11 research outputs found

### Monte Carlo simulations and ODE solutions of the market shares for symmetric appeals.

<p>The Monte Carlo simulations involved 10<sup>4</sup> realizations of the system for nine different set of parameters (grey lines). In all cases we used <i>q</i><sub>1</sub> = 1, the appeal used for both products were the same (<i>A</i><sub>1</sub> = <i>A</i><sub>2</sub>) and both products start with zero purchases, <i>d</i><sub>1</sub>(<i>t</i> = 0) = <i>d</i><sub>2</sub>(<i>t</i> = 0) = 0. Although the Monte Carlo simulations produce discrete dots in the (<i>d<sub>T</sub></i>, <i>MS</i><sub>2</sub>) space, we plot each simulation with straight lines that link consecutive dots to follow trajectories easily.</p

### Monte Carlo simulations and ODE solutions of the market shares for symmetric appeals.

<p>The Monte Carlo simulations involved 10<sup>4</sup> realizations of the system for nine different set of parameters (grey lines). In all cases we used <i>q</i><sub>1</sub> = 1, the appeal used for both products were the same (<i>A</i><sub>1</sub> = <i>A</i><sub>2</sub>) and both products start with zero purchases, <i>d</i><sub>1</sub>(<i>t</i> = 0) = <i>d</i><sub>2</sub>(<i>t</i> = 0) = 0. Although the Monte Carlo simulations produce discrete dots in the (<i>d<sub>T</sub></i>, <i>MS</i><sub>2</sub>) space, we plot each simulation with straight lines that link consecutive dots to follow trajectories easily.</p

### Market share of product 2 (<i>MS</i><sub>2</sub>) as a function of <i>Q</i><sub>2</sub> and <i>A</i><sub>2</sub>, for different values of <i>A</i><sub>1</sub> and , assuming <i>q</i><sub>1</sub> = 1.

<p>Market share of product 2 (<i>MS</i><sub>2</sub>) as a function of <i>Q</i><sub>2</sub> and <i>A</i><sub>2</sub>, for different values of <i>A</i><sub>1</sub> and , assuming <i>q</i><sub>1</sub> = 1.</p

### Monte carlo simulations and ODE solutions of the market shares for asymetric appeals.

<p>The Monte Carlo simulations involved 10<sup>4</sup> realizations of the system for nine different sets of parameters (grey lines). In all cases, <i>q</i><sub>1</sub> = 1 and both products start with zero purchases, i.e., <i>d</i><sub>1</sub>(<i>t</i> = 0) = <i>d</i><sub>2</sub>(<i>t</i> = 0) = 0. Although Monte Carlo simulations produce discrete dots in the (<i>d<sub>T</sub></i>, <i>MS</i><sub>2</sub>) space, we plot each simulation with straight lines that link consecutive dots to follow trajectories easily.</p

### The Unpredictability of the Rankings for the Unpredictability Measure of [8].

<p>The measure of unpredictability <i>u</i><sub><i>i</i></sub> for song <i>i</i> is defined as the average difference in market share <i>m</i><sub><i>i</i></sub> for that song between all pairs of realizations, i.e., </p><p></p><p></p><p><mi>u</mi><mi>i</mi></p><mo>=</mo><p><mo>âˆ‘</mo></p><p><mi>j</mi><mo>=</mo><mn>1</mn></p><mi>W</mi><p></p><p><mo>âˆ‘</mo></p><p><mi>k</mi><mo>=</mo><mi>j</mi><mo>+</mo><mn>1</mn></p><mi>W</mi><p></p><mo stretchy="false">âˆ£</mo><p><mi>m</mi></p><p><mi>i</mi><mo>,</mo><mi>j</mi></p><p></p><mo>âˆ’</mo><p><mi>m</mi></p><p><mi>i</mi><mo>,</mo><mi>k</mi></p><p></p><mo stretchy="false">âˆ£</mo><mo>/</mo><p></p><p><mi>W</mi><mn>2</mn></p><p></p><p></p><p></p>, where <i>m</i><sub><i>i</i>,<i>j</i></sub> is the market share of song <i>i</i> in world <i>j</i>. The overall unpredictability measure is the average of this measure over all <i>n</i> songs, i.e., <p></p><p><mi>U</mi><mo>=</mo></p><p><mo>âˆ‘</mo></p><p><mi>j</mi><mo>=</mo><mn>1</mn></p><mi>n</mi><p></p><p><mi>u</mi><mi>i</mi></p><mo>/</mo><mi>n</mi><mo>.</mo><p></p><p></p>. The left bar depicts the results for the first setting, while the right bar depicts the results for the second setting.<p></p

### The Inequality of Success for the Rankings.

<p>The success of song <i>i</i> is defined as the market share <i>m</i><sub><i>i</i></sub> for that song, i.e., </p><p></p><p></p><p><mi>m</mi></p><p><mi>i</mi><mo>,</mo><mi>k</mi></p><p></p><mo>=</mo><p><mi>D</mi></p><p><mi>i</mi><mo>,</mo><mi>k</mi></p><p></p><mo>/</mo><p><mo>âˆ‘</mo></p><p><mi>j</mi><mo>=</mo><mn>1</mn></p><mi>n</mi><p></p><p><mi>D</mi></p><p><mi>j</mi><mo>,</mo><mi>k</mi></p><p></p><p></p><p></p> for a given world <i>k</i>. The success inequality is defined by the Gini coefficient <p></p><p><mi>G</mi><mo>=</mo></p><p><mo>âˆ‘</mo></p><p><mi>i</mi><mo>=</mo><mn>1</mn></p><mi>n</mi><p></p><p><mo>âˆ‘</mo></p><p><mi>j</mi><mo>=</mo><mn>1</mn></p><mi>n</mi><p></p><mo stretchy="false">âˆ£</mo><p><mi>m</mi></p><p><mi>i</mi><mo>,</mo><mi>k</mi></p><p></p><mo>âˆ’</mo><p><mi>m</mi></p><p><mi>j</mi><mo>,</mo><mi>k</mi></p><p></p><mo stretchy="false">âˆ£</mo><mo>/</mo><mn>2</mn><mi>n</mi><p><mo>âˆ‘</mo></p><p><mi>j</mi><mo>=</mo><mn>1</mn></p><mi>n</mi><p></p><p><mi>m</mi></p><p><mi>j</mi><mo>,</mo><mi>k</mi></p><p></p><p></p><p></p>, which represents the average difference of market share for all songs. The figure depicts a boxplot with whiskers from minimum to maximum. The results for the first setting are in black, while the results for the second setting are in red and dashed line.<p></p

### The Quality <i>q</i><sub><i>i</i></sub> (gray) and Appeal <i>A</i><sub><i>i</i></sub> (red and blue) of song <i>i</i> in the two settings.

<p>The settings only differ in the appeal of songs, and not in the quality of songs. In the first setting, the quality and the appeal for the songs were chosen independently according to a Gaussian distribution normalized to fit between 0 and 1. The second setting explores an extreme case where the appeal is negatively correlated with the quality used in setting 1. In this second setting, the appeal and quality of each song sum to 1.</p