The power of game theory lies in mechanism design (aka reverse game theory), which changes rules of the game to make players play in an intended way. Leonid Hurwicz, Eric Maskin and Roger Myerson are recognized 2007 Nobel Memorial Prize "for having laid the foundations of mechanism design theory". {Hurwicz1960, 1972}

## Concepts

Institution, in the sense of institutional arrangements (rules of the game) rather than organizations (players), defines the legal actions of each player and the consequences of possible outcomes, which is mathematically equivalent to a function from the (legal) strategy space to outcomes, referred to as game-form in game theory or mechanism in economics. A game-form is not a game, in that an outcome describes what happened which need not have simple mathematical formulation, while payoffs are how players are rewarded given the outcome. Mechanism design separates game-form from payoff functions and private information such as endowments and technologies: legislators have full control of the former but no control of the latter.

When you write the rules of the game you take into account that the players will try to cheat. {Hurwicz}

The true strategy space is the set of all feasible actions, which is a superset of the (legal) strategy space. Legal strategies are those prescribed by the mechanism governing the system; other feasible actions are illegal. The true game has the true strategy space and true outcome.

Implementation is an essential part of an institution, which means to have the money and the information to run the institutions, and there is legislation authorizing this. Enforcement of an institution is successful if the outcomes of the true game ensure that illegal strategies are (weakly) dominated by legal strategies. More broadly, implementation of an institution is successful if the equilibrium outcomes of the true game correspond to those envisaged by the legislation. Nash equilibrium is neither self-enforcing nor self-implementing.

Informationally decentralized {Hurwicz1972}

## General results

1. If a social choice rule is implementable, it must be monotonic.
2. A social choice rule is implementable if it is monotonic, no individual has veto power, and there are more than two players.

Social choice rule is a correspondence between states and strategy space: $$f \subseteq \Theta \times S$$. Mechanism $$g$$ implements social choice rule $$f$$ in Nash equilibrium if the Nash equilibrium outcomes coincide with social optimum outcomes under all states: $$f(\theta) = \text{NE}_g(\theta), \forall \theta$$. A social choice rule is monotonic if a social optimum in one state remains optimum in another as long as it does not decrease in any player's preference: $$\forall \theta \in \Theta, a \in f(\theta)$$, if $$u_i(a, \theta) \ge u_i(b, \theta) \Rightarrow u_i(a, \theta') \ge u_i(b, \theta')$$, $$\forall i \in N, b \in A$$, then $$a \in f(\theta')$$.

Information asymmetry {Arrow1963, Akerlof1970} can take the form of either hidden information (incomplete information) or hidden action (imperfect information). As of hidden information, adverse selection refers to the situation when the type of product is hidden from one party in a transaction. As of hidden action, moral hazard refers to the situation when individuals take greater risks when the cost is shared with other parties than taken alone.

The revelation principle: {Myerson1979, 1982, 1983, 1984} incentive-compatible direct mechanisms

In mechanism design, a process is incentive-compatible {Hurwicz2006} if all of the participants fare best when they truthfully reveal any private information asked for by the mechanism.

“Information efficient”

Samuelson's conjectures: {Samuelson1954, Hurwicz1972}

1. Truthful revelation of preferences is not a Nash equilibrium in the Lindahl game.
2. There may be no successful implementation for public goods under decentralization.

Principal-agent (委托-代理) problem is a class of games with information asymmetry where one player (the principal) attempts to offer incentives to the other (the agent) to encourage the agent to act in the principal's best interest.

## Specialized results

The greed process {Hurwicz1960} has the desired optimality properties for all environments from which so-called external economies or diseconomies are absent; but it lacks the stability properties known to hold for perfect competition at least in certain special cases.

The perfectly competitive market is the only efficient mechanism if there are: large numbers of buyers and sellers so that no single agent has significant market power; and no significant externalities, that is, an agent's consumption, production, and information does not affect others' production or consumption. {Maskin 2007 Nobel Price Lecture; Hammond1979, Jordan1982} However, mechanisms improving the market are generally possible if either assumption is violated. {Clarke1971, Groves1973; Laffont1985}

Optimal mechanism

Efficient mechanism (Vickrey-Clarke-Groves mechanism) {Vickrey1961, Clarke1971, Groves1973}