Frank Cowell: Design Basics DESIGN BASICS MICROECONOMICS Principles and Analysis Frank Cowell 1 Almost essential Welfare Basics Games: equilibrium Almost essential Welfare Basics Games: equilibrium Prerequisites July 2015

Frank Cowell: Design Basics Overview 2 A parable Social choice again Mechanisms Design Basics An introduction to the issues The design problem July 2015

Frank Cowell: Design Basics 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 3 July 2015

Frank Cowell: Design Basics Alf, Bill, Charlie and the taxi 4 preference Alf Bill Charlie 10:00 11:00 12:00 13:00  Alf’s preferences  Bill’s preferences  Charlie’s preferences  Taxi firm’s proposed time #1  Taxi 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 2015

Frank Cowell: Design Basics 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 5 July 2015

Frank Cowell: Design Basics Overview 6 A parable Social choice again Mechanisms Design Basics A link with the fundamentals of welfare economics The design problem July 2015

Frank Cowell: Design Basics 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 2015

Frank Cowell: Design Basics Social states and preferences 8 July 2015

Frank Cowell: Design Basics A reminder 9 July 2015

Frank Cowell: Design Basics Social-choice function 10 July 2015

Frank Cowell: Design Basics Social-choice function: properties 11 July 2015

Frank Cowell: Design Basics Monotonicity: example 12 x1x1 h x2x2 h B(   ; v)   h’s indifference curve under v(∙)  Better-than set for v and the state  *  h’s indifference curve under v (∙)  Better-than set for v and the state  *  So, if v h (   ) ≥ v h (  ) then v h (   ) ≥ v h (  )  If  is monotonic, then if   is the chosen point under [v] then   is also chosen point under [v]  Here state is an allocation July 2015  “Better-than” is used as shorthand for “Better-than-or-just-as-good-as-”

Frank Cowell: Design Basics Social-choice function: result 13 July 2015

Frank Cowell: Design Basics A key property of the SCF 14 July 2015

Frank Cowell: Design Basics Social-choice function: another result  Note that the monotonicity property is powerful: if  is monotonic then  cannot be manipulable  From this and the previous result a further result follows suppose  has more than two elements for each h any strict ranking of elements of  is permissible then a Paretian, non-manipulable SCF  must be dictatorial  This result is important connects the idea of misrepresentation and social choice introduces an important part of the design problem 15 July 2015

Frank Cowell: Design Basics 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 16 July 2015

Frank Cowell: Design Basics Overview 17 A parable Social choice again Mechanisms Design Basics The problem of implementation The design problem July 2015

Frank Cowell: Design Basics 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 18 July 2015

Frank Cowell: Design Basics Implementation  Is the SCF consistent with private economic behaviour? Yes if the  picked out by  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 19 July 2015

Frank Cowell: Design Basics 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  20 July 2015

Frank Cowell: Design Basics 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  given a strategy profile s := [s 1, s 2, s 3,…] social state is determined as  =  (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 (  ), …. 21 July 2015

Frank Cowell: Design Basics Mechanism  Consider this more formally  A mechanism consists of the set of strategy profiles S and an outcome function  from S to the set of social states .  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 22 July 2015

Frank Cowell: Design Basics Implementation: detail  Is the SCF consistent with private economic behaviour?  Mechanism is a (strategy-set, outcome-function) pair (S;  ).  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  * =  (s *1, s *2, s *3,…)  Is this  * the one that the designer would have wished from the social-choice function  ? 23 a formal statement July 2015

Frank Cowell: Design Basics Dominant-strategy implementation  Consider a special interpretation of equilibrium  Take a particular social-choice function   Suppose there is a dominant-strategy equilibrium of the mechanism (S;  (∙)): [s *1 (∙), s *2 (∙), s *3 (∙),…]  Suppose also it is true that  (s *1 (v 1 ), s *2 (v 2 ), s *3 (v 3 ),…) =  (v 1, v 2, v 3,…)  Then mechanism (S;  (∙)) weakly implements the  in dominant strategies 24 July 2015

Frank Cowell: Design Basics Direct mechanisms 25 July 2015

Frank Cowell: Design Basics 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?   is truthfully implementable in dominant strategies if s *h (v h ) = v h, 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 26 another key result July 2015

Frank Cowell: Design Basics Revelation principle 27 July 2015

Frank Cowell: Design Basics The revelation principle 28  () =  (s *1 (),s *2 (), …)  () [s *1 (),s *2 (), …] S   ()  Agents select strategies  Outcome function yields social state  The combined effect  Direct mechanism simply requires declaration of [v ] July 2015

Frank Cowell: Design Basics Direct mechanisms: manipulability  Reinterpret manipulability in terms of direct mechanisms: if all, including h, tell the truth about preferences:  =  (v 1,…, v h, …, ) if h misrepresents his preferences but others tell the truth:  =  (v 1,…, v h, …, )  How does the person “really” feel about  and  ? if v h (  ) > v h (  ) there is an incentive to misrepresent information if h realises then clearly  is manipulable  What type of SCF would be non-manipulable? need to characterise a class of  central issue of design 29 July 2015

Frank Cowell: Design Basics Overview 30 A parable Social choice again Mechanisms Design Basics Allowing for human nature The design problem July 2015

Frank Cowell: Design Basics 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 31 another key result July 2015

Frank Cowell: Design Basics Gibbard-Satterthwaite 32 July 2015

Frank Cowell: Design Basics Onward from the G-S result 33 July 2015

Frank Cowell: Design Basics 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  is Nash-implementable then it is monotonic  But Nash-implementation is itself limited economically interesting cases may still require dictatorial  34 July 2015

Frank Cowell: Design Basics 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 35 July 2015