Lloyd Shapley’s Value
By Michael Lieberman
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.
Basically, the Shapley Value is the average expected marginal contribution of one player after all possible combinations have been considered. This has been proven to be the fairest approach to allocate value. A ‘player’ can be a product sold in a store, an item on a restaurant menu, a party 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.
The Shaply Value shows up in several popular marketing research techniques. Below, we will set out the ABCs of Shapley Values, and then show its most common uses in marketing research.
Shapley Value ABCs
Here’s the 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 point-value in the game is 10.
As the chart below illustrates, when the order of entry is A B C, A’s and B’s contribution is 4; C’s is 2. However, in the second round of the game, A’s contribution is 3, while 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, we arrive at the Shapley Value.
Now we will see several common applications of the Shapley Value in marketing research. The Value is quite useful: it yields the highly equitable solutions and thus provides several vital research measures.
Shapley Value – Regression and Brand Equity
Let’s say that a major automobile company has a public relations disaster. In order to regain trust in their brand equity, the company commissions a series of regression analyses to gauge how buyers are viewing their type of vehicle. However, what they really want to know is how American auto buyers view trust.
The disaster is fresh, so our company would like a composite of which values go into ‘Is this a Company I Trust’ across industry. Thus, it 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 Shapley Values of the major components. These findings would normally be handed over to the public relations team to begin damage control.
Shapley Value – Product Design
The Shapley concept of relative importance comes from product design, where we are able to piece together components in any way we wish. Products are bundles of attributes, and attributes are collections of levels. We’ll take a typical conjoint study for a product design.
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 a hypothetical energy drink that are presented one at a time to respondents, who rate their preferences for all the product configurations.
In a conjoint study, relative importance is defined as the percentage contribution of each attribute. We sum the effects of all the attributes to get the total variation, and then we divide the effect of each attribute by the total variation to get the percent contribution. The attribute with the largest percent contribution is where we have the most leverage. This is, in effect, the Shapley Value. For our energy drink client, the Shapley Values for three different customer bases are shown below.
Changing the number of ounces in a bottle impacts most heavily the likelihood of purchase. Price is way up there too, with a Shapley 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 will demonstrate how to use a Shapley 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 a revenue decline.
Perhaps a supermarket chain, Gigantic Market, wishes to determine the maximum number of laundry soaps it should display. The first thing to do is deploy a Maximum Difference (MaxDiff) choice exercise. For purposes of illustration, let’s say that Gigantic is trying to decide which of 28 brands to carry.
We take our 28 brands and divide them into 7 questions of 4 products. That way, each respondent sees each brand once. Below is a sample question from the MaxDiff.
Of the laundry brands shown below, which are most likely to purchase and which are you least likely to purchase?
- Cold Power
The beauty of this analysis is that we can create many different splits (a split is the 7-choice question) 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 it is not usually necessary to do so.
The MaxDiff exercise yields a data structure in which 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 of all brand coefficients equals 0 for each respondent. Thus, some are positive and some are negative.
In a nutshell, 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. That is the Shapely Value.
In the table below we see the Shapley Value for each of the 28 brands. Those in blue are positive. Those in red are negative.
Once the Shapley value is calculated, we simply choose 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 would like to sell 14 brands.
We would tell Gigantic Supermarket to 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.
The Shapley Value makes a positive allocation of items or value to that which generates positive revenue. How will this help a marketing research professional? In maximizing flows. The conditions under which the Shapley value makes a positive allocation exclusively to items or value involved in maximizing flows is of extreme interest to our clients, and thus to us.