Luck titles guide


 * Please note: the following analyses describe the expected values for tickets won and games lost at the various games of chance. By nature of statistical statements, even after hours of playing there is no guarantee that the actual, individual results will be exactly as or even close to what has been calculated here. If you are convinced you have found a flaw in the calculations, please present the flaw as precisely as possible and discuss it in the article's talk page before changing the article.

Overview
The Unlucky and Lucky titles are account-based titles which are obtained through various in-game activities:


 * Playing the Shing Jea Boardwalk games Nine Rings and Rings of Fortune. These games last for certain special events and will return in the future.
 * Opening chests with a lockpick. Retaining the lockpick adds points to the Lucky title; breaking it adds points to the Unlucky title. This is a permanent feature of the game.
 * Using a Four-Leaf Clover and having 15% DP removed from the character awards 4 Lucky points. The clovers were part of the Lucky weekend event.
 * Participating and winning in the ring game during the finales of the Wintersday 2006 and Wintersday 2007 event awarded Lucky points.

Boardwalk games analysis

 * Nine Rings
 * The amount of tickets gained is statistically equal in every location, but more games are lost on corners (awarding more Unlucky points and the same amount of Lucky points), so the following analysis will be based on playing on a corner ring.
 * Rings of Fortune
 * All locations are equally advantageous.

Achieving one title only

 * If one wants to achieve the Unlucky title with the least amount of Gold spent, Rings of Fortune is optimal.
 * Otherwise, Nine Rings is better. Notably, it is faster for either title.

Achieving both titles at the same time

 * It is beneficial to play both games in order to reach both titles of the same tier at the same time.

Lockpick analysis
A Lockpick has a certain chance of breaking after opening a chest. If a lockpick breaks, it will add [250 * retention rate points] to the Unlucky title; on the other hand, if the lockpick is retained, it will add 250 points to the Lucky title. The numbers are the same in Normal and Hard Mode.

The rate at which (Un)Lucky points are gained depends on the number of chests opened per unit time and the chance of retaining lockpicks. The chests-per-unit-time depends on the difficulty of the area, the amount of chests there and the skill of the player/team. It cannot be quantified and will be omitted in this analysis.

Note that all calculations involving gold are based on a lockpick price of 1,500. Lockpicks can be bought for 1,200 from Discount Merchants. For discounted lockpicks, all costs are to be reduced accordingly.

The following formulas are relevant to the lockpick analysis:


 * Let R be the chance to retain a lockpick after use. (See Lockpick for details on how to calculate R). R is a number between 0 and 1; it is not the percentage. (1-R) is then the lockpick's chance of breaking. All numbers involving R have to be taken as statistical averages.


 * Unlucky points per key = $$250R$$
 * Unlucky points per chest = $$250R(1-R)$$
 * Price per key = 1500
 * Price per Unlucky point = $$(Price per key) / (Unlucky points per key) = 1500/ (250R) = 6 / R$$


 * Lucky points per chest = $$250R$$
 * Price per chest = $$1500(1-R)$$
 * Price per Lucky point = $$(Price per chest)/(Lucky points per chest) = (1500(1-R))/(250R) = 6 *(1-R)/R$$

For the purpose of calculating in the Treasure Hunter title with the Lucky titles and when they will change R when they "level", here are some additional formulas:


 * Chests opened per key = $$1/(1-R)$$ = Treasure Hunter points per key if chest is "high end"
 * Unlucky points per key = $$250R$$
 * Lucky points per key = $$(Lucky points per chest) * (chests opened per key) = 250R / (1-R)$$

Keys needed for next title = points needed for next title / points per key

These results mean the following:
 * The price of Unlucky points is inverse to the retention rate, decreasing as the retention rate increases.
 * The highest rate of Unlucky points per chest is achieved when your rate is 50%.
 * Reaching the maximum Unlucky title via lockpicks alone would require spending 12,000, if you only opened chests at a 50% retention rate.
 * The price of Lucky points decreases quadratically as R increases, because the price per chest drops and at the same time the number of Lucky points per chest increase.
 * start and end values:
 * At the minimum R = 0.1, 1000 Lucky points cost 54,000 on average.
 * At R=0.98, 1000 Lucky points cost 122 (Note that for R to equal .98, the Lucky title would already have to be maxed.)


 * For illustration, these are the most extreme prices to reach the maximum Lucky title (2,500,000 points), starting from zero Lucky points. These calculations take into account improving retention rates from increasing Lucky and Treasure hunter title rank:
 * At Treasure hunter rank 7 and only picking Ascalonian chests on Normal mode, on average 10,600 chests have to be opened and 887 have to be spent.
 * At Treasure hunter rank 0 and only picking Hard mode chests, on average 26,300 chests have to be opened and 24,450 have to be spent.

Comparison to Nine Rings
Question: a level 20 character with rank 0 Treasure Hunter wants to farm Lucky points. Which method is more favorable, playing Nine Rings or farming Ascalonian chests on Normal mode?


 * Nine Rings can be played while AFK and requires no user action whatsoever beyond buying tickets beforehand.
 * Nine Rings awards 3454 Lucky points per hour. At 65% base retention rate, this equals opening one chest every $$60 minutes * 250 * 0.65 / 3454 = 2.8$$ minutes. This figure already includes breaking lockpicks.
 * At 3048 per hour and 3454 points per hour, each Lucky point costs 0.882 at Nine Rings, whereas at R=0.65, each point costs 6*(1-R)/R = 3.2 for chest farming. Chest farming for Lucky points becomes cheaper at R > 0.87, not considering chest drops and discounted lockpicks. Assuming discounted lockpicks and 100 value per drop, break-even occurs at R > 0.83.
 * Unlucky points cost 12.5 each for Nine Rings and 60 each for chest farming.
 * Chests can be run all the time, while Nine Rings is only available during special events.
 * Chest farming has added value in the sense that it also increases the treasure hunter (assuming high end chests) and wisdom titles (from identifying gold chest drops). It also potentially gives useful item drops and equipment modifications.

The best method depends mostly on the chance to retain the lockpick and therefore mostly on the Treasure Hunter and Lucky title track. Chest running is more cost-efficient at >85% retention rates and is available all year long but requires farming with all the tediousness involved, while Nine Rings awards a lot more points per day and can be played without interaction but is only rarely available.