Last Night on Earth combat probabilities

Last Night on Earth is a game I really enjoy playing. It's got zombies, a cheesy theme, interesting randomness, deep strategy and cool combat. The combat is complicated enough though, that when playing, I'm not exactly sure the chances of the various outcomes. To help myself out, I'll take a look at the basic combat probabilities and see if I can create a generalized rule to make it easy to consider the situation. For this exercise, I'll ignore the effects of weapons, like the chainsaw, and potentially consider looking at them in the future. I would also like to see the effect "reroll a die" cards might have on the probabilities, but again, in the future maybe.

Combat occurs whenever a zombie and a hero occupy the same space. The base combat rules have the zombie rolling 1 die and the hero 2. If any of the hero's die are greater than any of the zombie's die, the hero "defends" against the zombie. If any of the zombie die are higher than all of the hero's die, then the hero takes a "wound". Finally, if the hero rolls doubles AND any die is higher than the zombie's, the hero "kills" the zombie. The possible outcomes of this type of combat are described with this R function.



To simulate the combat as described above, we simply use fight(1, 2).

 > fight(1, 2)  
 $counts  
 results  
 defend  kill wound   
   110   15   91   
   
 $ratios  
 results  
   defend    kill   wound   
 0.50925926 0.06944444 0.42129630   
   
 >   

So, roughly, a single zombie has a 42% chance to inflict a wound on the hero. The odds are in the hero's favor, but just barely and the hero has a dismal chance to kill the zombie. Now, one of the things that makes the combat so interesting is that player's can add extra die to their rolls using event or weapon cards like Faith. In this example, the hero is now rolling 3 fight die instead of 2 while the zombie is still only rolling 1.

 > fight(1, 3)  
 $counts  
 results  
 defend  kill wound   
   510  345  441   
   
 $ratios  
 results  
   defend   kill   wound   
 0.3935185 0.2662037 0.3402778   
   
 >   

The zombie's chance of inflicting a wound is reduced to 1:3 and the hero gains a much greater chance to kill the zombie. Clearly, the hero getting that extra fight die makes combat a bit more rational for the hero. But what about when the zombie gains an extra die.

 > fight(2, 2)  
 $counts  
 results  
 defend  kill wound   
   450   55  791   
   
 $ratios  
 results  
   defend    kill   wound   
 0.34722222 0.04243827 0.61033951   

The extra die increases a zombie's chances to inflict a wound by 50% over just one die. Big difference. One zombie attacking the player is now very dangerous. How about +1 die for each.

 > fight(2, 3)  
 $counts  
 results  
 defend  kill wound   
  2262  1405  4109   
   
 $ratios  
 results  
   defend   kill   wound   
 0.2908951 0.1806842 0.5284208   
   

Still very much in the zombie's favor. So, essentially that 1 extra die is much more powerful to a zombie than the hero.

Now we have a good idea of the probabilities for simple combat, but let's see about making that generalized rule I mentioned at the start. The question I want to know most is how many zombies should I concentrate on a hero to pressure them. Its rare than the zombie has an extra die, at least without the expansions, but common for the player to have 1 or more extra die. So in this example, I'm going to calculate the outcomes of the zombie(s) inflicting at least one wound to the hero with a varying number of combat die. To do this, I'm going to use this function to calculate probability of at least one wound using a varying number of zombies and varying number of hero combat die.

 > df <- expand.grid(numZombie = 1:6, heroDie = 2:4)  
 > df$woundChance <- apply(df, 1, function(x) { cumProb(1, x[[1]], fight(1, x[[2]])$ratios[[3]])})  
 > head(df)  
  numZombie heroDie woundChance  
 1     1      2  0.4212963  
 2     2      2  0.6651020  
 3     3      2  0.8061933  
 4     4      2  0.8878433  
 5     5      2  0.9350945  
 6     6      2  0.9624390  
 >   

Numbers are sometimes hard to read and that's where charts serve us nicely.



Pretty, but let's make it even easier. A simple linear model should net us a nice easy to remember rule, much like the 2/4 rule in hold em.

 lm <- lm(woundChance ~ numZombie + heroDie, data=df)  
 > summary(lm)  
   
 Call:  
 lm(formula = woundChance ~ numZombie + heroDie, data = df)  
   
 Residuals:  
    Min    1Q  Median    3Q   Max   
 -0.09623 -0.06060 0.01660 0.05243 0.08463   
   
 Coefficients:  
        Estimate Std. Error t value Pr(>|t|)    
 (Intercept) 0.525251  0.065099  8.068 7.75e-07 ***  
 numZombie  0.109486  0.008815 12.421 2.70e-09 ***  
 heroDie -0.066075  0.018437 -3.584 0.00271 **   
 ---  
 Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1  
   
 Residual standard error: 0.06387 on 15 degrees of freedom  
 Multiple R-squared: 0.9176,     Adjusted R-squared: 0.9067   
 F-statistic: 83.56 on 2 and 15 DF, p-value: 7.379e-09  

After all this, what do we have?
In combat, there is a 50-50 chance of wounding the hero. Each zombie in the fight increases the odds by 10%. Each hero die reduces it by ~6%.

Using this simplified rule, 2 zombies attacking a hero with 2 die has a (50% + (2 * 10%) - (2 * 6%)) 58% chance. Our table above states the chance to be 66%. Looking at the residual plot clearly illustrates the exponential nature of our model, but I want easy math to make the calculations on the fly. Adding +/- 5% will help me to generally consider the odds and while not perfect, it's better than what I had to start with.