And that means some of the players who could have got more individually will have to content with the lesser pay offs in, the larger interest of the group.