Lloyd Shapley’s Value

The Shapley value, named in honor of Lloyd Shapley, who introduced it in 1953, is a solution concept in cooperative game theory. To each cooperative game it assigns a unique distribution of a total surplus generated by the coalition of all players.

In laymen’s terms, the Shapley Value is the average expected marginal contribution of one player after we’ve looked at all possible combinations. This has been proven to be the fairest approach to allocate value. A ‘player’ can be products sold in a store, items in a restaurant menu, parties injured in a car accident, or a group of investors in a large real estate deal. It is employed in economic models, product line distribution, procurement measures for embassies and industry, market mix models, and calculations for tort damages.

There are several common marketing research techniques in use these days that constitute a Shapley Value. In this piece, are going to explain the Shapley Value, then display its most common uses in marketing research.

Shapley Value Explained

Let’s take a simplest case of the Shapley Value. Let’s say there are three players, A, B, and C. When they enter a game, they add points to the score. The total points in the game is 10.

As the chart below shows, when the order of entry is A B C, A and B’s contribution is 4; C’s is 2. However, in the second round of the game, A’s contribution is 3, B’s is 5.

In total there are six possible different orders of entry. If we play all six, and then take the average contribution of each player, that is the Shapley Value.

This article is now going to explore several common application of the Shapley Value in marketing research. It is a useful notion, as the Shapley Value yields the most equitable solutions and thus yields several vital measures when conducting studies.

Shapley Value – Regression and Brand Equity

In this example a major automobile company has a major public relations disaster due to computer code. In order to regain trust in their brand equity, they commissioned a series of regression analyses to get a better gauge on how are viewing their type of vehicle. However, what they most interested in is how American auto buyers generally view trust.

The technique is fairly common. Given that the news is fresh, our company would like a composite of which values go into ‘Is a Company I Trust’ across industry. To accomplish this they surveyed ten of its major competitors on various elements of automobile purchase. We then stack the data into one dataset and run a Shapley regression. What we hope to see are the major components of Trust.

Not surprisingly family safety is the leading driver of Trust. However, we now have Shapely values of the major components. These findings would normally be fed to the public relations team to begin damage control.

Shapley Value – Product Design

The Shapely concept of relative importance comes from product design where we are able to piece together components any way we want. Products are bundles of attributes, and attributes are collections of levels. A good example is a typical conjoint study for a product design.

For example, an energy drink company may be thinking of how best to configure a package with attributes like number of cans in a bundle, size of ounces in a can, amount of caffeine, flavor, and price. By systematically varying these attribute levels according to an experimental design, they can generate descriptions of hypothetical energy drink that are presented one at a time to respondents who provide ratings of their preferences for all the product configurations.

In a conjoint study relative importance is defined as percentage contribution. We sum the effects of all the attributes to get total variation, and then we divide the effect of each attribute by the total variation to get percent contribution. The attribute with the largest percent contribution is where we have the most leverage. This is, in effect, the Shapely Value. For our energy drink client, the Shapley Values for three different customer bases are shown below.

Changing the ounces in a bottle will impact the likelihood of purchase the most. Not surprisingly, price has a Shapely Value of around 25%. Flavor and strength (caffeine) are really secondary factors in purchase intent, but they still matter.

Shapley Value – Attribute Attrition/Maximizing Product Lines

In our final example, we are going to display how to use a Shapely Value to maximize product lines displayed in a store. Adding the right combination of new items will grow your business; introducing the wrong new items will result in no growth or even cannibalize your top performers leading to revenue declines.

As an example, let’s say a supermarket chain, Gigantic Supermarket, is trying to determine the maximum line of laundry soaps it should display. To accomplish this, we are first going to deploy a Maximum Difference choice exercise. In our example, let’s assume that Gigantic Supermarket is trying to determine which of 28 brands to carry.

Given that we have 28 brands, we divide into 7 questions of 4 products so each respondent sees each brand once. Below is an example question from the MaxDiff.

Of the laundry brands shown below, which are most likely to purchase and which are you least likely to purchase?

1.     Woolite

2.     Wisk

3.     Cold Power

4.     Daz

The power of this analysis is that we can create as many different splits (a split is the 7 choice questions) in random order so that each respondent sees a different set of questions. This is performed using a random-design Excel macro. If the sample is, say, 2000, we may design 200 splits so that each is seen 10 times. We could, if requested, design a split for each respondent, but this is often not necessary.

After completion of the MaxDiff exercise, we have a data structure where we can calculate a Bayesian coefficient using logistic regression for each brand for each respondent. The coefficients are then normalized across each respondent. That is, the sum all brands coefficients equals 0 for each respondent. Thus, some are positive and some are negative.

To put it simply, have had the odds of purchase for each brand for each respondent—the likelihood or purchase. If we take the average across the entire sample of the coefficients, we get the average contribution of each brand to the store. In other words, the Shapely value.

In the table below we see the Shapely values for each of the 28 brands. Those in blue are positive. Those in red are negative.

It is common sense, once the Shapley value is calculated, to choose only those brands which add a positive revenue stream to the product line. Those in red that are near 0 such as Surf and Persil may be added to the inventory if Gigantic Supermarket would like to sell 14 brands.

Our advice to Gigantic Supermarket, stock those brands in blue. To maximize product placement, we would then suggest a TURF analysis. A full explanation is beyond the scope of this article.

Conclusion

In marketing, the Shapley value makes a positive allocation of items or value to that generates positive revenue. A marketing research professional needs to understand conditions under which the Shapley value makes positive allocation to only items or value involved in items or value maximizing flows.