Math Hulk XI: Broodlord vs Powersword Sergeant

In the final major calculation of my Math Hulk series, we look at the broodlord verses the powersword sergeant. This is a complicated calculation to do exactly and to develop a decision tree for especially when both parry and guard are active. To review the rules, the broodlord has mighty blow which allows him to roll 3 dice and add his high and low to get his combat value. Parry allows the powersword sergeant to force the broodlord to reroll his highest die while guard allows the sergeant to reroll his own attack die. The order is to roll the dice then decide whether to parry see that result and then decide to guard or not.

The reason this gets complicated is that fact that having the broodlord reroll an attack die depends on what all three of his dice currently are. The former high die being rerolled can now become either the high die, rejected die or the low die. For example if the broodlord currently has 7 with a 6, 6, and 1 rerolling the high die does nothing since he cannot roll higher than 6 or lower than 1 so this die just becomes the rejected one. If he has a 7 with a 5, 5, 2 then 1/6 of the time he will get to 8 (6, 5, 2), 4/6 he will stay at 7 (5, 5-2, 2) and 1/6 he will go down to 6 (5, 2, 1). Since when to reroll also depends on what the sergeant has (never reroll wins and sometimes reroll ties) just using parry of the Broodlord involves about 330 possible combinations.

To make determining whether to parry easier on myself I tried to optimize the ratio of wins to loses for the sergeant. Of course without Guard you parry every time you lose regardless of whether it can help. This just leaves determining whether to parry in ties. If you always hold a tie as the baseline you get 18.85% win, 18.90% tie, and 62.24% lose. While this seems pretty bad remember that the normal powerfist terminator is at win 5.56%, tie 5.25 %, and die 88.19%. So you are doing 3 times better. Since you are at pretty low survival chances for each successive combat using parry in some ties might give you a better chance of killing the broodlord now compared to another attack. Being aggressive in trying to kill the broodlord from ties yields win 23.97%, tie 11.93, and lose 64.09%. The first rule is never reroll a double high die with a 1 like 6, 6, 1 or 3, 3, 1 since you can gain nothing and just increase the chances of dying. Other than that reroll all ties on 7 and 6. For ties on 5 do not reroll a 3,3,2 tie or a 2,2,2 for 4. Never rerolls ties on 3 or 2 with parry. These little things get very complicated but make small changes to the ratios. The basic rule to fight by with just parry is look at his results if 2 of the 6 outcomes on the rerolled die gets you a win have him reroll less than that probably hold the tie.

Now on to parry and guard, if you take the simplest approach which starts with holding all ties and then rerolling all loses with parry and then if you are still losing use guard the numbers are 26.35% win, 25.82% tie, and 47.83% loss. While this is even better than just using parry it is far from optimal from a killing the broodlord point of view. Fighting for ultimate survival where you reroll no ties and do not use parry in cases of the Broodlord starting with a pretty low score yields 29.08% win, 23.67% tie, and 47.27% loss. Going for the highest win to loss ratio gets 34.29% wins, 16.20% ties, and 49.51% loss. You are losing about 2% more but also winning 5% more. This result is from not parrying broodlord total results of 4 or less, essential only parrying results of 5 if he has a 1, and not parrying totals less than seven with a double high and a 1. There are a couple of more little adjustments but they are all in the margins. On the guard side it is rerolling 2-4 results on a tie but holding 5-7. You can get a better kill percent by rerolling on the 5 but the death percent goes up enough that the ratio is worse.

These results make the power sword sergeant on guard essentially as good at fighting the broodlord as the thunderhammer sergeant so about as good as you could expect as the broodlord is the stealers big buy and should not go down easy to anyone in close combat.

Now that the main series is done I will try to get all the results plotted up in a final summary post. Might go back and add some additional little things to earlier posts also.

Math Hulk X: Close Combat Powersword Sergeant vs Genestealer

The powersword sergeant in close combat with a genestealeris a pretty interesting problem from a game theory point of view once you combine parry and guard into one decision tree. To start simply, we will start with just normal combat and parry. The sergeant should of course parry anytime he is losing if he is not on guard. The question comes when to parry in cases of a tie. The cutoff point is about 5. If the stealer has a 5 or better on the high die use the parry otherwise hold the tie. Unfortunately sometime parrying will do you no good if the genestealer has a second die that is the same as his high die. In this case you do not want to parry on a tie since it gains you nothing and has a chance of killing you. Using these cut offs the sergeant with powersword will win 46.72%, tie 17.05% and die 36.23%. These odds are still good enough for the stealer player to throw stealers at him since a dead marine is not replacable unlike a dead stealer in most missions.

When both Parry and Guard are active things get really complicated. You must first decide to parry. As always you take your win if you have already won. Next you need to consider whether having him reroll his high die will on average make it harder or easier to beat him. If his high die is 6 have him reroll everytime, if his high die is 5 and he does not have double 5 also have him reroll. If you are losing on a 4 and he does not have a double 4 have him reroll but hold a tie on a 4. If he is below a 4 I would hold that result lose or tie and try to get him with guard. Now comes the guard decision. If you are losing reroll automatically as normal. For Ties of 2 or 3 I would reroll, above 4 and above hold the tie. These rules will get a win for the marine 63.10%, tie 18.75%, and the marine dies 18.15%.

These results make the powersword sergeant about the same in combat as the normal marine with lightening claws but he has the added bonus of being able to shoot also.

Math Hulk IX: Broodlord vs Librarian

This is the battle of the big boys. The broodlord packs a big close combat punch but the Librarian is unbeatable if he has enough psi points left. The quick result is that the librarian needs on average 3.58 psi points on the front to win with guard dropping it down to 2.87. On the flank the win costs 4.53 with guard 3.79 but remember that just gets the win in that combat which allows you to turn and face the broodlord. The guard cut offs to optimize psi point use was found to be reroll any die 3 or less for a tie or loss.

If you librarian gets caught in a room by the broodlord he can attack your side then if he has 2 action points(1 point move diagonally and turn and 1 point to attack) left attack on another side. If you have command points you can keep turning you facing which might enter an interesting game of Cat and Mouse as the broodlord tries to get around you and you keep you facing at him.

Averages do not really tell the whole store here since the number of points it might cost to beat the broodlord can vary from 0 to 12 so I have prepared a plot of the distribution. These distributions are pretty broad and flat with approximately the same chances of it costing between 0 and 5 to 7 points all near the same depending on the status of the librarian.



Now when the librarian is getting low on points he becomes much more vulnerable so I have prepared a chart showing the cummulative probabilities of the librarian getting a win based on how many points he has available. He is in pretty good shape at about 6 with 80% wins regardless of guard or flank but below that the broodlord can start to have a chance.

Math Hulk VIII: Broodlord Close Combat Part 1

The broodlord is a terror in close combat. The mighty blow ability which allows him to add his high and low die together makes his average score a 7. Since that is better than most of the marine best score they are in trouble. The breakdown for the broodlord is 12 (0.46%), 11 (2.78%), 10 (6.02%), 9 (11.11%), 8 (17.13%), 7 (25%), 6 (17.13%), 5 (11.11%), 4 (6.02%), 3 (2.78%), and 2 (0.46%). These numbers say that against a normal marine the marine will die 62.5% of the time regardless of how well he rolls or with guard or without.

Against a normal marine or the flank of the lightening claw the marine with win 5.56%, tie 5.25 %, and die 88.19%. Pretty grim. Adding in guard where you reroll a tie if you have any chance of winning yields, 9.23% win, 7.70% tie, and dead marine 83.06%. Still pretty much a dead marine. A guarding marine has about a 1 in 6 chance of surviving the first combat.

A normal Sergeant(not actually in the game) or an attack on a librarian without any psi points left, they do a little better since they still get their +1 to the die. In normal combat marine will win 11.80%, tie 10.4%, and die 77.78%. Adding is guard were you reroll on ties less than 6 is a big help with winning now 18.36%, tying on 14.30% and dying 67.34%. Now the sergeant attack on guard will survive 1/3 the time.

Next we will look at the lightening claws. How much does 2P6+1 help? Without guard the lightening claws will be winning 11.81%, tying 10.42%, and losing 77.78%. Guard is good for the claws since you can always save you tying die and reroll the other to go for the win. This gets the claws up to 18.84% win, 15.162% tie, and 66% dead. You might notice that these are very close the sergeants being attacked on the flank even though the lightening claw has an extra die. This makes sense due to the fact that the broodlord is 7 or above 62.5% of the time so more than have the time you have no hope to win that combat no matter how well you roll.

For the last part of this post I am going to look at the broodlord verses the thunderhammer/stormshield sergeant. Here things are interesting. The Block ability removes one of the dice from the broodlord so now he just rolls 2 and adds them. This results in the exact same average value for the broodlord of 7 but changes the distribution substantially.



As you can see from the figure this essential broadens the distribution. It increases the probabilities for values for which you can never beat the broodlord but also increase the probability for values where you can win more easily. This coupled with the +2 to his die actually makes the thunderhammer win 25.92%, tie 11.57% and lose .625%. Throw in guard where you reroll ties on a 3-5, the thunderhammer wins 35.49%, ties 13.89%, and dies 50.62%. Still not very good but 3 times better than the normal marine.

I know there were a lot of numbers in the post so here is the summary figure shown in classic traffic light style for the marine player. That is a whole lot of red for dead blood angels.

Math Hulk VII: Librarian verses Stealers

Now, the librarian should never lose a combat while he has psi points left since he can of course not use them once he is dead. He can choose to spend them after seeing the dice so there is not even risk involved. Stealers will be attacking him just to sap away points limiting what else you can do with him(like smite the broodlord). Basic combat percentages are win 34%, tie 16.6%, and lose 49.4%. This is better than a standard marine since he has the +1 but still not very good compared to the close combat specialists.

So the first question is do you spend your points up to get the win or the tie. Average number of points spent to get a tie is 1.08 while the win is 1.74. A healthy jump but I would think you would want to go with the win since just getting the second tie costs you 2.16 on average then so the win straight away is cheaper in the long run. You might go for the tie if the stealer is out of action points and will block additional stealers from getting to you this turn, but that is a general rule of thumb for most of the marines.

What about the Librarian on Guard? The question here is when should he reroll his die (tie or loss) and when should he just go for the win straight away with points. The optimal decision for conserving psi points on average is to buy your way to victory if you have a combat score of 5 or 6 (die 4 and 5 +1) since the stealer also has a 5 or 6 in these cases. You should also buy your way out of a 4-4 tie but other than that reroll. So you would reroll any die of 2 or 1 and a 3 if you are currently losing the fight. This decision tree will get to a 1.12 psi points for the victory on average. If you are playing for the tie the psi point average can be as low as .6.

When rerolling here, I would remind you that this is averages and if you are low enough on psi points such that you could roll a result you could not buy your way to victory and are sitting on a point where you can buy yourself out off currently, I would probably take the sure thing. These reroll decisions are for use when you have the psi points to gamble and want to try to get the most use out of everyone. Never gamble the life of your librarian to try to save those last few points.

Math Hulk VI: Thunderhammer Terminator verses Genestealer

The thunderhammer terminator is pretty good in close combat as you would expect with the Block ability stripping a die from the stealer and a +2 to his attack roll he starts in a pretty good place. A thunderhammer terminator will win combat 58.3%, tie 14.8%, and lose 26.9%. So in this combat the terminator only dies 1 in 4 times.

Adding in guard makes him really survivable. Again a decision is required about rerolling ties but since their can only be a tie on 3-6 outcomes it is less likely event. Ofcourse on a 3 (roll of 1+2 bonus) you would reroll since you cannot get lower. Using that as the starting point the guarding thunderhammer terminator wins 71.6%, draws 17.4%, and will lose only 11.0%. Looking at these numbers tells us that a thunderhammer terminator on guard should not reroll a tie unless he rolled a 1. Any die roll which results in a chance to lose (1/6 = 16.7%) is worse than just letting the stealer attack you again. You might try a reroll if you are tied on a 4 and you need that stealer out of the way so you can go forward.

How should genestealers deal with the thunderhammer terminator if he is so good? If the terminators need to get someplace while you have unlimited stealers I would just use 1-2 stealers to block his path and use you blips to try to come at them a different way. You have a much better chance if he is forced to attack you.

Math Hulk V: Close Combat Genestealer vs Lightening Claw Terminator

For this installment of Math Hulk, we look at the close combat battle between a genestealer and the terminator with lightening claws. Stealer gets his roll 3 and pick highest while lightening claw gets to roll 2 and take highest +1. Remember that this bonus die and +1 only comes into by for the terminators front space. So attacking from the flank or rear causes this terminator to fight just like a regular powerfister. The plus 1 makes a big difference since now no matter how well the stealer rolls, a 6 from the terminator wins the fight. The probabilities for the terminator is 30.6% 7, 25% 6, 19.4% 5 , 13.9% 4, 8.3% 3, and 2.8% 2. These compare to for the stealer of 6 (42.1%), 5 (28.2%), 4 (17.1%), 3 (8.8%), 2 (3.2%), and 1 (0.5%).

Combining these two sets results in 52.8% terminator wins, 19.2% tie, and 27.9% stealer wins. So the lightening claw terminator is twice as good as the powerfist ones in close combat.

Now if you place the lightening claw terminator on guard, how does he perform? Unlike the powerfist terminator this is no need to worry about whether to take your reroll or not. If you won you have no need of the reroll, if you lost you always take it, and if you tied you just save the tied die and reroll the other die which is lower or equal to it. This makes the lightening claw terminator on guard with 64.8% win, 18.5% tie, and 16.7% lose. This is pretty good. Essential as good as a stealer attacking a normal terminator.

Tactics for the lightening claws are all about using guard or attacking in close combat since you have no ranged weapon to go on overwatch with. Most of the time going to guard is probably the best bet unless the stealer is isolated and can get around you without killing you. If there is an isolated stealer with an exposed flank it might work out to attack him since this event is most likely to happen in a room where the stealer can avoid your close combat death dealer anyway or attack you from the flank.