Prerequisites Almost essential Welfare Basics Games equilibrium DESIGN

Prerequisites Almost essential Welfare Basics Games: equilibrium DESIGN BASICS MICROECONOMICS Principles and Analysis Frank Cowell April 2018 Frank Cowell: Design Basics 1

Overview Design Basics A parable An introduction to the issues Social choice again Mechanisms The design problem April 2018 Frank Cowell: Design Basics 2

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 April 2018 Frank Cowell: Design Basics 3

preference Alf, Bill, Charlie and the taxi §Blue: Alf’s preferences §Red: Bill’s preferences §Green: Charlie’s preferences §Taxi firm’s proposal #1: 12: 45 §Taxi firm’s proposal #2: 15 Alf Bill Charlie § 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? 10: 00 April 2018 11: 00 12: 00 13: 00 Frank Cowell: Design Basics 4

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 April 2018 Frank Cowell: Design Basics 5

Overview Design Basics A parable A link with the fundamentals of welfare economics Social choice again Mechanisms The design problem April 2018 Frank Cowell: Design Basics 6

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 April 2018 Frank Cowell: Design Basics Begin with purpose 7

Social states and preferences § April 2018 Frank Cowell: Design Basics 8

A reminder § April 2018 Frank Cowell: Design Basics 9

Social-choice function § April 2018 Frank Cowell: Design Basics 10

Social-choice function: properties § April 2018 Frank Cowell: Design Basics 11

Monotonicity: example §Here state is an allocation §solid line: h’s indifference curve under v(∙) §Blue: Better-than set for v and the state * §dotted line: h’s indifference curve under v(∙) §Pink: Better-than set for v and the state * x 2 h B( *; v) B( *; v ) §“Better-than” is used as shorthand for “Better-than-or-just-as-good-as-” §So, if vh( *) ≥ vh( ) then vh( *) ≥ vh( ) · * §If G is monotonic then, if * is the chosen point under [v], then * is also chosen point under [v] x 1 h April 2018 Frank Cowell: Design Basics 12

Social-choice function: result § April 2018 Frank Cowell: Design Basics 13

A key property of the SCF § April 2018 Frank Cowell: Design Basics 14

Social-choice function: another result § Note that the monotonicity property is powerful: • if G is monotonic • then G cannot be manipulable § From this and the previous result a further result follows • suppose Q has more than two elements • for each h any strict ranking of elements of Q is permissible • then a Paretian, non-manipulable SCF G must be dictatorial § This result is important • connects the idea of misrepresentation and social choice • introduces an important part of the design problem April 2018 Frank Cowell: Design Basics 15

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 April 2018 Frank Cowell: Design Basics 16

Overview Design Basics A parable The problem of implementation Social choice again Mechanisms The design problem April 2018 Frank Cowell: Design Basics 17

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 April 2018 Frank Cowell: Design Basics 18

Implementation § Is the SCF consistent with private economic behaviour? • yes if the picked out by G 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 ? • if so, the social state is implementable § There are many possible mechanisms April 2018 Frank Cowell: Design Basics 19

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 April 2018 Frank Cowell: Design Basics 20

Design: basic ingredients § The agents’ strategy sets S 1, S 2, S 3, …. • collectively write S : = S 1 S 2 S 3 … • each element of S is a profile [s 1, s 2, s 3, …] § The outcome function g • given a strategy profile s : = [s 1, s 2, s 3, …] • social state is determined as = g (s) § Agents’ objectives • a profile of preferences [v] : = [v 1, v 2, v 3, …] • once the outcome is determined • get utility payoffs v 1( ), v 2( ), v 3( ), …. April 2018 Frank Cowell: Design Basics 21

Mechanism § Consider this more formally § A mechanism consists of • the set of strategy profiles S • and an outcome function g from S to the set of social states Q. § 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 April 2018 Frank Cowell: Design Basics 22

Implementation: detail § Is the SCF consistent with private economic behaviour? § Mechanism is a (strategy-set, outcome-function) pair (S; g) § Agents’ behaviour: • given their preferences [v 1, v 2, v 3, …] • 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 • * = g(s*1, s*2, s*3, …) § Is this * the one that the designer would have wished from the social-choice function G? a formal statement April 2018 Frank Cowell: Design Basics 23

Dominant-strategy implementation § Consider a special interpretation of equilibrium § Take a particular social-choice function G § Suppose there is a dominant-strategy equilibrium of the mechanism (S; g (∙)): [s*1(∙), s*2(∙), s*3(∙), …] § Suppose also it is true that g(s*1(v 1), s*2(v 2), s*3(v 3), …) = G(v 1, v 2, v 3, …) § Then mechanism (S; g (∙)) weakly implements the G in dominant strategies April 2018 Frank Cowell: Design Basics 24

Direct mechanisms § April 2018 Frank Cowell: Design Basics 25

Truthful implementation § An 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? § G 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 April 2018 Frank Cowell: Design Basics another key result 26

Revelation principle § April 2018 Frank Cowell: Design Basics 27

The revelation principle §Agents select strategies §Outcome function yields social state §The combined effect S ] g( • , … ) • *2 ( §Direct mechanism simply requires declaration of [v] ) ), s 1 ( • * [s Q G( • ) = g (s*1( • ), s*2( • ), …) April 2018 Frank Cowell: Design Basics 28

Direct mechanisms: manipulability § Reinterpret manipulability in terms of direct mechanisms: • if all, including h, tell the truth about preferences: = G(v 1, …, vh, …, ) • if h misrepresents his preferences but others tell the truth: = G(v 1, …, vh, …, ) § How does the person “really” feel about and ? • if vh( ) > vh( ) there is an incentive to misrepresent information • if h realises then clearly G is manipulable § What type of SCF would be non-manipulable? • need to characterise a class of G • central issue of design April 2018 Frank Cowell: Design Basics 29

Overview Design Basics A parable Allowing for human nature Social choice again Mechanisms The design problem April 2018 Frank Cowell: Design Basics 30

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 another key result April 2018 Frank Cowell: Design Basics 31

Gibbard-Satterthwaite § April 2018 Frank Cowell: Design Basics 32

Onward from the G-S result § April 2018 Frank Cowell: Design Basics 33

Nash implementation § How 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 G is Nash-implementable then it is monotonic § But Nash-implementation is itself limited • economically interesting cases may still require dictatorial G April 2018 Frank Cowell: Design Basics 34

Summary § An 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 April 2018 Frank Cowell: Design Basics 35