Holmström's theorem
Encyclopedia
In economics
, Holmström's theorem is an impossibility theorem attributed to Bengt R. Holmström
proving that no incentive system for a team of agents can make all of the following true:
Thus a Pareto-efficient system with a balanced budget does not have any point at which an agent can not do better by changing their effort level, even if everyone else's effort level stays the same, meaning that the agents can never settle down to a stable strategy; a Pareto-efficient system with a Nash equilibrium does not distribute all revenue, or spends more than it has; and a system with a Nash equilibrium and balanced budget does not maximise the total profit of everybody.
The Gibbard-Satterthwaite theorem
in social choice theory
is a related impossibility theorem dealing with voting system
s.
agents whose preference
functions are strictly concave
and increasing, and also additively separable in money and effort. Then, under an incentive system that distributes exactly the output among the team members, any Nash equilibrium is Pareto inefficient.
Rasmusen studies the relaxation of this problem obtained by removing the assumption that the agents are risk neutral (Holmström: "linear in money").
The economic reason for Holmström's result is a "Sharing problem". A team member faces efficient incentives if he receives the full marginal returns from an additional unit of his input. Under a budget-balanced sharing scheme, however, the team members cannot be incentivized this way. This problem would be circumvented if the output could be distributed n times instead of only once. This requires that the team members promise fixed payments to an "Anti-Sharer", as demonstrated by Kirstein and Cooter. However, if one of the team members takes over the role of the Anti-Sharer, this player has no incentive whatsoever to spend effort. The article derives conditions under which internal Anti-Sharing induces the team members to spend more effort than a budget-balanced sharing contract.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, Holmström's theorem is an impossibility theorem attributed to Bengt R. Holmström
Bengt R. Holmström
Bengt Robert Holmström is the Paul A. Samuelson Professor of Economics at M.I.T. He is a Finnish citizen and belongs to the Swedish speaking minority in Finland....
proving that no incentive system for a team of agents can make all of the following true:
- Income equals outflow (the budget balances),
- The system has a Nash equilibriumNash equilibriumIn game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
, and - The system is Pareto efficient.
Thus a Pareto-efficient system with a balanced budget does not have any point at which an agent can not do better by changing their effort level, even if everyone else's effort level stays the same, meaning that the agents can never settle down to a stable strategy; a Pareto-efficient system with a Nash equilibrium does not distribute all revenue, or spends more than it has; and a system with a Nash equilibrium and balanced budget does not maximise the total profit of everybody.
The Gibbard-Satterthwaite theorem
Gibbard-Satterthwaite theorem
The Gibbard–Satterthwaite theorem, named after Allan Gibbard and Mark Satterthwaite, is a result about the deterministic voting systems that choose a single winner using only the preferences of the voters, where each voter ranks all candidates in order of preference...
in social choice theory
Social choice theory
Social choice theory is a theoretical framework for measuring individual interests, values, or welfares as an aggregate towards collective decision. A non-theoretical example of a collective decision is passing a set of laws under a constitution. Social choice theory dates from Condorcet's...
is a related impossibility theorem dealing with voting system
Voting system
A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....
s.
Statement of the theorem
Suppose there is a team of n > 1 risk neutralRisk neutral
In economics and finance, risk neutral behavior is between risk aversion and risk seeking. If offered either €50 or a 50% chance of each of €100 and nothing, a risk neutral person would have no preference between the two options...
agents whose preference
Preference
-Definitions in different disciplines:The term “preferences” is used in a variety of related, but not identical, ways in the scientific literature. This makes it necessary to make explicit the sense in which the term is used in different social sciences....
functions are strictly concave
Concave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...
and increasing, and also additively separable in money and effort. Then, under an incentive system that distributes exactly the output among the team members, any Nash equilibrium is Pareto inefficient.
Rasmusen studies the relaxation of this problem obtained by removing the assumption that the agents are risk neutral (Holmström: "linear in money").
The economic reason for Holmström's result is a "Sharing problem". A team member faces efficient incentives if he receives the full marginal returns from an additional unit of his input. Under a budget-balanced sharing scheme, however, the team members cannot be incentivized this way. This problem would be circumvented if the output could be distributed n times instead of only once. This requires that the team members promise fixed payments to an "Anti-Sharer", as demonstrated by Kirstein and Cooter. However, if one of the team members takes over the role of the Anti-Sharer, this player has no incentive whatsoever to spend effort. The article derives conditions under which internal Anti-Sharing induces the team members to spend more effort than a budget-balanced sharing contract.