If you live in this Digital Age, chances are you have come across some applications of the Shapley Value. The two most notable use cases of the Shapley Value in everyday business include marketing attribution and (ride-sharing) fare attribution.
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. (Wikipedia)
Marketing use case
Take Marketing as an example. Nowadays, consumers are active on multiple apps and platforms. So naturally, when running an ads campaign, you don't just run on one but multiple channels: Facebook, Twitter, Instagram, ... you name it. This means, before purchasing your product, a customer may have been exposed to your ads on multiple, or one, or none of the channels that you have run ads on.
So, how do you determine the importance of each of these interrelated channels in driving sales in order to allocate your already-tight marketing budget wisely?
One possible answer: the Shapley Value.
Shapley Value can help you attribute a final outcome to multiple players who participate in producing that outcome collectively. The agents (or players) can be anything you want to divide the outcome to. Whereas the outcome can be made up of anything from monetary to non-monetary contribution (e.g., bonus money or marketing conversion rate). In our case, the players are the ad channels while the outcome is the conversion number.
Shapley Value can help you attribute a final outcome to multiple players who participate in producing that outcome collectively.
Calculate Shapley Value
1 - List out the possible Coalition of the set of agents.
If you market your product through 3 channels {Facebook, Tiktok, Google} (let abbreviate it as {FB, TT, GG}), the number of coalitions for a set of 3 is =3! or 6. They are the number of all possible paths that your customers can possibly take before they purchase your products.
2 - Fill out the marginal contribution of each agent based on the order of each agent in the specific coalition.
3 - As a rule of thumb, the marginal contribution of an agent i in a coalition is calculated by the value of everything until i (i included) - everything before i (i excluded).
4 - Sum the marginal contributions, and divide it by the number of all possible coalitions
Putting the Formula into context
What we just did is relatively simple calculations, including 3 steps:
Calculate the marginal contribution of each coalition for each agent i
Sum the marginal contributions for each agent i
Divide it by the number of total possible coalitions (i.e., we assume all coalitions have an equal chance of occurring)
These steps are reflected in the Shapley Value formula though in a more daunting way, especially if you come into contact with the formula before establishing an intuition for it.
Why don’t we just use simple subtraction?
Given: (same numbers as above)
v({FB}) = 100
v({TT}) = 125
v({FB,TT}) = 270
Why v({FB}) ≠ v({FB,TT})-v({TT})?
This is because the marginal impact of a player may vary in different coalitions. In our case, the contribution of TT channel when used individually may be different compared to when it is used in combination with another channel - FB.
If you run a campaign with FB and TT, there are 3 possible paths that your potential customers can take:
See only TT ads then convert
See only FB ads then convert
See both ads, either in FB-TT or TT-FB order, then convert
In the (1) and (2) cases, we can attribute 100% of the impact to the channel - 100 and 125, respectively. However, in the (3) case, the allocation of collective impact is most likely not going to be attributed 100% to any of the channels but rather a number lower than 100%.
It is this consideration of the additional contribution to a collective result that sets this method apart from traditional Rule-based attribution in Marketing, which completely disregards the complementary impacts of other channels and credits an outcome, says sales conversion, wholly to the first touch or last touch channel.
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