# Perfect play compared to normal play in UNO and other games

I was wondering about perfect play in UNO. UNO is a pretty simple game, and generally most humans are able to do pretty well at it. But how effective would we be if we played perfectly?

The other day, my friend was one seat ahead of my other friend in the rotation of turns (turn order), and he played a wild card to attemp to stop my other friend from winning (he had one card left). He eventually decided to pick green, saying "there have been lots of green cards left, so it's not as likely that (my other friend) will have a green card as his last".

That got me thinking. There's a lot of open information in UNO; one knows their own hand and every card that has been played, as well as the size of the deck and their opponent's hand. Therefore, it should be possible to easily calculate probabilities, if one remembers everything and is a computer. Then, when having a choice of options, one could evaluate the probabilities of these, make the optimal decision, and become a perfect player.

So, my question is: It should be possible to quantify how much of an advantage a perfect player actually has in terms of win probability. What are the win probabilities of two players, A and B, in a two-player game of UNO played accordingly to the rules from 2006 (before they added the crazy "swap hands" cards and such)?

Definitions:

• These are one-game matches. It doesn't matter by how much you lose, I just want the probability of winning. No 500 point rule.
• Player A is our perfect player. Player A has a perfect memory, can calculate probability essentially instantaneously, and always chooses the correct play considering the information available.
• Player B is an average player. If Player B has a play, he plays it. If player B has a choice of two cards, he picks one at random (unless one of them is an action card). If one or more of Player B's possible plays is an action card, he chooses the play (a play is the series of actions that B takes before A gets to play) that removes the most cards (or if tied, randomly) from his hand; B can see combos, like skip-skip-reverse-green 5, and will play those if it removes the most cards from his hand. When B is faced with whether or not to challenge a draw four, he decides randomly (50/50 chance). A does not know how B decides to challenge or not.
• Both players never forget to call UNO when down to one card. Both players never make card-play suggestions.
• There is a 50% chance that A will get to go first, and a 50% chance B will go first.
• Neither player ever resigns, and no time limits.

Please let me know of things that need explaining, as I'm sure I missed something.

• I am not sure this will be easily solvable as at the start of the game there is about 35% of the deck that is playable off a 1-9 card and 31% for a 0 card. I would also say that having player B pick a random card when having a choice would make him a below average player at best since a random play could leave him in a bad position if it leaves the pile in a color where he has no cards versus one where he has a lot of them. – Joe W Oct 6 '18 at 22:27