Myerson-Satterthwaite theorem
Encyclopedia
The Myerson-Satterthwaite theorem is an important result in mechanism design
and the economics of asymmetric information
, due to Roger Myerson
and Mark Satterthwaite
. Informally, the result says that there is no efficient way for two parties to trade a good when they each have secret and probabilistically varying valuations for it, without the risk of forcing one party to trade at a loss.
Formally, the theorem applies if a prospective buyer A has a valuation
, and the prospective seller B has an independent valuation 
, such that the intervals 
and 
overlap, and the probability densities
for the valuations are strictly positive on those intervals. Under those conditions, there is no Bayesian
incentive compatible
social choice function that is guaranteed in advance to produce efficient
outcomes and guarantees buyers and sellers non-negative returns
regardless of
and 
.
The Myerson-Satterthwaite theorem is among the most remarkable and universally applicable negative results in economics — a kind of negative mirror to the fundamental theorems of welfare economics
. It is, however, much less famous than those results or Arrow's earlier result on the impossibility of satisfactory electoral systems
.
Mechanism design
Mechanism design is a field in game theory studying solution concepts for a class of private information games...
and the economics of asymmetric information
Information asymmetry
In economics and contract theory, information asymmetry deals with the study of decisions in transactions where one party has more or better information than the other. This creates an imbalance of power in transactions which can sometimes cause the transactions to go awry, a kind of market failure...
, due to Roger Myerson
Roger Myerson
Roger Bruce Myerson is an American economist and Nobel laureate recognized with Leonid Hurwicz and Eric Maskin for "having laid the foundations of mechanism design theory." A professor at the University of Chicago, he has made contributions as an economist, as an applied mathematician, and as a...
and Mark Satterthwaite
Mark Satterthwaite
Mark A. Satterthwaite is an economist at the Kellogg School of Management at Northwestern University in Evanston, Illinois. He is currently A.C. Buehler Professor in Hospital & Health Services Management, Professor of Strategic Management & Managerial Economics, and chair of the Management &...
. Informally, the result says that there is no efficient way for two parties to trade a good when they each have secret and probabilistically varying valuations for it, without the risk of forcing one party to trade at a loss.
Formally, the theorem applies if a prospective buyer A has a valuation
, and the prospective seller B has an independent valuation , such that the intervals and overlap, and the probability densitiesProbability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
for the valuations are strictly positive on those intervals. Under those conditions, there is no Bayesian
Bayesian game
In game theory, a Bayesian game is one in which information about characteristics of the other players is incomplete. Following John C. Harsanyi's framework, a Bayesian game can be modelled by introducing Nature as a player in a game...
incentive compatible
Incentive compatibility
In mechanism design, a process is said to be incentive-compatible if all of the participants fare best when they truthfully reveal any private information asked for by the mechanism. As an illustration, voting systems which create incentives to vote dishonestly lack the property of incentive...
social choice function that is guaranteed in advance to produce efficient
Pareto efficiency
Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.Given an initial allocation of...
outcomes and guarantees buyers and sellers non-negative returns
Participation constraint (mechanism design)
In game theory, and particularly mechanism design, participation constraints or rational participation constraints are said to be satisfied if a mechanism leaves all participants at least as well off as they would have been if they hadn't participated....
regardless of
and .The Myerson-Satterthwaite theorem is among the most remarkable and universally applicable negative results in economics — a kind of negative mirror to the fundamental theorems of welfare economics
Fundamental theorems of welfare economics
There are two fundamental theorems of welfare economics. The first states that any competitive equilibrium or Walrasian equilibrium leads to a Pareto efficient allocation of resources. The second states the converse, that any efficient allocation can be sustainable by a competitive equilibrium...
. It is, however, much less famous than those results or Arrow's earlier result on the impossibility of satisfactory electoral systems
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
.