Shapley Value Definition and Example of How It Is Applied

What Is the Shapley Value?

The Shapley value is a solution concept used in game theory that involves fairly distributing both gains and costs to several actors working in a coalition. Game theory is when two or more players or factors are involved in a strategy to achieve a desired outcome or payoff.

The Shapley value applies primarily in situations when the contributions of each actor are unequal, but each player works in cooperation with each other to obtain the gain or payoff.

The Shapley value ensures each actor gains as much or more as they would have from acting independently. The value obtained is critical because otherwise there is no incentive for actors to collaborate. Shapley value–which is named after Lloyd Shapley–has many applications, including business, machine learning, and online marketing.

Understanding Shapley Values

In game theory, a game can be a set of circumstances whereby two or more players or decision-makers contribute to an outcome. The strategy is the gameplan that a player implements while the payoff is the gain achieved for arriving at the desired outcome.

Essentially, the Shapley value is the average expected marginal contribution of one player after all possible combinations have been considered. Shapley value helps to determine a payoff for all of the players when each player might have contributed more or less than the others. Shapley value has numerous applications whereby the players could instead be factors needed to achieve the desired outcome or the payoff.

While not perfect, this has proven a fair approach to allocating value. In this situation, "fair" means that the Shapley value satisfies four conditions:

1. All the gains from cooperation are distributed among the players—none is wasted.
2. Players who make equal contributions receive equal payoffs.
3. The game cannot be divided into a set of smaller games that together achieve greater total gains.
4. A player that makes zero marginal contribution to the gains from cooperation receives zero payoff.

How Shapley Values Are Applied

A famous example of the Shapley value in practice is the airport problem. In the problem, an airport needs to be built in order to accommodate a range of aircraft that require different lengths of runway. The question is how to distribute the costs of the airport to all actors in an equitable manner.

The solution is simply to spread the marginal cost of each required length of runway among all the actors needing a runway at least that long. In the end, actors requiring a shorter runway pay less, and those needing a longer runway pay more; however, none of the actors pay as much as they would have if they had chosen not to cooperate.

Shapley values help with marketing analytics. A company selling its product on its 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.

In theory, a player can be a product sold in a store, an item on a restaurant menu, a party injured in an auto accident, or a group of investors in a lottery ticket fund. The Shapley value can be applied in economic models, product line distributions, procurement measures for embassies and industries, market mix models, and calculations for tort damages. Strategists are continuously discovering new methods to use the solution.

The Bottom Line

In game theory, the Shapley value takes into consideration how all players contribute a different amount and, therefore, helps determine a payoff for all players when this happens. Shapley values have wide applications, from airport design to market analytics.