Accepted Manuscript A general impossibility theorem and its application to individual rights Jianxin Yi, Yong Li PII: S0165-4896(16)30016-6 DOI: http://dx.doi.org/10.1016/j.mathsocsci.2016.03.009 Reference: MATSOC 1856 To appear in: Mathematical Social Sciences Received date: 30 June 2014 Revised date: 7 March 2016 Accepted date: 24 March 2016 Please cite this article as: Yi, J., Li, Y., A general impossibility theorem and its application to individual rights. Mathematical Social Sciences (2016), http://dx.doi.org/10.1016/j.mathsocsci.2016.03.009 This is a PDF file of an unedited manuscript that has been accepted for publication. Asa service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. *Highlights (for review) Research highlights > We prove that it is impossible to find an incentive compatible social choice mechanism with finite social welfare losses > The impossibility also occur even if we replace incentive compatibility with approximately incentive compatibility. > We discuss the compatibility problems between incentive and individual rights. *Manuscript Click here to view linked References A general impossibility theorem and its application to individual rights☆ Jianxin Yia*1, Yong Lib aSchool of Mathematic Sciences, South China Normal University, Guangzhou 510631, China bUQ Business School, The University of Queensland, Brisbane 4072, Australia ______________________________________________________________________________ Abstract In this paper, we generalize Green-Laffont’s (1979) impossibility theorem to the following form: in quasi-linear environments, when the set of each agent’s types is sufficiently rich, we can not find mechanisms that allow bounded deviations from the decisive efficiency, incentive compatibility and budget-balance at the same time. Hence, it is impossible to find an incentive compatible mechanism with minimum social welfare losses. Furthermore, we discuss the compatibility problems between incentive and individual rights in a quasi-linear environment (see Sen, 1970a, 1970b; Deb et al., 1997). Specifically, some new impossibility results are established. JEL Classification: C79; D82; D71. Keywords:Impossibility theorem; Incentive compatibility; Efficiency; Budget-balance; individual rights. _______________________________________________________________________________ 1. Introduction A theme of mechanism design is to find mechanisms compatible with individual incentives that simultaneously satisfy efficient decisions and other requirements, such as balanced transfers and the voluntary participation of the individuals. In a quasi-linear environment where the utility of an agent relies on the results of social choice, his private information and money transfer, the VCG (Vickrey, 1961; Clarke, 1971; Groves, 1973) mechanism can satisfy decisive efficiency together with incentive compatibility. Noticeably, when the set of each agent’s type is sufficiently rich, the VCG mechanism does not meet the budget balance (Green and Laffont, 1979).2 This means, if the environment is closed and there is no external funding inflow, then there is no Pareto optimal and incentive compatible mechanism, because a Pareto optimal mechanism requires decisive efficiency and budget balance simultaneously. The natural questions emanating from Green and Laffont’s impossibility theorem thus are: i) given a Pareto inefficient but incentive compatible social choice mechanism, can we estimate the size of the unavoidable social welfare losses? or ii) in a class of incentive compatible mechanisms, does a mechanism exist that can minimize the social welfare losses? Studies on the questions above mainly follow two lines. One strand of literature insists on budget balance and incentive compatibility conditions to discuss the estimation of the efficiency loss and how to minimize it (see, Serizawa, 1996; Moulin, 1999; ☆ We are very grateful to the referees for their comments which helped us to improve the paper considerably. * Corresponding author. E-mail address: [email protected] (J. Yi); [email protected] (Y. Li) 1 This study is supported by NNSF of China (No. 11371155). 2 For other related results, see, Walker (1980), Hurwicz and Walker (1990), Beviá and Corchón (1995). 1 Moulin and Shenker, 2001; Olszewski, 2004; and Juarez, 2008). The other strand of literature, under the requirements of decisive efficiency and incentive compatibility, attempts to estimate and minimize the budget imbalance (see, Deb and Seo, 1998; Danilov and Sotskov 2002; Cavallo, 2006; Guo and Conitzer, 2009; Mehta et al, 2009; Moulin, 2009, 2010; Yengin, 2012; You, 2015). In this paper, we aim to discuss the questions above in a more general framework. Different from the positive findings in previous studies, our answer is negative. Firstly, we propose a general impossibility theorem. That is, if the set of possible types for each agent is sufficiently rich, there is no social choice mechanism satisfying the following three conditions at the same time: i) approximative decisive efficiency, ii) approximative incentive compatibility and iii) budget boundedness. Our impossibility theorem significantly generalizes the classical impossibility theorems proposed by Green and Laffont (1979). Hence, if there is no source of outside funding for the agents, then there is no social choice mechanism satisfying both incentive compatibility and approximative Pareto optimality if the set of each agent’s types is sufficiently rich. It implies that, for an incentive compatible social choice mechanism, its unavoidable social welfare (absolute) losses are infinite. Specifically, a budget-balanced and incentive-compatible mechanism must have infinite efficiency loss and a decisive-efficient and incentive-compatible mechanism must have infinite budget waste.3 Thus, among the incentive compatible mechanisms, it is meaningless to look for a social choice mechanism with minimum social welfare losses. Another strand of literature related with this paper is on the auctions with financial constrains (see, Che and Gale 1998, Laffont and Robert, 1996, Maskin, 2000). An auctioneer wishes to sell several identical or heterogeneous indivisible items to agroup of potential bidders. Each bidder has private value and budget constraint. Therefore they can not pay more than their “budget” regardless of their valuation. Dobzinski, et al.(2012), Lavi and May (2012) show that any deterministic auction can not simultaneously satisfy individual-rationality, incentive-compatibility, Pareto-efficiency, and no-positive-transfers. It is straight forward that an auction with financial constrains is a mechanism with bounded budget. Moreover, our impossibility theorem is more general because it claims that the impossibility can also occur even if the incentive compatibility is weakened to allow a certain degree of manipulation, that is, approximative incentive compatibility. Approximative incentive compatibility has attracted more and more attention from the early work of Roberts and Postlewaite (1976) to Archer, et al. (2003), Schummer (2004), Kothari et al. (2005), Zou et al. (2010), Birrell and Pass (2011), Carroll (2011), and Lubin and Parkes (2012). From the view of limited rationality, the concept of approximative incentive compatibility performs better in approaching reality (Radner, 1980). In the end, it must be pointed that, by replacing incentive compatibility with Bayesian incentive compatibility, the expected externality mechanism can ensure budget balance, decisive efficiency and Bayesian incentive compatibility (d’Aspremont and Gérard-Varet, 1979). However, Myerson and Satterthwaite (1983) demonstrate that a mechanism with decisive efficiency, Bayesian incentive compatibility and budget balance must violate individual rationality constraints.4 Thus, similarly, an interesting question is whether a mechanism exists that allows bounded deviations from the Bayesian incentive compatibility, efficiency, budget balance and individual rationality constraints.5 This question is beyond the 3 This does not exclude the research in some special cases and the studies on the mechanisms with relatively minimum social loss. 4 Schweizer (2006) explores the scope of universal possibility and impossibility theorems. 5 Williams (1999) discusses the relation among efficiency, Bayesian incentive compatibility and the VCG 2 scope of this paper. Second, we use our impossibility theorem (Theorem 1) to give some new impossibility theorems on individual rights and incentive compatibility. Ever since Sen's (1970b) seminal work, the subject of individual libertarian rights has attracted much attention from economists (see Sen, 1970a, 1970b; Deb et al., 1997). The main focus of Mechanism design is whether one can design an incentive compatible mechanism, which can satisfy other requirements, such as efficiency, egalitarianism, individual rationality and so on. However, to the best of our knowledge, it is rare to see literature on the relation between incentive compatibility and individual rights. We argue, it is not a trivial ignorance because we find if a mechanism satisfies a kind of individual rights but is not incentive compatible, then an agent’s strategic misrepresentation can in fact affect