If you have a medieval game with a rock paper scissor dynamic to it whereby horse beat the archers, archers beat the spear men and spear men beat the horses.

If you both have equal resources to spend on an army and he sends a mix of 33% horse 33% archer and 33% spear men at you, what is the correct counter army?

My hunch is 33% spear and 66% horses but I’m just not exactly sure.

Maybe there is no counter and the best you can do is neutralise it with a mirrored army.

Assuming it's random which soldier from the opposing army each soldier from your army faces, it doesn't matter what you do. Regardless of the ratio in your army, each of your soldiers will have a 1/3 chance of winning, losing, or fighting to a draw.

Obviously it does matter to some extent what you do; for example, you cannot choose a 100% army of one type; at some point, if you choose all horses, then run into one spear guy, and he decimates the entire army. So, the win condition is theoretically that if you can wipe out a particular segment of your opponent's army without ever having one of your segments wiped out, you win (the RPS equivalent is that if your opponent can never throw Rock, you can just throw scissors until your opponent has no rock and no paper, and then you just throw scissors every throw until he's wiped out).

Is there a way to give yourself a better chance of wiping out a segment? eg, if this is a real battle and the horses are charging at the front of the line, can you overweight your army to spear men while hiding your archers and horses at the back of the field, unlikely to get engaged? Like..say you use 90% spear, 5% archers, 5% horses, but you throw exclusively spear men for the first battles. If your opponent is overweighted with horses in their early battles, then you can win.

In the end, there's not enough details about the game to answer.

It’s the latest age of empires rts game haha.

Yeah I figured this was correct. If you can get the balance right then you can essentially create an army that can’t be countered but only neutralised at best in a long term average kind of way

Maybe there's some details implied but not specified that might change this but I see no reason why if all your forces lose equally and win equally they shouldn't be spread equally in response. It seems pretty obvious that that must be the Nash equilibrium. The interesting question would be how to exploit deviations. Which it wouldn't surprise me if that were pretty easy and you just move on their ratios one, i.e. if they have 50% rocks 25% paper 25% scissors you should have 50% papers 25% scissors 25% rocks. But again it depends on how these battles play out exactly.

This is RPS, you just changed the components but the dynamic is the same. The attackers are playing GTO, so the optimal response is to also play GTO and mirror the attackers, 33/33/33.