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DRN gains at equal damage and protection

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Jul 28th, 2014
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  1. ALT PROOF
  2.  
  3. +----------------------------------------------------TL;DR---------------------------------------------------------+
  4. +When damage and protection are equal, the DRN will benefit the attacker by about 1.42 points of damage on average.+
  5. +------------------------------------------------------------------------------------------------------------------+
  6.  
  7. The purpose of this document is to show how the DRN can benefit an attacker when damage and protection are equal. When damage and protection are vastly different, the DRN has no overall effect because the effects of negative rolls negate the effects of positive rolls, the average difference is exactly zero. This is not the case when damage and protection are equal, a DRN roll may only result in no additional damage (nothing), or a positive value. So the average bonus becomes non-zero!
  8.  
  9. This sheet uses 2d5 rolls to model the DRN, the exploding dice have been abstracted as the average damage bonus of a 1d5 vs. 1d5 roll against the other unit with one of the dice set at '5' (natural six implies a '5' on the previous roll), plus the average damage bonus of an opposed 1d5 roll with your die as any number (the most recent roll), modified by the probability of rolling a natural six in the first place, or 1/6.
  10.  
  11. The odds of rolling a particular value for a 2d5 are:
  12.  
  13. 2: 1/25
  14. 3: 2/25
  15. 4: 3/25
  16. 5: 4/25
  17. 6: 5/25
  18. 7: 4/25
  19. 8: 3/25
  20. 9: 2/25
  21. 10: 1/25
  22.  
  23. Now let's look at the possible combinations of a 2d5 vs 2d5 Protection/Damage DRN roll, assuming both stats are the same. There are (9*9) or 81 values that may result from a total of (5*5)*(5*5) or 625 unique rolls.
  24.  
  25. Prot,Dam|AtkrBonus|Odds
  26.  
  27. 2,2= 0 1/625
  28. 2,3= 1 2/625
  29. 2,4= 2 3/625
  30. 2,5= 3 4/625
  31. 2,6= 4 5/625
  32. 2,7= 5 4/625
  33. 2,8= 6 3/625
  34. 2,9= 7 2/625
  35. 2,10= 8 1/625
  36. Subtotal: 25/625
  37.  
  38. 3,2= 0 2/625
  39. 3,3= 0 4/625 <--note: there are two different ways to roll a '3' in a 2d5 roll,
  40. 3,4= 1 6/625 1,2 and 2,1 so 2*2 is 4 different combinations.
  41. 3,5= 2 8/625
  42. 3,6= 3 10/625
  43. 3,7= 4 8/625
  44. 3,8= 5 6/625
  45. 3,9= 6 4/625
  46. 3,10= 7 2/625
  47. Subtotal: 50/625
  48.  
  49. 4,2= 0 3/625
  50. 4,3= 0 6/625
  51. 4,4= 0 9/625
  52. 4,5= 1 12/625
  53. 4,6= 2 15/625
  54. 4,7= 3 12/625
  55. 4,8= 4 9/625
  56. 4,9= 5 6/625
  57. 4,10= 6 3/625
  58. Subtotal: 75/625
  59.  
  60. 5,2= 0 4/625
  61. 5,3= 0 8/625
  62. 5,4= 0 12/625
  63. 5,5= 0 16/625
  64. 5,6= 1 20/625
  65. 5,7= 2 16/625
  66. 5,8= 3 12/625
  67. 5,9= 4 8/625
  68. 5,10= 5 4/625
  69. Subtotal: 100/625
  70.  
  71. 6,2= 0 5/625
  72. 6,3= 0 10/625
  73. 6,4= 0 15/625
  74. 6,5= 0 20/625
  75. 6,6= 0 25/625
  76. 6,7= 1 20/625
  77. 6,8= 2 15/625
  78. 6,9= 3 10/625
  79. 6,10= 4 5/625
  80. Subtotal: 125/625
  81.  
  82. 7,2= 0 4/625
  83. 7,3= 0 8/625
  84. 7,4= 0 12/625
  85. 7,5= 0 16/625
  86. 7,6= 0 20/625
  87. 7,7= 0 16/625
  88. 7,8= 1 12/625
  89. 7,9= 2 8/625
  90. 7,10= 3 4/625
  91. Subtotal: 100/625
  92.  
  93. 8,2= 0 3/625
  94. 8,3= 0 6/625
  95. 8,4= 0 9/625
  96. 8,5= 0 12/625
  97. 8,6= 0 15/625
  98. 8,7= 0 12/625
  99. 8,8= 0 9/625
  100. 8,9= 1 6/625
  101. 8,10= 2 3/625
  102. Subtotal: 75/625
  103.  
  104. 9,2= 0 2/625
  105. 9,3= 0 4/625
  106. 9,4= 0 6/625
  107. 9,5= 0 8/625
  108. 9,6= 0 10/625
  109. 9,7= 0 8/625
  110. 9,8= 0 6/625
  111. 9,9= 0 4/625
  112. 9,10= 1 2/625
  113. Subtotal: 50/625
  114.  
  115. 10,2= 0 1/625
  116. 10,3= 0 2/625
  117. 10,4= 0 3/625
  118. 10,5= 0 4/625
  119. 10,6= 0 5/625
  120. 10,7= 0 4/625
  121. 10,8= 0 3/625
  122. 10,9= 0 2/625
  123. 10,10= 0 1/625
  124. Subtotal: 25/625
  125.  
  126. Total: 625/625 (as it should be)
  127.  
  128. Let's look at the odds for a damage bonus on the 2d5 roll, before exploding dice are taken into consideration.
  129.  
  130. Chance of +0: 355/625 56.8%
  131. Chance of +1: 80/625 12.8%
  132. Chance of +2: 68/625 10.88%
  133. Chance of +3: 52/625 8.32%
  134. Chance of +4: 35/625 5.6%
  135. Chance of +5: 20/625 3.2%
  136. Chance of +6: 10/625 1.6%
  137. Chance of +7: 4/625 0.64%
  138. Chance of +8: 1/625 0.16%
  139.  
  140. Pre-Explosion 2d5 Average: 1.1328
  141.  
  142. Now let's abstract the die explosions. We can represent bonus dice with a 1d5 vs 1d5 roll because there is an equal chance for either the attacker or defender getting a bonus die (it is symmetrical) and they share a probability coefficient, thus we can lump them together into one average. Muh algebra.
  143.  
  144. Prot,Dam | AtkrBonus | Odds
  145. 1,1= 0 1/25
  146. 1,2= 1 1/25
  147. 1,3= 2 1/25
  148. 1,4= 3 1/25
  149. 1,5= 4 1/25
  150. 2,1= 0 1/25
  151. 2,2= 0 1/25
  152. 2,3= 1 1/25
  153. 2,4= 2 1/25
  154. 2,5= 3 1/25
  155. 3,1= 0 1/25
  156. 3,2= 0 1/25
  157. 3,3= 0 1/25
  158. 3,4= 1 1/25
  159. 3,5= 2 1/25
  160. 4,1= 0 1/25
  161. 4,2= 0 1/25
  162. 4,3= 0 1/25
  163. 4,4= 0 1/25
  164. 4,5= 1 1/25
  165. 5,1= 0 1/25
  166. 5,2= 0 1/25
  167. 5,3= 0 1/25
  168. 5,4= 0 1/25
  169. 5,5= 0 1/25
  170.  
  171. Chance of +0: 15/25 60%
  172. Chance of +1: 4/25 16%
  173. Chance of +2: 3/25 12%
  174. Chance of +3: 2/25 8%
  175. Chance of +4: 1/25 4%
  176.  
  177. [Average AtkrBonus for 1d5 vs 1d5 bonus die]: +0.8
  178.  
  179. In addition to the generating an extra 1d5 roll, a bonus die ensures the previous roll will have at least one '5.'. This modifies the total possible outcomes and thus the average.
  180.  
  181. Prot,Dam | AtkrBonus | Odds
  182.  
  183. (assuming the attacker got the bonus die)
  184. 1,5= 4 1/10
  185. 2,5= 3 1/10
  186. 3,5= 2 1/10
  187. 4,5= 1 1/10
  188. 5,5= 0 1/10
  189.  
  190. (assuming the defender got it)
  191. 5,1= 0 1/10
  192. 5,2= 0 1/10
  193. 5,3= 0 1/10
  194. 5,4= 0 1/10
  195. 5,5= 0 1/10
  196.  
  197. Again, we can lump these together into one average.
  198. [Average AtkrBonus for natural 6 vs 1d5]: +1
  199.  
  200. Now that we have those two averages, we can add them together (1 + 0.8) to get a cumulative benefit of 1.8 for the attacker in the event of a bonus die. But we need to factor in the probability coefficient for bonus dice actually occurring. Because the exploding dice are recursive, you can have a potentially infinite number of bonus dice, but the likelihood of this is infinitesimally small. I've only gone three iterations deep.
  201.  
  202. (1.8 - 1.1328)(2/6) + (1.8 - 0.8)(2/6/6) + (1.8 - 0.8)(2/6/6/6) = 0.287214814
  203. + 1.1328 (this is the average value for 2d5 vs 2d5) = 1.420014814
  204.  
  205. This is much harder to explain, but I will say that the subtraction done within the leftmost factors are to avoid redundantly adding to the overall bonus since the rolls are done not in parallel but serially, I've included those for clarity. The rightmost factors represent increasingly improbable levels of die recursion.
  206.  
  207. [Average Attacker Damage Bonus]: 1.42
  208.  
  209. So there you have it. You can expect the DRN to benefit the attacker by about 1.42 if damage and protection are equal, and you can expect those gains to reduce with any inequality in those two stats as negative modifiers become possible.
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