A coffee shop wants to bring back a group of customers that are socially connected to each other. It wants to offer them a great discount as an incentive to return. The coffee shop wants to empower an individual trusted by all of the group to share this offer around with other group members. The more he or she successfully shares the offer (i.e. the person he or she shares the offer with, uses it), then the less he or she (the sharer) pays for their own coffee.
so in this scenario you have the coffee shop on one side of the dillema and the group on the other side. To bring about a solution, I (a company) introduces a third party trusted by all sides to negotiate a (price) balance
Can we use game theory to model this?
To me it looks like a non-standard version of a Principal-Agent model, where the Principal, the shop owner, decides on the payment scheme for the agent, the trusted party. The general assumption is that the shop can't tell how hard the agent works (how much effort is expended) and ends up basing compensation on output, on how many people show up at the coffee shop. That's the main result, but in specific cases you may be able to calculate exactly what that compensation scheme should be (e.g. linear or logarithmic in the number of people who show up?). A game theoretic approach to the general problem is in section 11.4 of Mailath and Samuelson Repeated Games and Reputations. An added twist would be to model how the trusted party interacts with the customers,e.g. how much would he have to work to get them to come back to the shop, and even if there are network effects (say, if a majority agree to come back, will they all come back?).