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General Guide by Karhumies

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League of Legends Build Guide Author Karhumies

Rune Purchase Optimization: Case of Flat AD Marks

Karhumies Last updated on September 7, 2013
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Why Write this Guide?

I have previously heard some arguments of "Never buy any runes until Tier 3 runes become available at Lv20" without any mathematical or statistical back-up. So, I decided to do some math analysis myself and compare the IP cost-effectiveness as well as the marginal utility of rune purchases.

Last updated: 7th September 2013


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Introduction of Flat AD Marks

Tier 1
Official Name: Lesser Mark of Attack Damage
Becomes available for purchase at: 0 Cumulative XP (Lv1)
Cost: 15 IP (Least Expensive)
Effect: + 0.53 AD (Least Effective)


Cumulative effect at Lv 20 (cumulative AD): 3.18 AD (Least effective in cumulative effect)
Cost-effectiveness (IP/AD): 28.30 IP/AD (Most cost effective)
Utility (AD/IP): 0.03533
Marginal utility at Lv 20, given that you could be purchasing Tier 3 runes: 0.00000 (used as a cut-off point in marginal utility discount analysis)

Tier 2
Official Name: Mark of Attack Damage
Becomes available for purchase at: 2926 Cumulative XP (Lv10)
Cost: 80 IP
Effect: +0.74 AD
Cumulative effect at Lv 20 (cumulative AD): +4.44 AD
Cost-effectiveness (IP/AD): 108.11 IP/AD
Utility (AD/IP): 0.00925
Marginal utility at Lv 20, given that you could be purchasing Tier 3 runes: 0.00000 (used as a cut-off point in marginal utility discount analysis)

Tier 3
Official Name: Greater Mark of Attack Damage
Becomes available for purchase at: 15676 Cumulative XP (Lv20)
Cost: 205 IP (Most expensive)
Effect: +0.95 AD (Most Effective)
Cumulative effect at Lv 20 (cumulative AD): 5.70 (Most effective in cumulative effect)
Cost-effectiveness (IP/AD): 215.79 IP/AD (Least cost effective)
Utility (AD/IP): 0.00463
Marginal utility at Lv 20: 0.00463


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Effectiveness Analysis (no consideration for IP cost)

Tier 1 - Lesser
6x Tier 1 AD Mark = 3.18 AD (90 IP)

Tier 2
6x Tier 2 AD Mark = 4.44 AD (480 IP)

Tier 3
6x Tier 3 AD Mark = 5.70 AD (1230 IP)


(NOTE for people inexperienced in reading statistics: In this graph, higher = better)

For comparison purposes:
Total cumulative cost at Lv 20 for always purchasing the most effective AD Marks as soon as they become available
3x Tier 1 at 15 IP each (at Lv1, Lv 4, and Lv 7)
+ 5x Tier 2 at 80 IP each (3 at Lv 10; 1 at Lv 11, Lv 14 and LV 17)
+ 6x Tier 3 at 205 IP each (6 at Lv 20)

-> Total cumulative cost at Lv 20:
45 IP for Tier 1 + 400 IP for Tier 2 + 1230 IP for Tier 3
= 1675 IP (5.70 AD at Lv 20)


Conclusion:
If IP cost was not a factor, you should buy as many Tier 1, Tier 2 and Tier 3 Marks as fits into your rune page as soon as they become available.
However, in reality, IP cost IS a factor. Therefore, see next section.


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Cost-effectiveness analysis

Tier 1 - Lesser
Cost / Effect: (15 IP / 0.53 AD)
-> 28.30 IP/AD

Tier 2
Cost / Effect: (80 IP / 0.74 AD)
-> 108.11 IP/AD

Tier 3 - Greater
Cost / Effect: (205 IP / 0.95 AD)
-> 215.79 IP/AD


(NOTE for people inexperienced in reading statistics: In this graph, <-to the left = better & higher = better)

Cost effectiveness comparison:
Tier 3 cost-effectiveness / Tier 1 cost-effectiveness
(215.79 IP/AD) / (28.30 IP/AD)
= 7.625...
-> Tier 1 Attack Damage Marks are more than 7,6 times more cost effective (IP/AD) than Tier 3 Attack Damage Marks.


For comparison purposes:
Total cumulative cost at Lv 20 for always purchasing the most cost-effective AD Marks as soon as they become available
6x Tier 1 at 15 IP each (at Lv1, Lv 4, and Lv 7, Lv 11, Lv 14 and Lv 17)
= 90 IP, (3.18 AD at Lv 20)


Conclusion:
If Marks were not limited (currently: maximum 9 Marks per rune page), you should never buy any Tier 2 or Tier 3 Marks of Attack Damage at any level. Because Tier 1 runes are the most cost effective.
However, since the Marks ARE limited, see next section.


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Discounted utility analysis of Flat AD runes

Calculating utility:

NOTE: Utility (AD/IP) is a variable which has no interest for us in itself; it is only used as a tool to meaningfully compare between the alternatives. Choosing to use a different function than my (AD/IP) for measuring utility will yield completely different results from my model.

Tier 1
Utility (AD/IP): 0.03533
Marginal utility at Lv 20 (see assumption #1): 0.00000 (used as a cut-off point in marginal utility discount analysis)

Tier 2
Utility (AD/IP): 0.00925
Marginal utility at Lv 20 (see assumption #1): 0.00000 (used as a cut-off point in marginal utility discount analysis)

Tier 3
Utility (AD/IP): 0.00463
Marginal utility at Lv 20: 0.00463

Assumption #1: At Summoner Lv 20, the marginal utility of Tier 1 and Tier 3 runes will become 0 because from that point on, you can purchase Tier 3 runes.
NOTE: This is an exaggeration, since Tier 1 and 2 runes DO have some marginal utility from 6 Marks -> 9 Marks ... but I am nevertheless going to make this assumption in my model. In practice, the utility of low tier Marks only becomes 0 when the rune page is maxed -> from that point on, utility can be gained only by upgrading the runes on the maxed rune page to higher tier runes.

Assumption #2: The first AD points gained from runes are not more valuable than the following AD points gained from runes.
NOTE: In practice, for last-hitting purposes, some players consider the first few AD points to have more in-game value than the following AD points gained from runes, because the first points are the most important for making last-hitting easier at low in-game levels. However, my model does not reflect this = my model does not have diminishing returns for AD from runes (Flat AD Marks).

Assumption #3: The decrease in utility of Tier 1 and Tier 2 AD Marks is a constant (not a variable). The decrease is illustrated by the utility line gradient in the graph.


(NOTE for people inexperienced in reading statistics: In this graph, higher = better)

Tier 3 line at levels lower than 20 is included in the graph as a reference for when it "makes more sense" to save IP for Tier 3 runes rather than buy Tier 1 or Tier 2 runes.

The point in the graph where Tier 1 line crosses Tier 3 line is when purchasing Tier 1 runes is no longer useful from cost-effectiveness and utility point of view. Because this happens sometime after LV 17 (LV17 = 11326 XP), finding out the exact point is not relevant from a practical application perspective.

For comparison purposes:
Total cumulative cost at Lv 20 for always optimizing the utility of AD Marks
6x Tier 1 at 15 IP each (at Lv1, Lv 4, Lv 7, Lv 11, Lv 14 and Lv 17)
+ 6x Tier 3 at 205 IP each (6 at Lv 20)

-> Total cumulative cost at Lv 20:
90 IP for Tier 1 + 1230 IP for Tier 3
= 1320 IP (5.70 AD at Lv 20)


Conclusions:

#1: In terms of cost-effectiveness and utility, you should buy Tier 1 flat AD Marks until LV 17, then buy Tier 3 Flat AD Marks from Lv 20 onwards. You should never buy Tier 2 Flat AD Marks.

#2: This particular result works only for Flat AD Marks. Other runes (e.g. Flat Magic Resistance Glyphs or Flat AD Seals) have different cost-effect ratios from Flat AD Marks = you need to calculate each rune type individually to achieve results.

#3: If you don't care about the cost-effectiveness or utility of runes, you should have ignored this guide instead of reading it.


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Comparison Between the Different Rune Purchase Strategies

Strategy A: Most Effective, regardless of IP cost
= Tier 1 at Lv 1-7, then Tier 2 at Lv 10-17, then Tier 3 at Lv20 (see first section, about effectiveness)

Strategy B: Most Cost-Effective, regardless of how effective
= Tier 1 only, Lv 1-20 (see second section, about cost-effectiveness)

Strategy C: Utility Maximization = Optimization
(kind of optimizing between strategies #1 and #2)
= Tier 1 first Lv 1-17, then Tier 3 at Lv 20

Strategy D: What the Guys on the Internet Tell You to Do
= Don't buy anything until Lv 20, then buy Max. Tier 3 runes at once



(NOTE for people inexperienced in reading statistics: The greater the area between the Strategy line and y=0, the better)

In this graph, we can see that Strategy A is the most effective without taking into account the IP cost.

How to take into account the IP cost:

Comparison between Strategy A and Strategy C:
(Area between the lines Strategy A and Strategy C) / (Price difference between Strategy A and Strategy C)
= (Strategy A area - Strategy C area) / (1675 IP - 1320 IP)
= ("not that much area", based on visual evaluation) / 355 IP
-> at the cost of spending extra 355 IP, not a lot has been gained in practice. In my opinion, not worth the IP cost to spend it on Tier 2 Flat AD marks.


Comparison between Strategy C and Strategy D:

(Area between Strategy C and Strategy D) / (Price difference between Strategy C and Strategy D)
= (Strategy C area - Strategy D area) / (1320 IP - 1230 IP)
= ("quite a lot of area", based on visual evaluation) / 90 IP
-> at the cost of spending extra 90 IP, quite a lot has been gained in practice. In my opinion, this much gain in AD between levels 1-19 is worth the relatively low IP price of 90 IP.

Also note: Strategy D is the least successful strategy between levels 1-19. This purchase strategy only begins to have any positive effect on game play after level 20.


Conclusion:
In my personal opinion, rune purchase Strategy C is the best strategy for purchasing Flat AD runes until level 20.

If you have a different opinion, we can have a discussion about this in the Comments section.


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References

League of Legends Wiki: Leveling Up - Cumulative XP Needed (table)
http://leagueoflegends.wikia.com/wiki/Experience_%28summoner%29#Leveling_up


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