Attack Bar (ATB)¶

Warning

This section is currently incomplete and under construction.

Summoners War is a turn based game. The attack bar (ATB) of a monster determines when it gets a turn. The attack bar can be manipulated in battle via numerous effects.

First a few definitions:

$\tau$ is the ticksize (see the section on speed ticks)
$B$ is the base speed of a monster.
$t$ is the Sky Tribe Totem value, e.g. usually 15% for a maxed totem.
$l$ is the SPD leaderskill value.
$b$ is the sum of all SPD stats on your runes.
$s$ is the value for Swift runes. $s = \begin{cases} 0.25 & \text{Monster is on Swift} \\ 0 & \text{else} \end{cases}$
$S_{combat}$ will be used for the combat speed a monster has in battle.

Speed¶

Without a SPD buff your monster has a total speed of

$S_{combat} = \lceil \left(B \cdot (1 + t + l + s) + b\right) \rceil$

$\lceil x \rceil$ means "round $x$ up to the nearest integer value".

If there is a SPD buff present then your monster has a speed of

$S_{combat} = \lceil \left(B \cdot (1 + t + l + s) + b\right) \cdot (1 + 0.3 \cdot (1 + a))\rceil$

where $a$ is the artifact value for "Increase SPD Effect +N%" if the monster has such an artifact equipped, otherwise it is 0.

Attention

When talking about a monster people usually take the value from their monster box, the base speed plus the green value. So for example if their page shows "111 + 189" then they will say "my Bernard is 300 SPD fast". This value is useless for anything except for bragging rights, if you want to actually do any calculations like speed tuning your monsters then you always need to use the combat speed.

Speed Ticks¶

A Summoners War battle works in ticks. During each tick, every monster on the battlefield gets a certain amount of Attack Bar that is proportionate to its combat speed value, determined by the tick size $\tau$ such that a monster gets $\tau \cdot S_{combat}$ ATB. In most areas of the game the tick size is 7%, the only exceptions are in Raids (4.5%) and RTA (1.5%).

If after this ATB increase no monster on the battlefield has an ATB value of 100% or more (ATB can go beyond 100% ("overflow")), nothing happens and the next tick starts. But if at least one monster has at least 100% ATB, then one monster will get a turn and then only once the monster finished moving will the next tick start.

The monster that will move is determined by a set of rules. In most cases it is simply the monster with the highest amount of ATB. If two monsters have the same amount of ATB (for example because they have the same SPD and thus both reach 100% at the same time), the monster that is placed more left will take priority. In case the tie is between the two left-most monsters of opposing teams, priority is given to the attacker (first pick in RTA).

Graphically this process can be depicted like this:

graph LR
  A[Battle Start] --> B[Tick: ATB for everyone];
  B --> C{Anyone at >= 100%?}
  C -->|Yes| D[Monster moves];
  D --> B;
  C ---->|No| B;

Notes

Extra turns from violent procs do not add another tick, the monster just moves again.
The graph is intentionally kept simple, special effects like Triana instantly getting a turn and thus fixing the turn order are not represented in order to not blow the graph up.

The number of ticks it takes the first monster to reach 100% ATB and thus get the first turn of the battle is determined by the speed (of the fastest monster) and the area in which the battle happens. Calculating the minimum speed needed to reach 100% ATB in $n$ ticks with a ticksize of $t_s$ is given by $\lceil\frac{100}{t_s \cdot n}\rceil$ .

Using this formula, one obtains the following values:

Normal (7%)Raid (4.5%)RTA (1.5%)

Ticks to reach 100% ATB	Minimum SPD needed
1	1429
2	715
3	477
4	358
5	286
6	239
7	205
8	179
9	159
10	143

Ticks to reach 100% ATB	Minimum SPD needed
1	2223
2	1112
3	741
4	556
5	445
6	371
7	318
8	278
9	247
10	223
11	203
12	186
13	171
14	159
15	149

Ticks to reach 100% ATB	Minimum SPD needed
1	6667
2	3334
3	2223
4	1667
5	1334
6	1112
7	953
8	834
9	741
10	667
11	607
12	556
13	513
14	477
15	445
16	417
17	393
18	371
19	351
20	334
21	318
22	304
23	290
24	278
25	267
26	257
27	247
28	239
29	230
30	223

As you can see, the individual tick brackets are much smaller for the smaller tick sizes.

Speed Tuning¶

Speed tuning generally means the process of fine-tuning your monsters' SPD values such that they move in a specific order and cannot be interrupted by enemy units by normal means. This typically requires at least one ATB boost.

Now in order to speed tune your team, we will assume that you have the fastest monster on the battlefield, that this monster can boost the ATB of the rest of your team by a boost amount $B$ , and that it will get its first turn after $n$ ticks. Further we will also designate the combat speed values of your monsters with $S_1, S_2, S_3, S_4$ (omit $S_4$ for GW/Siege), with the number indicating the order in which you want the monsters to move. Since we want to assure that your team will be speed tuned even in the most difficult case, we will assume the fastest enemy unit to be only slightly slower or even exactly as fast as your fastest unit, so $S_{enemy} = S_e \approx S_1$ .

After your first monster moved, every monster on the battlefield will have gotten $n + 1$ ticks of ATB. Since turn order is determined by the amount of ATB of each monster, it's easy to see that in order for your team to get the second turn (in the next tick directly, otherwise it would be possible to cut in) the following inequality needs to hold:

$(n + 1)\tau \cdot S_2 + B \geq (n + 1)\tau \cdot S_e$

Since after every turn all monsters get an ATB tick, one can trivially extend this equation to the general case of the $i$ th monster needing to fulfill this inequality for all $i > 0$ :

$(n + i)\tau \cdot S_{i + 1} + B \geq (n + i)\tau \cdot S_e$

Now shifting some things around, one can express this as

$S_{i + 1} \geq S_e - \frac{B}{(n + i)\tau}$

Now since we assumed the enemy to be as fast as your fastest monster, we can substitute $S_e$ with $S_1$ and then divide by $S_1$ to obtain the fraction of your fastest monster's SPD that your other monsters need to be speed tuned:

$r_i \equiv \frac{S_{i + 1}}{S_1} = 1 - \frac{B}{S_1(n + i)\tau}$

Since $n$ itself is directly determined by $S_1$ , all you need to calculate how fast your monsters need to be is the SPD of your first moving monster (the booster), and the amount of boost you get from it.

As one can directly see, the later monsters need higher speed values in order to speed tuned than the ones moving directly after the booster since $\lim_{i \to \infty} r_i = 1$ . This means that in order to maintain the proper turn order, your slowest monster determines how much speed everyone needs. This means that you only need to calculate the ratio for the last monster ( $r_3$ in Arena etc., $r_2$ for GW/Siege) and then just make the monsters in between faster by like 1 SPD each for the correct order.

Example

Taking a 300 SPD Tiana (reminder: combat speed, not what you see on the monster page) as an example in arena, then $n = 5$ and $B = 30%$ , which results in the following ratios for your other monsters:

$\begin{align} r_1 &= 1 - \frac{30}{300 \cdot 6 \cdot 0.07} \approx 0.7619 \\ r_2 &= 1 - \frac{30}{300 \cdot 7 \cdot 0.07} \approx 0.7959 \\ r_3 &= 1 - \frac{30}{300 \cdot 8 \cdot 0.07} \approx 0.8214 \end{align}$

As described above you also need to maintain the proper turn order, so while technically the first two monsters don't need to have that much SPD to be speed tuned to Tiana, they still need to be faster than your slowest monster, which needs the highest fraction of Tiana's SPD in order to be speed tuned. This means all of your monsters have to be faster than ~82% of Tiana's SPD.

Tip

While you can of course calculate the necessary ratios by hand, there are tools available that will do the speed tuning for you. SW Tools and SWOP being the most popular choices.

Leo¶

Leo (Wind Dragon Knight) has a special passive:

The attack speed of all enemies and allies is limited to your attack speed. This effect does not affect enemies with the same skill or Boss Monsters. Decreases the enemy's Attack Bar by 15% with an attack. [Automatic Effect]

This passive puts an upper limit (his own SPD) on all SPD values in the battle when they are used for a calculation. This means that speed scaling skills will deal less damage than usual (since Leo is typically built with no speed), but it also means that the amount of ATB a monster gets in each tick is limited. Typically Leo is the slowest monster on the battlefield and thus everyone will be brought down to his SPD value and then everyone's ATB will tick at the same rate.

Now this last part is what makes Leo so effective. In addition of limiting everyone's SPD to his own, he also introduces a few other special rules that affect the turn order. In a battle with Leo, Leo will always go first. Then after him the turn order is determined by ATB status as usual, with the exception of ties now being broken by the original SPD value instead of placement order.

Couple this with Leo reducing the ATB of an enemy on hit, it's pretty easy to see why he works so well on Arena Offence. A typical Leo battle works like this:

Leo is on the field and thus limits everyone to his own SPD
Every monster reaches 100% ATB in the same tick and everyone has the same amount of ATB
By the rules of Leo, Leo moves first
Leo ideally reduces the ATB of the fastest enemy monster
Your fastest monster moves and uses a skill to boost the ATB of your team
Since your other monsters now have more ATB than the enemy, they can move uninterrupted, even though they would normally be much slower

So if you have a Leo on the field, you only need one very fast monster (that can boost ATB and ideally also has at least an ATK buff) and can bring two super slow but heavy hitting units whereas without Leo your damage dealers would still need a decent amount of SPD in order to be speed tuned. A natural setup for this is Leo Bastet/Megan Lushen Lushen or variants of this.