An Efficient Securely Implementable Allocation Rule in Linear Production Economies

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(1)61. An Efficient Securely Implementable Allocation Rule in Linear Production Economies＊ NISHIZAKI Katsuhiko†‡ Abstract In linear production economies with classical preferences, Nishizaki (2018b) showed that the combination of strategy-proofness and non-bossiness (Satterthwaite and Sonnenschein, 1981) is equivalent to secure implementability (Saijo, , and Yamato, 2007) under Pareto-efficient rules. In fact, the equal budget free choice rule (Maniquet and Sprumont, 1999), that is a strategy-proof, non-bossy, and Paretoefficient rule in the environments, is securely implementable, as shown in this paper. JEL Classification : C72, D51, D52, D61, D71.. 1. Introduction. This paper studies securely implementable rules in linear production economies with classical preferences. Securely implementability (Saijo, , and Yamato, 2007) is defined as double implementability in dominant strategy equilibria and Nash equilibria1) . This requirement is equivalent to the combination of strategy-proofness and the rectangular property (Saijo, and Yamato, 2007). Strategy-proofness requires that the truthful revelation is a weakly dominant strategy for the agent under the rule. The rectangular property requires that if each agent ＊This paper is a product of research that was financially supported by designated research projects of the Research Institute, Momoyama Gakuin University, (2016) “Secure Implementability of Uniform Mechanism” and (2017) “Preliminary Study on New Developments in Mechanism Design Theory based on the Interdependence among Decision-Makings” and the Research Institute for Socionetwork Strategies, Kansai University, (20152017) “Implementation Theory and Rationality : Secure Implementation Reconsidered,” supported by Matching Fund Subsidy from MEXT (Ministry of Education, Culture, Sports, Science and Technology). Any errors in this paper are entirely the responsibility of the author. †Graduate School of Economics, Momoyama Gakuin University, 1 1 Manabino, Izumi, Osaka 5941198, Japan. TEL : +81 725 54 3131 (main phone number). FAX : +81 725 54 3202. E-mail : ka-nishi @andrew.ac.jp ‡Research Institute for Socionetwork Strategies, Kansai University, 3 3 35, Yamate, Suita, Osaka, 564 8680, Japan. TEL : +81 6 6368 1228. FAX : +81 6 6330 3304. 1) See Saijo, , and Yamato (2007) for a formal definition of secure implementability. Keywords : Secure Implementation, Dominant Strategy Implementation, Nash Implementation, StrategyProofness, Linear Production Economy..

(2) 62. 桃山学院大学総合研究所紀要. 第45巻第１号. does not gain and lose by changing the agent’s revelation, then the allocation does not change by the revelations of all the agents under the rule. Although secure implementability is practically appealing in preventing strategic manipulations due to Cason, Saijo, , and Yamato (2006), previous literature illustrated the difficulty of finding securely implementable rules with desirable properties2). Contrary to such illustration, in linear production economies with classical preferences, this paper shows that the equal budget free choice rule (Maniquet and Sprumont, 1999), that satisfies strategy-proofness, Paretoefficiency, and equal treatment of equals, is securely implementable3). Because the equal budget free choice rule satisfies non-bossiness (Satterthwaite and Sonnenschein, 1981) in addition to strategy-proofness and Pareto-efficiency, this result supports Nishizaki’s (2018b) result showing that the combination of strategy-proofness and non-bossiness is equivalent to secure implementability under Pareto-efficient rules in linear production economies with classical preferences4). The remainder of this paper is organized as follows. Section 2 introduces the model presented here and Section 3 the properties of rules related to secure implementability. Section 4 demonstrates the result of this paper.. 2. Model. Similar to Maniquet and Sprumont (1999) and Nishizaki (2018b), this paper considers linear production economies with agents and divisible and private goods. Let be the set of goods. For each and each be the set of agents and , let be consumption of good for agent and . be consump tion for agent . Let . be an allocation. In the model presented here, a good can. be transformed into another good by a technology that exhibits constant return to scale. For simplicity, let

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(4) be the set of feasible allocations. A preference of an agent is represented by a binary relation defined on . For each , let. be a preference for agent and ’ s indifference preference induced by . This be the agent paper assumes that each preference is classical, that is, continuous, strictly monotonic, and strictly convex. For each , let be the set of such preferences for agent . Let . be a profile of preferences and be the set of profiles of preferences. For each , let 2) See Saijo, , and Yamato (2003, 2007), Mizukami and Wakayama (2005, 2017), Fujinaka and Wakayama (2008, 2011), Berga and Moreno (2009), Bochet and Sakai (2010), Nishizaki (2012, 2013, 2014, 2018a, 2018b), and Kumar (2013) for theoretical results on secure implementability. 3) In convex production economies with classical preferences, there are also possibilities of secure implementability with desirable properties due to Saijo, , and Yamato (2007) and Kumar (2013). 4) The proof of the result presented here shows that equal treatment of equals is not necessary to imply secure implementability in the environments..

(5) An Efficient Securely Implementable Allocation Rule in Linear Production Economies. 63. and be the set of be a profile of preferences other than agent profiles of preferences other than agent . In addition, for each , , let be a profile of preferences other than agents and . Agents collectively choose a feasible allocation according to a rule. Let ： be a rule that associates a feasible allocation with a profile of preferences 5). For each and each , let be the consumption for agent at the allocation . . 3. Properties of Rules. Saijo, , and Yamato (2007, Theorem 1) characterized securely implementable rules by strategy-proofness and the rectangular property (Saijo, , and Yamato, 2007). Strategy-proofness requires that the truthful revelation is a weakly dominant strategy for the agent under the rule. The rectangular property requires that if each agent does not gain and lose by changing the agent’s revelation, then the allocation does not change by the revelations of all the agents under the rule.. Definition 1. The rule satisfies strategy-proofness if and only if for each

(6) and each

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(8) ,

(9) . . Definition 2. The rule satisfies the rectangular property if and only if for each

(10) , if

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(13) for each , then

(14) . . 4. Result. Maniquet and Sprumont (1999) introduced the equal budget free choice rule in linear production economies. Let

(15) be the equal budget set. For each and each , let

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(17) for each

(18) be the set of maximizers of in the equal budget set . Note that the set of maximizers of each preference in the equal budget set is singleton because preferences are classical.. Definition 3. The rule is the equal budget free choice rule if and only if for each and each ,

(19) . . In the model presented here, Maniquet and Sprumont (1999, Theorem 2) characterized the equal budget free choice rule by strategy-proofness that is necessary for secure implementability, 5) In this paper, a rule is defied as a direct revelation mechanism associated with a social choice function. This means that a rule is equivalent to a social choice function..

(20) 64. 桃山学院大学総合研究所紀要. 第45巻第１号. Pareto-efficiency, and equal treatment of equals6). The following proposition shows that the equal budget free choice rule also satisfies the rectangular property that is necessary for secure implementability.. Proposition. The .

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(22) f satisfies the

(23) .

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(26). . Proof. Let be such that for each . Because satisfies . strategy-proofness and each agent’s most preferred consumption in the agent’s budget set is unique, this implies that for each . . Together with the uniqueness of each agent’s most preferred consumption in the agent’s budget set, this implies that for each . . By (1), we find that . By the uniqueness of each agent’s most preferred . . . consumption in the agent’s budget set, we also find that . . for each . . . These imply that . . By (2) and the uniqueness of each agent’s most preferred consumption in the agent’s budget set, we find that . By the uniqueness of each agent’s most preferred consumption in the agent’s budget set, we also find that . . and . . for . each . . . These imply that . . By (3) and (4), we find that . By sequentially replacing by . . . . for each . . in the above manner, we find that .. □. Note that the above proof shows that “unequal” budget free choice rules, that do not satisfy equal treatment of equals, also satisfy the rectangular property. Together with characterizations of securely implementable rules and the equal budget free choice rule, the above proposition implies the following corollary.. Corollary. The . .

(27) .

(28) is .

(29) . . 6) The rule satisfies Pareto-efficiency if and only if for each and each , if for each , then for each . The rule satisfies equal treatment of equals if and only if for each and each , if , then . .

(30) An Efficient Securely Implementable Allocation Rule in Linear Production Economies. 65. In linear production economies with classical preferences, Nishizaki (2018b) showed that the combination of strategy-proofness and non-bossiness (Satterthwaite and Sonnenschein, 1981) is equivalent to secure implementability under Pareto-efficient rules7) . Because the equal budget free choice rule satisfies non-bossiness in addition to strategy-proofness and Pareto-efficiency, the above corollary supports Nishizaki’s (2018b) result8).. References Berga, D. and B. Moreno (2009) “Strategic Requirements with Indifference : Single-Peaked versus SinglePlateaued Preferences,” Social Choice and Welfare 32, pp. 275 298. Bochet, O. and T. Sakai (2010) “Secure Implementation in Allotment Economies,” Games and Economic Behavior 68, pp. 35 49. Cason, T., T. Saijo, T. , and T. Yamato (2006) “Secure Implementation Experiments : Do StrategyProof Mechanisms Really Work?” Games and Economic Behavior 57, pp. 206235. Fujinaka, Y. and T. Wakayama (2008) “Secure Implementation in Economies with Indivisible Objects and Money,” Economics Letters 100, pp. 91 95. Fujinaka, Y. and T. Wakayama (2011) “Secure Implementation in Shapley-Scarf Housing Markets,” Economic Theory 48, pp. 147 169. Kumar, R. (2013) “Secure Implementation in Production Economies,” Mathematical Social Sciences 66, pp. 372 378. Maniquet, F. and Y. Sprumont (1999) “Efficient Strategy-Proof Allocation Functions in Linear Production Economies,” Economic Theory 14, pp. 583 595. Mizukami, H. and T. Wakayama (2005) “Bossiness and Implementability in Pure Exchange Economies,” RIMS Kokyuroku 1461, pp. 126 140. Mizukami, H. and T. Wakayama (2017) “New Necessary and Sufficient Conditions for Secure Implementation,” Economics Letters 152, pp. 7678. Nishizaki, K. (2012) “Secure Implementation in Queueing Problems,” Theoretical Economics Letters 2, pp. 561 565. Nishizaki, K. (2013) “An Impossibility Theorem for Secure Implementation in Discrete Public Good Economies,” Economics Bulletin 33, pp. 300 308. Nishizaki, K. (2014) “An Equivalence of Secure Implementability and Full Implementability in Truthful Strategies in Pure Exchange Economies with Leontief Utility Functions,” Review of Economic Design 18, pp. 73 82. Nishizaki, K. (2018a) “Securely Implementable Social Choice Functions in Divisible and Nonexcludable Public Good Economies with Quasi-Linear Utility Functions,” Journal of Public Economic Theory 20, pp. 424 436. Nishizaki, K. (2018b) “Secure Implementability under Pareto-Efficient Rules in Linear Production Economies with Classical Preferences,” Research in Economics 72, pp. 379 383. 7) The rule satisfies the non-bossiness if and only if for each and each , if , then . 8) By definition, we find that consumption for each agent is not changed by changing another agent’s revelation under the equal budget free choice rule. This implies that the equal budget free choice rule satisfies non-bossiness..

(31) 66. 桃山学院大学総合研究所紀要. 第45巻第１号. Saijo, T., T. , and T. Yamato (2003) “Secure Implementation : Strategy-Proof Mechanisms Recons idered,” RIETI Discussion Paper 03 E 019. Saijo, T., T. , and T. Yamato (2007) “Secure Implementation,” Theoretical Economics 2, pp. 203229. Satterthwaite, M. A. and H. Sonnenschein (1981) “Strategy-Proof Allocation Mechanisms at Differentiable Points,” Review of Economic Studies 48, pp. 587 597. （2019年 3 月28日受理）.

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