WH spam strategy

Each player spams Wormhole (WH) as fast as possible.
After all WHs have been used, all players move to a brand new game.
They should now be able to purchase more WHs than the previous game.

All this doesn't take into account levels from killing mobs

The strategy exploits the following flaws:

growth created by more than 1000 active players
growth by jumping 10 levels at every 100th level

Both flaws result in the player having more badge points to spend the next game as result of reaching a higher max level.

$$\text{badge points} = \text{max level} : 10$$$$\text{number of wormholes} = \lfloor\text{badge points} \div 100\rfloor$$

We can calculate the number WHs we will have after N games, given the number of active players, and number of WHs per player at the start.

$$ \text{wormholes at start} \times (\frac{\text{number of active players}}{1000} + \frac{\text{number of active players}}{10000})^{\text{Nth game}-1} = \text{number of wormholes at Nth game} $$

Let's calculate the number WHs each player will have at the start of the second game, given 1000 active players and 10 WHs each for the first game.

$$ 10 \times (\frac{1000}{1000} + \frac{1000}{10000})^{2-1} = 11 $$$$ 10 \times (1+0.1)^1 = 11 $$

Now with 1500 active players.

$$ 10 \times (\frac{1500}{1000} + \frac{1500}{10000})^{1} = 16.5 $$

What about 10th game?

$$ 10 \times (\frac{1500}{1000} + \frac{1500}{10000})^{10-1} = 906.47 $$

Let's plot this to visualize it better.



In [1]:

    
%pylab inline
import matplotlib.pyplot as plt









    



Populating the interactive namespace from numpy and matplotlib



In [2]:

    
n_wormholes = 10
n_games = 20

def calc(n_active, n_game, multiplier=1.0):
    return n_wormholes * multiplier *(n_active/1000.0 + n_active/10000.0)**(n_game-1)



In [3]:

    
title = "WH spam stategy (starting w/ %d WHs for each player)" % n_wormholes

plt.figure(figsize=(8,4), dpi=72, facecolor='w')
plt.title(title)

x =  range(1,n_games + 1)

for n_active in reversed([910] + range(1000,1600,100)):
    y = map(lambda i: calc(n_active, i), x)
    plt.plot(x, y, label='active players: %d' % n_active)

plt.yscale('log')    
plt.ylabel("number of WHs at start")
plt.xlabel("N game")
plt.xticks(x)
plt.legend(bbox_to_anchor=(1, 1), loc=2)
plt.grid(True)
plt.show()

That's crazy and great, but remember WH has 60s cooldown.



In [4]:

    
plt.figure(figsize=(8,4), dpi=72, facecolor='w')

for n_active in reversed([910] + range(1000,1600,100)):
    y = map(lambda i: (calc(n_active, i)) / 60, x)
    total = sum(y)
    for i, n in reversed(list(enumerate(y))):
        total, y[i] = total - y[i], total
        
    plt.plot(x, y, label='active players: %d' % n_active)

plt.title("60s on every WH use")
plt.ylabel('hours to N game')
plt.yscale('log')
plt.yticks(map(lambda x: 10**x, range(-1,6)), map(lambda x: 10**x, range(-1,6)))
plt.xlabel("N game")
plt.xticks(x)
plt.legend(bbox_to_anchor=(1, 1), loc=2)
plt.grid(True)
plt.show()

Nobody has that kind of time. Fortunatelly, there is Like New (LN) ability which resets all cooldowns in the current lane. If we spend say 10% of our badges towards Like New (LN) and activate it based on a chance, we can remove a large portions of those 60s cooldowns on wormholes. This in turn should send us climbing at warp speed.



In [5]:

    
chance_one_per_tick = 0.99999

x2 = range(100,1600,50)
y2 = map(lambda n_actives: 1-((1-chance_one_per_tick) ** (1.0/n_actives)), x2)

title = "Achieving %0.3f%% chance to use at least one per server tick" % (chance_one_per_tick * 100.0)
title += "\nbased on number of active users\n"
plt.title(title)

plt.plot(x2, y2)
plt.ylabel("each player use chance per tick")
plt.xlabel("Active players")
plt.grid(True)
plt.show()

Based on the ladderboard and this graph, using 0.10 should give us good results for lone wolves. When chosing this value, we should consider the ratio of WHs to LNs we use for buying badges and adjust accordingly. We could buy less LNs and have a lower chance in order to minimize the LH overlap. Having more overlap could potentially result in faster stacking of WHs.

In conclusion, WH stategy is extremely overpowered if executed correctly.

Ideal strategy

Using the 10 times multiplier from Gold Helm level, this is the ideal candidate to spam as many black holes as possible. All players end up in the same lane, which makes LNs use much more effective. Combined with the stacking LHs this could result in huge leaps. Ideally, DPS should be low on Gold Helm levels to allow as many as possible WHs uses.