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

Question

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.

Answer 1

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.

Answer 2

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.