Milgrom - Chapter One

Overview of Mechanism Design

Mechanism: the control of the designer.

A mechanism consists of rules that govern what the participants are permitted to do and how these permitted actions determine outcomes.

Environment: the world beyond the designer's control.

1. A list of participants.
2. A list of possible outcomes.
3. A list of participants' types (private information such as preferences and beliefs).

The most commonly studied mechanisms in economics are resource allocation mechanisms.

1. The outcome is an allocation of resources.

Mechanism theory evaluates alternative designs based on their comparative performance. Formally, performance is the function that maps environments into outcomes.

The goal of mechanism design analysis is to determine what performance is possible and how mechanisms can best be designed to achieve the designer’s goals.

An auction is a mechanism to allocate resources among a group of bidders.

Formal Definitions

An environment

A tuple $(N, \Omega, \Theta)$, where

1. $N = \{1, ..., n\}$ is the set of (potential) participants. If the mechanism designer itself is also included, we have $N = \{0,...,n\}$.

2. $\Omega$ is the set of possible outcomes.

3. $\Theta$ is the set of all possible type profiles that the modeler considers, where each $\vec{t} = \{t^1, ..., t^n\} \in \Theta$ is a vector of types; $t^i$ is the type for the $i$th participant, $i \in N$.

The utility. Given a type profile $\vec{t} \in \Theta$ and an outcome $\xi \in \Omega$, a participant $i$ receives a payoff: $u^i : \Theta \times \Omega \rightarrow \mathbb{R}$. In general, $i$'s utility depends on the types of other players, though as pointed out by Milgrom, in many case the utility only depends on a player's own type.

Uncertainty about other's type. Designer often assumes that participants do not know the exact types of others, specified by a conditional probability distribution $\pi^i(\vec{t} | t^i)$ for each player $i$. Such distributions are often assumes to come from a common prior $\pi$. This assumption is known as the Harsanyi doctrine.

A mechanism (strategic form)

A pair $(\mathcal{S}, \omega)$, where

1. $\mathcal{S}$ is the set of all possible strategy profiles of the participants.

2. $\omega : \mathcal{S} \rightarrow \Omega$ is a function that maps profiles to outcomes.

Given a mechanism and a realization $\vec{t}$ of types, one can define a game of the form $(N, S, U(\cdot | \vec{t}))$, where

1. $N$ is the set of players (participants) from the environment.

2. $\mathcal{S}$ is the set of strategy profiles from the mechanism.

3. $\vec{t}$ is a type vector from the environment.

4. $U(\cdot | \vec{t})$ is a payoff function.

This payoff function $U(\cdot | \vec{t})$ takes an action profile as the argument, where

Here, $u^i(\cdot, \cdot)$ is the utility function from the environment, $\sigma \in \mathcal{S}$ is a particular action profile, and $\omega(\cdot)$ maps this profile to an outcome in $\Omega$. As of writing this note, I don't understand the necessity of this additional function $U^i$. So far, it seems to be equivalent to the definition of $u^i$.

Given a mechanism $(\mathcal{S}, \omega)$, if the game theoretic solution concept forecasts that a particular strategy profile $\sigma$ will be played, then one can use that forecast to predict and evaluate the performance of the mechanism. The forecasted outcome will be $\xi(\vec{t}) = \omega(\sigma)$. The function $\xi(\cdot)$ mapping type profiles to outcomes is the performance function corresponding to the mechanism.

Does it mean that an action profile $\sigma$ can be entirely determined by a given type vector $\vec{t}$, which then tells us what outcome to expect from $\omega$?