Shapley Values Concept

4 min readAug 17, 2022

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What are Shapley Values?

Game theory is the branch of mathematics studying how and why people make decisions (Specifically, it is “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers”; source: Wikipedia). A famous example of game theory is the prisoner’s dilemma
The Shapley value is a solution concept used in game theory that involves fairly distributing both gains and costs to several actors working in coalition. Basically, “the Shapley value is the average expected marginal contribution of one player after all possible combinations have been considered”
For example, you can use Shapley Values to measure each players’ importance in a basketball’s team. To do this, we would check the average marginal contribution across every possible sequence in which the players could have been added to the team

Shapley Values application example

“Shapley values help with marketing analytics. A company selling their product on their website will likely have different touchpoints, which are ways for customers to engage with the company and drive them to ultimately buy their product. For example, a company might have various marketing channels to attract potential customers, such as social media, paid advertising, and email marketing campaigns. The Shapley value can be applied here, assigning each marketing channel as “players,” with the “payoff” being the purchase of the product. By assigning values to each channel, Shapley value analysis can help determine what channels get the credit for the online purchase.”

Step by step process

Let’s say that Amy, Bob and Claire share a taxi to come back to their respective homes which are all on the same way / direction:

it costs Amy 6 pounds to get home
it costs Bob 12 pounds to get home
it costs Claire 42 pounds to get home

Framework and characteristics

In a cooperative game you need players and coalitions that we define as :

each group of players is called a coalition and the coalition of all players is called the grand coalition. For example, a coalition could be Amy and Bob catching the taxi by themselves
recall that ⊆ means subset as in {7, 15} ⊆ {7, 13, 15, 21}

each coalition is matched to a real number. For example, if Amy takes the taxi by herself, the fare will be 6 pounds. If Bob and Claire take a taxi together, the value of the coalition will be 42 pounds

Fair distribution

In order to be fair with everybody, we need to respect the following conditions:

Efficiency: all the gains from cooperation are distributed among the players — none is wasted.
Null player: a player that makes zero marginal contribution to the gains from cooperation receives zero payoff
Symmetry: players that make equal contributions receive equal payoffs
Additivity: the game cannot be divided into a set of smaller games that together achieve greater total gains

It has been shown that there is only one way to distribute the values in order to respect those conditions and that is the Shapley Value

Shapley Value

Given a characteristic function game val (previously labelled as G), the Shapley value of a player j is given by:

Shapley value is a distribution
the right part represent the difference between value achieved by sub team and the realized value when a player j is added