Friday, 7 June 2013

non-linear costs

Following on from last post I'm thinking about the relationship between the numeric cost of an object and how difficult it is to actually acquire in practise. Double the cost of something, does that make it exactly twice as hard to pay? In real life you'd want the answer to be close to yes (though it isn't really; we have feedback effects) for convenience, but games don't need to be convenient - a large part of what games provide is unnecessary obstacles and obscurity. It's easier to intuitively think about linear relationships: if the difficulty of paying costs is non-linear that makes things harder to evaluate, interesting.

Okay first concept is feedback loops. If you can spend resources to increase the rate at which you gain resources (positive feedback), then buying sufficiently expensive things is going to have sub-linear difficulty. Usually games that do this balance it out with an incentive to buy things early - e.g. rushing in an RTS vs. building your economy - because otherwise getting the steepest growth curve will eventually dominate everything else pretty easily. Similarly negative feedback loops make it harder to buy expensive things. In Settlers of Catan when a 7 is rolled any player with more than 7 cards must discard half of them, so there's a risk to accumulating lots: on average your effective wealth ends up slightly less than your apparent wealth.

In Magic you can play one land per turn, a linear constraint. But in practise this is sub-linear, because you can't rely on always having a land card to play. This gives a fuzzy threshold at which spells become harder to cast - you're very likely to be able to cast a 2-mana spell by turn 2, less likely to cast a 4-cost spell by turn 4, and quite unlikely to cast an 8-cost by turn 8 (ignoring mana-acceleration powers).
However since lands in play can be reused each turn, the total amount you can pay grows triangularly - by turn 4 you can have paid a total of 1+2+3+4=10 mana if the individual costs are small enough. (Note that cards are a resource too, which is linearly constrained in turns, so casting one 4-cost spell is cheaper than 2 2-cost because it saves a card.)

Dominion's price structure is quite complex. Because your hand is limited to five cards each turn, buying things that cost more than 5 is disproportionately hard as it requires not just more money but money in larger denominations (or a way to draw more at once). Also what you're able to buy on the first two turns is significant: you can definitely buy something costing ≤4 on the first or second turn, something costing 5 you have a 1/6 chance of being able to buy, and ≥6 is impossible (without a few specific expansion cards).
Also there's a default limit of buying one card per turn, which means with 4 money you can't buy two 2-costs - unless you've played a card that allows extra buys. So a lot of the time the actual numeric cost doesn't matter if it's below the threshold of 5; it only makes a difference if you're somehow very short on money or you have extra buys. This is the reasoning for the Chapel costing only 2 even though it's one of the most powerful cards in the game - it's not something you generally want multiple copies of so its power is not significantly increased by being so cheap.

@ostroffj mentioned that the alert costs (the number of enemies spawned when you siphon a wall tile) in 868whatever behave a bit like Dominion's acquisition costs. It's hard to evaluate because the cost of a number of enemies is eventually converted into the currencies of turns/hp/energy/credits but the exact rates depend on which enemy types spawn and where, the layout of walls, how you choose to deal with them, and other random elements. But in general below 5 you can usually handle unharmed (4 is on the edge) while more than that will usually demand a real cost. And then, like Dominion, you have a limited ability to acquire things and the the distinctions between prices matter if you try to take more than one at a time; 2+2 stays comfortably sub-threshold but 4+4 is highly dangerous.

Mmm still trying to think of examples with more fundamentally different ways of complicating costs.

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