Offer Games and Non-Market-Clearing Nash Equilibria: a Biform Game Analysis and Agent-Based Simulation Study

Game theoretic studies of market equilibria are usually noncooperative, but every exchange is a cooperative interaction, so this approach is unavoidably incomplete. One mixed cooperative-noncooperative approach that allows for this is the "biform game" approach, in which agents choose noncooperatively among a set of potential cooperative games. This paper explores a biform market game of many buyers and sellers. The game has many Nash equilibria, some of which dominate others in which potential gains from trade are not realized. The introduction of positive costs of transaction reduces the number of Nash equilibria by a large proportion, but there remain plural equilibria, some of which dominate others, including an equilibrium in which no-one trades. The paper then reports agent-based simulations of the biform game model. There are two types of agents, potential buyers and sellers. The agents are boundedly rational learners who learn to make the polychotomous choice of a trading partner by experience in the form of accumulated past utility. There are100 agents of each type who are situated on a toroidal grid of 10x10, and only propose trades to agents of the other type within the von-Neumann neighborhood of their own cell. This serves to reduce the dimension of what would otherwise be a polychotomous choice of very large dimension. In the simulations, some agents fail to trade either because of random errors or because there are no available trading partners in the potential trader's von Neumann neighborhood. In the absence of transaction costs, however, this category declined to what appears to be a practical minimum, so that we may say that, so far as possible, Pareto-dominated Nash equilibria were eliminated. Moreover, modest transaction costs changed this result only slightly. However, as transaction costs rose, failure to trade from all causes increased to a majority of potential traders (with transaction costs at 40% of gains from trade) and beyond. This suggests that the refinement by eliminating Pareto-dominated equilibria will only be applicable when transaction costs are modest. Future research suggested by the paper but beyond its immediate scope would include a generalization to allow agents endogenously to determine their "neighborhoods" of potential trading partners, exploration of the impact of the average size of the neighborhood, and exploration of the impacts of different degrees of rationality. In discussion, the paper suggests the biform game model as and interpretation of the concept of "involuntary unemployment" and makes some suggestions as to policies for dealing with unemployment that might, in the light of the simulations, be relatively effective.