Focus: Shapley value

Game theory is a fascinating topic codifying and quantifying all sorts of interactions between stakeholders in a game. The most popular setup is the prisoner's dilemma but there is much more to it. Today, we will cover the Shapley value as I recently stumbled across this original yet relatively unknown concept.

Problem at stake:

Example 1: In a salary negotiation, the employee presents their skills and what they bring to the company. But how much are these skills worth?

Example 2: In a joint venture, each founding company contributes expertise. What’s a fair way to distribute ownership or shares?

Example 3: Two or three telecom companies want to build a fiber network that would benefit all. What’s a fair way to split the costs?

When we think about these problems, most of us might guess: “You should ask for an X% raise because you deserve it,” but there is actually a theory for this.

Let’s explore the theory behind it.

Solution:

This is known as a cooperative game, and there is only one breakdown function that meets a few conditions (more on this later).

The main idea: “Player A's fair reward is the average of their marginal contributions to the different coalitions leading to the final setup,” where:

For a game with 3 players (A, B, C), we define:

• Final setup: the final set of stakeholders S {A, B, C}

• Coalition: a subset of S

• Marginal contribution: Adding A to {A, B} is: Value {A, B, C} – Value {B, C}

Now, let’s introduce some values:

• V(A) = 12

• V(B) = 8

• V(C) = 2

• V(A, B) = 22

• V(A, C) = 15

• V(B, C) = 11

• V(A, B, C) = 23

The Shapley value of A is calculated as:

Generalization

From there, you can intuit the general formula where n! is the total number of permutations :

Where K \ A notes the coalition K without A.

What current problem can you apply the Shapley value to?