[JoshCole]

Just to give a sense of hardness, Stratego's game tree is infinitely larger than Go's game tree, because in imperfect information you select a continuous strategy vector over action space whereas in Go you select a discrete action. Meanwhile Stratego is also infinitely more complex than Go because in Go there are less moves with every move, but in Stratego moves don't monotonically decrease the remaining game length.

Beyond those two infinities of greater complexity, Stratego is imperfect information, so it is played relative to the infostate, not the state. In Stratego there are thirty three pieces with unknown information on the first move. We can definitely get a lower upper bound by applying abstraction via domain knowledge, but just so I don't have to deal with the complexity I'll state that there are at most 8683317618811886495518194401280000000 different states associated with the infostate of your first move. Meanwhile, on your first move in Go, you are in at most 1 state.

In practice though the average length of a game of Go is ~200ish, the average length of Stratego ~400ish.

Much like chess, I would expect optionality to be an important strategic consideration in Stratego. So branching factors are likely selected for such that they get higher by good agents. In contrast to something like Go, the branching factor would tend to diminish over time.

Mostly sharing this because I think most people are bad at reasoning about complexity - our mind is really good at making complex things seem simple and simple things seem complex because we can actually deal with the complexity of simple things in their full complexity but dealing with the full complexity of the complicated is intractable. I imagine a lot of people's mind latch onto the things like "we get information so playing well means memorization" and don't pay as much focus to the dizzying complexity; but playing well isn't memorizing. Playing well is playing perfectly according to your uncertainty and it just so happens that as part of doing that you sometimes reduce your uncertainty by scouting.

[spywaregorilla]

This is all technically true, but it's also misleading I feel. Chess and Go have a lot of states, and you need all of them. The entropy of the games are enormous. Move that bishop and the significance of every piece on the board could change. Stratego has verrrryy low entropy. You can only move one piece one space at a time aside from scouts. You can also enter scenarios where the game just simplifies on some dimensions immensely to which some fairly dumb algorithms can win the game. If the enemy loses their top dude and their spy, your top guy is invincible against anything that has ever moved, which isn't victory assured but its probably relatively easy to write an ai to win from that point on.

Maybe this is just me but when I thought about stratego I'm pretty sure I only thought about a few units at a time. Moving all of them is a bad idea because of bombs anyway. Your mental model of the game does not consider the state of units in the back row of either side. It probably doesn't even consider all of the units in the front.

I also suspect a lot of optimal play is just doing dumb shit back and forth to force your opponent into making the more aggressive move in a lot of cases without technically forfeiting which most humans likely wouldn't do but ai would like to do if not constrained otherwise.

But I agree a simple tree search on possible moves is unlikely to work very well until the end game.

[JoshCole]

If I'm understanding you correctly, you're saying that the rules of Stratego let you apply an abstraction rule which drastically simplifies the game. I agree you can do this. I think this point is remarkably similar to the technique of blueprint abstraction.

Of course, we ought to be focused on the situations which lead to these nice situations - which leads us back to the imperfect information situations which lead to those positions we can reason about.

Interestingly, you can invert your point and get a similar rule more generally. The trick is to backward induction on the abstracted category transitions. So your point is actually so valid that is valid even in situations where your examples don't apply. I hope you can see I'm strong manning here. I agree with you that this should be done and is critical to making things tractable.

There is an issue though, at least in the generalized case, which is that perfect information is much easier than imperfect information.

See, in perfect information games when you do the backward induction step you actually just flat out solve the game. Meanwhile, in imperfect information games, this backward induction step merely makes solving the game more tractable. Chess endgames are the backward induction version of your forward induction from the rules abstraction, but the abstraction rule is so hard to determine we wouldn't usually think of it that way. Notice that chess is hard enough that we don't have endgame tables that go all the way back to the start of the game? Yet the average game length in Stratego is hundreds of moves longer and there are more then double the number of pieces.

I still think your point is right and someone who pushes hard enough in this direction would manage to tackle the complexity. So I basically agree with you. But if you take standard approaches like counterfactual regret minimization and throw it at the game without adjusting them for the fact that the problem is "hard" then they just wont terminate.

[spywaregorilla]

I guess I have two points. Perfect information stratego is not a hard game at all. There's still a lot of moves one can make but perfect play is easy to calculate. Near perfect play is probably easy enough even for an amateur player. My gut says many games will converge on this state (early) if you have an unbeatable piece.

Without perfect information, the number of states is still mostly a red herring, because the differences between the states are immaterial. The moves aren't super important. If you decide to move frontline unit A against the frontline unit on the opposite side of the field, there may be several moves between that but you've only made one noteworthy decision. Could be bad intuition. I think a more abstract model here would do better and be far simpler with some minor tactical move prioritization or whatever.

[JoshCole]

> Could be bad intuition.

This actually does seem to be bad intuition to me, because your intuitive explanation isn't being expressed in terms of strategies. You wouldn't want to "attack this frontline unit" but would want to have a probability distribution over your potential options. This is similar how you wouldn't want to "play rock" in RPS, but would instead want to play {R: 1/3, P: 1/3, S: 1/3}.

In a certain sense the thing that is "different" about imperfect information is that you shouldn't play as if you are making only one decision. You should play as if you are in multiple different game states at the same time.

FWIW, I don't think detracts from your point about it being possible to simplify the game, just your reasoning explaining the "why" we ought to seems off to me.

[spywaregorilla]

> This actually does seem to be bad intuition to me, because your intuitive explanation isn't being expressed in terms of strategies. You wouldn't want to "attack this frontline unit" but would want to have a probability distribution over your potential options. This is similar how you wouldn't want to "play rock" in RPS, but would instead want to play {R: 1/3, P: 1/3, S: 1/3}.

My point is that your decision to attack is one that takes 3-5 turns with no meaningful possible positional maneuvering. So a game may have 200 turns but many fewer "decisions". There's not much "area control" like in many other strategic boardgames. It's more bluffing about your original setup choices. Every battle is reduced to a bet of whether your unit is higher or not and if it is, well you don't have a lot of agency on that. Especially if attacking.

[TemplateRex]

It's true that many move sequences can be collapsed to a single maneuver. However, the area control remark is inaccurate: controlling or occupying the 3 alleys is very important and requires careful positional play. Getting the right "lane parity" to attack pieces frontally is crucial here, as is pincer movements around the lake with multiple power pieces. You seem to view Stratego as a superficial and repetitive game, but high level play goes considerable beyond a few bluffs and mechanical exchanges.