Holmström's theorem

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:

# Income equals outflow (the budget balances),
# The system has a Nash equilibrium, and
# The system is Pareto efficient.

Thus a Pareto-efficient system with a balanced budget has a point at which an agent can do better by changing their effort level, if everyone else's effort level stays the same; 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 could allow some agents to do better without hurting any others.

The Gibbard-Satterthwaite theorem in social choice theory is a related impossibility theorem dealing with voting systems.

tatement of the theorem

Suppose there is a team of "n" > 1 risk neutral 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 [Eric Rasmusen, "Moral Hazard in Risk-Averse Teams", "The RAND Journal of Economics" 18, no. 3 (1987), pp. 428–435.] studies the relaxation of this problem obtained by removing the assumption that the agents are risk neutral (Holmström: "linear in money").

References

* Bengt Holmström, "Moral Hazard in Teams", "The Bell Journal of Economics" 13, no. 2 (1982), pp. 324–340.