# Hammer of Math: Probabilities in Conquest

This week’s Hammer of Math takes a look at some of the probabilities behind the core rules of Para Bellum’s Conquest: The Last Argument of Kings.

One of the nice things about writing for a game website is that it gives me an opportunity to be exposed to a variety of games and experiences. Often the games you play are heavily driven by your community and FLGS. Necromunda might be popular in one region and ignored in another, one store may be all about AoS and another purely 40k, and then you have the competitive versus casual angle. One such game that I’ve heard a bit about but haven’t had a chance to explore yet is Conquest: The Last Argument of Kings which you can download the rules for here. Rank and File is a genre that I hadn’t played since GW had Fantasy with square bases, so it was cool to see something new. And boy does this game look fun.

### Core Rolls

Rolls in Conquest are d6-based and revolve around target numbers, but your goal is to always roll equal to or under the target number. Yes, sixes are bad. Yes, it will take some time to accept that. A roll of 6 is always a failure and a roll of 1 is always a success, so the minimum probability of success is 17% and the maximum is 83%. Toss in a re-roll and the minimum probability is 31% and the maximum is 97%.

### Reinforcements

One of the things that really sets Conquest apart from other wargames is that there is no deployment; every Regiment starts off the table and must arrive as Reinforcements. Regiments are divided into three categories that determine the earliest that they can possibly arrive; Light, Medium, and Heavy. During the Reinforcement phase players select one Regiment that can arrive to automatically do so; all of the others must be rolled for.

• Light Regiments can arrive on Turn 1 on a d6 roll of 1-2, Turn 2 on a D6 roll of 1-4, and automatically on Turn 3.
• Medium Regiments can arrive on Turn 2 on a d6 roll of 1-2, Turn 3 on a D6 roll of 1-4, and automatically on Turn 4.
• Heavy Regiments can arrive on Turn 3 on a d6 roll of 1-2, Turn 4 on a D6 roll of 1-4, and automatically on Turn 4.

What this means is that every Regiment outside of the one you select has a 33% chance of arriving the first round they can, a 66% chance of arriving the second, and a 22% chance of not being available until the automatic turn.

### Command Decks and Priority

Conquest is a highly reactive game; in addition to not having full control of what Regiments will be available when, you also alternate activating Regiments by stacking every unit into a Command Deck and selecting the top card one by one. The first player to activate a Regiment is based on roll-off called the Priority Roll. Whoever rolls lowest on the Priority Roll has to go first. The player with the smaller Command Deck has a slight advantage; they can choose to add or subtract 1 from their Priority Roll. So what is the probability that the player with the smaller Command Deck can have the outcome they desire?

For a given Priority Roll there are three possible outcomes; the player gets what they want, the player doesn’t, and there’s a re-roll. If neither player get to modify their roll then the probability of success is a flat 50%. If the player does have the option to modify the roll, then they can either succeed naturally or with a tie, fail without any chance to modify the roll, or have a roll that’s within one of the opposing roll and modify their result to force a re-roll. Regardless of what the definition of success is, the player with the modifier has a 58% chance of succeeding and a 14% chance of forcing a re-roll. Including the option to force the re-roll to be re-rolled, and so forth, the end result is that the player with the smaller Command Deck has a 68% chance of getting the result they want.

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