# How does UNO determine how often do +4,+2 & other special cards occur in a deck/game?

For UNO-like card games, what factors would one consider when they wanted to set a reasonable proc rate of receiving a special card (for example in UNO's case: +4,+2, color-wheel, reverse, skip)? What factors should I consider when designing a similar game?

I understand probabilities etc. are involved in this. I would appreciate if someone can demonstrate the mathematics/concepts behind it.

• Should this be migrated to Board & Card Games Stack Exchange? I don't see anything here that is specific to video game development. Mar 9, 2018 at 16:20
• @Thunderforge This site is called Game Development. The 'video' part is optional.
– Mast
Mar 9, 2018 at 18:38
• @Mast and the 5 upvoters... The Help Center for this site disagrees with you: "Game Development Stack Exchange is for professional and amateur video/computer game developers"... Mar 9, 2018 at 21:17
• @JPhi1618 Sounds like a good Meta question, actually! Do you want to write it up, or should I?
– anon
Mar 10, 2018 at 3:03
• @JPhi1618 I will note that there's nothing in this question which implies it isn't for a new computer or mobile based card game. There's certainly plenty of those. Just because it isn't explicitly about the digital part doesn't mean that it isn't going to be a digital game. Mar 11, 2018 at 1:45

tl;dr: Math is useful, but it can not replace playtesting.

1. Card economy. You don't want the game to take forever, so you need to make sure that over time players will have a net loss of cards. So you need to make sure that the cards picked up by "take X" cards do not outweight the natural rate of losing cards. This is one aspect where trying to find a solution based on math makes sense. We could now analyze the exact math for UNO here, but that might be enough material for a university homework assignment and of little use for you unless you want to make an UNO clone.
2. Balance luck vs. skill. The more powerful the special cards and the easier they are to play, the more luck-based does the game become. For example, many people who play UNO ignore an important rule regarding the +4 card. You can only play it when you can not play any other card. Following that rule correctly makes that card a lot less powerful during the early- and mid-game and thus makes the game less luck-based.

Children usually prefer more luck-based games while adults often prefer more skill-based games. So the ideal balance in this regard is a question of target demographic.

3. But most importantly, playtests, playtests and more playtests. Even the best game designer can not design a fun game without actually playing it. Play it over and over again with different decks and different rule variations to figure out what works best in achieving your game design goals.

• Don't just play a certain combination once. Play it multiple times to make sure that your results are not just due to chance or bad play.
• Don't always play to win. Have testplayers vary their playstyle from match to match. Try strategies you don't expect to work and see where it leads to. Even if that strategy eventually loses, the results might still surprise and inspire you.
• Don't always playtest it with the same people. Try to bring in new players from time to time to bring in a fresh perspective and make sure that your rules are still accessible to newcomers.

Then analyze your test games and pinpoint phases which weren't fun. Think of how you can modify the game to fix or avoid these phases. Implement your changes and do yet more playtests.

• @Anko "What factors did they consider" seems answered enough without going into a detailed mathematical treatment of the particular details of how the factors were considered. Mar 9, 2018 at 11:45
• Doesn't making a +4 card "only usable when nothing else can be played" make it more luck-based? You lose the strategy aspect of deciding "when to play it" :O Mar 9, 2018 at 13:44
• @BlueMoon93 Paradoxically it doesn't. It means that +4 cards can only be played during the endgame. That means when you reached the endgame there is a better chance for all players that they have a +4 card they weren't able to play yet. It also creates a new strategic goal: Intentionally try to get rid of some colors in order to obtain more opportunities to play your +4. Mar 9, 2018 at 13:51

In the case of UNO, I find it highly unlikely that there was any deep consideration of the underlying mathematics. Consider this image of the UNO deck of cards (taken from the Wikipedia page on UNO):

It contains 108 cards. That's an interesting number. Why? Because it's twice the number of cards in a standard deck of playing cards - 54 (13 of each suit, plus 2 jokers) - moreover, the cards appear in 4 "suits" (colours in the final game). I strongly suspect therefore that UNO was developed playing with two sets of playing cards and, in fact, the ratios of the different cards reflect the distributions that could be easily created from these decks rather than any deep consideration of design principles.

• While it is indeed conspicuous that UNO could be played with two conventional 54 card decks (as long as at least one rank has a different design), I don't think it is fair to deny that the creators had any originality. For example, why is it +2 and +4 and not +3 and +6? Why do only half of the "pick a color" cards also have the addtional +4 effect? And what's with those 0 cards? Why didn't they use those 4 excess cards for something more interesting? Mar 9, 2018 at 13:32
• @Philipp: I'm not denying them any creativity, I'm saying that the game was not carefully constructed according to any underlying mathematical principles. Mar 9, 2018 at 13:36
• Uno derives a lot of its fun by being so random that it's anybody's game until it's over. Games like that don't really need to be balanced, so the things mentioned by @Philipp wouldn't change the game dramatically. An even more extreme example is the Monopoly card game, which has a few cards that are almost instant-win cards. Mar 9, 2018 at 16:54
• To be fair, Uno is simply a commercially successful variant of much older games such as Crazy Eights and Mau Mau that were played with a standard deck of cards. Mar 10, 2018 at 9:19
• @JoshCaswell Why not? OP can use the same tactic.
– Mast
Mar 10, 2018 at 16:47