Uno games can go on forever. How long is an Uno game likely to take? Does including more players make the game longer, or does it make it shorter? (I think there’s a reasonable argument for either guess.)
This sounds like the perfect way to squirm out of a college course requirement by writing an independent paper about Uno!
Most people play Uno with a casual strategy, and it’s not hard to program a computer to imitate that. My laptop can simulate one million Uno games in about five minutes. Then it tells me how long the games took. The results look like this.
Red = 2-player games; Green = 3-player games; Blue = 4-player games
First Conclusion: adding players makes the game take longer. Not a surprise.
But there are two ways to count how long an Uno game takes. You could count total number of turns or you could count rounds — that is, times that play circles around the table. (Rounds = Turns / Number of Players.) Here’s the same data, counted in rounds.
Red = 2-player games; Green = 3-player games; Blue = 4-player games
The curves have changed places: adding players makes for shorter Uno games if we count rounds. This makes sense: more players means more chances that someone is holding a winning hand.
The Conclusion: Uno games with more players tend to take longer in terms of actual turns, but it takes fewer rounds of play for someone to win.
Going First Helps. The first player has a small but measurable advantage. In fact, each player has a 0.5% better chance of winning than the player who plays after him. (This works for big games, too. I ran simulations up to 12 players.)
What’s with those peaks? The first peak, at 7 rounds, corresponds to Player 1 holding a perfect hand, going out at the earliest possible turn. The secondary peak, at 11 rounds, corresponds to the same scenario, but Player 1 was also hit by a Draw 4 card. This can be confirmed by simulating games using a deck without Draw 4 cards in it.
Red = standard deck; Blue = deck without Draw 4 or Wild Draw 4 card
So, were you successful in getting out of the History of Math requirement? This is a nice little simulation. Did you use a regulation deck, with shuffling after it ran out? (I see you ran some simulations with no Draw 4 cards.)
BTW, I like the layout of the new website. Very clean, very nice.
Yes, the paper did the trick.
I used a regulation deck (drawing with “no replacement”). For four players or more, I had to use multiple decks because the game could actually get stuck with all the cards in players’ hands and no cards to draw. Unlikely in a real game, but when you simulate 1,000,000 games these things happen. Of course, expanding the deck to fix this messes with the odds a little. As the deck becomes large, it resembles a deck drawn “with replacement.” I studied this for awhile – with lots of comparison simulations – and the effect is really small: you can simulate with however many decks you’d like without distorting the outcome.