Game Theory and Multiple Objectives

Sounds like some variant of asymmetric play were each player has control over their path to victory. I can think of plenty of games which do it (netrunner, mtg, 7 wonders). You could also look at Eurogames in general with their multiple point generating objectives. Changing how many points each objective gives assigns various weights to how valuable that objective is.

>>54716642
However, this weighting and in-the-moment selection of path is hard to encapsulate. One could very easily change their path at a certain point and be at a completely different score towards that new objective. Meaning that applying something like Minmax wouldn't work just out of the box.
Looking at Eurogame, it seems like in many cases, the winner is decided by some combined score total, since there's a lack player elimination. This would be a single-objective game.
A game which allows for player elimination bears multiple routes to that end, each valuing different actions more or less. In the end, using some average weight for these actions across all possible objectives seems like a bad decision which will poorly represent every victory route, rather than represent each fairly well.

I'm very familiar with a lot of the games you mentioned themselves, but I'm more looking for a general theory surrounding the evaluation of games at the abstract level, simply having multiple possible objectives to which actions contribute variably to each.

Bump from the 7th page.

>>54716474
> It wouldn't make sense to value these on the same metric if your deck is a Mill deck anyways, so giving the opponent extra HP has no impact on your strategy.
Well, if that's strictly true, then you're not looking at multiple objectives, are you? You only need a metric for your Mill strategy. The problem arises because it's not strictly true.

I don't know any of the underlying math, but intuitively it seems as if the decision you're trying to make on any given turn is "Do I attempt to maximize the number of cards my opponent discards, or do I attempt to maximize the amount of damage I can deal?" That decision can only be made in terms of a common metric— although a Pareto frontier does seem appropriate, yes. The most intuitive metric would be "number of turns to victory given a list of victory conditions"; if your opponent has five cards and 15 health, and you're in a position to mill them for ten or do ten damage, you should always opt to mill, because you win that turn, whereas you don't win if you opt for damage. ("If I do this, do I win, given this list of victory conditions?" is probably the dumbest viable approach.)

As I think about it, it seems to me that you can separate the case where your per-turn options are static from the case where your options change every turn— i.e., the case where you know that next turn you will need to decide between milling for X or doing Y damage is just a simple case of the problem where you have a probability P1 or milling for X next turn and a probability P2 of doing Y damage next turn.

The other thing that occurs to me that there is a qualitative resemblance between this problem and that of risk analysis (I think?); in the former, you're asking something like "Which move brings me closest to a common condition (victory)?" whereas the latter is concerned with (again, I think) the question "Which distribution of resources minimizes the probability of a common condition (failure of the system)?"

>>54719037
(cont'd)

There is almost certainly some good work out there that concerns this question, but you probably need to know the Magic Words to find it— there's probably some name that describes this type of problem which is apparent if you know the field, but is otherwise hard to guess.

>>54716474
In the game tree only the end states determine the value of a position, all earlier moves have their relative values determined by the end states that they lead to.

A win is a win, no matter how it is performed.

(of course this ignores heuristics that are needed because the full game tree is usually humongous)

>>54719037
Rephrasing the problem as a "turns until victory" does work in games with early elimination. It's definitely something I'll consider. Doing so causes the game to have a single objective when framed like that.

Game theory and economics are always at each other's throats, but you can model each as each other to a high degree.

>>54719073
I would love to find those magic words, lol. It would have helped before me posting this thread because I tried so hard to find anything at all.

>>54719139
In the case of short game trees, yeah, you can just vie for high win probability. My question is more about really fucking huge game trees, like TCG or TTGs which must have an associated heuristic for each action/turn.

My goal with these kinds of inquiries is to solve a problem I see cropping up in many games that offer diverse player character creation. In many cases, those games consist only of "Kill" and lack additional modes of victory. This causes a convergence in character specialization or the trivialization thereof which leads to only certain builds being capable of success, or respectively a lack of builds which are incapable of success.

To avoid this, the idea I had was to increase action diversity. In cases where a pure fighter has no magic capabilities, fighting a physical immune enemy, there should be actions available to them which subvert the usual "Kill" victory in favor of alternate objectives.

However, by granting multiple avenues of victory, I open up the problem of balancing options so no one option is too strong. (i.e. the player now simply tanks hits and does nothing else until he activates Special Ritual #5 which kills everyone always).

However, I want there to be moments where the combat history up to a certain point has lead to certain actions being stronger than others. For example, in the case of TCG against a Mill deck, an ability which allows you to reshuffle your deck can have so much more utility than in a general game (cont'd 1/?)

>>54719459
(2/2)

So how would I evaluate encounters/duels/etc. to make sure that nothing's out of place before doing a playtest? One issue I have with the usual playtest philosophy is that often times, the playtesters will overlook an overpowered strategy that would have been obvious if some game tree analysis was done.
Emergent strategies sometimes arise from games like these such as "Mill them down to 15 cards and then stop. From that point on, Mill is only beneficial to the opponent and Kill or Tank is the better strategy from this point forward."
The only way I can think of capturing all these things during the design is to do game tree analysis, but I've only ever done it for things like board games in College.

Is there better way do this sort of analysis?

>>54716474
Wouldn't the mathematical optimization start at deck building? There's probably a theoretical deck with optimized matchups against a given meta

It would be the option with the maximized return towards your secondary objective based off what you sacrifice for your primary.

>>54719459
>My goal with these kinds of inquiries is to solve a problem I see cropping up in many games that offer diverse player character creation. In many cases, those games consist only of "Kill" and lack additional modes of victory. This causes a convergence in character specialization

I'm working with this right now. Your thread is a godsend.

>>54719956
Thanks. If you find anything before this falls off the last page, feel free to share it here.
You've probably gotten more headway than I have because I've only just started trying to solve this problem. I can't really find a good angle of tackling it yet.

>>54719549
I do not know much about TCGs, but as I understand it, the starting health points and the number of cards in the deck are both constants known in advance, and can be known throughout the game. So can't it be a simple function? I mean, a 60 card deck/52-53 card library and 20 health points means I need to mill almost 3 cards for every 1 damage I would need to deal to win. Once you start a strategy of this kind, it doesn't seem like you would switch from it, because you would sort of be starting over, but a few rounds behind.

>>54720249
Ah, but the opponent draws a card a turn. Many decks will draw more, since drawing is good.
If you can drag the game out three turns, that's the same as dealing at least 1 damage. If, of course, you're running mill.
And some decks pack lifegain but not mill protection, or vice versa.
And some decks use their 'yard as a resource, others use their life as a resource.

Basically, there's a shitton of variables.

>>54720555
Then if you are talking about testing a deck you have built, a simple function can show you under what conditions you can win, as opposed to the various tree of ways you can lose. Then what you want in deck could look like a Pareto efficiency.

But if you are talking about realtime strategy during a game, like whether to ignore mill cards in hand to play damage delivering cards instead, it would probably wouldn't even be a function, just a simple constant along the lines of, "If I fall below n health points, it is favorable to try to play this card". Favorability would treat named resources like health points and deck size the same as unnamed resources like the opportunity cost of playing a card. If you try to think in terms of the nearly infinite configurations of individual cards, you're looking at a computer problem. But if you think in terms of the resources being manipulated, you limit the number of variables, and I think you just haven't isolated a few of the resources yet.