There is a famous saying in economics: When everyone pursues his or her own interests, there is an invisible hand that brings the market to equilibrium. However, this is not always the case. Here is an example. Consider a market with two firms $f_1, f_2$, and two workers $w_1, w_2$. Firm $f_1$ derives $6$ from hiring both workers and does not want to hire only one worker. Firm $f_2$ values each worker at $4$ in the absence of the other but does not want to hire both. Both workers care only about their wages. There is no equilibrium in this market. If firm $f_1$ hires both workers at an expenditure of no more than 6, then there must be a worker earning a wage no more than 3. This worker will go to firm $f_2$ since firm $f_2$ is willing to pay a wage between 3 and 4, then the other worker will be fired by firm $f_1$ and come to compete for firm $f_2$'s offer. The competition will lead to a market wage of less than 3, and we then return to the beginning: Firm $f_1$ will hire both workers at an expenditure of no more than 6.

The existence of equilibria is a central question in economics. Recent studies found that tropical geometry is useful in this problem in which firms' maximum profits are tropical Laurent polynomials. Baldwin and Klemperer (2019) used tools from tropical geometry to find a necessary and sufficient condition for the existence of an equilibrium when there are two firms. Tran and Yu (2019) showed that its generalization to an arbitrary number of firms is equivalent to Oda's conjecture in algebraic geometry. (Firms and workers are respectively called agents and goods in the above two papers.)

My question is about a related discrete market. This market does not concern money. Each worker has a preference ordering over firms, and each firm has a preference ordering over subsets of workers. For instance, consider a market with two firms $f_1, f_2$, two workers $w_1, w_2$, and the following preference orderings. \begin{equation} \begin{aligned} &f_1: \{w_1,w_2\}\succ\emptyset \qquad &w_1: &\quad f_1\succ f_2\\ &f_2: \{w_1\}\succ\{w_2\}\succ\emptyset \qquad &w_2: &\quad f_2\succ f_1 \end{aligned} \end{equation} This market has no equilibrium either. If firm $f_1$ hires both workers, worker $w_2$ will go to firm $f_2$; then, firm $f_1$ will drop worker $w_1$; and then, firm $f_2$ will drop worker $w_2$ and hire worker $w_1$; but we then return to the beginning: both workers will go to firm $f_1$. In the literature of economics, the latter discrete market shares many similar results on the existence of equilibria with the former market with money. However, in the discrete market, a necessary and sufficient condition for the existence of an equilibrium has not been found even in a market with only two firms. We cannot use tropical geometry in the discrete market. My question is: What kind of tools may be useful for studying the existence of equilibria in the discrete market? What kind of tools may play a similar role as tropical geometry has played in the market with money?

Reference

Baldwin and Klemperer(2019). Understanding Preferences: ``Demand Types'', and the Existence of Equilibrium with Indivisibilities. Econometrica, 87, 867-932.

Tran and Yu(2019). Product-mix auctions and tropical geometry. Mathematics of Operations Research, 44(4), 1145-1509.