Can certain tactics give a players an edge at Uno? The following simulations play five players, one of which uses a tactic that presumably gives an edge, in 10,000 independent games of Uno.
Control: all random
This is important so we can compare how much of an edge plain old randomness can have.
| player | tactic | edge |
|---|---|---|
| 1 | random | -0.30% |
| 2 | random | +0.04% |
| 3 | random | +0.34% |
| 4 | random | -0.22% |
| 5 | random | +0.11% |
| run-outs | +0.03% | |
edge is calculated by games won - ( number of games / number of players) as a percentage of all the games.
run-out is when there are no more cards to draw, the came cannot continue. I'm displaying them as positive because when you add all the numbers up they should total 0%.
So in this case, out of 10,000 games, player 2 came out ahead 0.04% of the time while player 5 fell behind 0.11% of the time, and nobody won 0.03% of the time (3 games) due to run-out. But really every player was within a small window of breaking even.
Choose the most abundant color
When playing a wild card you have a choice of which color to set for the next player to play. Reason dictates if you choose the most abundant color in your hand you are more likely to decrease the size of your hand. What happened when facing four random players against one who picks the most abundant color in their hand?
| player | tactic | edge |
|---|---|---|
| 1 | abundant color | +0.07% |
| 2 | random | -0.06% |
| 3 | random | -0.08% |
| 4 | random | -0.22% |
| 5 | random | +0.22% |
| run-outs | +0.07% | |
Wow, if player 1 just picked a color at random it probably wouldn't have made a difference. Keep in mind the random players don't even consider their own hand and might choose a color they don't have.
Play draw cards as soon as possible
If you have draw cards, play them soon right? Hit 'em hard!
| player | tactic | edge |
|---|---|---|
| 1 | eager draw | -0.76% |
| 2 | random | +0.36% |
| 3 | random | +0.13% |
| 4 | random | +0.40% |
| 5 | random | -0.18% |
| run-outs | +0.05% | |
hmm, I don't see anything noteworthy here. If you must infer something, infer that out of 10,000 games this tactic slightly loses.
Play draw cards as late as possible.
Don't play the draw cards soon, play them late. If draws are stacked you will be safe and it makes others more likely to have more cards when you have less.
| player | tactic | edge |
|---|---|---|
| 1 | lazy draw | +1.70% |
| 2 | random | -1.30% |
| 3 | random | -0.07% |
| 4 | random | -1.35% |
| 5 | random | +0.98% |
| run-outs | +0.04% | |
We broke 1%, but player 5 was close and is only random. I don't consider it that significant.
In the spirit of full disclosure ...
These simulations are far from complete. The human interaction aspects like calling uno when a player has 1 card left, seeing the faces of other's cards accidentally, drawing more cards than needed to play, cheating, etc. are not simulated and probably effect the game.
I didn't consider the ability for a player to look at the size of another player's hand. These are key factors when I play skip or reverse cards. So i really didn't test any tactics that specifically use skips or reverses. Does anyone know a better way to play skips or reverses?
All my simulations test a strategy against 4 random players. The combinatorics of all possible tactics with possible games with 2 to 7 players is beyond my motivation for this casual exercise.
Externalities: does a strategy effect other players without giving the user a significant edge or handicap? I have no idea but if you are qualified to answer these questions please let me know.
Also, I could have made errors, I'm only human.
Is the case closed?
Will these tactics will give you an edge? Nope, I think we are fooled by randomness when applying them.
Have I made it less fun? A bit. Then again maybe one can focus on the harder to simulate, human aspects of the game with the peace of mind that the above tactics are safe to not consider.
Will I ever play Uno again? Sure, its fun foiling other people's tactics and seeing the looks on their faces. It is also a great way to kill time, the average number of rounds of these 40,000 games was about 24.
If you have a tactic you think has an edge, or have found a flaw in my reasoning let me know I'll check it out, or grab the code!
Why did I do this? To get an edge in Uno.
