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<p>Since <spanclass="math inline">\(2p \geq p^2\)</span> when <spanclass="math inline">\(p \in [0,1]\)</span>, you get a nice boost to your chance of success.
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<p>Since <spanclass="math inline">\(p \geq p^2\)</span> when <spanclass="math inline">\(p \in [0,1]\)</span>, you get a nice boost to your chance of success compared to the one-shot probability <spanclass="math inline">\(p\)</span>.
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This can be illustrated with the following chart:</p>
<p>The charts for <strong>advantage</strong> and <strong>disadvantage</strong> look <em>somewhat</em> symmetrical.
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What the fuss is about?</p>
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<h2id="relative-effect">Relative effect</h2>
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<p>Things start to get interesting when we look at the <em>relative</em> effects of advantage and disadvantage<ahref="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a>.</p>
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<p>Things start to get interesting when we look at the <em>relative</em> effects of advantage and disadvantage<ahref="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a>.</p>
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<p>First, let’s look at the change in the chance of success relative to the <em>one-shot</em> chance of success:</p>
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<p><spanclass="math display">\[
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\mathbb{P}(\mathrm{success})_\mathrm{adv} / p = 2 - p
<p>In the best/worst-case scenario, when a one-shot chance of success is <spanclass="math inline">\(1\)</span> out of <spanclass="math inline">\(20\)</span>, having <strong>advantage</strong> would <em>almost</em> double your chances, but with <strong>disadvantage</strong> you’d be <spanclass="math inline">\(20\)</span> times less likely to succeed!</p>
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<p>This is an interesting asymmetry in what — at first glance — supposed to be symmetric game mechanics.
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I’m not sure if it’s particularly useful outside of the world of Dungeons & Dragons, but the next time I play Baldur’s Gate 3, I’ll be more serious about picking <em>disadvantage</em>-inducing spells to debuff enemies rather than just throwing fireballs at them.</p>
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<p>In the best/worst-case scenario, when a one-shot chance of success is <spanclass="math inline">\(1\)</span> out of <spanclass="math inline">\(20\)</span>, having <strong>advantage</strong> would <em>almost</em> double your chances, but with <strong>disadvantage</strong> you’d be <spanclass="math inline">\(20\)</span> times less likely to succeed!
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Overall, <strong>disadvantage</strong> has a disproportionally large impact on the odds of success compared to the effect of <strong>advantage</strong>.
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This is true for the whole range of one-shot probabilities <spanclass="math inline">\(p \in (0, 1)\)</span>, with the gap becoming visible at <spanclass="math inline">\(p\)</span> around <spanclass="math inline">\(\frac{3}{4}\)</span> and apparent at <spanclass="math inline">\(p \leq \frac{1}{2}\)</span>.</p>
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<p>This is an interesting asymmetry in what — at first glance — supposed to be a symmetric game mechanics.
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I’m not sure if it’s particularly useful outside of the world of Dungeons & Dragons, but the next time I play Baldur’s Gate 3, I’ll be more serious about picking <em>disadvantage</em>-inducing spells to debuff enemies rather than just throwing fireballs at them.<ahref="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a></p>
<liid="fn1"><p>The animated charts were produced with <ahref="https://github.com/3b1b/manim">manim</a> library.<ahref="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
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<liid="fn2"><p>Just don’t mind the case when <spanclass="math inline">\(p = 0\)</span> which is not particularly interesting but would complicate the formulas.<ahref="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
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<liid="fn1"><p>Just don’t mind the case when <spanclass="math inline">\(p = 0\)</span> which is not particularly interesting but would complicate the formulas.<ahref="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
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<liid="fn2"><p>The animated charts were produced with <ahref="https://github.com/3b1b/manim">manim</a> library.<ahref="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
@@ -85,9 +85,11 @@ Plotting these charts side-by-side, there's much less symmetry than originally a
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## Conclusion
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In the best/worst-case scenario, when a one-shot chance of success is $1$ out of $20$, having **advantage** would *almost* double your chances, but with **disadvantage** you'd be $20$ times less likely to succeed!
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Overall, **disadvantage** has a disproportionally large impact on the odds of success compared to the effect of **advantage**.
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This is true for the whole range of one-shot probabilities $p \in (0, 1)$, with the gap becoming visible at $p$ around $\frac{3}{4}$ and apparent at $p \leq \frac{1}{2}$.
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This is an interesting asymmetry in what --- at first glance --- supposed to be symmetric game mechanics.
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I'm not sure if it's particularly useful outside of the world of Dungeons & Dragons, but the next time I play Baldur's Gate 3, I'll be more serious about picking *disadvantage*-inducing spells to debuff enemies rather than just throwing fireballs at them.
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This is an interesting asymmetry in what --- at first glance --- supposed to be a symmetric game mechanics.
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I'm not sure if it's particularly useful outside of the world of Dungeons & Dragons, but the next time I play Baldur's Gate 3, I'll be more serious about picking *disadvantage*-inducing spells to debuff enemies rather than just throwing fireballs at them.[^manim]
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