Cooperative Purchasing with General Discount: A Game Theoretical Approach
In some situations, sellers of certain commodities usually provide price discounts for large orders according to a decreasing unit price function. Buyers of such commodities can cooperate and form purchasing groups to benefit from these price discounts. A natural way to allocate the corresponding co...
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| Format: | Article |
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MDPI AG
2022-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/22/4195 |
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| author | Jose A. García-Martínez Ana Meca G. Alexander Vergara |
| author_facet | Jose A. García-Martínez Ana Meca G. Alexander Vergara |
| author_sort | Jose A. García-Martínez |
| collection | DOAJ |
| description | In some situations, sellers of certain commodities usually provide price discounts for large orders according to a decreasing unit price function. Buyers of such commodities can cooperate and form purchasing groups to benefit from these price discounts. A natural way to allocate the corresponding cost reductions is the equal price rule. We analyze this situation as a cooperative game. We show that when the decreasing unit price function is linear, the equal price rule coincides with the Shapley value and the nucleolus of the cooperative game. However, some buyers may argue that the equal price rule is not acceptable because it favors those who buy just a few units of the product. This can be more problematic when the decreasing unit price function is nonlinear: In that case, the equal price rule loses some of its good properties and it no longer matches the Shapley value or the nucleolus. Unlike the linear case, in this nonlinear case, the Shapley value and nucleolus do not assign the same price to all agents, so there are different price rules. However, they have a computability problem, as both are very laborious to calculate for a large number of agents. To find a suitable alternative, we first study the properties that a different price rule should have in this situation. Second, we propose a family of different price rules that hold those properties and are easy to calculate for a large number of agents. This family of different price rules provides buyers (companies, institutions, consumers, etc.) with an easy-to-implement method which ensures stability in cooperative purchasing. |
| first_indexed | 2024-03-09T18:10:35Z |
| format | Article |
| id | doaj.art-246babfc7f67402298276dbe0121e568 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T18:10:35Z |
| publishDate | 2022-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-246babfc7f67402298276dbe0121e5682023-11-24T09:07:32ZengMDPI AGMathematics2227-73902022-11-011022419510.3390/math10224195Cooperative Purchasing with General Discount: A Game Theoretical ApproachJose A. García-Martínez0Ana Meca1G. Alexander Vergara2Departamento de. Estudios Económicos y Financieros, Universidad Miguel Hernández de Elche, 03202 Elche, SpainI. U. Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, 03202 Elche, SpainInteruniversity Doctorate in Economics (DEcIDE), I. U. Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, 03202 Elche, SpainIn some situations, sellers of certain commodities usually provide price discounts for large orders according to a decreasing unit price function. Buyers of such commodities can cooperate and form purchasing groups to benefit from these price discounts. A natural way to allocate the corresponding cost reductions is the equal price rule. We analyze this situation as a cooperative game. We show that when the decreasing unit price function is linear, the equal price rule coincides with the Shapley value and the nucleolus of the cooperative game. However, some buyers may argue that the equal price rule is not acceptable because it favors those who buy just a few units of the product. This can be more problematic when the decreasing unit price function is nonlinear: In that case, the equal price rule loses some of its good properties and it no longer matches the Shapley value or the nucleolus. Unlike the linear case, in this nonlinear case, the Shapley value and nucleolus do not assign the same price to all agents, so there are different price rules. However, they have a computability problem, as both are very laborious to calculate for a large number of agents. To find a suitable alternative, we first study the properties that a different price rule should have in this situation. Second, we propose a family of different price rules that hold those properties and are easy to calculate for a large number of agents. This family of different price rules provides buyers (companies, institutions, consumers, etc.) with an easy-to-implement method which ensures stability in cooperative purchasing.https://www.mdpi.com/2227-7390/10/22/4195cooperative purchasingprice discountsequal price ruledifferent price rulesgame theory |
| spellingShingle | Jose A. García-Martínez Ana Meca G. Alexander Vergara Cooperative Purchasing with General Discount: A Game Theoretical Approach Mathematics cooperative purchasing price discounts equal price rule different price rules game theory |
| title | Cooperative Purchasing with General Discount: A Game Theoretical Approach |
| title_full | Cooperative Purchasing with General Discount: A Game Theoretical Approach |
| title_fullStr | Cooperative Purchasing with General Discount: A Game Theoretical Approach |
| title_full_unstemmed | Cooperative Purchasing with General Discount: A Game Theoretical Approach |
| title_short | Cooperative Purchasing with General Discount: A Game Theoretical Approach |
| title_sort | cooperative purchasing with general discount a game theoretical approach |
| topic | cooperative purchasing price discounts equal price rule different price rules game theory |
| url | https://www.mdpi.com/2227-7390/10/22/4195 |
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