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How a “fractal tiebreak” made Catchup deeper

July 21, 2014

fractal-chess

Sometimes the best way to learn about game design is to examine one little issue like it’s a diamond under a jeweler’s lense. That’s what I’ll do with this post.

Here I discuss a problem that came up during the design of my game Catchup (coming out on August 7 for iPhones and iPads), which, as I’ve emphasized many times on this site, is one of the best games I’ve designed.

Background

You can read the rules of Catchup here, but the idea is players take turns placing stones of different colors on the spaces of a hex grid, each trying to create the largest contiguous group of stones in her color by the time the board fills.

A game of Catchup, in early mid-game

The trick is that if you take or advance the lead on your turn by creating a bigger group than has come before, your opponent gets to place an extra stone on her next turn. This keeps each player from making one big clump of stones in the middle and forces careful thought about how to time group growth – if you grow your groups too quickly, you give your opponent too many extra hexes, but if you grow your groups too slowly, your opponent can cut your groups off from one another so they can never connect together.

That’s all you need to know to understand the dilemma I’m about to describe.

The Dilemma

Early versions of the game had the following rule:

You may not take your turn such that at the end of it, the players’ largest groups are the same size.

I’d included this rule to prevent ties, which would be too frequent otherwise.

But fixing problems by banning actions players feel like they should naturally be allowed to do is a bad idea. I knew this at the time, but I was in a bit of denial, probably because I didn’t know how to fix it. That was my dilemma.

The Solution: A Fractal Tiebreak

At around this time I was playing a lot of Reiner Knizia‘s ingenious game…Ingenious. One of the bits I’d come to love about it is the tiebreak mechanism, which I’ve come to call a “fractal tiebreak”.

The game Ingenious

The game Ingenious

A fractal tiebreak is a series of nested, tiebreaking win conditions, all with exactly the same form, and all replicating the form of the game’s overall goal.

In Ingenious’ case, each player has a bunch of different point categories, and the goal is to have a higher score in your lowest-scoring category than your opponent does in hers’. If there’s a tie, players compare their second-lowest scoring categories, and so on, until they come to a pair with different scores, and whoever scores higher wins.

I realized I could add something similar (but conceptually simpler) to Catchup: if the players’ largest groups end up the same size, players compare their second-largest groups, and so on, until they came to a pair which weren’t the same size. Whoever owns the larger wins.

By adding this fractal tiebreak I could dump the rule against placing stones such that the two players’ groups were the same size. This fix was straightforward and intuitive and had the added bonus that, if the board you play on has an odd number of spaces, ties are impossible.

I was proud of that rule to begin with, but in retrospect, I’ve come to think of it as my niftiest Catchup design maneuver. The reason, which I didn’t appreciate until after I became skilled at the game, is that it made Catchup deeper, but that extra depth is hidden out of view such that new players won’t be troubled by it or even realize it’s there – a critical feature given I wanted the game to be inviting and unintimidating.  This was a complete and completely pleasant accident.

Why the fractal tiebreak makes Catchup deeper

As I’ve mentioned, it’s common for the players’ largest groups to end up the same size. It becomes even more common as players become more skilled. This fact, combined with the fractal tiebreak, forces players to maximize not just their largest groups, but their second-largest groups, and so on.

As a result, more stones on the board matter, and players must focus on more areas of the board. There’s more pressure to develop a “whole-board” strategy.

This whole-board strategy entails choices about how to deploy your stones which are equal parts complicated and agonizing. Here’s why:

Imagine you’re playing a game of Catchup and you can see the players’ largest groups will likely end up the same size, so the game will be decided on a tie-break. So you start trying to ensure your second-largest group ends up bigger than your opponent’s second-largest group.

But when you do, you realize something more confounding: you don’t know how to ensure your second-largest group is larger than your opponent’s without simultaneously making your largest group smaller than your opponent’s – because a stone added to your second-largest group is a stone not added to your largest.

Each stone you place must somehow bring you closer to achieving both goals simultaneously, but they conflict. Whether you can solve this riddle (there are a couple of different general kinds of solutions) depends on patterns established back in the beginning of the game which seemed inconsequential at the time.

Anyway, when you realize this, that’s when the real thinking about Catchup starts. You realize every little thing matters, all the way back to the very first turns of the game, and so you start thinking about every little thing.

So far, in nearly all games of Catchup I’ve played in or watched, only the largest and second-largest groups are in play, which is complicated enough.

But I’ve also played in 4 games decided by the third-largest groups, and I’ve even played in one game (out of more than 1000) decided by the fourth-largest groups (I lost). I suspect such endings will become more common as players get more skilled. The thinking required to win such battles will be ferociously complex (assuming no one figures out how to break the game before then).

But, thankfully, inexperienced players should and will remain blissfully unaware of this complexity. Most players need only worry about making the largest group, and the other stuff will slowly dawn on those who play enough to start losing frequently on tie-break.

Nick Bentley

fractal chess image via fdecomite

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8 Comments
  1. gamesb00k permalink

    This looks like a really cool game; and the rules you have developed definitely deserve the label “elegant”.

    • nickbentley1000 permalink

      Thanks! I like the cut of your jib!

      • gamesb00k permalink

        Have printed out the rules…

  2. Just saw the feedback on finalists for the 1000 Year game contest. Apparently they liked some games less because they used hexagonal boards – and hexagons are a recent “fad” and therefore not likely to be used in a 1000 years (?). I find that singularly annoying – not to mention short-sighted – because they are a very valid and extremely useful alternative to square tiling, particularly for games that need to preserve equality in spatial terms. Surely the fact that games in the past were too simplistic to require them, or that game designers overlooked the honeycombs that must have been right under their very noses, cannot bias our view on their usage in the future?!

    [end rant, sorry]

    I also wondered if you’d seen this page – http://xahlee.info/math/go_board_variations.html – where he shows that a triangular grid works just as well as a hexagonal one. This may be easier to construct for the PnP folks. Oh, and if you’d entered this design for the competition they could hardly have complained, because triangles are certainly used in games like “Fox and Geese” – an abstract classic.

    • nickbentley1000 permalink

      Re: Thousand Year Game Design contest – I agree. It made it feel like Catchup got eliminated on a technicality instead of on the merits, which is no fun. I got a bunch of emails after that from people telling me Catchup should have won. But you know, it wasn’t my contest, and griping about it just comes off as sour grapes.

      I have seen (and played) Go on a triangular board, and agree it works. The nice thing about Catchup is you can play it on any tessellation, regular or irregular. I should’ve emphasized this fact more for the contest, and argued that it raises the chance of Catchup’s survival.

  3. Out of interest, did you ever try it on a square grid? If so, how did you judge “adjacency”?

    • nickbentley1000 permalink

      I did. It works! I defined adjacency as orthogonal only. Defense seemed too hard with orthogonal+diagonal adjacencies. It was a long time ago though.

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