Almost essential Welfare Basics Games: equilibriumdarp.lse.ac.uk/presentations/MP2Book/OUP/DesignBasi… · · 2017-07-29Almost essential. Welfare Basics. Games: equilibrium. Prerequisites

Frank Cowell: Design Basics

DESIGN BASICSMICROECONOMICSPrinciples and AnalysisFrank Cowell

1

Almost essentialWelfare BasicsGames: equilibrium

Prerequisites

July 2017


Overview

2

A parable

Social choice again

Mechanisms

Design Basics

An introduction to the issues

The design problem

July 2017


A parable Think through the following everyday situation

• Alf, Bill and Charlie have appointments at the same place but different times

• they try to book taxis, but there’s only one available• so they’ll have to share!

What is the decision problem?• do they care about being early/late?• do they care about the others’ objectives?• clearly a joint problem with conflicting interests

Consider a proposed solution• if taxi firm suggests an efficient pickup time – accept• otherwise ask for the earliest preferred time by A,B,C• look at this in a diagram

3July 2017


Alf, Bill, Charlie and the taxi

4

pref

eren

ce

Alf

Bill Charlie

10:00 11:00 12:00 13:00

Alf’s preferencesBill’s preferencesCharlie’s preferencesTaxi firm’s proposed time #1Taxi firm’s proposed time #2

12:45 is inefficient – everyone would prefer an earlier time. So they’d ask for 11:00 instead

12:15 is also inefficient. But Charlie would prefer it to 11:00. So why not pretend it’s efficient? Why not pretend his first choice is 12:15?

July 2017


The approach Some questions:

• what properties should a taxi rule satisfy?• would Alf, Bill or Charlie want to misrepresent preferences?• could we find a problem of manipulation?

Manipulation (sometimes “cheating” or “chiselling”):• an important connection with the issue of efficiency• rules might be inefficient because they provide wrong incentives

Design problem:• find a rule so that individuals choose a socially desirable outcome• but will only do so if it is in their private interests• what is “socially desirable”?

Need to examine the representation of choices • build on the analysis from social welfare• and reuse some results

5July 2017


Overview

6

A parable

Social choice again

Mechanisms

Design Basics

A link with the fundamentals of welfare economics

The design problem

July 2017


Agenda Basic questions

• purpose of design• informational context• strategic setting

Purpose• modelling group objectives• need a review of social choice

Information• agents may have private information• so need to allow for the possibility of misrepresentation

Strategy• a connection with game-theoretic approaches• so need to review concepts of equilibrium

7

Begin with purpose

July 2017


Social states and preferences Social state: θ

• a comprehensive description• of all relevant features of the economy in question

Set of all social states: Θ Preferences vh(∙)

• a “reduced form” version of agent h’s utility function• utility of agent h given social state θ is vh(θ)• preference profile is an ordered list, one for each agent: [v1, v2, v3,…]• a list of functions, not utility levels

Set of all preference profiles: 𝕍𝕍

8July 2017


A reminder Constitution

• a mapping from 𝕍𝕍 to set of all v(∙)• given a particular set of preferences for the population• the constitution should determine a specific v(∙)

Properties• Universality• Pareto Unanimity• Independence of Irrelevant Alternatives• Non-Dictatorship

Arrow theorem• if there are more than two social states then there is no constitution

satisfying the above four properties• a key result

Use this reminder to introduce a new concept9July 2017


Social-choice functionA social choice function Γ

• a mapping 𝕍𝕍 → Θ• given a particular set of preferences for the population• picks out exactly one chosen element from Θ

Note that argument of the SCF is same as for constitution• a profile of preferences [v]• a list of utility functions

But that it produces a different type of “animal”• the constitution uses [v] to yield a social ordering• the SCF uses [v] to yield a social state

10July 2017


Social-choice function: properties Three key properties of an SCF, Γ : Γ is Paretian if

• given a θ∗ such that vh(θ∗) ≥ vh(θ), for all h and all θ∈Θ,• then θ∗ = Γ(v1, v2, v3,…)

Γ is monotonic if• given any [v] and [v] ∈ 𝕍𝕍 such that

“vh(θ∗) ≥ vh(θ)” implies “vh(θ∗) ≥ vh(θ)”• then “θ∗ = Γ(v1, v2, v3,…)” implies “θ∗ = Γ(v1, v2, v3,…)”

Γ is dictatorial if • there is some agent whose preferences completely determine θ

11July 2017


Monotonicity: example

12

x1h

x2h

B(θ∗; v)B(θ∗; v )

• θ∗

h’s indifference curve under v(·)Better-than set for v and the state θ*h’s indifference curve under v(∙)Better-than set for v and the state θ*

So, if vh(θ∗) ≥ vh(θ) then vh(θ∗) ≥ vh(θ)

If Γ is monotonic, then if θ∗ is the chosen point under [v] then θ∗ is also chosen point under [v]

Here state is an allocation

July 2017

“Better-than” is used as shorthand for “Better-than-or-just-as-good-as-”


Social-choice function: resultAssume: 1. Θ has more than two elements2. Γ is defined for all members of 𝕍𝕍3. Γ is Paretian and monotonic Then Γ must also be dictatorial

A counterpart of the Arrow result on constitutions

13July 2017


A key property of the SCF Γ is manipulable if there is a profile [v] ∈ 𝕍𝕍 such that

• for some h and some other utility function vh(∙)• vh(θ) > vh(θ)• where θ = Γ(v1,…, vh, …, )• and θ = Γ(v1,…, vh, …, )

Significance is profound:• if Γ is manipulable then some agent h should realise• that if h misrepresents his preferences but others tell the truth• then h will be better off

An incentive to misrepresent information?• does not imply that there is some h who can manipulate• implies that, under some circumstances, there is an h who could manipulate

14July 2017


Social-choice function: another resultNote that the monotonicity property is powerful:

• if Γ is monotonic • then Γ cannot be manipulable

From this and the previous result a further result follows• suppose Θ has more than two elements• for each h any strict ranking of elements of Θ is permissible• then a Paretian, non-manipulable SCF Γ must be dictatorial

This result is important • connects the idea of misrepresentation and social choice• introduces an important part of the design problem

15July 2017


Social-choice function: summary Similar to the concept of constitution

• but from the set of preference profiles to the set of social states

Not surprising to find result similar to Arrow• introduce weak conditions on the Social-choice function• there’s no SCF that satisfies all of them

But key point concerns link with information• misrepresentation and manipulability are linked• important implication for design problem

16July 2017


Overview

17

A parable

Social choice again

Mechanisms

Design Basics

The problem of implementation

The design problem

July 2017


Forward from social choice Social choice is just the first step

• SCF describes what is desirable• not how you achieve it

The next step involves achievement• reconcile desirable outcomes with individual incentives• the implementation problem • underlies practical policy making

Requires the introduction of a new concept• a mechanism

18July 2017


Implementation Is the SCF consistent with private economic behaviour?

• Yes if the θ picked out by Γ is also• the equilibrium of an appropriate economic game

Implementation problem: find an appropriate mechanism• mechanism is a partially specified game of imperfect information• rules of game are fixed• strategy sets are specified• preferences for the game are not yet specified

Plug preferences into the mechanism:• does the mechanism have an equilibrium?• does the equilibrium correspond to the desired social state θ ?• if so, the social state is implementable

There are many possible mechanisms

19July 2017


Mechanism: example The market is an example of a mechanism Suppose the following things are given:

• resource ownership in the economy • other legal entitlements• production technology

Mechanism consists of institutions and processes determining• incomes• production allocations • consumption baskets

Once individuals’ preferences are specified• market maps preferences into prices• price system yields a specific state of the economy θ

20July 2017


Design: basic ingredients The agents’ strategy sets S1, S2, S3,….

• collectively write S := S1×S2×S3×…• each element of S is a profile [s1, s2, s3,…]

The outcome function γ• given a strategy profile s := [s1, s2, s3,…] • social state is determined as θ = γ (s)

Agents’ objectives • a profile of preferences [v] := [v1, v2, v3,…]• once the outcome θ is determined • get utility payoffs v1(θ ), v2(θ ), v3(θ ), ….

21July 2017


Mechanism Consider this more formallyA mechanism consists of

• the set of strategy profiles S• and an outcome function γ from S to the set of social states Θ.

The mechanism is an almost-completely specified game. All that is missing is the collection of utility functions

• these specify the objective of each agent h • and the actual payoff to each h

Once a particular profile of utility functions is plugged in:• we know the social state that will be determined by the game• and the welfare implications for all the economic agents

22July 2017


Implementation: detail Is the SCF consistent with private economic behaviour?Mechanism is a (strategy-set, outcome-function) pair (S; γ).Agents’ behaviour:

• given their preferences [v1, v2, v3,…] • use the mechanism as the rules of the game• determine optimal strategies as the profile [s*1, s*2, s*3,…]

The outcome function• determines social from the profile of strategies • θ* = γ(s*1, s*2, s*3,…)

Is this θ* the one that the designer would have wished from the social-choice function Γ?

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a formal statement

July 2017


Dominant-strategy implementation Consider a special interpretation of equilibrium Take a particular social-choice function Γ Suppose there is a dominant-strategy equilibrium of the

mechanism (S; γ (∙)):[s*1(∙), s*2(∙), s*3(∙),…]

Suppose also it is true that γ(s*1(v1), s*2(v2), s*3(v3),…) = Γ(v1, v2, v3,…)

Then mechanism (S; γ (∙)) weakly implements the Γ in dominant strategies

24July 2017


Direct mechanisms For exposition consider a very simple mechanism

• the direct mechanism

Map from profile of preferences to states• involves a very simple game.• the game is “show me your utility function”• enables direct focus on the informational aspects of implementation

For a direct mechanism• strategy sets are just sets of preferences S = 𝕍𝕍• so the outcome function and the social-choice function are the same:

γ(v1, v2, v3,…) = Γ(v1, v2, v3,…)• the mechanism is effectively just the SCF

25July 2017


Truthful implementationAn SCF that encourages misrepresentation is of limited use Is truthful implementation possible?

• will people announce their true attributes?• will it be a dominant strategy to do so?

Γ is truthfully implementable in dominant strategies if• s*h(vh) = vh, h = 1,2,…• is a dominant-strategy equilibrium of the direct mechanism

Specifying a dominant strategies is quite strong• we insist that everyone finds that “honesty is the best policy”• irrespective of whether others are following the same rule • irrespective of whether others are even rational

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another key result

July 2017


Revelation principle Take a social-choice function Γ Suppose that mechanism (S;γ) can weakly implement Γ

• for any [v]∈ 𝕍𝕍:• (S;γ) has at least one equilibrium [s*1(v1), s*2(v2), s*3(v3),…]• such that θ* = γ(s*1(v1), s*2(v2), s*3(v3),…) = Γ(v1, v2, v3,…)

Now consider a direct mechanism • maps profiles from 𝕍𝕍 to social states in Θ

We can always get truthful implementation of Γ in dominant-strategies • vh, h = 1,2,…is a dominant-strategy equilibrium of the direct mechanism• θ* = Γ(v1, v2, v3,…)

Formally stated the result is: • If Γ is weakly implementable in dominant strategies by mechanism (S;γ) then

Γ is truthfully implementable in dominant strategies using direct mechanism (𝕍𝕍; Γ)

27July 2017


The revelation principle

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Γ(•) = γ (s*1(•),s*2(•), …)

S

𝕍𝕍 ΘΓ(•)

Pick a preference profile [v] from 𝕍𝕍Agents select strategies Outcome function yields social state

The combined effect

Direct mechanism simply requires declaration of [v]

July 2017


Direct mechanisms: manipulability Reinterpret manipulability in terms of direct mechanisms:

• if all, including h, tell the truth about preferences: θ = Γ(v1,…, vh, …, )

• if h misrepresents his preferences but others tell the truth:θ = Γ(v1,…, vh, …, )

How does the person “really” feel about θ and θ? • if vh(θ) > vh(θ) there is an incentive to misrepresent information • if h realises then clearly Γ is manipulable

What type of SCF would be non-manipulable?• need to characterise a class of Γ• central issue of design

29July 2017


Overview

30

A parable

Social choice again

Mechanisms

Design Basics

Allowing for human nature

The design problem

July 2017


The core of the problem Focus on a coherent approach to the implementation problem How to design a mechanism so that agents truthfully reveal

private information They only do so if it is in their private interests to act this way Take a standard form of implementation

• mechanism has equilibrium in dominant strategies

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another key result

July 2017


Gibbard-Satterthwaite The G-S result can be stated in several waysA standard versions is:

• if the set of social states Θ contains at least three elements;• and the SCF Γ is defined for the set 𝕍𝕍 of all possible preference profiles• and the SCF is truthfully implementable in dominant strategies • then the SCF must be dictatorial

Closely related to the Arrow theoremHas profound implications for design

• misinformation may be endemic • may only get truth-telling mechanisms in special cases

32July 2017


Onward from the G-S result The generality of the result is striking

• one could expect the phenomenon of market failure• crucial to the issues of design

Way forward? Try to relax one part of G-S resultNumber of states

• choice problems where Θ has just 2 elements?• see presentation on public goods and projects

All types of preferences• restricted attention to a subclass of 𝕍𝕍 ?• see presentation on contract design

Truth telling as dominant strategy• consider a less stringent type of equilibrium?• examine this now

33July 2017


Nash implementationHow to induce truth-telling?Dominant strategy equilibrium is demanding

• requires everyone to tell truth• irrespective of what others do

Nash equilibrium is weaker• requires everyone to tell truth• as long as everyone else does so• “I will if you will so will I”

An important implementation result:• if a social choice function Γ is Nash-implementable then it is monotonic

But Nash-implementation is itself limited• economically interesting cases may still require dictatorial Γ

34July 2017


SummaryAn issue at the heart of microeconomic policy-making:

• Regulation• Allocations with pure public goods• Tax design

Mechanism gives insight on the problems of information• may be institutions which encourage agents to provide false information• mechanisms may be inefficient because they provide wrong incentives

Direct mechanisms help focus on the main issue• use the revelation principle

G-S result highlights pervasive problem of manipulability

35July 2017

Documents

Almost essential Welfare Basics Games: equilibriumdarp.lse.ac.uk/presentations/MP2Book/OUP/DesignBasi… · · 2017-07-29Almost essential. Welfare Basics. Games: equilibrium. Prerequisites