With increasing marginal costs, the agents impose a negative externality on each other; with decreasing marginal costs, the agents impose a positive externality on each other (see example below).
This cost-sharing method has several advantages: However, it has a disadvantage: This is a measure of fairness: no agent should suffer too much from the negative externality.
Hence, it makes sense only when such transfers are impossible (for example, with cable TV or telephone services).
As a typical example, consider two agents, Alice and George, who live near a water-source, with the following distances: Suppose that each kilometer of water-pipe costs $1000.
The goal is to find a truthful mechanism that will induce the agents to reveal their true willingness-to-pay.
A cost-sharing problem is defined by the following functions, where i is an agent and S is a subset of agents: A solution to a cost-sharing problem is defined by: A solution can be characterized by: It is impossible to attain truthfulness, budget-balance and efficiency simultaneously; therefore, there are two classes of truthful mechanisms: A budget-balanced cost-sharing mechanism can be defined by a function Payment(i,S) - the payment that agent i has to pay when the subset of served agents is S. This function should satisfy the following two properties: For any such function, a cost-sharing problem with submodular costs can be solved by the following tatonnement process:[6] Note that, by the population-monotonicity property, the price always increases when people leave S. Therefore, an agent will never want to return to S, so the mechanism is truthful (the process is similar to an English auction).
[6]: Proposition 1 The mechanism can select the Payment function in order to attain such goals as fairness or efficiency.
When agents have equal apriori rights, some reasonable payment functions are: The above cost-sharing mechanisms are not efficient - they do not always select the allocation with the highest social welfare.
But, when the payment function is selected to be the Shapley value, the loss of welfare is minimized.
Therefore, the Marginal Cost Pricing mechanism selects to serve both agents.