Thursday, December 07, 2006

Mathematics & Game Design, Part Two

In October I published an article called Mathematics & Game Design, Part One. Therein I wrote about some of the problems that appear when games make their math too obvious. But in closing I stated, "Though I think math should largely be hidden from players, designers absolutely need to think about it."

In this article I want to look at the other side of that coin by examining two games that had potential, but where I was let down by the lack of mathematical analysis applied to the games--which ultimately resulted in them not standing up to, at least, my play (though in both cases they would have stood up to more casual play, if players did not think about the math).

Desert Bazaar

Desert Bazaar is the first game in Brian Yu's pushout of designer games for Mattel, an admirable goal in and of itself. The gameplay is pretty simple: each turn you either draw 1-4 resource cards (probably 2+ on average) or you play as many resource cards as you want to build tents.

Tents are represented by tiles. Each one shows what resources are needed to build the tent. It always takes three resources to build a tent, sometimes one each of three types, sometimes two of one type and one of another. However there's a catch: each tent also represents a type of resource, and if you build a new tent adjacent to an old tent with the right type of resource, you don't have to play that card. This can reduce the tent building cost from 3 cards to 2 cards (often), 1 card (sometimes) or 0 cards (infrequently).

Tents are arranged into encampments, by the by, which are just groups of connected tents. This has some effects on placement, but notably whenever you form a new encampment, you're clearly not getting to use any resources from adjacent tents, because there aren't any.

And, building tents gives you victory points: 3 points for the first tent in an encampment (2 immediately, 1 when the encampment finishes or the game ends), 2 points for the last, and 1 for each additional.

At the most basic level, the game is thus about the most efficient use of cards to achieve victory points. The fewer cards you use, and the more victory points that you get as a result, the better you're doing. The following chart details this in short, by outlining the math of building the first through seventh tent in an encampment:

TentMin. CardsMax. CardsPointsRatio for Min.Ratio for Max.
#1
3 cards
3 cards
3 points
1 card/point
1 card/point
#2
2 cards
3 cards
1 point
2 cards/point
3 cards/point
#3
1 card
3 cards
1 point
1 card/point
3 cards/point
#4
0 cards
3 cards
1 point
0 cards/point
3 cards/point
#5
0 cards
3 cards
1 point
0 cards/point
3 cards/point
#6
0 cards
3 cards
1 point
0 cards/point
3 cards/point
#7
0 cards
3 cards
2 points
0 cards/point
1.5 cards/point

Now at first Desert Bazaar looked to me like a game which allowed for a lot of cleverness, where you could build a bunch of tents each taking advantage of one or two nearby tents and come out way ahead. But, that turned out not to be the case. In building tents you're always limited to a set of no more than 4 tents to choose from, and thus it becomes increasingly hard to take advantage of more than 1 or 2 adjacent tents per build, and thus you might occasionally get a 0 card/point build, your average ratio on a turn where you tried to build a lot was likely to be 1 card/point or higher.

Unfortunately the ratio of just building a standalone tent was 1 card/point. Thus, it was almost always better to build a standalone tent, than to try and be clever at all. After I realized this in a second game, my standard strategy became: build a standalone tent, build a second standalone tent, build a tent between them, using both their colors, and then if possible build a fourth tent which took advantage of two or three of the tent colors. The result was a ratio of 1 card/point or slightly better, and since two of the tents were standalone, I had more tents to choose from to try and get a set that that would support each other.

Perhaps a better way to explain this problem is that the margin for cleverness was too narrow. At first it looked like any move you came up with which allowed for a better than 3 card/point ratio was good, but in truth it became clear that you needed better than a 1 card/point ratio to win, and thus something was only clever if you got into the tight (and difficult) range of 0-1.

If this game had been looked at mathematically, I'd betting that first-time tent would have been worth 2 points, not 3. The math here is really simple, but it made the difference between a game that worked for serious players, and one that did not.

Take Stock

The math for Take Stock, a stock market game by Z-Man Games is more complex, but I could tell from my first play that it didn't work quite right, and I knew it was because the author (Simon Hunt) didn't do his math homework.

This is one of several games that uses stock cards for dual purposes: to mark the value of stocks or to increase your own personal portfolio. You can always play any stock to increase the value of the stock (but with a no more than 4 point jump at a time) or to gain stock certificates for yourself. At the end of a round you multiply the final value of the stock times your number of shares to get your score

There's a direct correlation between stock value and stock certificate count: 2-4 are worth no certificates; 5-7 are worth 1; 8-10 are worth 2; and 11-12 are worth 3.

Now the game greatly depends upon plenty of cards being played to each stock. First, the number of cards in a stock ultimately (and artificially) constrains how many certificates each player can play, which in turn marks the end of a round. Second, playing an "11" or "12" can end a round early. Either of these is important to make sure the game doesn't drag. Unfortunately the math of the game is set up in such a way as to greatly discourage people from playing the higher value cards.

Consider, for example, the value of playing an "11", the lower valued "stop" card. Since a stock value can jump no more than 4 points, this means the current value must be "7" to "10". Now keep in mind that this "11" also has a value of 3 stock certificates.

The following chart shows how much you'd earn from playing that "11" to increase the value of the stock, based upon the current value, and the current number of certificates you have:

Current Value1 Cert.2 Certs.3 Certs.4 Certs.5 Certs.6 Certs.
Value 7
+4
+8
+12
+16
+20
+24
Value 8
+3
+6
+9
+12
+15
+18
Value 9
+2
+4
+6
+8
+10
+12
Value 10
+1
+2
+3
+4
+5
+6

Now compare this to the fact that laying down that same card as 3 certificates gives you 21 points for value 7 (3*7), 24 for value 8, 27 for value 9, or 30 for value 10. Therefore, the only time you'd ever want to play a value 11 as a value (absent being able to catch a lot of people out by ending the round early, which really isn't that likely) is when you're making the maximal leap (from 7 to 11) and when you have a 6 certificates already out, which is a pretty hugely unlikely number. This one good play is marked in bold italics.

As with Desert Bazaar the margin for success is so narrow as to be almost non-existent.

This describes the problem at the top end, but it generally exists for playing all of the higher cards as values rather than certificates. Here's a similar chart for playing an "8", which is the first of the two certificate cards.

Current Value1 Cert.2 Certs.3 Certs.4 Certs.5 Certs.6 Certs.
Value 4
+4
+8
+12
+16
+20
+24
Value 5
+3
+6
+9
+12
+15
+18
Value 6
+2
+4
+6
+8
+10
+12
Value 7
+1
+2
+3
+4
+5
+6

Now the value of an "8" certificate in these cases is: +8 (for 4); +10 (for 5); +12 (for 6); or +14 (for 7). Again the good plays are marked in bold italics. I generally did not consider the break-evens good plays because they also increase the value for all your opponents.

The chart is better, but there's still a pretty scant set of options for profitably increasing the value of a stock rather than just greedily keeping the certificate to yourself: only a third of the chart.

Now as a mentioned, the total number of value cards played to a stock define the number of cards that you can play as stock certificates and thus when the round ends. Generally you need 4-7 value cards on one stock to be able to finish a round. Just looking at the above numbers you can see that becomes a very tough battle. People are mostly willing to play the 1-7, but if any numbers at all are skipped (and some will be because there's only one of each and a whole deck of cards) then it becomes harder and harder to finish out a round. It becomes a sort of Free Rider's Paradox, where no one has any incentive to expend the resource that will help every one.

Here the problem is a lot larger and more integral than in Desert Bazaar. I think the relation between stock values and stock certificates is almost totally broken, and that undercuts
the game for anyone playing seriously.

Conclusion

Last time I said that I, as player, don't want to see the math in a game. But I always want the designers to have carefully considered it if the game has a truly mathematical basis, and I think Desert Bazaarand Take Stock were both games that were hurt by their lack of mathematical care, to the point where as a serious gamer I can not play them.

It somewhat explains what Reiner Knizia, a professor of mathematics, can produce such well-designed games: even when the math isn't showing its underlying basis is very important.

6 comments:

Anonymous said...

So, for Desert Bazaar, does simply changing the number of points for the first tent to 2 (1 immediately and 1 at the end) fix the game, or does it introduce other problem?

Shannon Appelcline said...

I'm guessing it would help out without causing problems, but I haven't tried it.

Coldfoot said...

I wonder if applying a little math would fix Monopoly?

I am specifically thinking of combustion rates of cardboard in a large bonfire as opposed to the more thorough, even destruction that can be accomplished with a kiln.

Alas, I don't know the combustion temperature of cardboard and the time needed for complete consumption.

Perhaps Knizia could help?

Fellonmyhead said...

I may be missing something, but didn't your analysis of the 11 card in Take Stock omit one important factor?

There is more than one market.

Shannon Appelcline said...

Each card only applies to one market, so I don't know what that has to do with choosing to play it.

Yes, any of the markets could reach the 5-value to close down trading, but the same limitations will apply to every single one.

Fellonmyhead said...

Shannon: Each card only applies to one market, so I don't know what that has to do with choosing to play it.

But that's the whole point; it's not likely to be used to close the market by raising the share price of a market in which you have an interest; hence if your interest is in oil you may wish to close the market by playing the 11 corn before a) anybody gets any more corn shares down or b) anyone else gets any more oil shares down.

Your calculations only consider the benefits in one market, not in all markets for all competitors. Calling this "almost broken" as a result seems to me a rather premature conclusion to reach. Please correct me if I'm wrong, but from what I can tell there are numerous situations in which this could be a more beneficial play than taking three shares in the relevant commodity.