# Cherry Picking: Consumer Choices in Swarm Dynamics, Considering Price and Quality of Goods

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## Abstract

**:**

## 1. Objectives and Plan of the Paper

## 2. Behavioral Dynamics of Prices

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- ${u}_{s}$, $s=1,\dots ,N$ corresponds to the first functional subsystem (sellers), where each s-firm expresses the price ${u}_{s}$ of the product (good) offered for sale.
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- ${w}_{b}$, $b=1,\dots ,M$ corresponds the second functional subsystem (buyers), where each b-buyer expresses the price ${w}_{b}$ that he/she accepts to pay.
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- The variables which define the activities within each FS are given by the vectors$$\mathbf{u}=({u}_{1},\dots ,{u}_{s},\dots ,{u}_{N})\phantom{\rule{14.22636pt}{0ex}}\mathrm{and}\phantom{\rule{14.22636pt}{0ex}}\mathit{w}=({w}_{1},\dots ,{w}_{b},\dots ,{w}_{M}),$$

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- Micro-micro interactions take place only across FSs, but not within the same FS. By these interactions, firms and customers adjust the price by direct contacts.
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- Macro-micro interactions take place within the same FS, but not across different ones. By these interactions, each seller adjusts her/his price according to the mean stream of sellers, while customers adjust the price accounting for the mean stream of buyers.

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- ${\eta}_{s}^{b}({u}_{s},{w}_{b})$ models the rate at which a seller s interacts with a buyer b;
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- ${\eta}_{b}^{s}({w}_{b},{u}_{s})$ models the rate at which a buyer b interacts with a seller s;
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- ${\mu}_{s}({u}_{s},{\mathbb{E}}_{s})$ models the micro-macro interaction rate between a seller s and her/his own FS;
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- ${\mu}_{b}({w}_{b},{\mathbb{E}}_{b})$ models the micro-macro interaction rate between a buyer b and her/his own FS;
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- ${\phi}_{s}^{b}({u}_{s},{w}_{b},{v}_{s},{z}_{b})$ denotes the micro-micro action, which occurs with rate ${\eta}_{s}^{b}$, of a buyer b over a seller s;
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- ${\phi}_{b}^{s}({w}_{b},{u}_{s},{z}_{b},{v}_{s})$ denotes the micro-micro action, which occurs with rate ${\eta}_{b}^{s}$, of a seller s over a buyer b;
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- ${\psi}_{s}({u}_{s},{\mathbb{E}}_{s})$ denotes the micro-macro action, which occurs with rate ${\mu}_{s}$ of the FS of sellers over a seller s;
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- ${\psi}_{b}({w}_{b},{\mathbb{E}}_{b})$ denotes the micro-macro action, which occurs with rate ${\mu}_{b}$ of the FS of buyers over a buyer b.

- The interaction rates for both micro-micro and macro-micro interactions asymmetrically decay with the distance between the interacting entities starting from the same rates ${\eta}_{0}$ and ${\mu}_{0}$. In addition, when M increases with respect to N the interaction rates $\eta $ and $\mu $ both decrease by the so-called sticking effect:$$\left\{\begin{array}{c}{\eta}_{s}^{b}\simeq {\eta}_{s}={\eta}_{0}exp\left(-\frac{\rho}{\epsilon}{u}_{s}\right),\hfill \\ {\displaystyle {\eta}_{b}^{s}={\eta}_{0}\phantom{\rule{4pt}{0ex}}exp\left(-\frac{1}{\epsilon}\phantom{\rule{4pt}{0ex}}\frac{|{u}_{s}-{w}_{b}|}{{w}_{b}}\right),}\hfill \end{array}\right.$$$$\left\{\begin{array}{c}{\mu}_{s}={\mu}_{0},\hfill \\ {\displaystyle {\mu}_{b}={\mu}_{0}\phantom{\rule{0.166667em}{0ex}}exp\left(-\frac{1}{\epsilon}\phantom{\rule{0.166667em}{0ex}}\frac{|{w}_{b}-{\mathbb{E}}_{b}|}{{w}_{b}}\right).}\hfill \end{array}\right.$$
- The actions $\phi $ and $\psi $ correspond to a dynamics of consensus driven by the difference between the seller and buyer prices, in the micro-micro interaction, and between the local price and the global one, in the micro-macro interaction. The following model of interaction is proposed$$\left\{\begin{array}{c}{\phi}_{s}^{b}=\alpha \phantom{\rule{0.166667em}{0ex}}{u}_{s}\mathrm{sign}({w}_{b}-{u}_{s}),\hfill \\ {\phi}_{b}^{s}=\beta \phantom{\rule{0.166667em}{0ex}}({u}_{s}-{w}_{b}),\hfill \end{array}\right.$$$$\left\{\begin{array}{c}{\psi}_{s}=\gamma \phantom{\rule{0.166667em}{0ex}}({\mathbb{E}}_{s}-{u}_{s}),\hfill \\ {\psi}_{b}=\kappa \phantom{\rule{0.166667em}{0ex}}({\mathbb{E}}_{b}-{u}_{b}),\hfill \end{array}\right.$$

## 3. Cherry Picking

#### 3.1. Modeling Consumers as Cherry Pickers

- Each buyer looks for the right seller (under the above-mentioned conditions), visiting and comparing the prices and quality offered by all of them.
- After choosing the right one, the buyer will compare their prices and buy if her/his reservation price is higher or equal than seller price (or not if it is not).
- If the buyer effectively makes the transaction, then her/his reservation price will go down (if not it will go up).
- Each seller is visited by every buyer. If they buy, then she/he will increase the price of the product (if not she/he will decrease it).

#### 3.2. Derivation of Model 1

- Type of buyer ${B}_{1}$, numbered from 1 to ${a}_{1}$, who always chooses the seller offering the highest quality product.
- Type of buyer ${B}_{2}$, numbered from ${a}_{1}+1$ to ${a}_{2}$, who always chooses the seller with the highest quality-price ratio.
- Type of buyer ${B}_{3}$, numbered from ${a}_{2}+1$ to M, who always chooses the seller with the lowest price.

**Remark**

**1.**

**Remark**

**2.**

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- ${s}_{{c}_{max}}=\underset{s\in \{1,\dots ,N\}}{arg\; max}{c}_{s}$ (for the sake of simplicity ${s}_{c}$) is the seller offering the highest quality.
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- ${s}_{{r}_{max}}=\underset{s\in \{1,\dots ,N\}}{arg\; max}\frac{{c}_{s}}{{w}_{s}}$ (for the sake of simplicity ${s}_{r}$) is the seller with the highest quality-price ratio.
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- ${s}_{{w}_{min}}=\underset{s\in \{1,\dots ,N\}}{arg\; min}{w}_{s}$ (for the sake of simplicity ${s}_{w}$) is the seller offering the lowest price.

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

#### 3.3. Derivation of Model 2

#### 3.4. Numerical Results

- Prices are assumed to be ordered numbers.
- Productive factors do not change (capital and labor, here represented by the number of sellers).
- It is assumed the absence of new seller entries or existent seller exits in or from the market.
- The Statements (2) and (3) consequence is that any automatic price control mechanism is missing; instead, allowing the entry and exit mechanism, if prices go too high new sellers (firms) enter in the market increasing the offer side and lowering the prices, and vice versa.
- Both in our construction and in reality—when price are exposed by the sellers (e.g., in the mall)—, buyers coordination is easier than that of the sellers, which ignore the reservation prices of the buyers (the max price that a buyer accepts to pay); sellers blindly react step by step to their successes (made a sale) or failures (no sale) in dealing.
- Consistently with (5), buyers very well coordinate their reservation prices because they see all the set of the sellers, which on turn receive the reactions of all the other buyers; sellers instead have to act on the basis of information collected observing buyer decisions without seeing their internal reservation prices; certainly, they have micro-macro (mean field) interactions with the other sellers.

#### 3.5. Numerical Results for Model 1

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- Seller Pareto market efficiency is the sum, at every time t, of ${P}_{s}-{I}_{c}$ calculated at every exchange at a selling price ${P}_{s}$ and for every seller with initial cost ${I}_{c}$, fixed from the beginning as $\frac{1}{10}$ of seller price.
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- Buyer Pareto market efficiency is the sum, at every time t, of ${R}_{p}-{P}_{s}$ calculated at every exchange at a selling price ${P}_{s}$ and for every buyer with reservation price ${R}_{p}$.
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- The total Pareto market efficiency is the sum of the two above.

#### 3.6. Numerical Results for Model 2

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Seller and buyer prices and mean prices for a short term $T=1000$. (

**a**) Buyer prices, (

**b**) seller prices, (

**c**) buyer mean price and (

**d**) seller mean price.

**Figure 3.**Red: buyers; blue: sellers; purple: total Pareto market efficiency. (

**a**) Pareto market efficiency with $\rho =0.1$, $\gamma =0.1$, short term. (

**b**) Buyer Pareto market efficiency with $\rho =0.1$, $\gamma =0.1$, short term. (

**c**) Pareto market efficiency with $\rho =0.1$, $\gamma =0.1$, long term.

**Figure 5.**Buyer prices trends for (

**a**) $\gamma =0.1$ and (

**b**) $\gamma =0.01$. In the first case prices range remains constant, in second case we can see macro-waves appearing.

**Figure 6.**Dynamics of buyer prices for different times intervals: (

**a**) [0,1500], (

**b**) [49,000,50,000], (

**c**) [0,150,000]. Each color represents a buyer type, namely green ${B}_{1}$, purple ${B}_{2}$, yellow ${B}_{3}$. Blue in (

**c**) is for seller prices that remain in the same constant interval, as in the previous case.

**Figure 7.**Comparing sellers prices trend for (

**a**) $\gamma =0.1$ and (

**b**) $\gamma =0.01$. Here with $\rho =0.1$ and $\eta =1$.

**Figure 8.**Red: buyers; blue: sellers; purple: total Pareto market efficiency. (

**a**) Pareto market efficiency with $\rho =0.1$, $\gamma =0.01$, medium term. (

**b**) Buyer Pareto market efficiency with $\rho =0.1$, $\gamma =0.01$, medium term.

**Figure 9.**Sellers (blue) and buyers (red) prices and mean prices for a short term $T=1000$, with $\rho =2$. (

**a**) individual prices (

**b**) buyer mean price and (

**c**) seller mean price.

**Figure 10.**Evolution of buyer prices for different time intervals: (

**a**) [0,1500], (

**b**) [49,000,50,000], (

**c**) [0,150,000]. Buyers are divided into 6 reservation qualities that, ordered from larger to lower, will be represented in black, red, cyan, yellow, green and magenta. Here, $\rho =0.5$, $\gamma =0.01$, $\eta =1$.

**Figure 11.**Two different simulations showing evolution of buyer prices in which a unique cluster appears in the medium term. Here with $\rho =0.5$, $\gamma =0.01$, $\eta =1$. (

**a**) buyers can organize both in three clusters. (

**b**) generalization of (

**a**).

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**MDPI and ACS Style**

Knopoff, D.; Secchini, V.; Terna, P.
Cherry Picking: Consumer Choices in Swarm Dynamics, Considering Price and Quality of Goods. *Symmetry* **2020**, *12*, 1912.
https://doi.org/10.3390/sym12111912

**AMA Style**

Knopoff D, Secchini V, Terna P.
Cherry Picking: Consumer Choices in Swarm Dynamics, Considering Price and Quality of Goods. *Symmetry*. 2020; 12(11):1912.
https://doi.org/10.3390/sym12111912

**Chicago/Turabian Style**

Knopoff, Damian, Valeria Secchini, and Pietro Terna.
2020. "Cherry Picking: Consumer Choices in Swarm Dynamics, Considering Price and Quality of Goods" *Symmetry* 12, no. 11: 1912.
https://doi.org/10.3390/sym12111912