I'm surprised they pick their conkers out of a bag. The whole fun when I was a kid was competing for who could find the toughest conker. Common cheating methods included putting it through the tumble dryer to dry it out (Mum didn't love that) or soaking in vinegar. If you're pulling conkers out of a bag I think each match is basically a coin flip, unless there's much more technique I'm missing?
As someone that played it over 60 years ago, there is quite a bit of technique involved - for example, aiming to hit the opponent's conker accurately and hard.
Wouldn't that newton's-second-law your own conker just as hard though? As the aggressor you get to choose the points of contact, which must be where the accuracy comes in. If you can strike your opponent downwards you're more likely to knock them off the string and lead to Stamps.
Conkers are usually oblate spheroids and the dangling one has its largest radius spot on top usually - thinking back I never thought to string one differently not sure if that was allowed. So the person taking the hit can aim to hit with the shortest radius section of their conker on the flat spot of the other. There's also the skill of an accurate hit - someone who misses a lot or hits away from center glancing blows is not going to win very much.