Corporative Game Theory and Shapley Values!
Game Theory is used to analyze the strategic interaction among two or more players in an environment where risk, rewards, and rules are involved. Corporative Game Theory, in-depth assumes that groups of players, called coalitions, are the primary units of decision-making, and may enforce cooperative behavior.
An Example — Suppose I’m setting up a new city on Mars with just 3 groups. Game theory is a tool that will help answer the following question for society — How are the three groups interacting with each other?
Once the above question has been answered, the society has the responsibility to rest power amongst the groups so that they can take decisions on key matters for the city.
How do you distribute power among the groups — The Shapley Value!
“In Game Theory, the Shapley value is a solution concept of fairly distributing both gains and costs to several actors working in coalition. The Shapley value applies primarily in situations when the contributions of each actor are unequal, but they work in cooperation with each other to obtain the payoff.”
In our example, the Shapley Value distributes power among the three groups based on their marginal contribution to society. In other words, the additive value that a group is bringing to the table — higher the marginal contribution, more the power rests with you!
Mathematical Front –
Mathematically, we can imagine the process happening something like this –
Let the society define contribution as money donated by each group A, B, and C in a day, and the total donation a city needs in a day is $90. So, if group A goes first and donates $80, group B which goes second doesn’t have to pay anything since the total donation by A and B can be just $80. The city still needs $10 which is paid by C, going last. Similarly, all permutations of the group are tried and at last — we take an average of the donation made by each group in every instance that makes up to be the final contribution of each group involved. In this case — Group A will come out to be the most powerful with Group C falling second.
Simple enough, right? But since, I’m not going to Mars anytime soon to set up a city, what are the other use cases for The Sharpley Value?
Variable Contribution — Machine Learning
One of the major asks out of a machine learning exercise is to determine each variable’s contribution to the prediction value. If we apply the same concept of The Shapley Value here as well using the below-mentioned formula, we can get the individual contribution of a field in prediction.
Post this, we can see the results something like this –
Here, the size_house variable in a simulation contributes the highest during computation of the dependent variable.
Also, we can see whether a particular variable is responsible for driving up the dependent variable or pulling it down.
Game Theory can be incredibly interesting for discussions and can come in handy for various use cases. To go deeper into the process, you can read further into the key sources to this article —
