Market equilibrium computation
Market equilibrium computation (also called competitive equilibrium computation or clearing-prices computation) is a computational problem in the intersection of economics and computer science. The input to this problem is a market, consisting of a set of resources and a set of agents. There are various kinds of markets, such as Fisher market and Arrow–Debreu market, with divisible or indivisible resources. The required output is a competitive equilibrium, consisting of a price-vector (a price for each resource), and an allocation (a resource-bundle for each agent), such that each agent gets the best bundle possible (for him) given the budget, and the market clears (all resources are allocated). Market equilibrium computation is interesting due to the fact that a competitive equilibrium is always Pareto efficient. The special case of a Fisher market, in which all buyers have equal incomes, is particularly interesting, since in this setting a competitive equilibrium is also envy-free. Therefore, market equilibrium computation is a way to find an allocation which is both fair and efficient. Definitions The input to the market-equilibrium-computation consists of the following ingredients: A set of resources with pre-specified supplies. The resources can be divisible (in which case, their supply is w.l.o.g. normalized to 1), or indivisible . A bundle is represented by a vector , where is the quantity of resource . When resources are indivisible, all xj are integers; when resources are divisible, the xj can be arbitrarily real numbers (usually normalized to [0,1]). A set of agents. For each agent, there is a preference relation over bundles, which can be represented by a utility function. The utility function of agent is denoted by . An initial endowment for each agent. In a Fisher market, the endowment is a budget of "fiat money" - a money that has no value outside the market, and thus does not enter the utility function. Since the agents come with money only, they are often called buyers. In an Arrow–Debreu market, the endowment is an arbitrary bundle ; in this model, agents can be both buyers and sellers. The required output should contain the following ingredients: A price-vector ; a price for each resource. The price of a bundle is the sum of the prices of the resources in the, so the price of a bundle is . An allocation - a bundle for each agent i. The output should satisfy the following requirements: The bundle should be affordable to i, that is, its price should be at most the price of agent i's endowment. In a Fisher market, this means that . In an Arrow-Debreu market, this means that . The bundle should be in the demand set of i: , defined as the set of bundles maximizing the agent's utility among all affordable bundles (regardless of supply), e.g., in a Fisher market: The market clears, i.e., all resources are allocated. The corresponding prices are called market-clearing prices. A price and allocation satisfying these requirements are called a competitive equilibrium (CE) or a market equilibrium; the