Cooperative Game Theory Introduction
What is Cooperative Game Theory?
In Cooperative Game Theory, players are allowed to work together, make agreements, and share rewards.
Cooperative means players can form groups.
Game Theory means we study the outcomes of choices.
The key idea is that players may be better off by cooperating than acting alone.
Cooperative doesn't mean the players are being nice or generous. They are still trying to get the best outcome for themselves.
They cooperate only because they can get more by working together than by working alone.
Coalitions
A coalition is a group of players who agree to cooperate.
- Singletons: A single player acting alone
- Sub-coalitions: Small groups formed within a larger game
- The Grand Coalition: When every player in the game joins forces
The big questions are:
- How much can a coalition earn?
- How should that reward be shared?
Example: Three Delivery Companies
Example: FastBox, QuickShip, and SureSend
Three delivery companies can choose to work alone or cooperate.
By sharing trucks and warehouses they can reduce costs.
The numbers below show the total profit (in thousands of dollars) each coalition can make:
| Coalition | Total Profit |
|---|---|
| {FastBox} | 40 |
| {QuickShip} | 30 |
| {SureSend} | 20 |
| {FastBox, QuickShip} | 90 |
| {FastBox, SureSend} | 75 |
| {QuickShip, SureSend} | 65 |
| {FastBox, QuickShip, SureSend} | 120 |
Notice how cooperating increases the total profit.
Notice: The Grand Coalition earns 120, which is much more than the 90 (40+30+20) they would earn if they all worked alone!
Sharing the Reward
If all three companies cooperate, they earn 120.
But how should that be split?
- Equally? (40, 40, 40)
- Based on contribution?
- So that no group wants to leave?
This is where cooperative game theory focuses: fair and stable sharing.
There are two famous ways to look at this:
1. The Core (Stability)
The core is the set of all possible reward splits where:
- The total adds up correctly
- No coalition can do better by breaking away
If a split is in the core, everyone is happy enough to stay cooperating.
If a group could leave and earn more, the agreement is not stable.
Example: if we offer FastBox and QuickShip a combined share of only 80, they will quit the group and form their own coalition to make 90. To keep them, we must pay them at least 90.
2. The Shapley Value (Fairness)
The Shapley Value is a famous mathematical way to split the profit based on each player's average contribution. It calculates how much extra profit a player brings to every possible combination of partners.
"A fair split is the glue that keeps cooperation from falling apart."
Example: Shapley Value for FastBox
If Fastbox ...
- starts the group alone, they bring 40
- joins QuickShip (30) the total becomes 90, FastBox's contribution is 90 − 30 = 60
- joins SureSend (20) the total becomes 75, FastBox's contribution is 75 − 20 = 55
- joins QuickShip and SureSend together (65) the total becomes 120. FastBox's contribution is 120 − 65 = 55
FastBox's fair share (the Shapley Value) is the average of all possibilities = 52.5.
Try calculating the others and you should get the full fair split of: FastBox: 52.5, QuickShip: 37.5, and SureSend: 30. Notice how this adds up exactly to 120.
Why Cooperative Games Matter
Cooperative Game Theory is used to solve real-world questions like:
- Business partnerships
- Cost sharing
- Political alliances
- Teamwork and collaboration
It helps answer questions like:
- Who should get how much?
- Is an agreement fair?
- Will cooperation last?
Working together can create more value, but sharing it fairly is the challenge!