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
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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.
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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 this kind of rights of his or others’ (see Examples 1 and 2 in Section 4). It follows from the Gibbard-Satterthwaite (Gibbard, 1973; Satterthwaite, 1975) theorem that there is conflict between individual libertarian rights and incentive compatibility. In this paper, we move forward to discuss the compatibility between incentive and individual libertarian rights and propose some new impossibility theorems. This paper is organized as follows: In Section 2, we present the social choice model. Section 3 exhibits our general impossibility results. Section 4 establishes some new possibility or impossibility results about incentive compatibility and individual rights and the Appendix contains proofs omitted from the text.
2. The model A set of agents is denoted by . A set of potential alternatives is denoted by set , which can be finite or infinite. For our purposes, we assume that .6 Every agent has private information, called his type. Let denote the type of agent , from a set of possible types . We denote a profile of types as and the product of type sets of all agents as . Let denote the utility of agent for the outcome given the type and the set of functions from to . For convenience, given , and , denotes the profile in which agent i has type if and if . Further, when , we shall write rather than . A decision rule is a function : , which a social planner or policy maker uses to assign a collective choice to each possible profile of the agents’ types . A decision rule is (decisive) efficient if for all and all . is dictatorial if a exists such that for all and all . 7 A transfer or payment is a vector function : . The function represents the payment that an agent receives (or makes if it is negative) based on the announcement of types . A transfer function is budget-balanced if for all .
mechanism. 6 represents the cardinal number of , i.e., the number of elements in the set . 7 We do not require , expecting that our model is suitable for allocating a set of indivisible objects and a fixed amount of a divisible good (e.g., money) to a set of agents ( see Svensson, 1983; Tadenuma and Thomson, 1991; Andersson et al., 2014).
3 A social choice function or mechanism consists of a decision rule and a transfer function . We assume that the agent i's preferences depend on the outcome , the type
, and the transfer payment in a quasi-linear manner: . A social choice function is incentive-compatible if, for all and all :
for all . We say is truthfully implementable if there exists a transfer such that is incentive compatible.
3. A general impossibility theorem
In a quasi-linear environment, for any efficient decision rule, a transfer exists,
, such that is incentive-compatible (Vickrey, 1961; Clarke,
1971; Groves, 1973). Under some appropriate conditions, we can require that , that is, there is no external funding inflow.8 This is the well-known VCG mechanism. Moreover, under certain mildly rich assumptions on the domain, the VCG mechanism is the only one that have these properties (see, Green and Laffont, 1977, 1979; Holmström, 1979; Suijs, 1996). However, Green and Laffont (1979) show that if the set of possible types for each agent is sufficiently rich then decisive efficiency, incentive compatibility and budget-balance cannot be achieved together. The precise statement is given in the form of the following theorem.
Green-Laffont Impossibility Theorem: Suppose that for each agent ,
; that is, every possible valuation function from to arises for some . Then there is no social choice function that is efficient, incentive compatible and budget balanced.
To propose our impossibility theorems, we introduce the following concepts, which aim to weaken the requirements for efficiency, incentive compatibility and budget-balance respectively. 9 Let and . We say that is -efficient if, for all , for all .
If is -efficient, then . This implies the maximum efficiency loss does not exceed . Let . We say that is -incentive compatible, if for all , all and
, .
-incentive compatibility means that every agent’s potential gain from misrepresentation is not larger than . We say is approximative incentive compatible if is -incentive compatible for some .
Let . We say that transfer is -bounded if for all . A transfer is budget bounded if it is -bounded for some . Obviously, is budget-balanced if and only if it is -bounded. Now we can state our main results as follows.
10 Theorem 1. Suppose that and for each agent , .
8 For example, is finite. 9 Here means for all and for some .
4 Then there is no social choice function that satisfies the following conditions: (1) is -efficient for some and some ;11 (2) is approximative incentive compatible; (3) is budget-bounded. Proof. See appendix.
By Theorem 1, is not budget-bounded if is efficient and incentive compatible. That is, we must have . Hence, it is meaningless to discuss minimizing
. Similarly, if is incentive compatible and balanced, then the efficient loss
.
According to Theorem 1, we know that, for any , a VCG mechanism does not exist to satisfy . Where and , or for simplicity, . Next, we move on to discuss a closed system in which the n agents have no outside source of financing. A transfer is therefore a function from to that assigns a element for every , where
. The outcome set of social choice mechanism is therefore rather than . An outcome
is -Pareto optimal (or -Pareto efficient) at if there is no outcome such that for every and for at least one . A mechanism satisfies -Pareto optimality if is -Pareto optimal at for all .
It is well known that is Pareto optimal if and only if and maximizes over . Hence, Green and Laffont’s impossibility theorem implies that there is no mechanism satisfying Pareto optimality and incentive compatibility. Applying Theorem 1, we have a stronger result as follows.
Corollary 1. Suppose that and for each agent , . For any , there is no mechanism satisfying -Pareto optimality and incentive compatibility.
Proof. We only need to prove if is -Pareto optimal, then there must
be and for all .
Suppose is -Pareto optimal. First, if , let , where . Then, for every ,
A contradiction. Second, if a exists such that
Let and
Then,
10 When , Kos and Messner (2013) characterize the boundaries of the set of transfers (extremal transfers) implementing a given allocation rule. 11 Here if and only if for all .
5
for all . A contradiction. □
4. Individual rights and incentive compatibility In this section, we investigate the problem of inconsistency between individual rights and incentive compatibility. We suppose a transfer is allowed. To remain as simple as possible, we suppose that is finite. A decision rule is said to be an affine maximizer rule (or affine-efficient) if there exist a vector of weights and a function such that for all ,
for all .
Here, we need to apply the classic Roberts’s theorem as follows. 12 Roberts’s (1979) Theorem. Suppose and for each agent . For any onto decision rule , if transfer exists such that is incentive compatible, then is an affine maximizer rule.13
Conversely, if is an affine maximizer rule with weights ， then generalized Groves transfers exist: for agent , for every ,
such that is incentive compatible and where is an arbitrary function of ,
Furthermore, it is easy to check that any affine maximizer rule must be -efficient for some . To see this, suppose that is a affine maximizer rule with respect to
and a , that is:
for all .
Let . Then is -efficient. Therefore, if is incentive compatible and , it follows that is -efficient with some weight vector
and some by applying Roberts’s theorem above.
4.1 Sen’s liberalism Amartya Sen (1970a, 1970b) is the first author to model the exercise of individual rights within the social choice context and point out the conflict between individual rights and weak Pareto principle. According to Deb, et al. (1997), agent is decisive for if, for any , whenever and whenever . satisfies Sen’s liberalism if for every i there is at least one pair , such that is decisive for .14
12 Carbajal et al., (2013) provide a generalization of Roberts’s theorem. 13 is onto decision rule if . 14 In his original presentation of the ‘liberal paradox’, Sen (1970a,b) used the concept of a social decision function whose range is restricted to social preference relations that generate a choice function.
6 Sen’s liberalism is a stronger requirement than non-dictatorship. Thus, it follows from the Gibbard-Satterthwaite’s theorem that there is the conflict between Sen’s liberalism and incentive compatibility. An interesting question is whether the conflict between them is eliminated if compensatory transfers are paid. To this question, we give a negative answer.
Theorem 2. Suppose , and for each agent . There is no social choice function that satisfies the following conditions: (1) ; (2) is incentive compatible; (3) satisfies Sen’s liberalism.
Proof. If exists that satisfies (1) and (2), then there exist and , such that is -efficient. We show that it is in conflict with Sen’s liberalism. To see this, let
be an individuals such that . For any , let be a profile such that
and for all .
Then, we have
for all . Since is -efficient, we obtain . Note , which is contrary to the condition (3): satisfies Sen’s liberalism. □
In fact, Theorem 2 is also valid if we replace Sen’s liberalism with minimal liberalism. Minimal liberalism requires the decisiveness over at least one pair of outcomes be given not to all individuals but to at least two individuals. As a comparison, Sen’s liberalism is a stronger condition. Theorem 2 turns to a possibility theorem if we weaken the condition of Sen’s liberalism to the concept of relative freedom as follows.
Let and . We say enjoys relative freedom if there exist , such that for any , we have both for some and for some . satisfies relative liberalism if every enjoys relative freedom.
Proposition 1. Suppose for each agent . If a decision rule is -efficient, then enjoys relative freedom if and only if has positive weight (i.e.,
.
Proof. Suppose has positive weight. Let be distinct alternatives. For any , choose
such that and , where is a negative number satisfying . Then, we have
.
Since is -efficient, it follows that . Similarly, we can prove that
for some . Hence, enjoys relative freedom. Conversely, if enjoys relative freedom, we can show that . Suppose to the contrary that . For any , let be chosen such that
7
for all ,
where . Then, for any , we have
for all .
Since is -efficient, we get . A contradiction. □
From Proposition 1, we know that VCG mechanism is incentive compatible and satisfies relative liberalism. However, we have a negative result below.
Theorem 3. Suppose , and for each agent . There is no social choice function that satisfies the following conditions: (1) ; (2) is incentive compatible and budget bounded; (2) satisfies relative liberalism.
Proof. If exists that satisfies (1) and (2), then there exist and , such that is -efficient. From Proposition 1, for all . This contradicts Theorem 1. □
4.2 Condition
The second concept on rights we consider is Condition (see, Deb, et al, 1997).
Given the decision rule , has a say for if there are such that
. has a say if has a say for some . satisfies Condition if each individual has a say (Deb, et al, 1997).
is a condition much weaker than liberalism and does not even exclude dictatorial decision rules. If satisfies Condition , it is possible to use balanced transfer to provide individual incentives for it. For example, consider a situation where there are three agents and is -efficient decision rule, that is,
for all and all .
To ensure that individual 3 has a say, we require that for all and all . Moreover, for every , we set the transfers of agents to be
, and .
Then satisfies Condition and is incentive compatible and budget balanced. A strong version of is that every agent i has a say for any . By analogy with the proof of Proposition 1, we have the following results:
Proposition 2.Suppose for each agent . If a decision rule is -efficient, then agent has positive weight if and only if has a say for any .
From Proposition 2, it is easy to see that the VCG mechanism satisfies incentive compatibility and a strong version of , that is, every individual has a say for any . However, such social choice functions must violate the requirement of bounded budget.
Theorem 4. Suppose , and for each agent .
8 There is no social choice function that satisfies the following conditions: (1) ; (2) is incentive compatible and budget bounded;
(3) Every agent has a say for any . Proof.This is an immediate consequence of Theorem 1 and Proposition 2. □
Note, in general, the right that has a say for any is not equivalent to that enjoys relative freedom if is onto and .
4.3 Condition The third rights concept we consider is also from Deb, et al, (1997).
Given the , agent can veto if a exists such that for any . satisfies Condition if every agent can veto some alternative.
Theorem 5. Suppose , and for each agent . There is no social choice function that satisfies the following conditions: (1) ; (2) is incentive compatible;
(3) satisfies Condition . Proof. It can be proved by analogy with the proof of Theorem 2.
Condition is a stronger requirement that causes the negative result above. We consider a condition but weaker than Condition . Agent has veto power if there is a such that for any , for some . satisfies veto power if every agent has veto power.
Obviously, that has a say for any implies that has veto power. Moreover, when the set of possible types for agent is sufficiently rich, that enjoys relative freedom implies that has veto power as well. However, the converse of the statements above are not correct.
Proposition 3. Suppose for each agent . If a decision rule is -efficient, then has veto power if and only if individual has positive weight. Proof. It can be proved by analogy with the proof of Proposition 1.
From Proposition 3, we know immediately that the VCG mechanism satisfies incentive compatibility and veto power. However, as a direct corollary of Theorem 1 and Proposition 3, we have the following negative result:
Theorem 6. Suppose , and for each agent . There is no social choice function that satisfies the following conditions: (1) ; (2) is incentive compatible and budget bounded; (3) satisfies veto power.
Using Theorem 6, we can generalize Gibbard-Satterthwaite’s Theorem to the quasi-linear environments. Let be a set of those agents who have veto power. is said to be internally
9 balanced if for all .
Corollary 2. Suppose , and . If a transfer exists such that is incentive compatible and internally balanced, then is dictatorial.15 Proof. Suppose is incentive compatible. Let be the set of those agents who have veto power. From Proposition 3 and Theorem 6, we obtain . Let . Then incentive compatibility and internal balance imply that. for all and .
Note . Hence the inequality above implies is dictatorial. □
In Corollary 2, if for all , we revert to Gibbard-Satterthwaite’s theorem (Gibbard, 1973; Satterthwaite, 1975).
Last, we illustrate two examples to end this section. These two examples show that a social choice function satisfying some individual right may lose these characteristics in practice due to strategic behaviors of selfish agents.
Example 1. Suppose a set of outcomes , a set of agents and sets of types of agents 1 and 2 are , , respectively. Define : and
: as follows:
1, , ,
, ,
Define the social choice function by , ,
, .
As shown in the examples above, it is easy to see 1 has a say for and 2 has a say for .
Thus, satisfies Condition . However, is not incentive compatible, because whether or not agent 1’s true type is or , his dominant strategy is always to select . Hence, 1 always selects no matter what others select. Accordingly, the set of agent 1’s types, in fact, reduces to
and the social choice function changes from to , the restriction of to 16 . While does not satisfy Condition . Since the agent 2 has no say for . This means agent 2 in fact has lost the rights of having a say.
Example 2. Suppose a set of outcomes , a set of agents and sets of types of agent 1 and 2 are , , respectively. Define : and
: as follows:
1, , ,
, ,
Define the social choice function by , ,
, .
It is easy to see satisfies Condition but does not satisfy incentive compatibility.
15 Similar results can be established following Theorem 3 and Theorem 4. 16 In general, given function and , the restriction of to is the function defined by for .
10
Whether or not agent 1’s true type is or , agent 1’s optimal strategy is always selecting no matter what others do. This makes the set of agent 1’s types actually degenerates to and the social choice function changes from to , the restriction of to . Note that agent cannot veto but can veto by choosing only. Thus does not satisfy . Hence, in seeking to maximize his own payoff, rational agent 1 actually relinquishes his veto rights.
Appendix
We first prove the following two preliminary lemmas.
Given : and , let for any
.
Lemma 1. Suppose that for each agent , . For any and , if is -efficient, then has no upper bound in .
Proof. To prove Lemma 1, we only need to prove that: for any positive number , there
are such that
For this purpose, given an and a number , such that
. From the assumptions of Lemma 1, there are such that for each ,
and . (A1)
Then we prove are what we need.
First, we prove that for any , if and . If , from (A1), we have
But for any
Since , it follows from -efficiency that . If , by -efficiency, it is easy to check that . Next, by simple calculations, we know
for all and
11
Hence, we finally have
□
Lemma 2. Suppose that for each agent , . For any
and any , if satisfies the following three conditions: (1) is -efficient; (2) is -incentive compatible; (3) is -budget bounded; then for all . Hence
has an upper bound in . Proof. To facilitate comprehension, we organize the proof into a number of small steps.
Step 1. If , then . Since is -incentive compatible, we have
and .
The two inequalities above imply .
Step 2: Let . Then for each , we have
17 for any and any . (A2)
To prove (A2) we consider two cases as follows.
Case 1. : then
.
Case 2. : Let be any positive number. By assumption, we can select , such that
. (A3)
We assert that . Otherwise, . According to -efficiency and (A3), we have
17 In fact, (1) and (2) suffice to guarantee the existence of this inequality.
12
.
This is a contradiction. Hence .
From step 1, we have . Therefore, by definition of and (A3), we have
(A4)
Note and is -incentive compatible, we have
. (A5)
Equations (A4) together with (A5), gives the inequality .
Let , then we have .
By the same method, . Therefore we finish our proof for
.
Step 3: for all .
For every given and any , it is easy to see that
From step 2, we have the following inequality
Note for all and . We have
for all .
Replacing with , we have
13
Summing on both sides for , we obtain
Finally, from (A6), we have
as required. □
The proof of Theorem 1: If exists, three conditions are satisfied in Theorem 1, that is, (1) is -efficient for some and some ; (2) is approximative incentive compatible; and (3) is budget-bounded. Then from Lemma 1, has no upper bound in . On the contrary, from Lemma 2, has an upper bound in . This is a contradiction.
14 References
Andersson, T., Ehlers, L., and Svensson, L.-G., 2014. Least manipulable Envy-free rules in economies with indivisibilities. Mathematical Social Sciences. 69, 43-49. Archer, A., Papadimitriou, C., Talwar, K. and Tardos, E., 2003. An approximate truthful mechanism for combinatorial auctions with single parameter agents. In Proceedings of the 14th ACM-SIAM Symposium on Discrete Algorithms (SODA 2003), Baltimore, Maryland, USA, 12–14 January. pp. 205–214. Beviá, C., and Corchón, L., 1995. On the generic impossibility of truthful behavior: a simple approach. Economic Theory. 6, 365–371. Birrell, E. and Pass, R., 2011. Approximately strategy-proof voting. In Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI 2011), Barcelona, Spain, pp. 67–72. Carbajal, C. C., McLennan, A., and Tourky R., 2013. Truthful Implementation and Preference Aggregation in Restricted Domains. Journal of Economic Theory. 148, 1074-1101. Carroll, G., 2011. The efficiency-incentive tradeoff in double auction environments. Working Paper. Cavallo, R., 2006. Optimal decision-making with minimal waste: strategy proof redistribution of VCG payments. International conference on autonomous agents and multi-agents systems (AAMAS) Hakodate, Japan. Che Y., and Gale I., 1998. Standard auctions with financially constrained bidders, The Review of Economic Studies. 65, 1–21. Clarke, E., 1971. Multipart pricing of public goods. Public Choice. 8, 19-33. Danilov, V. I., and Sotskov, A. I., 2002. Social Choice Mechanisms. Springer-Verlag, Berlin. d’Aspremont, C., and Gérard-Varet, L. A., 1979. Incentives and incomplete information. Journal Public Economics. 11, 25–45. Deb, R., Pattanaik, P. K., and Razzolini, L., 1997. Game forms, rights, and the efficiency of social outcomes. Journal of Economic Theory. 72, 74-95. Deb, R., and Seo, T. K., 1998. Maximal surplus from the pivotal mechanism: a closed form solution. Review of Economic Design. 3, 347–357. Dobzinski, S., Lavi, R., and Nisan, N., 2012. Multi-unit auctions with budget limits. Games and Economic Behavior. 74, 486–503. Gibbard, A., 1973. Manipulation of voting schemes. Econometrica. 41, 587-601. Green, J. R., and Laffont, J. J., 1977. Characterization of satisfactory mechanism for the revelation of preferences for public goods. Econometrica. 45, 427-438. Green, J. R., and Laffont, J. J., 1979. Incentives in Public Decision Making. Amsterdam, North-Holland. Groves, T., 1973. Incentives in teams. Econometrica. 41, 617-631. Guo, M., and Conitzer, V., 2009. Worst case optimal redistribution of VCG payments in multi-unit auctions. Games and Economic Behavior. 67, 69–98. Holmstrom, B., 1979. Groves scheme on restricted domain. Econometrica. 47, 1137-1144. Hurwicz, L., and Walker, M., 1990. On the generic nonoptimality of dominant-strategy allocation mechanisms: a general theorem that includes pure exchange economies. Econometrica. 58, 683-704.
15 Juarez, R., 2008. The worst absolute loss in the problem of the commons: random priority versus average cost. Economic Theory. 34, 69-84. Kos, N. and Messner, M., 2013. Extremal Incentive Compatible Mechanisms. Journal of Economic Theory. 148, 134-164. Kothari, A., Parkes, D. C., and Suri, S., 2005. Approximately strategyproof and tractable multi-unit auctions. Decision Support Syst.(Spec. Issue). 39, 105–121. Laffont J., Robert J., 1996. Optimal auction with financially constrained buyers, Economics Letters. 52, 181–186. Lavi R., and May M., 2012. A note on the incompatibility of strategy-proofness and Pareto-optimality in quasi-linear settings with public budgets. Economics Letters. 115, 100–103. Lubin, B., and Parkes, D. C., 2012. Approximate strategyproofness. Current Science (Special section: Game Theory). 103, 1021-1032. Maskin E., 2000. Auctions, development, and privatization: efficient auctions with liquidity-constrained buyers, European Economic Review. 44, 667–681. Mehta, A., Roughgarden, T. and Sundararajan, M., 2009. Beyond Moulin mechanisms. Games and Economic Behavior. 67, 125-155. Moulin, H.,1999. Incremental cost sharing: Characterization by coalition strategyproofness. Social Choice and Welfare. 16, 279-320. Moulin, H., 2009. Almost budget-balanced VCG mechanisms to assign multiple objects. Journal of Economic Theory. 144, 96–119. Moulin, H., 2010. An efficient and almost budget balanced cost sharing method. Games and Economic Behavior. 70, 107-131. Moulin, H., and Shenker, S., 2001. Strategyproof sharing of submodular costs: Budget balance versus efficiency. Economic Theory. 18, 511–533. Myerson, R. B., and Satterthwaite, M. A., 1983. Efficient mechanisms for bilateral trading. Journal of Economic Theory. 28, 265-281. Olszewski, W., 2004. Coalition strategy-proof mechanisms for provision of excludable public goods. Games and Economic Behavior. 46, 88-114. Radner, R., 1980. Collusive behavior in noncooperative epsilon-equilibria of oligopolies with long but finite lives. Journal of Economic Theory. 22, 136-154. Roberts, D. J., and Postlewaite, A.,1976. The incentives for price-taking behavior in large economies. Econometrica. 44, 115-128. Roberts, K., 1979. The characterization of implementable choice rules. In: Laffont, J.-J. (Ed.), Aggregation and Revelation of Preferences. North-Holland, Amsterdam, pp. 321-348. Satterthwaite, M. A., 1975. Strategy-proofness and Arrow’s conditions: Existence and correspondence theorem for voting procedure and social welfare functions. Journal of Economic Theory. 10, 187-217. Schweizer, U., 2006. Universal possibility and impossibility results, Games and Economic Behavior. 57, 73–85. Schummer, J., 2004. Almost-dominant strategy implementation: Exchange economies. Games and Economic Behavior. 48, 321-336. Sen, A. K., 1970a. Collective Choice and Social Welfare. Holden-Day, San Francisco. Sen, A. K., 1970b. The impossibility of a Paretian liberal. The Journal of Political Economy. 72,
16 152-157. Serizawa, S., 1996. Strategy-proof and individually rational social choice functions for public good economies. Economic Theory. 7, 501-512. Suijs, J., 1996. On incentive compatibility and budget balancedness in public decision making. Economic design. 2,193-209. Svensson, L.-G., 1983. Large indivisibles: an analysis with respect to price equilibrium and fairness. Econometrica. 51, 939-954 Tadenuma, K. and Thomson, W., 1991. No-envy and consistency in economies with indivisible goods. Econometrica. 59, 1755-1767. Vickrey, W., 1961. Counterspeculation, auctions and competitive sealed tenders. Journal of Finance. 16, 8-37. Walker, M., 1980. On the nonexistence of a dominant strategy mechanism for making optimal public decisions. Econometrica. 48, 1521-1540. Williams, S., 1999. A characterization of efficient, Bayesian incentive compatible mechanisms. Economic Theory. 14, 155-180. Yengin, D., 2012. Egalitarian-equivalent Groves mechanisms in the allocation of heterogenous objects. Social Choice and Welfare. 38, 137–160. You J. S., 2015. Optimal VCG mechanisms to assign multiple bads. Games and Economic Behavior. 92, 166–190. Zou, J., Gujar, S., and Parkes, D. C., 2010. Tolerable manipulability in dynamic assignment without money. In Proceedings of the 24th AAAI Conference on Artificial Intelligence (AAAI 2010), Atlanta, Georgia, USA, pp. 947–952.
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