# Shapley Value: Definition – Overview – Applications – Examples

## What Is the Shapley Value?

The Shapley value is a concept drawn from cooperative game theory.  It is used to assign a value to a player’s participation which corresponds to the eventual result of the game. Assigning value includes equitably allocating both advantages and costs among the numerous individuals in a coalition. When two or more players or components are involved in a strategy to obtain a desired outcome or payout, this is referred to as game theory.

The Shapley value is most applicable when the contributions of each actor are uneven.  Yet, each person works together in cooperation to achieve the gain or payout. The Shapley value assures that each actor gets as much as or more than they would if they were acting on their own. The value obtained is crucial since there is no motivation for actors to interact otherwise. The Shapley value, named after Lloyd Shapley, has a wide range of applications, including business, machine learning, and internet marketing.  For his efforts and contribution, Lloyd Shapley won the Nobel Prize in Economics in 2012

### What is Game Theory

Game theory is a theoretical framework for situations among competing players. It is the science of optimal decision-making of independent and competing actors in a strategic setting. A “Game” is any situation in which there are several decision-makers, and each of them wants to optimize their results. The optimizing decision will depend on the decisions of others. The game identifies the players’ identities, preferences, and available strategies and how these strategies affect the outcome. Game Theory attempts to define these situations in mathematical terms and determine what would happen if every player acts rationally. Cooperative game theory assumes that groups of players, called coalitions, are the primary units of decision-making, and may enforce cooperative behavior. Consequently, cooperative games can be seen as a competition between coalitions of players, rather than between individual players.

## Shapley Value – A Closer Look

A game, according to game theory, may be defined as a set of conditions in which two or more participants or decision-makers contribute to an outcome. The strategy is the game plan that a player follows, whereas the payout is the profit earned as a result of achieving the intended goal. The Shapley value is essentially the average predicted marginal contribution of one player after all potential combinations have been explored. When each participant may contribute more or less than the others, the Shapley value assists in determining a payout for all of the players. Shapley value offers a wide range of applications.  For example, where the players might instead be elements required to accomplish the desired outcome or payment. While not perfect, this has shown to be a fair method to value allocation. In this case, “fair” indicates that the Shapley value meets four criteria:

1. Total distribution – All the gains from cooperation are distributed among the players—none is wasted.
2. Equal pay for equal play – Players that make equal contributions receive equal payoffs.
3. No sub-games – The game cannot be divided into a set of smaller games that together achieve greater total gains.
4. Zero pay for zero play – A player that makes zero marginal contribution to the gains from cooperation receives zero payoffs.

In principle, a player may be a commodity sold in a store, an item on a restaurant menu, a person injured in a car accident, or a cooperative of investors to a lottery ticket fund. The Shapley value is useful in economic models, product line distributions, procurement measures for embassies and industry, market mix models, and tort damage estimates. Strategists are always coming up with fresh ways to apply Shapley values and game theory methodology.

## Shapley Value – Examples

The airport problem is a well-known application of the Shapley value. Consider an airport that is being built to handle a variety of aircraft.  However, each requires a different length of the runway to take off and land. The dilemma is how to disperse the airport’s costs to all stakeholders fairly and equitably. The answer is simply to distribute the marginal cost of each necessary length of runway among all actors who demand at least that length of the runway. In the end, actors requiring a shorter runway pay less, while those requiring a longer runway pay more. However, none of the participants pay as much as they would have if they had not cooperated.

### Shapley Values in Business and Marketing

Shapley value analysis can assist in determining the values for various components.  However, in practice, an estimate is required in assigning those values, which can lead to inaccuracies. For example, Shapley values can help with marketing analytics. A firm selling its product on its website would most likely have many touchpoints.  Each of these points is a method for buyers to interact with the company and eventually purchase their goods. Moreover, the corporation may use a variety of marketing channels to attract potential consumers.  These might include social media, paid advertising, and email marketing campaigns. The Shapley value may be used here.  For example, each marketing channel acts as a “player” and the “payoff” is the purchase of the goods. Value analysis can assist in determining which channels receive credit for an online transaction by assigning values to each channel.

### Shapley Values in Machine Learning

Shapley values are used in everyday life to divide a cost or reward equitably among a group of people who may not have equal influence on the result. In machine learning models, Shapley values can appropriately allocate impact to characteristics that may not have equal influence on machine predictions.

## Up Next: What Is a Trimmed Mean?

A trimmed mean is a form of averaging in which a specified fraction of the greatest and smallest values are removed before computing the mean. The trimmed mean is calculated after deleting the given outlier data using a conventional arithmetic averaging algorithm. The use of a trimmed mean reduces the impact of outliers or data points on the tails. These extreme data points may have an unfair effect on the traditional or arithmetic mean. Trimmed means are used in economic data reporting to smooth out the findings and provide a more accurate image.

An arithmetic mean is a straightforward mathematical average of two or more values.  Whereas, the trimmed mean assists in reducing the impact of outliers on the computed average. The trimmed mean is best suited for data with substantial, unpredictable deviations or distributions that are strongly skewed.