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

Abstract

<p>The measure of unpredictability <i>u</i>_<i>i</i> for song <i>i</i> is defined as the average difference in market share <i>m</i>_<i>i</i> 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>∑</mo></p><p><mi>j</mi><mo>=</mo><mn>1</mn></p><mi>W</mi><p></p><p><mo>∑</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">∣</mo><p><mi>m</mi></p><p><mi>i</mi><mo>,</mo><mi>j</mi></p><p></p><mo>−</mo><p><mi>m</mi></p><p><mi>i</mi><mo>,</mo><mi>k</mi></p><p></p><mo stretchy="false">∣</mo><mo>/</mo><p></p><p><mi>W</mi><mn>2</mn></p><p></p><p></p><p></p>, where <i>m</i>_{<i>i</i>,<i>j</i>} 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>∑</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