## Probability of Konquest

I really like the KDE game, Konquest. It reminds me of another game I like, KDice. KDice has a handy table reference to tell what the odds of an attack succeeding are. Calculating the odds of winning a confrontation in Konquest is, undeniably, more complex.

First, a little about the calculation of a winner in Konquest. There are two sides to every confrontation, an attacker and a defender. Each has a number of ships, and each has a “Kill Percentage”, a measure of the probability that they will destroy an enemy ship. The game conducts the battle by having each side “roll” in turn to determine if an opposing ship is destroyed. First the defender rolls to see whether they destroyed a ship or not, then the attacker. When one side is destroyed, the remaining side is the victor.

It’s pretty easy to determine the probability of successfully attacking an undefended planet. Likewise, there’s no battle if no fleet attacks. So we’ll start with a 1 v 1 battle where both sides have a 50% kill percentage. The ways in which the battle could be resolved are:

d kill, a doesn’t roll = d wins 0.5000000

d miss, a kills = a wins 0.2500000

d miss, a miss, d kill = d wins 0.1250000

d miss, a miss, d miss, a kill = a wins 0.0625000

d miss x N, a miss x N, d kill = d wins (1-dkp)^N * (1-akp)^N * dkp

d miss x N+1, a miss x N, a kill = a wins (1-dkp)^N+1 * (1-akp)^N * akp

N doesn’t have to get very large before the probability of those outcomes quickly reaches zero.

Rounding to 4 digits, I found that the odds of a 1v1 matchup going 22 rolls or more is 0%.

Adding together the probabilities of all outcomes, I got 66.6667% chance that the defense would hold against the attacker.

So there’s only a 1 in 3 chance of an attacker winning an otherwise even matchup. First strike is a valuable advantage in Konquest.

If the attacker sends forces with a better kill percentage, say 60% against a defending force of 50%, I calculate the odds of attacker winning at 43.75%.

If the attacker doubles his forces, 2 attackers versus 1 defender, the situation changes a bit. The ways in which the battle could be resolved are:

d kill, a kill = a wins 0.250000

d kill, a miss, d wins the 1v1 = d wins 0.166667

d kill, a miss, a wins 1v1 = a wins 0.083334

d miss, a kill = a wins 0.250000

and for each of the above outcomes, there could be N misses before them:

d miss x N, a miss x N, d kill, a kill = a wins

d miss x N, a miss x N, d kill, a miss, d wins the 1v1 = d wins

d miss x N, a miss x N, d kill, a miss, a wins the 1v1 = a wins

d miss x N, a miss x N, d miss, a kill = a wins

This drops the percentage significantly, but it’s midnight and this post is due, so check back later to see how much.

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