Many modern video games feature an achievements system. Achievements are typically awarded for completing specific tasks; typically achieving all of these requires doing substantially more than just what is required to finish the game. In many cases, it’s not even possible to complete all of the achievements on one playthrough.

In a previous post, I wrote about Slay the Spire, an RPG where combat is driven by a deckbuilding mechanic. The game also features an achievements system. I finished getting 100% completion on these achievements this week – this took quite a long time. If I wanted to start from the beginning and accomplish these as quickly as possible, I would need to plan carefully. The main issue here is that achievements are not independent. For example, there are chains of achievements where the final achievement subsumes all previous ones (e.g. “have 99 block, have 999 block” or “unlock ascension mode, finish ascension 10, finish ascension 20”). Furthermore, some achievements can be achieved using the same strategy (e.g. “play 25 cards in a single turn” and “defeat a boss before it takes a turn”, which can both be achieved by building an infinite combo). Other achievements can often be significant handicaps (e.g. “beat the game with a deck that has no uncommons or rares”, “beat the game with only 1 relic”) and I thus think it’s a lot harder to combine these with other achievements.

The game is played in distinct runs, and a majority of the achievements involve performing specific feats on individual runs. We can model this by identifying strategies that unlock one or more achievements. If we try to use the fewest number of distinct approaches, this becomes what’s called a set cover problem. There are achievements and strategies with each being non-empty and a subset of . We also require that , otherwise not all achievements are achievable. The goal is to identify some set of strategies we need to deploy such that . Trivially, picking everything is possible, but it’s very unlikely that that’s going to be minimal.

This problem is known to be NP-hard; the decision version (does there exist a solution using strategies or fewer?) is NP-complete. Clearly, we can obtain a solution that is worst-case exponential in by iterating through each possible subset of , checking if it satisfies the criteria and remembering the best solution we’ve seen so far. There’s a little dynamic programming trick we can do to memoise partial achievement states (e.g. I can get to the state where I have the first 8 achievements and no others using two runs with one strategy but three with another; I’ll always pick the first). This achieves a solution exponential in but linear in , which is an improvement. We can also try to pre-process the input for and a little, by eliminating any achievements that are subsumed by harder achievements, and by eliminating any strategies that are strictly dominated by others.

If this is unacceptable, we can either try to use approximation algorithms (for example, a standard ‘greedy algorithm’ which picks the subset with the most outstanding items doesn’t always fare too poorly). We can also attempt to leverage constraint solving systems; these don’t deal with the exponentiality of the problem directly, but they often have good heuristics to deal with this.

That said, this isn’t a very good model, as some strategies are riskier to implement than others. For example, it may be theoretically possible to combine “beat the game with one relic” and “beat the game without uncommon or rare cards”, but that seems unlikely. For example, if each of these individually has a 1% chance of being completed but attempting them together has just a 0.1% chance, doing the achievements separately will take an expected 200 runs while the combined approach has an expectation of 1000.

A natural extension to this is to consider the weighted version of the problem. We change the formulation slightly, so each strategy has a weight and we seek to minimise the total weight of the strategies we pick. If we assign a strategy a given probability of success , since runs are (mostly) independent, the expected number of runs it takes to attain a successful outcome would be following standard geometric distribution results. The dynamic programming based algorithm still works if we keep track of weights instead of number of sets.

This is better, but there are still a few issues with the model. Firstly, partial successes haven’t been considered in this model; we might try a risky strategy for three achievements and fail, but still complete two of them. We could try to account for these by figuring out some kind of state transition between achievement states – it wouldn’t be a set cover problem any more. We would be building some kind of a discrete-time Markov chain on-the-fly.

Modelling this as a set cover problem also seems to imply that our algorithm is offline; that is, it formulates a strategy and then executes it. However, Slay the Spire has a significant luck element, especially if one is trying for specific achievements, and thus an offline algorithm doesn’t work so well. For example, one achievement requires “play 10 shivs on a single turn” and another “have an enemy reach 99 poison”; the cards and relics that support these are quite different, and depending on what cards are offered as the game progresses, one achievement or the other could become easy or nigh-impossible. If both achievements remain to be unlocked, it seems suboptimal to commit early on.

I had a weekend break in Zurich to meet up with a friend who works there. We often play board games when we meet, so I brought Hanabi and FUSE (very compact games, suitable for travelling – especially since I almost always travel HLO) but my friend was thinking of getting something different. We thus headed down to a game shop, and picked up The Game: Extreme (Board Game Geek).

The rules are fairly simple. There is a deck of 98 cards – one copy each of the integers from 2 to 99, inclusive. Players will have some number of cards in their hand (eight for a solo game, seven for 2 players), and play them to the table; the game ends in a loss if a player must play, but is unable to (or if all cards are played, which is a win). It is a cooperative game; players all win or lose together. The rulebook also mentions a half-win if players finish with ten cards or fewer remaining.

There are four communal piles that players can play cards to. On two of these, cards played should be numerically increasing; on the other two, they should be numerically decreasing. However, one may also go against the direction of a pile – this can only be done if the jump is exactly 10. Thus, for example, if an increasing pile has a 57 on top, one may play a 59 (increasing) or a 47 (ten less), but may not play a 49 (not increasing, and not ten less). Of course, if a player has both the 59 and 49, the player may play the 59, after which the 49 becomes legal.

One can probably write some search-based algorithm to figure out what the best possible outcome is for a given hand and individual pile, or even set of piles given some fitness function. There’s also a straightforward dynamic programming optimisation, and for certain classes of fitness functions there might be better principles one can leverage.

Players must typically play two cards on their turn, though they may play more (for example, if they have a chain of cards that involves several reverse jumps) while there are still cards in the deck; once that is emptied, they only need to play one. Once they have played all of their cards, they draw back to their hand size and play passes to the next player.

The Extreme version adds special effects to some of the number cards. These may either restrict the current player’s turn (requiring that they stop immediately, play exactly three cards, or immediately cover up the card that was played) or affect all players while they are exposed (requiring players keep quiet, play all the cards in their turn to the same pile, not do reverse jumps or only draw 1 card at the end of their turn). Apart from the stop card, all of these make the rules for playing more restrictive (and thus I’d say this is somewhat harder than the base game).

The solo game doesn’t generally seem too difficult as one can plan and execute sequences of reverse jumps without interference from others. The keep-quiet special effect no longer really matters, and the stop cards actually become positive as they allow for drawing cards back up earlier. I’m not sure how one might formulate an algorithm for determining optimal moves, but I can generally see the following principles:

• Play the minimum number of cards needed per turn, and then draw back up. This lets you see more options, which is useful. Normally in a multiplayer game you may want to play more than the minimum to ensure that no one interferes with your plan, but in single player no one will anyway.
• Plan and aggressively use reverse jumps. These often get thwarted in multiplayer games (you might draw a card that allows a reverse jump, but someone might need to play on the relevant pile before it gets to your turn).

Playing with more than one player gets trickier, as communication between players is permitted but precise numerical information is not. I would say things that imply precise numerical information (e.g. “I can jump this pile”) are also bad. There are edge cases which are probably acceptable (e.g. “it makes no difference to me; which of these would you prefer I play to?” with an up-65 and down-67 suggesting I hold the 66). Still, talk of “small/large jumps” and actively saying “stop” after interesting cards are played is very useful.

Cooperating well involves looking ahead to other players’ turns as well; it’s not always optimal to, on each player’s turn, individually minimise “damage” to the piles. There have been times where being forced to jump a pile by 30 or so, I’ve encouraged my teammate to play more cards there since it would be destroyed on my turn anyway. We only played with two players, but I imagine the dynamics get more interesting with larger groups.

The Game: Extreme is a very light game; while there is some minor logic involved it’s not particularly complex. It’s fairly fun and witnessing successful chains can be enjoyable; the special effect cards can also cause annoying obstacles (while blocking reverse jumps is probably the most damaging, the keep-quiet one is interesting as it changes the game for a while). It’s certainly a fun thing to play while passing time and discussing other things, and hard enough to provide a good challenge without being frustrating.