~ when game theory fails in real world applications ~
Game Theory, as a mathematical science, requires very confined variables in order to come to a workable outcome. These confined variables often don't reflect the complexity of the real world and the Shapley Value is one product of Game Theory fails to apply in reality.
So, this is the formula to find the Shapley value. Yes, you read right, this is the formula that is meant to help... HELP!.. us to allocate resources.
Now, I know you're probably really looking forward to a 25 page explanation of the Shapely value, but! You're out of luck.
Why? Well, not just because it's over-complicated, but because it is only applicable when the participants are essentially psychopaths with no sense of shared humanity!!!
What's your alternative? I hear you say.
Let's say we have 3 rabbits sharing a [rabbit] taxi. If each rabbit were to travel solo, it would cost:
But the joint ride will cost only $30 together. To me, there are two simple ways of splitting the bill:
Each rider pays the same proportion of the joint cost that they would have paid of the total of the solo fares.
I think most people would call this fair, EXCEPT in the case where the group is more concerned with the get-together itself and the required energy and being there is a big enough reward, in which case...
They might choose to split the fare evenly since they are all getting the same reward: spending time with friends.
The Shapley Value is an example of where Game Theory can fall down - where something that works in theory, is completely impractical to apply and will almost always be superceded by a concern outside of the purely theoretical context. In our example, the influence of the friendship dynamic, cultural expectations and desired outcomes are all more influencial motivating factors. That is to be expected, given the trivial nature of the interaction, but I would suggest that even in the highest stakes situations, like a geo-political negotiation or multi-national business merger there will always be greater forces at play.
It's a reminder to bear in mind that theoretical models are tools that need to be used in the context of a world that isn't always purely rational and that sometimes those "irrational" aspects of humanity, like empathy, friendship and a sense of shared responsibility, are beautiful and can actually make negotiations easier.