2. Frank Cowell: Microeconomics Design Basics MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Welfare Basics Almost essential Welfare Basics Prerequisites October 2006

3. Frank Cowell: Microeconomics Overview... A parable Social choice again Mechanisms Design Basics AN introduction to the issues The design problem

4. Frank Cowell: Microeconomics A parable Think through the following everyday situation 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? 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 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…

5. Frank Cowell: Microeconomics Alf, Bill, Charlie and the taxi 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?

6. Frank Cowell: Microeconomics The approach Some questions: 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”): Manipulation (sometimes “cheating” or “chiselling”):  an important connection with the issue of efficiency  rules might be inefficient because they provide wrong incentives Design problem: 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 Need to examine the representation of choices  build on the analysis from social welfare…  …and reuse some results

7. Frank Cowell: Microeconomics Overview... A parable Social choice again Mechanisms Design Basics A link with the fundamentals of welfare economics The design problem

8. Frank Cowell: Microeconomics Agenda Basic questions Basic questions  purpose of design  informational context  strategic setting Purpose Purpose  modelling group objectives…  …need a review of social choice Information Information  agents may have private information…  …so need to allow for the possibility of misrepresentation Strategy Strategy  a connection with game-theoretic approaches…  …so need to review concepts of equilibrium Begin with purpose

9. Frank Cowell: Microeconomics Social states and preferences Social state:  Social state:   a comprehensive description…  …of all relevant features of the economy in question Set of all social states:  Set of all social states:  Preferences Preferences v h (∙)  a “reduced form” version of agent h’s utility function  utility of agent h given social state is  utility of agent h given social state  is v h (  )   preference profile is an ordered list, one for each agent: [v 1, v 2, v 3,…]   a list of functions, not utility levels Set of all preference profiles: V

10. Frank Cowell: Microeconomics A reminder Constitution Constitution   a mapping from V to set of all v(∙)  given a particular set of preferences for the population…  …the constitution should determine a specific  …the constitution should determine a specific v(∙) Properties Properties  Universality  Pareto Unanimity  Independence of Irrelevant Alternatives  Non-Dictatorship Arrow theorem 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 concept… Use this reminder to introduce a new concept…

11. Frank Cowell: Microeconomics Social-choice function  A social choice function    a mapping V    given a particular set of preferences for the population…  …picks out exactly one chosen element from  Note that argument of the SCF is same as for constitution 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” 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

12. Frank Cowell: Microeconomics Social-choice function: properties Three key properties of an SCF,  : Three key properties of an SCF,  :  is Paretian if…  is Paretian if…  given a such that for all h and all   given a   such that v h (   ) ≥ v h (  ), for all h and all   then =  ()  then   =  (v 1, v 2, v 3,…)  is monotonic if…  is monotonic if…  given any [] and [] such that  given any [v] and [v]  V such that “”“” “v h (   )≥v h (  )” implies “v h (   )≥v h (  )”  then “=  ()” implies “=  ()”  then “   =  (v 1, v 2, v 3,…)” implies “   =  (v 1, v 2, v 3,…)”  is dictatorial if  is dictatorial if  there is some agent whose preferences completely determine  there is some agent whose preferences completely determine 

13. Frank Cowell: Microeconomics Monotonicity: example 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  *   Clearly, 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

14. Frank Cowell: Microeconomics Social-choice function: result Suppose has more than two elements… Suppose  has more than two elements… … and that  is defined for all members of. … and that  is defined for all members of V.  Then, if  is Paretian and monotonic…  …  must also be dictatorial A counterpart of the Arrow result on constitutions A counterpart of the Arrow result on constitutions

15. Frank Cowell: Microeconomics A key property of the SCF  is manipulable if there is a profile [v] such that…  is manipulable if there is a profile [v]  V such that…  …for some h and some other utility function v h (∙)  v h (  ) > v h (  )  where  =  (v 1,…, v h, …, )  and  =  (v 1,…, v h, …, ) Significance is profound: Significance is profound:  if  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? An incentive to misrepresent information?  does imply that there is some h who can manipulate…  …but it does mean that, under some circumstances there is an h who could manipulate

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

17. Frank Cowell: Microeconomics Social-choice function: summary Similar to the concept of constitution 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 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 But key point concerns link with information  misrepresentation and manipulability are linked  important implication for design problem

18. Frank Cowell: Microeconomics Overview... A parable Social choice again Mechanisms Design Basics The problem of implementation … The design problem

19. Frank Cowell: Microeconomics Forward from social choice Social choice is just the first step Social choice is just the first step  SCF describes what is desirable…  …not how you achieve it The next step involves achievement 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 Requires the introduction of a new concept  a mechanism

20. Frank Cowell: Microeconomics Implementation Is the SCF consistent with private economic behaviour? 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 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: 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 There are many possible mechanisms

21. Frank Cowell: Microeconomics Mechanism: example The market is an example of a mechanism The market is an example of a mechanism Suppose the following things are given: Suppose the following things are given:  resource ownership in the economy  other legal entitlements  production technology Mechanism consists of institutions and processes determining Mechanism consists of institutions and processes determining  incomes…  production allocations …  consumption baskets Once individuals’ preferences are specified Once individuals’ preferences are specified  market maps preferences into prices…  …price system yields a specific state of the economy 

22. Frank Cowell: Microeconomics 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 (  ), ….

23. Frank Cowell: Microeconomics 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

24. Frank Cowell: Microeconomics Implementation: detail Is the SCF consistent with private economic behaviour? 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  ? a formal statement

25. Frank Cowell: Microeconomics 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 (∙),…] If 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

26. Frank Cowell: Microeconomics Direct mechanisms For exposition consider a very simple mechanism For exposition consider a very simple mechanism  The direct mechanism Map from profile of preferences to states 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 For a direct mechanism  strategy sets are just sets of preferences S = V  so the outcome function and the social-choice function are the same  (v 1, v 2, v 3,…) =  (v 1, v 2, v 3,…)  the mechanism is effectively just the SCF

27. Frank Cowell: Microeconomics An SCF that encourages misrepresentation is of limited use An SCF that encourages misrepresentation is of limited use Is truthful implementation possible? 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  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 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 Truthful implementation another key result

28. Frank Cowell: Microeconomics Revelation principle Take a social-choice function  Suppose that mechanism (S;  ) can weakly implement    for any [v]  V …   … (S;  ) has at least one equilibrium [s *1 (v 1 ), s *2 (v 2 ), s *3 (v 3 ),…]…   …such that  * =  (s *1 (v 1 ), s *2 (v 2 ), s *3 (v 3 ),…) =  (v 1, v 2, v 3,…) Now consider a direct mechanism Now consider a direct mechanism  maps profiles from to social states in .  maps profiles from V to social states in . We can always get truthful implementation of  in dominant-strategies  v h h = 1,2,…  v h h = 1,2,…is a dominant-strategy equilibrium of the direct mechanism    * =  (v 1, v 2, v 3,…) Formally stated the result is:  )  If  is weakly implementable in dominant strategies by the mechanism (S;  ) then  is truthfully implementable in dominant strategies using the direct mechanism ( V ;  )

29. Frank Cowell: Microeconomics The revelation principle  () =  (s *1 (),s *2 (), …)  () [s *1 (),s *2 (), …] S V   ()   Pick a preference profile [v] from V   Agents select strategies   Outcome function yields social state   The combined effect   Direct mechanism simply requires declaration of [v]

30. Frank Cowell: Microeconomics Direct mechanisms: manipulability Reinterpret manipulability in terms of direct mechanisms: 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  ? 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? What type of SCF would be non-manipulable?  need to characterise a class of   central issue of design

31. Frank Cowell: Microeconomics Overview... A parable Social choice again Mechanisms Design Basics Allowing for human nature… The design problem

32. Frank Cowell: Microeconomics The core of the problem Focus on a coherent approach to the implementation problem Focus on a coherent approach to the implementation problem How to design a mechanism so that agents truthfully reveal private information 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 They only do so if it is in their private interests to act this way Take a standard form of implementation Take a standard form of implementation  mechanism has equilibrium in dominant strategies another key result

33. Frank Cowell: Microeconomics Gibbard-Satterthwaite The G-S result can be stated in several ways The G-S result can be stated in several ways A standard versions is: A standard versions is:  if the set of social states  contains at least three elements; ...and the SCF  is defined for the set V 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 theorem Closely related to the Arrow theorem Has profound implications for design Has profound implications for design  Misinformation may be endemic  May only get truth-telling mechanisms in special cases

34. Frank Cowell: Microeconomics Onward from the G-S result The generality of the result is striking 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 result Way forward? Try to relax one part of G-S result 1. Number of states  choice problems where  has just 2 elements?  see presentation on public goods and projects 2. All types of preferences  restricted attention to a subclass of V ?  see presentation on contract design 3. Truth telling as dominant strategy  consider a less stringent type of equilibrium?  examine this now… Public Goods Contracts Market power

35. Frank Cowell: Microeconomics Nash implementation How to induce truth-telling? How to induce truth-telling? Dominant strategy equilibrium is demanding Dominant strategy equilibrium is demanding  requires everyone to tell truth…  …irrespective of what others do Nash equilibrium is weaker 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: An important implementation result:  If a social choice function  is Nash-implementable then it is monotonic But Nash-implementation is itself limited But Nash-implementation is itself limited  Economically interesting cases may still require dictatorial.

36. Frank Cowell: Microeconomics Summary An issue at the heart of microeconomic policy-making: 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 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 Direct mechanisms help focus on the main issue  use the revelation principle G-S result highlights pervasive problem of manipulability G-S result highlights pervasive problem of manipulability