Design Basics

1. Prerequisites Almost essential Welfare Basics Design Basics MICROECONOMICS Principles and Analysis Frank Cowell October 2006

2. Overview... Design Basics A parable AN introduction to the issues Social choice again Mechanisms The design problem

3. 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…

4. Alf, Bill, Charlie and the taxi • Alf’s preferences • Bill’s preferences preference Alf • Charlie’s preferences • Taxi firm’s proposed time #1 Bill • Taxi firm’s proposed time #2 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 12:00 13:00 11:00

5. 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

6. Overview... Design Basics A parable A link with the fundamentals of welfare economics Social choice again Mechanisms The design problem

7. 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 Begin with purpose

8. Social states and preferences • Social state: q • a comprehensive description… • …of all relevant features of the economy in question • Set of all social states: Q • Preferences vh(∙) • a “reduced form” version of agent h’s utility function • utility of agent h given social state q is vh(q) • 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: V

9. A reminder • 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 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 concept…

10. Social-choice function • A social choice function G • a mapping V Q • given a particular set of preferences for the population… • …picks out exactly one chosen element from Q • 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

11. Social-choice function: properties • Three key properties of an SCF, G : • G is Paretian if… • given a q* such that vh(q*) ≥vh(q), for all h and all qQ, • then q* = G(v1, v2, v3,…) • G is monotonic if… • given any [v] and [v]  V such that “vh(q*)≥vh(q)” implies “vh(q*)≥vh(q)” • then “q* = G(v1, v2, v3,…)” implies “q* = G(v1, v2, v3,…)” • G is dictatorial if • there is some agent whose preferences completely determine q

12. x2 h x1 h Monotonicity: example • Here state is an allocation • h’s indifference curve under v(∙) • Better-than set for v and the state q* B(q*; v) B(q*; v) • h’s indifference curve under v(∙) • Better-than set for v and the state q* • Clearly, if vh(q*)≥vh(q) thenvh(q*)≥vh(q)) • q* • If G is monotonic, then if q* is the chosen point under [v] then q* is also chosen point under [v]

13. Social-choice function: result • Suppose Q has more than two elements… • … and that G is defined for all members of V. • Then, ifGis Paretian and monotonic… • … G must also be dictatorial • A counterpart of the Arrow result on constitutions

14. A key property of the SCF • G is manipulable if there is a profile [v]V such that… • …for some h and some other utility function vh(∙) • vh(q) > vh(q) • where q = G(v1,…, vh, …,) • and q= G(v1,…, vh, …,) • Significance is profound: • if G 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 imply that there is some h who can manipulate… • …but it does mean that, under some circumstances there is an h who could manipulate

15. 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 hany strict ranking of elements of Qis permissible • then a Paretian, non-manipulableSCFGmust be dictatorial • This result is important • connects the idea of misrepresentation and social choice • introduces an important part of the design problem

16. 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

17. Overview... Design Basics A parable The problem of implementation… Social choice again Mechanisms The design problem

18. 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

19. Implementation • Is the SCF consistent with private economic behaviour? • Yes if the q 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 q? • If so, the social state is implementable • There are many possible mechanisms

20. 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 q

21. 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 g • given a strategy profile s := [s1, s2, s3,…] … • … social state is determined as q = g (s) • Agents’ objectives • a profile of preferences [v] := [v1, v2, v3,…] • once the outcome q is determined… • …get utility payoffs v1(q ), v2(q ), v3(q ), ….

22. 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

23. 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 [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 • q* = g(s*1, s*2, s*3,…) • Is this q* the one that the designer would have wished from the social-choice function G? a formal statement

24. Dominant-strategy implementation • Consider a special interpretation of equilibrium… • Take a particular social-choice functionG • Suppose there is a dominant-strategy equilibrium of the mechanism (S; g (∙)): [s*1(∙), s*2(∙), s*3(∙),…] • If it is true that • g(s*1(v1), s*2(v2), s*3(v3),…) = G(v1, v2, v3,…) • Then mechanism (S; g (∙)) weakly implements the G in dominant strategies

25. 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 = V • so the outcome function and the social-choice function are the same g(v1, v2, v3,…) = G(v1, v2, v3,…) • the mechanism is effectively just the SCF

26. 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) = vhh = 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 another key result

27. Revelation principle • Take a social-choice function G • Suppose that mechanism (S;g) can weakly implement G • for any [v] V… • … (S;g) has at least one equilibrium [s*1(v1), s*2(v2), s*3(v3),…]… • …such that q* = g(s*1(v1), s*2(v2), s*3(v3),…) = G(v1, v2, v3,…) • Now consider a direct mechanism • maps profiles from V to social states in Q. • We can always get truthful implementation of G in dominant-strategies • vhh = 1,2,…is a dominant-strategy equilibrium of the direct mechanism • q* = G(v1, v2, v3,…) • Formally stated the result is: • If Gis weakly implementable in dominant strategies by the mechanism (S;g) thenGis truthfully implementable in dominant strategies using the direct mechanism (V;G)

28. The revelation principle • Pick a preference profile [v] fromV • Agents select strategies • Outcome function yields social state S • The combined effect • Direct mechanism simply requires declaration of [v] [s*1(•),s*2(•), …] g(•) Q V G(•) G(•) = g (s*1(•),s*2(•), …)

29. Direct mechanisms: manipulability • Reinterpret manipulability in terms of direct mechanisms: • if all, including h, tell the truth about preferences: q = G(v1,…, vh, …,) • if h misrepresents his preferences but others tell the truth: q = G(v1,…, vh, …,) • How does the person “really” feel about q and q? • if vh(q) > vh(q) 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

30. Overview... Design Basics A parable Allowing for human nature… Social choice again Mechanisms The design problem

31. 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

32. Gibbard-Satterthwaite • The G-S result can be stated in several ways • A standard versions is: • if the set of social states Q contains at least three elements; • ...and the SCF G 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 • Has profound implications for design • Misinformation may be endemic • May only get truth-telling mechanisms in special cases

33. 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 result • Number of states • choice problems where Q has just 2 elements? • see presentation on public goods and projects • All types of preferences • restricted attention to a subclass of V ? • see presentation on contract design • Truth telling as dominant strategy • consider a less stringent type of equilibrium? • examine this now… Market power Public Goods Contracts

34. 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 .

35. 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