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voters are free to reveal any ranking, they The Theory of Mechanism can also misrepresent their true ranking. Clearly their report will depend on how Design: An Overview they believe others will vote. A natural question is whether a voting rule can be designed where voters can be induced to Arunava Sen reveal their true ranking irrespective of their beliefs regarding the voting behaviour Three US-based economists have he Nobel Prize in economics for of others. Unfortunately this does not hold been awarded the Nobel Prize 2007 was awarded to Leonid for either the Plurality Rule or rules based for economics for 2007 for laying THurwicz (University of Minnesota, on the Majority Principle. US), Eric Maskin (Institute of Advanced To see this consider (for simplicity) the the foundations of mechanism Studies, Princeton, US) and Roger Myerson case where there are are three voters 1, 2 design theory. A description (University of Chicago, US) for “having and 3 and exactly three proposals a, b and and discussion of this theory, laid the foundations of mechanism design c. Suppose that voter 1’s true ranking is a its importance and the work of theory”. What is this theory and why is it better than b better than c, 2’s true rank important? What are the contributions ing is b better than c better than a while 3’s the award-winning economists. of the recipients of this year’s prize? true ranking is c better than a better than Mechanism design could help This article briefly attempts to address b. If these rankings were announced, a tie policymaking in a number of these questions. would result according to both the Plurality areas, one potential area is in Rule (since there is exactly one voter for 1 Introduction whom each of a, b and c is first ranked) drawing up the rules for land Consider the following example on the de and any rule based on the Majority Principle acquisition. sign of voting rules, which is a particular (since a majority of voters prefer a to b, b aspect of the theory of mechanism design. to c and c to a). Assume without any loss of Suppose that there are N voters in a com generality that the outcome in this situa mittee who have to choose one of K tion is a. Then voter 2 who believes that 1 proposals. Each voter has an opinion on and 3 will vote truthfully, will be better the proposals, which can be expressed in off by lying and reporting c better than b the form of a ranking of the proposals. For better than a, thereby obtaining c (for instance, voter j may prefer proposal a to both the Plurality Rule or a rule based on proposal b to proposal c and so on. A “voting the Majority Principle) which is better rule” is a method for aggregating individual than a, the outcome that would occur if opinions into an outcome. It determines she voted truthfully, according to her the chosen proposal at any given profile of true ranking. Note that lying leads to rankings of voters – the chosen proposal non-optimal outcomes. can be thought of as the “the optimal” pro As a result, a host of questions arise posal in that situation. An example of a which are addressed by mechanism design voting rule is the familiar “Plurality Rule” theory. Are there other rules where voters where the proposal selected is the first- are induced to vote truthfully? Can the ranked candidate for the largest number of truth-telling requirement be weakened voters. Another well known class of voting sensibly? What happens if monetary com rules is based on the “Majority Principle” pensation or randomisation is permitted? which picks the proposal that beats all other What if voters had better information proposals in pairwise majority contests about the true rankings of other voters? whenever such a proposal exists.1 Is there any benefit in designing more complex communication systems than Revealing Information simply requiring voters to reveal their A natural assumption in a setting where rankings directly? I am grateful to Parikshit Ghosh and Debasis individuals vote is that each voter’s opinion Mishra for their comments. is known only to the voter. Moreover vot Devising a Mechanism ing is the act of revealing this information. In general terms, mechanism design theory Arunava Sen ([email protected]) is at the Indian If voter opinions were commonly known, is concerned with resource allocation Statistical Institute, New Delhi. voting itself would be superfluous. Since (understood very broadly) in multi-agent
8 december 8, 2007 Economic & Political Weekly commentary environments. The key feature of the represents the collective will of all the a game of complete information and a so problem is that the determination of an agents. The goals of the planner are cap lution concept such as Nash equilibrium, “optimal” allocation depends on informa tured by a “Social Choice Correspondence” is appropriate.5 In the incomplete informa- tion which agents possess privately. In (SCC) F which is a mapping from the set Θ tion setting, each agent has only partial order to achieve an optimal allocation, to the set of non-empty subsets of A. Thus, information regarding the state. this private information must therefore be for any θ∈Θ, F(θ)⊂A are the alternatives Typically, θ comprises an N tuple (θ1,.., θN) elicited from the agents. However, agents which are socially optimal in the state θ. where θi is the private information of are sophisticated in the sense that they The crux of the problem is that the plan agent i. In general θ1,.., θN are random vari recognise that they may (depending on ner does not know the realisation of the ables whose joint distribution is commonly beliefs that they have about the information state of the world, i e, does not know known (by all agents and the planner) but revealed by the other agents) be served which θ∈Θ has occurred. Agents however θi is observed only by agent i. In the pri- better by lying rather than by telling the have better information about the state vate values model (to which I shall mainly truth. Computing the optimal allocation and in particular, jointly know the θ. In confine my attention), θi determines i’s from incorrect information may entail se other words, if their information is pooled, preferences Ri(θi). Here, (G,θ) is a game of rious errors; hence the challenge is to de then it identifies θ exactly (I shall try and incomplete information and there are two vise a mechanism or a procedure for com make these notions more precise very solution concepts which are natural. The municating the information of agents such shortly). The planner must induce the first is dominant strategy equilibrium where that the outcome is an optimal allocation agents to reveal their information. She each agent has an optimal message which even when these agents behave strategi does so by constructing a mechanism G gives him a better outcome no matter what 6 cally. Mechanism design theory can there which consists of a message set Mi for messages are sent by the other agents. fore be thought of as a theory of the design each agent i and an outcome function The second is the weaker requirement of institutions or the design of the rules of g:M1×M2×...×MN → A. of Bayes-Nash equilibrium which is a strat interactions amongst fully strategic agents Each agent i sends a message mi from egy N-tuple from which no agent, regard in order to achieve desirable outcomes. his message set; the outcome function less of the information he receives, can
A critical step in the development of a then specifies an outcome g(m1,..,mN) as a deviate unilaterally and obtain a higher theory is the formulation of a conceptual function of the messages received. The expected pay-off. A more complicated framework that allows consideration of rele mechanism is commonly known to all the model is the interdependent values model vant issues at a high degree of generality. agents. A pair (G,θ), θ∈Θ therefore defines in the incomplete information setting,
For instance, progress in game theory was a game. Consider a solution concept E ap where each agent i observes θi but whose possible only after von Neumann and propriate for this game and let E(G,θ)⊂A preferences depend not just on but also on
Morgenstern introduced the idea of normal denote the set of equilibrium outcomes ac θi the entire vector θ, i e, we have Ri(θi). and extensive form representations. This cording to E. We say that the mechanism G I now give a number of applications which permitted a unified analysis ofall strategic implements F (according to E) if there ex illustrate the general framework outlined interaction between players including in its ists a mechanism G such that E(G,θ)=F(θ) above and describe some important results. ambit for instance, interaction as disparate for all θ∈Θ.4 Moreover, the SCC F is imple- as that in chess and that in oil cartels. In mentable if there exists a mechanism 3 complete Information this respect, the foundations of mechanism which implements it. If a SCC is imple Adjudication Problems: The most cele design theory were laid by Hurwicz (1960, mentable, the planner can put the mecha brated example is the so-called King 1972). The basic model ,which I outline nism which implements it into operation Solomon problem from the Book of Kings below, is both simple and versatile. and be completely certain that no matter in the Old Testament of the Bible. Two which state occurs, the outcomes which women claim to be the mother of a child. 2 the Formal Model obtain will be optimal for that state. The Solomon orders the child to be cut in half There are N agents, a set of feasible alter primary goal of mechanism design theory and for each woman to be given a piece. natives or outcomes or allocations A and is to determine the SCCs that are implement One woman then withdraws her claim and an abstract set of “states of the world” Θ. able; in certain cases it is also to identify asks for the child to be given to the other Every realisation of a state θ∈Θ endows SCCs within the implementable class that woman. Solomon gives the child to her on each agent i with a preference ordering satisfy additional optimality properties. the ground that only the true mother would
Ri(θ) over the elements of A with the follow give up her child than to see it killed. Structure of Information ing interpretation: for all a,b∈A, aRi(θ)b Consider the following plausible formu implies a is “at least as good as b” accord Before proceeding further, an important lation of this situation. The two agents are 2, 3 ing to Ri(θ). There is another agent issue regarding the structure of informa woman 1 and 2 referred to as W1 and W2 called the “planner” or the “principal” who tion which also bears on the solution respectively and the planner is King has the authority to pick an alternative from concept E used, must be clarified. In the Solomon. The set A={a,b,c} where a is the
A. As we shall see from vatious examples, complete information setting, the state θ is alternative where the child is given to W1, the planner need not be a “real” person; commonly known to all agents. For any b the alternative where the child is given she can be a fictitious individual who mechanism G, the game (G,θ) is therefore, to W2 and c the alternative where the child
Economic & Political Weekly december 8, 2007 9 commentary is cut in half. The set Θ={θ,φ} where θ is may change the g function after receiving observable. The set of alternatives A is the the state where W1 is the true mother and their messages, they will change their set of allocations amongst the agents of φ the state where W2 is the true mother. In behaviour accordingly. In general, the the aggregate endowment Σω i. A state θ i each state both women most prefer the failure of the planner to commit to a specifies a preference ordering Ri(θ) for alternative which gives the child to her but mechanism can have bad consequences. each agent over allocations. The Walra- the true mother prefers giving up the child The question therefore remains; can the sian SCC W associates the set of perfectly to the other woman than to see it cut in Solomonic SCC be implemented? It turns competitive allocations to each state θ, i e, half while the exactly the reverse holds for out that in the form that it has been pre for each θ, W(θ)⊂A are the allocations which can be decentralised by a price the impostor. Hence a R1(θ)bR1(θ)c, bR2(θ) sented here, it cannot. There is a literature mechanism when preferences are given cR2(θ)a while aR1(φ)cR1(φ)b and bR2(φ) which examines this issue and related ad aR (φ)c. Solomon’s objective is to give the judication problems in detail. by (R1(θ),...,RN(θ)) and endowments by 2 ( ,..., ).7 Can W be implemented? What child to the true mother in each state, i e, ωi ωN else can be implemented? How much re by the SCC F(θ)=a and F(φ)=b. This is a Resource Allocation Problems: A classic distribution is feasible? Answers to these complete information problem because in debate in economic theory in the 1930s and other questions can be found by ap each state both women know exactly who and 1940s centred on the possibility of plying a very general result of Maskin. the real mother and the impostor are. “rational economic calculation” in social Solomon’s mechanism can be construed ist economies. Lange (1936) and Lerner A General Solution: Maskin (1977, 1999) as follows: M1 = M2 = Θ and the outcome (1944) claimed that socialist economies provided a very general solution to the function g:Θ×Θ→A is defined byg (θ,θ)=a, could replicate market outcomes with the complete information implementation g(φ,φ)=b, g(θ,φ)=g(φ,θ)=c. In other words, central planning agency (CPA) playing the problem. He proved that provided that both women are asked to reveal the state; role of the fictitious Walrasian auctioneer. N≥3, a SCC F is implementable only if F if they agree, the outcome is the F-optimal This view was disputed for instance, by satisfies a property which is now known alternative for that state while if they disa Hayek (1945) who not only emphasised as Maskin Monotonicity (MM). Conversely, gree, the outcome is c. the computational difficulties involved but if F satisfies MM and additionally, a weak Does this mechanism implement F? also pointed out that relevant information requirement called No Veto Power (NVP),8 Alas no! In state θ the unique Nash equi would be widely dispersed amongst agents then it is implementable. Moreover Maskin librium is for both W1 and W2 to send the in the economy who would not necessarily explicitly constructed a mechanism which message φ leading to the outcome b. Note have the correct incentives to reveal this implements it. that unilateral deviation by either W1 or information to the CPA. MM says the following. Suppose that al W2 leads to c which both W1 and W2 prefer Consider the following highly simpli ternative a∈F(θ). Consider another state φ less than b according to their preferences fied and stylised version of this problem. which has the property that for agents i, in state θ. The message pair (θ,θ) leading There are N agents who have endowments all alternatives b which are worse than a to outcome a is not a Nash equilibrium be ωi of L commodities which are publicly according to Ri(θ) are also worse than a cause W2 can deviate unilaterally to φ and obtain c which she prefers to a and neither
(θ,φ) are equilibria because W1 will devi ate. Analogous arguments establish that the unique Nash equilibrium in state φ is the message pair (θ,θ) leading to a. Thus the Solomonic mechanism achieves the exact opposite of what it sets out to do – it always leads to the child being given to the impostor. How then did Solomon achieve the right outcome? By using a trick that mechanism design theory does not approve of. Note that the true state can always be deduced from the equilibrium outcome. If both women send (φ,φ), then the true state is θ while observing (θ,θ) is equivalent to the true state being φ. Solomon thus infered the true state from the equi librium message pair and changed the g function ex post to make g(θ,θ)=b and g(φ,φ)=a. The two agents in this case were successfully fooled by the planner; how ever if agents anticipate that the planner
10 december 8, 2007 Economic & Political Weekly commentary according to Ri(φ); then we must have depends on the solution concept E. If E is cent. If voter preferences are single-peaked, a∈F(φ). MM is a condition which is gener dominant strategy equilibrium, then the then median rules are strategy-proof. For ally easy to verify. For instance, W satisfies incentive compatibility condition is called instance, suppose that there are five voters MM and can therefore be implemented. strategy-proofness and requires each and their declared rankings have the follow Other SCCs which can be implemented are agent to reveal his information truthfully ing peaks (in percentages): 2, 2, 5, 7 and the Pareto-efficient SCC9 and subcorre irrespective of his beliefs about the reports 10. Then the chosen outcome is 5 per cent. spondences of the envy-free SCC.10 of other agents. If E is Bayes-Nash equilib There is a rich theory of the design of voting rium, the incentive compatibility is re rules on restricted preference domains. 4 Incomplete Information ferred to as Bayes-Nash incentive compat Unlike the complete information case, ibility and requires an agent to be truthful Public Good Provision: There are N citi agents have exclusive information in the if it yields an expected pay-off higher than zens who have to decide whether or not to incomplete informaton model. The plan that obtained by lying, assuming that all undertake a public project (say, building a ner can therefore no longer rely on agents other agents are telling the truth. bridge). The set of alternatives A={0,1} where 0 and 1 represent the alternatives to verify each other’s reports.11 This intro “don’t build” and “build” respectively. duces extra constraints on implementa Voting: This has already been discussed Each citizen i has a valuation of the project tion as we shall see. earlier. There are N voters who have to θi∈Θi which is private information. Assume An important tool in the analysis of im elect one of the candidates in the set A. that the project costs c and that this is plementation in this model is “The Revela Voter j’s private information is his ranking shared equally if the project is undertaken. c tion Principle”. Consider the model where of all the candidates. Voting consists of re Thus the utility of citizen i is θ – — if 1 is i N a state θ is an N-tuple (θ1,...,θN) and θ1 is vealing his ranking and a voting rule selected and 0 otherwise. An efficient agent i’s private information with θ∈Θ . elects a candidate based on the revealed i i decision rule d(q1,..,qN) is a SCF whose Suppose a social choice function (SCF)12 f rankings of all voters. Here the planner is value is 1 if Σ θ >c and 0 if Σ θ
Economic & Political Weekly december 8, 2007 11 commentary (interpreted as revelation of private infor whose objective is to maximise social wel forced to sell at prices below their valua mation) and two decisions based on these fare. A regulatory mechanism elicits cost tions. It is possible that efficiency, voluntary bids. The first is the decision whether information from the monopolist and participation and Bayes-Nash incentive trade takes place and the second is the specifies an output target and a tax/subsidy. compatibility are mutually incompatible. price at which trade occurs if it occurs. Baron and Myerson (1982) derive the In that case what is the second-best rule, The question is whether a Bilateral Trade optimal mechanism which is the mecha the one which violates these requirements Rule exists which satisfies the following nism that maximises expected social wel the “least”? Is there a role for an outside requirements: (i) Bayes-Nash incentive fare subject to incentive compatibility and agency such as the government to set re compatibility, (ii) ex post efficiency, i e, a participation constraint for the monopo serve prices, force sale in some circum recommends trade only if nb ≥ ns, (iii) in list. The mechanism design approach to stances and redistribute appropriately? duces seller and buyer to participate vol regulation thus provides a sound theoreti untarily, and (iv) balancedness, i e, the cal framework for the investigation of a 6 a Brief Guide to the Literature seller receives what the buyer pays. Myer classical problem in economics. There is a huge literature on mechanism son and Satterthwaite (1983) show that design straddling several branches of such a rule does not exist. 5 concluding Remarks economic theory such as social choice Mechanism design theory is an active area theory, public economics, auction the Auction Design: There are N buyers for a of ongoing research. A particularly interest ory, the theory of contracts and so on. single object. Each buyer i has a valuation ing set of questions relate to the design of A brief survey of the issues as well as an ni for the object which is private informa combinatorial auctions. In these models, extensive reading list is provided by the tion. Assume that these valuations are the seller has multiple objects to sell and Nobel Foundation in connection with the random variables distributed independ buyers have valuations for all subsets of 2007 Economics prize (available at http:// ently. There are several well known auc objects. For instance, the government of nobelprize.org/nobel−prizes/economics/ tions such as the first and second price India is currently engaged in selling spec laureates/2007/press.html). sealed bid auctions, dynamic auctions trum licences for FM radio broadcasting Jackson (2001) and Maskin and such as the English and Dutch auctions across the country and there may be Sjöström (2002) survey the complete in and so on. What procedure to sell the ob synergies for buyers in acquiring parti formation implementation literature. ject should the seller choose in order to cular combinations of licenses. Or they Accounts of the mechanism design pro maximise her expected revenue? may be interested in selling off the rights to blem in the voting context can be found Applying the Revelation Principle, it suf modernise various airports. What pro in Moulin (1983) and Barberá (2001). fices to restrict attention to direct mecha cedure should they follow? A fundamental An excellent recent book on auctions nisms where a bid is solicited from each theoretical question which remains open including its mechanism design aspect is buyer. An auction is a rule which specifies is the design of expected revenue maxi Vijay Krishna’s Auction Theory while Green (as a function of the bids received) the mising auctions in this setting. Important and Laffont’s 1979 book Incentives in Public bidder who gets the object (the seller is computational issues also arise in the mul Decision Making surveys the classical issues allowed to retain the object) and the price tiple object model. For instance, it is typi in public good provision quite comprehen various bidders pay. The problem is there cally computationally difficult to deter sively. For “state of the art” accounts on com fore of determining an auction which (i) is mine efficient allocations. Do procedures binatorial auctions and algorithmic mech Bayes Nash incentive compatible, (ii) in exist that are computationally easy but anism design dealing with computational duces voluntary participation from the approximately efficient? issues, readers may refer to Crampton et al bidders, (iii) maximises expected revenue A potential application of mechanism (eds) (2006) and Nisan et al (eds) (2007). subject to (i) and (ii). This is a problem of design theory of topical interest in India, considerable mathematical complexity but is the issue of land acquisition for SEZs. Notes in a classic paper, Myerson (1981) provided Consider a model with N farmers each of 1 An example of such a rule is the “Dodgson Rule” de vised by Rev Charles Dodgson better known as Lewis a comprehensive solution to it. The general whom have exactly one unit of land to sell Caroll, the author of Alice in Wonderland. solution is subtle; however in the special but whose valuations are private informa 2 An important class of problems arises when θ deter mines the set A itself but this leads to complications case where the buyers are ex-ante sym tion.18 There is a buyer who has a positive which I wish to avoid here. 3 More formally, Ri(θ) is a binary relation on the ele metric, i e, the ni’s are identically distrib valuation only for all N units of land, i e, only ments of A satisfying the properties of completeness, uted and its distribution satisfies a certain if all farmers sell. Moreover this valuation is reflexivity and transitivity. 4 A weaker requirement would be ∅≠E(G,θ)⊂F(θ) for regularity condition, the optimal auction also private information for the buyer. What all θ∈Θ. * * 17 5 The message vector (m 1,...m N) is a Nash equilibrium is a second price auction with a reserve is the “best” trading rule in this situation? * * * if g ( m 1 ,...m N ) R i( θ ) g ( m i , m –i) for all mi∈ Mi and all i. ^ price set by the seller. An obviously desirabale property is effi 6 For each θi, the message mi(θi) is a dominant strategy for i if g(m^ (θ ),m ) R (θ )g(m , m ) for all m ∈M and ciency which would require sale to take i i –i i i i –1 i i m–i∈ M–i. Regulation and Auditing: There is a single place only if the buyer’s valuation exceeds 7 Let us assume that such allocations exist for all states. There are of course, well known conditions that ensure firm (a monopolist) in the market for a the sum of the farmers’ valuations. Another this. certain product whose costs are private requirement could be voluntary participation 8 NVP is trivially satisfied in the resource allocation problem provided agents like more of any commodity information. The planner is a regulator which would ensure that farmers are not to less, i e, if Ri(θ) is “increasing”.
12 december 8, 2007 Economic & Political Weekly commentary
9 This is the correspondence which picks the set of all References Krishna, V (2002): Auction Theory, Academic Press. Pareto-efficient allocations in a state. Lange, O (1936): ‘On the Economic Theory of Socialism’, 10 It is possible to implement the SCC which selects d’Aspremont, C and L-A Gerard-Varet (1979): ‘Incentives and Review of Economic Studies, 4. Incomplete Information’, Journal of Public Economics, Walrasian allocations from an equal division of the Lerner, A P (1944): The Economics of Control, MacMillan, 11, 25-45. aggregate endowment. Such allocations are envy- New York. free in the sense of Foley (1967). Barberá, S (2001): ‘An Introduction to Strategy-Proof Maskin, E (1977, 1999): ‘Nash Equilibrium and Welfare Op 11 In fact this is possible only if N≥3 in the complete in Social Choice Functions’, Social Choice and Welfare, 18, 619-53. timality’, paper presented at the summer workshop of formation case. If there are two agents 1 and 2 and the Econometric Society in Paris, June 1977, Published θ Baron, D and R Myerson (1982): ‘Regulating a Monopolist they send the equilibrium messages for states and in 1999 in Review of Economic Studies, 66, 23-38. φ respectively, then it is impossible for the planner to with Unknown Costs’, Econometrica, 50, 911-30. verify whether θ is the true state and 2 is deviating or Crampton, P, Y Shoham and R Steinberg (eds) (2006): Maskin, E and T Sjöström (2002): ‘Implementation whether the true state is φ and 1 is deviating. Combinatorial Auctions, MIT Press. Theory’ in K Arrow, A K Sen and K Suzumura (eds), Handbook of Social Choice and Welfare, Volume 1, 12 A SCF is a singleton-valued SCC. Foley, D (1967): ‘Resource Allocation and the Public Sec Elsevier Science, Amsterdam. 13 There are at least three distinct candidates which are tor’, Yale Economic Essays, 7, 45-98. elected at different voter rankings. Gibbard, A (1973): ‘Manipulation of Voting Schemes: A Moulin, H (1983): The Strategy of Social Choice, Advanced Textbooks in Economics (C J Bliss and M D Intrilliga 14 This is the well known Samuelson-Lindahl rule for General Result’, Econometrica, 41, 587-602. tor (eds)), North-Holland, Amsterdam. Pareto efficiency. Green, J and J-J Laffont (1979): Incentives in Public Deci- 15 William Vickrey recieved the Nobel Prize for econom sion Making, North Holland, Amsterdam. Myerson, R (1981): ‘Optimal Auction Design’, Mathematics ics in 1995. Hayek, F (1945): ‘The Use of Knowledge in Society’, Ameri- of Operations Research, 6, 58-73. can Economic Review, 35, 519-30. Myerson, R and M Satterthwaite (1983): ‘Efficient Mecha 16 Suppose y is the level of a public good and xi is the monetary compensation received the agent i. The Hurwicz, L (1960): ‘Optimality and Informational Ef nisms for Bilateral Trade’, Journal of Economic Theor y, 28, 265-81. utility function U(y,xi) is quasi-linear if it is of the ficiency in Resource Allocation Processes’ in Arrow, form Ui(y,xi)=hi(y)+xi. Karlin and Suppes (eds), Mathematical Methods in the Nisan, N, T Roughgarden, E Tardos and V Vazirani (eds) 17 A second price auction is one where the highest bid Social Sciences, Stanford University Press. (2007): Algorithmic Game Theory, Cambridge Univer der gets the object but pays a price equal to the second – (1972): ‘On Informationally Decentralised Systems’ in sity Press. highest bid. C B McGuire and R Radner (eds), Decentralisation and Satterthwaite, M (1975): ‘Strategyproofness and Arrow’s 18 It is easy to accommodate the case where a farmer Organisation, North Holland, Amsterdam. Conditions: Existence and Correspondence Theorems does not want to sell his land at any price by assuming Jackson, M (2001): ‘A Crash Course in Implementation for Voting Procedures and Social Welfare Functions’, that his valuation in this case is infinite. Theory’, Social Choice and Welfare, 18, 655-708. Journal of Economic Theory, 10, 187-217.
Hiren Gohain ([email protected]) is a well known writer, literary theorist and social critic, who lives in Assam.
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