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General Tabletop Discussion
Level Up: Advanced 5th Edition (A5E)
Minor Advantage and Minor Disadvantage
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<blockquote data-quote="NotAYakk" data-source="post: 8100348" data-attributes="member: 72555"><p>I'm trying to break that asymmetry where the same roll is better with Handicap than Edge.</p><p></p><p>Here is another stab at it:</p><p><strong>Edge</strong>: 2d20, take the higher one unless both are odd; then take the lower one.</p><p><strong>Handicap</strong>: 2d20, take the lower one unless both are even; then take the higher one.</p><p></p><p>Edge is then 75% advantage 25% disadvantage, and vice versa for Handicap.</p><p></p><p>Math:</p><p>[spoiler]</p><p>With a base probability of success of P, advantage is 1-(1-P)^2 or 2P-P^2. Disadvantage is P^2.</p><p></p><p>K advantage and (1-K) disadvantage is then K(2P-P^2) + (1-K)P^2 = 2KP - KP^2 + P^2 - KP^2 = 2KP + (1-2K)P^2.</p><p></p><p>At K=0.5 this is P, aka no advantage or disadvantage.</p><p></p><p>At K=1 this is 2P-P^2, aka advantage.</p><p>At K=0 this is P^2, aka disadvantage.</p><p></p><p>d/dK is 2P - 2P^2 = 2P(1-P). This is always positive, and maximal at P=0.5 (where it is 1/2) and zero at P=1 and 0.</p><p></p><p>What this means is that for any case besides certain failure/success, Advantage > Edge > Normal > Handicap > Disadvantage in terms of probability of success.</p><p></p><p>By having double-odd be bad on Edge, it doesn't block crits; your crit chance is doubled. Your natural 1 chance is halved over a normal roll.</p><p>With advantage, your natural 1 chance is basically eliminated (1 in 400).</p><p>Similarly, for Handicap, your natural 1 chance is doubled (just like disadvantage), and your crit chance is halved over a normal roll.</p><p>With disadvantage, your crit chance is basically eliminated (1 in 400).</p><p></p><p>This assumes crit on a 20; on a 19-20 the math changes a bit.</p><p></p><p>Chance of a 20: 1- .95 * .95 = 9.75%.</p><p>Chance of a 19 but not a 20: Die 1 lands on an even value that isn't 20 (9/20 chance) times die 2 lands on 19 (1/20), and vice versa (*2), for 18/400, plus double 19 (1/400).</p><p>Total: 14.5%</p><p></p><p>With real advantage it would be 19%.</p><p>With a normal roll it would be 10%.</p><p></p><p>Crit on a 18-20 with edge.</p><p>Same for 20 and 19.</p><p>For an 18 but not 19 or 20, it is die 1 on an 18, plus die 2 from 1 to 18, plus die 2 on 18 and die 1 from 1 to 17, for 35/400.</p><p>Total is 23.25%.</p><p></p><p>With real advantage it is 27.75%</p><p>With a normal roll it is 15%</p><p></p><p>again, about half way in between.</p><p>[/spoiler]</p><p>So a neutral roll is 50% advantage/50% disadvantage on this scale.</p><p></p><p>I think I like this one better. For a given roll, going up the Disadvantage -> Handicap -> (Skip Neutral) -> Edge -> Advantage always makes the roll better (well, no worse). And unless you land on Neutral, the same roll can be interpreted as any of those.</p></blockquote><p></p>
[QUOTE="NotAYakk, post: 8100348, member: 72555"] I'm trying to break that asymmetry where the same roll is better with Handicap than Edge. Here is another stab at it: [B]Edge[/B]: 2d20, take the higher one unless both are odd; then take the lower one. [B]Handicap[/B]: 2d20, take the lower one unless both are even; then take the higher one. Edge is then 75% advantage 25% disadvantage, and vice versa for Handicap. Math: [spoiler] With a base probability of success of P, advantage is 1-(1-P)^2 or 2P-P^2. Disadvantage is P^2. K advantage and (1-K) disadvantage is then K(2P-P^2) + (1-K)P^2 = 2KP - KP^2 + P^2 - KP^2 = 2KP + (1-2K)P^2. At K=0.5 this is P, aka no advantage or disadvantage. At K=1 this is 2P-P^2, aka advantage. At K=0 this is P^2, aka disadvantage. d/dK is 2P - 2P^2 = 2P(1-P). This is always positive, and maximal at P=0.5 (where it is 1/2) and zero at P=1 and 0. What this means is that for any case besides certain failure/success, Advantage > Edge > Normal > Handicap > Disadvantage in terms of probability of success. By having double-odd be bad on Edge, it doesn't block crits; your crit chance is doubled. Your natural 1 chance is halved over a normal roll. With advantage, your natural 1 chance is basically eliminated (1 in 400). Similarly, for Handicap, your natural 1 chance is doubled (just like disadvantage), and your crit chance is halved over a normal roll. With disadvantage, your crit chance is basically eliminated (1 in 400). This assumes crit on a 20; on a 19-20 the math changes a bit. Chance of a 20: 1- .95 * .95 = 9.75%. Chance of a 19 but not a 20: Die 1 lands on an even value that isn't 20 (9/20 chance) times die 2 lands on 19 (1/20), and vice versa (*2), for 18/400, plus double 19 (1/400). Total: 14.5% With real advantage it would be 19%. With a normal roll it would be 10%. Crit on a 18-20 with edge. Same for 20 and 19. For an 18 but not 19 or 20, it is die 1 on an 18, plus die 2 from 1 to 18, plus die 2 on 18 and die 1 from 1 to 17, for 35/400. Total is 23.25%. With real advantage it is 27.75% With a normal roll it is 15% again, about half way in between. [/spoiler] So a neutral roll is 50% advantage/50% disadvantage on this scale. I think I like this one better. For a given roll, going up the Disadvantage -> Handicap -> (Skip Neutral) -> Edge -> Advantage always makes the roll better (well, no worse). And unless you land on Neutral, the same roll can be interpreted as any of those. [/QUOTE]
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