Problem 5: Mathematically Challenged
Dungeons and Dragons' 4th edition has a game mechanic known as "skill challenges," in which players work together to defeat a challenge that requires using certain in-game skills. Each player's character has a "skill modifier" that represents how good they are at the skill in question, and the challenge has a "difficulty class" (DC) that represents how hard the challenge is. (In the game, there may be more than one skill that can be used for a particular challenge, but this is irrelevant to our analysis because each character can simply use whichever skill he is best at.)
Players go in a circle (order decided on by the players), each one making a skill check. A skill check is made by rolling a 20-sided die and adding it to the skill modifier. If the result is equal to or greater than the DC, the check is successful. For example, if the skill modifier is 7 and the DC is 15, the player needs an 8 or more on the die to earn a success. You continue around the table until 8 successes or 4 failures have been achieved. If the players get to 8 successes first, they win the skill challenge; otherwise, they lose the skill challenge.
A player can also spend his turn to "aid another." This means making a skill check against DC 10. This check does not count as a success or failure. Instead, if it fails, nothing happens and you move on to the next person, while if it succeeds, the next player gets a +2 bonus to his next skill check.
Consider a skill challenge with a DC of 15 and 4 players with skill modifiers of 8, 6, 4, and 2 respectively.
(a) What is the optimal way to use "aid another" to maximize the chance of winning?
(b) Suppose that we want to make "aid another" less powerful by increasing the difficulty of the aid another check from 10 to a higher number. In this particular challenge, how high would we have to raise the difficulty to make never using aid another the optimal strategy?
(Author's Note: This problem has some minor differences from how skill challenges actually work in D+D, but these differences do not change the overall conclusion.)
Answer is here.