# Cooperative Purchasing with General Discount: A Game Theoretical Approach

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## Abstract

**:**

## 1. Introduction

## 2. Related Literature

## 3. Model

Properties of P |

1. Class ${C}^{2}$ at $(0,+\infty )$: there exists $P\u2033\left(q\right)$ at all points of $(0,+\infty )$ and it is continuous. |

2. Decreasing: for all $q>0,{P}^{\prime}\left(q\right)<0.$ |

3. Convex: for all $q>0,P\u2033\left(q\right)\ge 0.$ |

4. Limited growth rate: for all $q>0,\left|{P}^{\prime}\left(q\right)\right|\le \frac{P\left(q\right)}{q}$ |

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 4. Equal Price Rule for CPL-Games

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

## 5. Balanced Different Price Rule for CPNL-Games

- 1.
**Monotonically decreasing through players (MDP)**. Given a $q\in {\mathbb{R}}_{+}^{n}$, if ${q}_{i}\le {q}_{j}$, then $\sigma ({q}_{i},{q}_{-i})\ge \sigma ({q}_{j},{q}_{-j})$ for all $i,j\in N.$- 2.
**Limited decrease through players (LDP).**Given a $q\in {\mathbb{R}}_{+}^{n}$, if ${q}_{i}\le {q}_{j}$, then $\sigma ({q}_{i},{q}_{-i}){q}_{i}\le \sigma ({q}_{j},{q}_{-j}){q}_{j}$ for all $i,j\in N.$- 3.
**Major-agents acceptability (MA).**For all $i\in {A}_{m}$, $\sigma ({q}_{i},{q}_{-i})<1$, and for all $i\in {A}_{nm}$, $\sigma ({q}_{i},{q}_{-i})>1$.- 4.
**Balanced weighting (BW).**${\sum}_{i\in {A}_{m}}\left(1-\sigma ({q}_{i},{q}_{-i})\right){q}_{i}={\sum}_{i\in {A}_{nm}}\left(\sigma ({q}_{i},{q}_{-i})-1\right){q}_{i}$.- 5.
**Non-major agents acceptability (NMA).**$\sigma ({q}_{1},{q}_{-1})\le \frac{P\left({q}_{{A}_{nm}}\right)}{P\left({q}_{N}\right)}.$- 6.
**Limited decrease in a player quantity (LDQ).**Let $(N,q,P)$ and $(N,{q}^{\prime},P)$ be two CPGD-models with $q=({q}_{i},{q}_{-i})$ and ${q}^{\prime}=({q}_{i}^{\prime},{q}_{-i}).$ If ${q}_{i}\ge {q}_{i}^{\prime},$ then$\sigma ({q}_{i},{q}_{-i})\ge \frac{P\left({q}_{N}^{\prime}\right){q}_{i}^{\prime}}{P\left({q}_{N}\right){q}_{i}}\sigma ({q}_{i}^{\prime},{q}_{-i}).$

- 1.
**Symmetry (SYM)**. If two agents i and j in a group are interchanged in the sense that $c(S\cup \{i\left\}\right)=c(S\cup \{j\left\}\right)$ for every $S\subset N\backslash \{i,j\}$, then ${\beta}_{i}\left(c\right)={\beta}_{j}\left(c\right).$ It means that equal agents in a group should pay equal costs. Indeed, $c(S\cup \left\{i\right\})=c(S\cup \left\{j\right\})\iff {q}_{i}={q}_{j}.$ Thus, $\sigma ({q}_{i},{q}_{-i})P\left({q}_{N}\right){q}_{i}=\sigma ({q}_{j},{q}_{-j})P\left({q}_{N}\right){q}_{j}.$- 2.
**Player motononicity (PMON)**. For all $i,j\in N$ s.t. ${q}_{i}\le {q}_{j}$, it holds that ${\beta}_{i}\left(c\right)\le {\beta}_{j}\left(c\right).$This holds by property 2 of function $\sigma .$- 3.
**Cost monotonicity (CMON)**. For all $i\in N$ s.t. ${q}_{i}\ge {q}_{i}^{\prime},$ it holds that ${\beta}_{i}\left(c\right)\ge {\beta}_{i}\left({c}^{\prime}\right),$ with $(N,c),(N,{c}^{\prime})$ being the CPGD-games corresponding to CPGD-models $(N,q,P)$ and $(N,{q}^{\prime},P)$ where $q=({q}_{-i},{q}_{i})$ and ${q}^{\prime}=({q}_{-i},{q}_{i}^{\prime})$. Satisfying this property means that if the number of units of the product to be purchased by one agent in a purchasing group remains the same or increases in comparison to a previous situation, then that agent should pay an equal or higher cost.This holds by property 6 of function $\sigma $.- 4.
**Fair ranking added cost (FRAC)**. If for two agents i and j in a group $c\left(N\right)-c(N\backslash \{i\left\}\right)\ge c\left(N\right)-c(N\backslash \{j\left\}\right),$ then ${\beta}_{i}\left(c\right)\ge {\beta}_{j}\left(c\right).$ Satisfying this FRAC property means that an agent with an equal or larger added cost (this is also called marginal costs) should pay an equal or larger cost.Indeed, $c\left(N\right)-c(N\backslash \left\{i\right\})\ge c\left(N\right)-c(N\backslash \left\{j\right\})\iff P\left({q}_{N\backslash \left\{i\right\}}\right){q}_{N\backslash \left\{i\right\}}\le P\left({q}_{N\backslash \left\{j\right\}}\right){q}_{N\backslash \left\{j\right\}}$, and by property 4 (limited growth rate) of function P, ${q}_{N\backslash \left\{i\right\}}\le {q}_{N\backslash \left\{j\right\}}\iff {q}_{i}\ge {q}_{j}$. Thus, by property 2 of function $\sigma $, FRAC holds.

**Theorem**

**1.**

**Proof.**

- (1)
- $S\subseteq {A}_{m}.$ Here, for all $i\in S$, $\sigma ({q}_{i},{q}_{-i})<1$ and so ${\sum}_{i\in S}\sigma ({q}_{i},{q}_{-i})P\left({q}_{N}\right){q}_{i}\le $ ${\sum}_{i\in S}P\left({q}_{S}\right){q}_{i}=c\left(S\right)$.
- (2)
- $S\subseteq {A}_{nm}$. We now prove that ${\sum}_{i\in S}\left(\sigma ({q}_{i},{q}_{-i})P\left({q}_{N}\right)-P\left({q}_{S}\right)\right){q}_{i}\le 0$. By P.5 (NMA) we know that $\forall i\in {A}_{nm},$$\sigma ({q}_{i},{q}_{-i})\le \frac{P\left({q}_{{A}_{nm}}\right)}{P\left({q}_{N}\right)}.$ Then, $\sigma ({q}_{i},{q}_{-i})P\left({q}_{N}\right)\le P\left({q}_{{A}_{nm}}\right).$ Take into account that $P\left({q}_{{A}_{nm}}\right)\le $$P\left({q}_{S}\right)$, for all $S\subseteq {A}_{nm},$ it is found that $\sigma ({q}_{i},{q}_{-i})P\left({q}_{N}\right)-P\left({q}_{S}\right)\le 0$ for any $i\in S\subseteq {A}_{nm}$. Hence, ${\sum}_{i\in S}\left(\sigma ({q}_{i},{q}_{-i})P\left({q}_{N}\right)-P\left({q}_{S}\right)\right){q}_{i}\le 0$
- (3)
- $S\cap {A}_{nm}\ne S$. By an argument similar to that above

## 6. The Family of $\mathbf{\alpha}$-Proportional Rules

**Proposition**

**5.**

- 1.
- For $\alpha =1$, all agents pay the equal price: ${\Theta}_{i}(c,1)={\u03f5}_{i}\left(c\right),$ for all $i\in N$.
- 2.
- For any $\alpha <1$,
- (a)
- For all $i\in \overline{L}$, ${\Theta}_{i}(c,\alpha )<{\u03f5}_{i}\left(c\right)$ and ${\Theta}_{i}(c,\alpha )$ decreases in α.
- (b)
- For all $i\in \overline{S}$, ${\Theta}_{i}(c,\alpha )>{\u03f5}_{i}\left(c\right)$ and ${\Theta}_{i}(c,\alpha )$ increases in α.
- (c)
- If there is $i\notin \overline{L}\cup \overline{S}$, then ${\Theta}_{i}(c,\alpha )={\u03f5}_{i}\left(c\right)$.

**Proof.**

## 7. Condition for an $\mathbf{\alpha}$-Proportional Rule to Be a BDP Rule

**Theorem**

**2.**

**Proof.**

- 1.
- (MDP) Take $i,j\in N$ s.t. ${q}_{i}\le {q}_{j}$ then, by property 2 (Decreasingness) of function P, it follows that $P\left({q}_{i}\right)\ge P\left({q}_{j}\right)$, and so ${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)\ge {\sigma}_{\alpha}\left({q}_{j},{q}_{-j}\right).$
- 2.
- (LDP) Take $i,j\in N$ s.t. ${q}_{i}\le {q}_{j}$. It can be shown that ${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right){q}_{i}\le {\sigma}_{\alpha}\left({q}_{j},{q}_{-j}\right){q}_{j}.$ Indeed, by property 4 (limited growth rate) of function P, it emerges that $\alpha P\left({q}_{N}\right){q}_{i}+(1-\alpha )P\left({q}_{i}\right){q}_{i}\le \alpha P\left({q}_{N}\right){q}_{j}+(1-\alpha )P\left({q}_{j}\right){q}_{j}$. Hence,$\frac{\alpha P\left({q}_{N}\right)+(1-\alpha )P\left({q}_{i}\right)}{{\sum}_{j\in N}[\alpha P\left({q}_{N}\right){q}_{j}+(1-\alpha )P\left({q}_{j}\right){q}_{j}]}{q}_{N}{q}_{i}<\frac{\alpha P\left({q}_{N}\right)+(1-\alpha )P\left({q}_{j}\right)}{{\sum}_{j\in N}[\alpha P\left({q}_{N}\right){q}_{j}+(1-\alpha )P\left({q}_{j}\right){q}_{j}]}{q}_{N}{q}_{j}.$
- 3.
- (MA) We now prove that for $\alpha <1,{\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)<1,$ for all $i\in {A}_{m},$ and ${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)>1,$ for all $i\in {A}_{nm}.$Indeed, as mentioned above, ${A}_{m}=\overline{L}$ and ${A}_{nm}=\overline{S}\cup \left(\overline{L}\cup \overline{S}\right)$. Thus, if $\alpha <1$, from point 2.a. of Proposition 5, we know that, for all $i\in {A}_{m}$, ${\Theta}_{i}(c,\alpha )<{\u03f5}_{i}\left(c\right)$, which is equivalent to ${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)P\left({q}_{N}\right){q}_{i}<P\left({q}_{N}\right){q}_{i}\iff {\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)<1$. Analogously, from point 2.b. and 2.c. of Proposition 5, it can be shown that, for all $i\in {A}_{nm}$, ${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)\ge 1$. Finally, note that if $\alpha =1$, then ${\Theta}_{i}(c,1)=\u03f5\left(c\right)$ and ${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)=1$ for all $i\in N$.
- 4.
- (BW) It is straightforward to prove that ${\sum}_{i\in {A}_{m}}\left(1-{\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)\right){q}_{i}=\left|{\sum}_{i\in {A}_{nm}}\left(1-{\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)\right){q}_{i}\right|.$Indeed,${\sum}_{i\in {A}_{m}}\left(1-{\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)\right){q}_{i}=\left|{\sum}_{i\in {A}_{nm}}\left(1-{\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)\right){q}_{i}\right|\iff $${\sum}_{i\in N}\left({\sigma}_{\alpha}({q}_{i},{q}_{-i})-1\right){q}_{i}=0\iff {\sum}_{i\in N}{\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right){q}_{i}={\sum}_{i\in N}{q}_{i}\iff $$\iff {\sum}_{i\in N}{\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right){q}_{i}={q}_{N}\iff {\sum}_{i\in N}\left(\frac{\alpha P\left({q}_{N}\right)+(1-\alpha )P\left({q}_{i}\right)}{{\sum}_{j\in N}[\alpha P\left({q}_{N}\right){q}_{j}+(1-\alpha )P\left({q}_{j}\right){q}_{j}]}{q}_{N}\right){q}_{i}={q}_{N}$$\iff {\sum}_{i\in N}\left(\frac{\alpha P\left({q}_{N}\right)+(1-\alpha )P\left({q}_{i}\right)}{{\sum}_{j\in N}[\alpha P\left({q}_{N}\right){q}_{j}+(1-\alpha )P\left({q}_{j}\right){q}_{j}]}\right){q}_{i}=1\iff \frac{{\sum}_{i\in N}\left[\alpha P\left({q}_{N}\right){q}_{i}+(1-\alpha )P\left({q}_{i}\right){q}_{i}\right]}{{\sum}_{j\in N}[\alpha P\left({q}_{N}\right){q}_{j}+(1-\alpha )P\left({q}_{j}\right){q}_{j}]}=1.$
- 5.
- (NMA) We show that there is always an ${\alpha}^{*}<1$ such that for any $\alpha \in [{\alpha}^{*},1)$, ${\sigma}_{\alpha}\left({q}_{1},{q}_{-1}\right)\le \frac{P\left({q}_{{A}_{nm}}\right)}{P\left({q}_{N}\right)}.$ Note that, as shown above, ${A}_{nm}=\overline{S}\cup {\left(\overline{L}\cup \overline{S}\right)}^{C}$.We first prove that for all $i\in {A}_{nm}$, ${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)$ is decreasing in $\alpha .$ Indeed, as ${\Theta}_{i}(c,\alpha )={\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)P\left({q}_{N}\right){q}_{i}$, thus $\frac{d\left({\Theta}_{i}(c,\alpha )\right)}{d\alpha}=\frac{d\left({\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)\right)}{d\alpha}P\left({q}_{N}\right){q}_{i}$. Therefore, $\frac{d\left({\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)\right)}{d\alpha}<0$ if and only if $\frac{d\left({\Theta}_{i}(c,\alpha )\right)}{d\alpha}<0,$ since $P\left({q}_{N}\right){q}_{i}>0$. The sign of the last derivative always holds for all $i\in \overline{S}$ (see point 2.b of Proposition 5). In addition, if there exists any $i\notin \overline{L}\cup \overline{S}$ then, by point 2.c of Proposition, $\frac{d\left({\Theta}_{i}(c,\alpha )\right)}{d\alpha}=0.$Now note that ${\sigma}_{\alpha}\left({q}_{1},{q}_{-1}\right)\le \frac{P\left({q}_{{A}_{nm}}\right)}{P\left({q}_{N}\right)}$ is equivalent to$$\frac{\alpha P\left({q}_{N}\right)+(1-\alpha )P\left({q}_{1}\right)}{{\sum}_{j\in N}[\alpha P\left({q}_{N}\right){q}_{j}+(1-\alpha )P\left({q}_{j}\right){q}_{j}]}}{q}_{N}\le {\textstyle \frac{P\left({q}_{{A}_{nm}}\right)}{P\left({q}_{N}\right)}}.$$This last inequality always holds for $\alpha =1$. Indeed, $\frac{P\left({q}_{N}\right)}{{\sum}_{j\in N}P\left({q}_{N}\right){q}_{j}}{q}_{N}=1<\frac{P\left({q}_{{A}_{nm}}\right)}{P\left({q}_{N}\right)},$ because of ${q}_{{A}_{nm}}<{q}_{N}$. Thus, ${\sigma}_{\alpha =1}\left({q}_{1},{q}_{-1}\right)=1$ and ${\sigma}_{\alpha}\left({q}_{1},{q}_{-1}\right)$ is decreasing in $\alpha $. Thus, only two different situations can occur: First, there is a root $\overline{\alpha}\in (0,1)$ such that ${\sigma}_{\overline{\alpha}}\left({q}_{1},{q}_{-1}\right)=\frac{P\left({q}_{{A}_{nm}}\right)}{P\left({q}_{N}\right)}$, i.e., (9) holds with equality, thus, ${\sigma}_{\alpha}\left({q}_{1},{q}_{-1}\right)\le \frac{P\left({q}_{{A}_{nm}}\right)}{P\left({q}_{N}\right)}$ for all $\alpha \in (\overline{\alpha},1)$. Second, there is no such $\overline{\alpha}$ that ${\sigma}_{\overline{\alpha}}\left({q}_{1},{q}_{-1}\right)=\frac{P\left({q}_{{A}_{nm}}\right)}{P\left({q}_{N}\right)}$. In that case, ${\sigma}_{\alpha}\left({q}_{1},{q}_{-1}\right)\le \frac{P\left({q}_{{A}_{nm}}\right)}{P\left({q}_{N}\right)}$, for all $\alpha \in (0,1)$.Assume that ${\alpha}^{*}=\overline{\alpha}$ if $\overline{\alpha}\in (0,1)$ and ${\alpha}^{*}=0$ otherwise. We conclude that there is always an ${\alpha}^{*}<1$, such that for any $\alpha \in [{\alpha}^{*},1$, ${\sigma}_{\alpha}\left({q}_{1},{q}_{-1}\right)\le \frac{P\left({q}_{{A}_{nm}}\right)}{P\left({q}_{N}\right)}$.
- 6.
- (LDQ) Take $(N,q,P)$ and $(N,{q}^{\prime},P)$ two CPGD-models with $q=({q}_{i},{q}_{-i}),{q}^{\prime}=({q}_{i}^{\prime},{q}_{-i}),$ and ${q}_{i}\ge {q}_{i}^{\prime}.$ It must be shown that ${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)P\left({q}_{N}\right){q}_{i}\ge {\sigma}_{\alpha}\left({q}_{i}^{\prime},{q}_{-i}\right)P\left({q}_{N}^{\prime}\right){q}_{i}^{\prime}.$ In fact, it must be proven that the function ${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)P\left({q}_{N}\right){q}_{i}$ is increasing in ${q}_{i}$, i.e.$$\frac{\partial \left({\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)P\left({q}_{N}\right){q}_{i}\right)}{\partial {q}_{i}}\ge 0.$$To simplify the proof, ${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)P\left({q}_{N}\right){q}_{i}$ can be rewritten as a function of ${\u03f5}_{i}\left(c\right)=P\left({q}_{N}\right){q}_{i}$, $c\left(\left\{i\right\}\right)=P\left({q}_{i}\right){q}_{i}$ and $c\left(N\right)=P\left({q}_{N}\right){q}_{N}$. In addition, to simplify the notation, we do not explicitly indicate that all the following derivatives are in regard to ${q}_{i}$; we denote them by ${\u03f5}_{i}^{\prime}\left(c\right)$, ${c}^{\prime}\left(\left\{i\right\}\right)$, and ${c}^{\prime}\left(N\right).$First we rewrite the function ${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)P\left({q}_{N}\right){q}_{i}$ as follows:${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)P\left({q}_{N}\right){q}_{i}=\frac{\alpha P\left({q}_{N}\right){q}_{i}+(1-\alpha )P\left({q}_{i}\right){q}_{i}}{{\sum}_{j\in N}[\alpha P\left({q}_{N}\right){q}_{j}+(1-\alpha )P\left({q}_{j}\right){q}_{j}]}P\left({q}_{N}\right){q}_{N}=\frac{\left(\alpha {\u03f5}_{i}\left(c\right)+(1-\alpha )c\left(\left\{i\right\}\right)\right)c\left(N\right)}{\alpha c\left(N\right)+(1-\alpha ){\sum}_{j\in N}c\left(\left\{j\right\}\right)}.$Denote by $f\left({q}_{i}\right)=\left(\alpha {\u03f5}_{i}\left(c\right)+(1-\alpha )c\left(\left\{i\right\}\right)\right)c\left(N\right)$ and $g\left({q}_{i}\right)=\alpha c\left(N\right)+(1-\alpha )$${\sum}_{j\in N}c\left(\left\{j\right\}\right)$, thus${\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)P\left({q}_{N}\right){q}_{i}=\frac{f\left({q}_{i}\right)}{g\left({q}_{i}\right)}$ and $\frac{\partial \left({\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)P\left({q}_{N}\right){q}_{i}\right)}{\partial {q}_{i}}=\frac{{f}^{\prime}\left({q}_{i}\right)g\left({q}_{i}\right)-f\left({q}_{i}\right){g}^{\prime}\left({q}_{i}\right)}{{\left(g\left({q}_{i}\right)\right)}^{2}}$, with$\frac{\partial \left({\sigma}_{\alpha}\left({q}_{i},{q}_{-i}\right)P\left({q}_{N}\right){q}_{i}\right)}{\partial {q}_{i}}>0\iff {f}^{\prime}\left({q}_{i}\right)g\left({q}_{i}\right)-f\left({q}_{i}\right){g}^{\prime}\left({q}_{i}\right)>0$In addition, it is known that${f}^{\prime}\left({q}_{i}\right)=\left(\alpha {\u03f5}_{i}^{\prime}\left(c\right)+(1-\alpha ){c}^{\prime}\left(\left\{i\right\}\right)\right)c\left(N\right)+\left(\alpha {\u03f5}_{i}\left(c\right)+(1-\alpha )c\left(\left\{i\right\}\right)\right){c}^{\prime}\left(N\right)$${g}^{\prime}\left({q}_{i}\right)=\alpha {c}^{\prime}\left(N\right)+(1-\alpha ){\sum}_{j\in N}{c}^{\prime}\left(\left\{j\right\}\right)$After some calculations, it can be shown that${f}^{\prime}\left({q}_{i}\right)g\left({q}_{i}\right)-f\left({q}_{i}\right){g}^{\prime}\left({q}_{i}\right)>0$⇔$\left(\alpha {\u03f5}_{i}^{\prime}\left(c\right)+(1-\alpha ){c}^{\prime}\left(\left\{i\right\}\right)\right)c\left(N\right)\left(\alpha c\left(N\right)+(1-\alpha ){\sum}_{j\in N}c\left(\left\{i\right\}\right)\right)$>$\left(\alpha {\u03f5}_{i}\left(c\right)+(1-\alpha )c\left(\left\{i\right\}\right)\right)(1-\alpha )\left(c\left(N\right){c}^{\prime}\left(\left\{i\right\}\right)-{c}^{\prime}\left(N\right){\sum}_{j\in N}c\left(\left\{j\right\}\right)\right)$Clearly, $\left(\alpha c\left(N\right)+(1-\alpha ){\sum}_{j\in N}c\left(\left\{j\right\}\right)\right)>\alpha {\u03f5}_{i}\left(c\right)+(1-\alpha )c\left(\left\{i\right\}\right)(1-\alpha )$, because $c\left(N\right)>{\u03f5}_{i}\left(c\right)$ and ${\sum}_{j\in N}c\left(\left\{j\right\}\right)>c\left(\left\{i\right\}\right)$.To end the proof, we prove that$$\left(\alpha {\u03f5}_{i}^{\prime}\left(c\right)+(1-\alpha ){c}^{\prime}\left(\left\{i\right\}\right)\right)c\left(N\right)>(1-\alpha )\left(c\left(N\right){c}^{\prime}\left(\left\{i\right\}\right)-{c}^{\prime}\left(N\right)\sum _{j\in N}c\left(\left\{j\right\}\right)\right)$$It is straightforward to show that (10) is equivalent to$\alpha {\u03f5}_{i}^{\prime}\left(c\right)c\left(N\right)>-(1-\alpha ){c}^{\prime}\left(N\right){\sum}_{j\in N}c\left(\left\{j\right\}\right)$,which always holds because ${\u03f5}_{i}^{\prime}\left(c\right)>0$ and ${c}^{\prime}\left(N\right)>0$. Note that, by property 4 (limited growth rate) of function P, it is straightforward to prove that ${c}^{\prime}\left(N\right)>0.$ Next, we show that ${\u03f5}_{i}^{\prime}\left(c\right)>0$,$${\u03f5}_{i}^{\prime}\left(c\right)={P}^{\prime}\left({q}_{N}\right){q}_{i}+P\left({q}_{N}\right)>{P}^{\prime}\left({q}_{N}\right){q}_{N}+P\left({q}_{N}\right)={c}^{\prime}\left(N\right)>0,$$

**Corollary**

**1.**

**Proof.**

## 8. Numerical Illustration

## 9. Conclusions, Limitations and Implications

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

EP | Equal price |

QDF | Quantity-discount function |

DP | Different price |

CPGD-model | Cooperative-purchasing model with general discount |

CPGD-game | Cooperative-purchasing cost game with general discount |

CPL-game | Cooperative-purchasing game with linear discount |

CPNL-game | Cooperative-purchasing game with non-linear discount |

MCP-situations | Maximum cooperative-purchasing situations |

MCP-games | Maximum cooperative-purchasing games |

MDP | Monotonically decreasing through players |

LDP | Limited decrease through players |

MA | Major-agents’ acceptability |

BW | Balanced weighting |

NMA | Non-major agents’ acceptability |

LDQ | Limited decrease in a player quantity |

SYM | Symmetry |

PMON | Player motononicity |

CMON | Cost monotonicity |

FRAC | Fair ranking added cost |

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$\mathit{P}\left({\mathit{q}}_{\mathit{i}}\right)=80+\frac{7000}{\sqrt{{\mathit{q}}_{\mathit{i}}}}$ | ||||||||
---|---|---|---|---|---|---|---|---|

No-Cooperate | Cooperate | |||||||

Agent | ${\mathit{q}}_{\mathbf{i}}$ | Individual | Equal Price | ${\left.\mathrm{BDP}(\mathbf{\alpha}-\mathrm{Proportional})\right|}_{\mathbf{\alpha}={\mathbf{\alpha}}^{*}=\mathbf{0}.\mathbf{368}}$ | ||||

Price | Cost | Price | Cost | Price | Cost | ${\mathbf{\sigma}}_{\mathbf{\alpha}}$ | ||

1 | 50 | 1069.9 | 53,497.5 | 192.7 | 9633.4 | 448.9 | 22,446.6 | 2.330 |

2 | 60 | 983.7 | 59,021.8 | 192.7 | 11,560.1 | 416.2 | 24,970.9 | 2.160 |

3 | 70 | 916.7 | 64,166.2 | 192.7 | 13,486.8 | 390.7 | 27,350.9 | 2.028 |

4 | 80 | 862.6 | 69,009.9 | 192.7 | 15,413.5 | 370.2 | 29,616.8 | 1.921 |

5 | 100 | 780.0 | 78,000.0 | 192.7 | 19,266.9 | 338.8 | 33,883.8 | 1.759 |

6 | 500 | 393.0 | 196,524.8 | 192.7 | 96,334.5 | 191.9 | 95,955.5 | 0.996 |

7 | 600 | 365.8 | 219,464.3 | 192.7 | 115,601.4 | 181.6 | 108,932.6 | 0.942 |

8 | 700 | 344.6 | 241,202.6 | 192.7 | 134,868.3 | 173.5 | 121453.6 | 0.901 |

9 | 800 | 327.5 | 261,989.9 | 192.7 | 154,135.2 | 167.0 | 133,613.4 | 0.867 |

10 | 900 | 313.3 | 282,000.0 | 192.7 | 173,402.1 | 161.6 | 145,478.2 | 0.839 |

$\mathit{P}\left({\mathit{q}}_{\mathit{i}}\right)=10+\frac{7000}{\sqrt{{\mathit{q}}_{\mathit{i}}}}$ | ||||||||
---|---|---|---|---|---|---|---|---|

No-Cooperate | Cooperate | |||||||

Agent | ${\mathit{q}}_{\mathbf{i}}$ | Individual | Equal Price | ${\left.\mathrm{BDP}(\mathbf{\alpha}-\mathrm{Proportional})\right|}_{\mathbf{\alpha}={\mathbf{\alpha}}^{*}=\mathbf{0}}$ | ||||

Price | Cost | Price | Cost | Price | Cost | ${\mathbf{\sigma}}_{\mathbf{\alpha}}$ | ||

1 | 50 | 999.9 | 49,997.5 | 122.7 | 6133.4 | 377.4 | 18,868.5 | 3.076 |

2 | 60 | 913.7 | 54,821.8 | 122.7 | 7360.1 | 344.8 | 20,689.2 | 2.811 |

3 | 70 | 846.7 | 59,266.2 | 122.7 | 8586.8 | 319.5 | 22,366.5 | 2.605 |

4 | 80 | 792.6 | 63,409.9 | 122.7 | 9813.5 | 299.1 | 23,930.3 | 2.438 |

5 | 100 | 710.0 | 71,000.0 | 122.7 | 12,266.9 | 267.9 | 26,794.7 | 2.184 |

6 | 500 | 323.0 | 161,524.8 | 122.7 | 61,334.5 | 121.9 | 60,957.8 | 0.994 |

7 | 600 | 295.8 | 177,464.3 | 122.7 | 73,601.4 | 111.6 | 66,973.2 | 0.910 |

8 | 700 | 274.6 | 192,202.6 | 122.7 | 85,868.3 | 103.6 | 72,535.3 | 0.845 |

9 | 800 | 257.5 | 205,989.9 | 122.7 | 98,135.2 | 97.2 | 77,738.5 | 0.792 |

10 | 900 | 243.3 | 219,000.0 | 122.7 | 110,402.1 | 91.8 | 82,648.4 | 0.749 |

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## Share and Cite

**MDPI and ACS Style**

García-Martínez, J.A.; Meca, A.; Vergara, G.A. Cooperative Purchasing with General Discount: A Game Theoretical Approach. *Mathematics* **2022**, *10*, 4195.
https://doi.org/10.3390/math10224195

**AMA Style**

García-Martínez JA, Meca A, Vergara GA. Cooperative Purchasing with General Discount: A Game Theoretical Approach. *Mathematics*. 2022; 10(22):4195.
https://doi.org/10.3390/math10224195

**Chicago/Turabian Style**

García-Martínez, Jose A., Ana Meca, and G. Alexander Vergara. 2022. "Cooperative Purchasing with General Discount: A Game Theoretical Approach" *Mathematics* 10, no. 22: 4195.
https://doi.org/10.3390/math10224195