I have a difficulty with a task, where $15$ producers compete for a multiple (hundreds) clients. The producers make the same product with different costs, and for each combination producer-client the specific transport cost exists.
The clients are price inelastic, i.e., they would buy the same amount of product with any price, but among several options they would prefer the one with the lowest price.
I would like to understand whether there is any way to obtain a set of equilibrium prices for each client using game theory in pure strategies? And if yes, how to approach this task?
Are there limits on how much each of the producers can make? I will ignore quantities otherwise this gets a lot more complicated.
For any given client you can compute the production + transport price for each of the producers. In equilibrium the cheapest producer would supply at second cheapest price 'minus 1 cent'. So you can check for each client how much they would have to pay, from whom they buy and how much profit the producer makes.
Edit: With production limits for producers there is not always a unique solution. One can build scenarios where Producer A can supply either customer X or Y but not both and is indifferent to which but depending on his choice he screws over either producer B or C. So I suspect at best one can have heuristic strategies for each producer that are often optimal but there exist situtations where they are clearly not optimal.