We consider mechanisms that provide traders the opportunity to exchange commodity i for commodity j, for certain ordered pairs ij. Given any connected graph G of opportunities, we show that there is a unique mechanism MG that satisfies some natural conditions of “fairness” and “convenience.” Let M(m) denote the class of mechanismsMG obtained by varying G on the commodity set {1, …, m}. We define the complexity of a mechanism M in M(m) to be a pair of integers τ(M), π(M) which represent the “time” required to exchange i for j and the “information” needed to determine the exchange ratio (each in the worst case scenario, across all i not equal to i ≠ j). This induces a quasiorder \preceq on M(m) by the rule
M \preceq M’ if τ(M) ≤ τ(M’) and π(M) ≤ π(M’).
We show that, for m > 3, there are precisely three \preceq-minimal mechanisms MG in M(m), where G corresponds to the star, cycle and complete graphs. The star mechanism has a distinguished commodity — the money — that serves as the sole medium of exchange and mediates trade between decentralized markets for the other commodities.
Our main result is that, for any weights λ, μ > 0; the star mechanism is the unique minimizer of λτ(M) + μπ (M) on M(m) for large enough m.
We consider abstract exchange mechanisms wherein individuals submit “diversified” offers in m commodities, which are then redistributed to them. Our first result is that if the mechanism satisfies certain natural conditions embodying “fairness” and “convenience” then it admits unique prices, in the sense of consistent exchange-rates across commodity pairs ij that equalize the valuation of offers and returns for each individual.
We next define certain integers τij, πij, and ki which represent the “time” required to exchange i for j, the “difficulty” in determining the exchange ratio, and the “dimension” of the offer space in i; we refer to these as time-, price- and message-complexity of the mechanism. Our second result is that there are only a finite number of minimally complex mechanisms, which moreover correspond to certain directed graphs G in a precise sense. The edges of G can be regarded as markets for commodity pairs, and prices play a stronger role in that the return to a trader depends only on his own offer and the prices.
Finally we consider “strongly” minimal mechanisms, with smallest “worst case” complexities τ = max τij and pi = max πij. Our third main result is that for m > 3 commodities that there are precisely three such mechanisms, which correspond to the star, cycle, and complete graphs, and have complexities (π,τ) = (4,2), (2,m − 1), (m2 − m, 1) respectively. Unlike the other two mechanisms, the star mechanism has a distinguished commodity — the money — that serves as the sole medium of exchange. As m approaches infinity it is the only mechanism with bounded (π,τ).
Consider agents who undertake costly effort to produce stochastic outputs observable by a principal. The principal can award a prize deterministically to the agent with the highest output, or to all of them with probabilities that are proportional to their outputs. We show that, if there is sufficient diversity in agents’ skills relative to the noise on output, then the proportional prize will, in a precise sense, elicit more output on average, than the deterministic prize. Indeed, assuming agents know each others’ skills (the complete information case), this result holds when any Nash equilibrium selection, under the proportional prize, is compared with any individually rational selection under the deterministic prize. When there is incomplete information, the result is still true but now we must restrict to Nash selections for both prizes.
We also compute the optimal scheme, from among a natural class of probabilistic schemes, for awarding the prize; namely that which elicits maximal effort from the agents for the least prize. In general the optimal scheme is a monotonic step function which lies “between” the proportional and deterministic schemes. When the competition is over small fractional increments, as happens in the presence of strong contestants whose base levels of production are high, the optimal scheme awards the prize according to the “log of the odds,” with odds based upon the proportional prize.
There are two sources of inefficiency of strategic equilibria (SE) in market mechanisms. The first is the oligopolistic effect, which occurs when an agent can single-handedly influence prices. With a continuum of agents we get “perfect competition” and this effect is, of course, wiped out. But the inefficiency of SE’s may nevertheless persist because agents are not “perfectly liquid,” i.e., the constraints of the mechanism are such that they cannot carry out arbitrary trades at the market prices. Our main result is that, if enough repeated rounds of trade are permitted within a single utility period, then the liquidity problem is overcome: SE outcomes turn out to be not only efficient but, in fact, Walrasian.
Existence of equilibrium is proved for an exchange strategic market game with complete markets. An example of equilibrium with inconsistent prices is given.