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This question concerns game theory and market equilibria which is rarely of focus here at QSE, but at the same time I believe this is a more appropriate place for such question rather than MSE.

There is only one good. There are $N$ consumers and $M$ producers.

  • Each consumer $i$ can buy at most $E_i$ of good from different producers in total. No matter how much he buys, $\xi_i$ is the highest price he may consider paying for the unit of good.
  • Each producer can sell at most $Q_i$ of good to different consumers in total. Now matter how much he sells, $r_i>0$ is the lowest price at which he may consider selling the unit of good.
  • The amount of good the consumer $i$ buys from the producer $j$ is $L_{ij}$, the corresponding price is denoted by $p_{ij}$.
  • Each consumer $i$ has a utility function $U_i(p_{i1},\dots,p_{iM},L_{i1},\dots,L_{iM})$ which he wants to maximize.
  • Each producer $j$ has a utility function $V_j(p_{1j},\dots,p_{Nj},L_{1j},\dots,L_{Nj})$ which he wants to maximize.

I am pretty sure that this problem is rather classical, and I am looking for the game-theoretical formulation of this problem. So far I do not wonder about the existence or uniqueness of Nash equilibrium, just about the formulation: what are the decision variables etc. Some references would also be useful. I think that in such case one may talk about a matrix of equilibrium prices that equalizes the demand and supply, however I am not sure how to approach this formally.

I would also be interested in an extension when all the players are consumers and producers at the same time. That is, there are $N+M$ players which of them having constraints $$ -E_i\leq\sum_{j=1}^{N+M}L_{ij}\leq Q_i $$ meaning that each player can buy from one counterparty and sell to another one.

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I'm not sure if this is correct, but this seems remarkably similar to the Lucas Tree Model to me, which has many extensions and renditions in game theoretic terms. You may have some success looking for game-theoretic problems derived from that model. –  EMS Oct 9 at 20:40

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There are several different ways you could formulate this problem in game theoretic terms. Hoping this is not too basic an answer for you : from what you write, the two canonical approaches would be to frame things in terms of Cournot oligopolies (firms simultaneously set quantities and prices result from the market clearing condition supply=demand) or Bertrand oligopolies (firms simultaneously set prices and quantities result from the market clearing condition supply=demand). You can find a lot of reference on these two models on google.

As you read these references, you will see that your assumptions do not really fit into either the Bertrand or the Cournot model. Your model is somewhat more complicated (and probably slightly under-specified if you want to get to any clearcut conclusion). In particular, to be able to frame your questions in terms of the canonical Bertrand or Cournot you would need

  • To be able to derive an aggregate demand function linking any price level with an aggregate quantity which would sell at this price. So you need to know more that the higher price agents are ready to pay. One way to do so would be to specify further the profile of consumer's utility function (then you may be able to derive consumers' optimal quantity for every price, and from this build an aggregate demand function).

  • To give up the idea that firm are not able to produce more than a certain quantity. In the canonical models, each firm must be able to produce as much as it wants, possibly covering the whole demand if it turned out to be optimal.

Now this does not mean that there are no ways to accommodate your current model in a formal setup more or less derived from the canonical Cournot and Bertrand. But this would certainly require quite a bit of work. Regarding you interest in capacities constraints for instance, you may want to read the part of section 12.C from Mas-Collel, Whinston and Green, Microeconomic theory which covers this issue.

Notice finally that in any situation which allows for different firms to eventually sell positive quantities at different prices, you will need to add hypothesis on the rationing mechanism. Unless all firms end up either selling at the same price, or selling zero, you will end up in a situation in which some consumers pay say $p$ whereas others pay $p' < p$ for the same good. Arguably the consumers who pay $p$ would like to pay $p'$ instead (assuming as you did that there is only one identical good).

Then either the situation does not last and everyone ends up buying from the producer which charges $p'$, or it lasts but then some firms face an excess demand, and there must be a rationing mechanism in your model specifying who is allowed to buy the cheap goods and who is not. You could assume that some consumer have a priority access to the production of some firms, or that the rationing takes place via a lottery, or anything else, but to close your model you will need to make it clear which rationing mechanism you choose.

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