I'm Morgan McGuire (@CasualEffects). I've been working on computer graphics and games for 20 years at great places including NVIDIA, Williams College, Brown University, and Activision.

See my home page for a full index of my blog posts, books, research, and projects.

Wednesday, November 5, 2014

McGuire House Rules for 2-Player Uno

Uno is a card game for "2 to 10" players that is currently owned by Mattel. Like other games that appeal to small children, the challenge is in learning to follow rules and recognize appropriate play and not in executing a strategy. The game appeals to its core audience because the rules are simple, play is fast, and a lack of strategy means that there really are no "good" or "bad" players. The first two properties are desirable in any game. The shallowness enforced by the third may limit the sting of losing for children, but also limits the audience.



The only way to play Uno truly well is to card-count a double deck. That doesn't match the target demographic, and isn't a style of play generally desirable for board games. Uno's mechanics also degenerate at the two player level, because the game becomes both zero-sum and concepts such as turn order no longer apply.

We've created two-player duel version of Uno to address these issues. I play it daily with a kindergartener and a second grader, and it engages each of us. The rule changes in our version simultaneously increase pace and strategic complexity, primarily by making all player information public. That also turns a game about learning to just follow rules into one in which players can also learn strategic thought and minimax search. This happens in two ways.

A five-year old's depiction of gameplay
The first way our game teaches is that a weaker player sees what choices a stronger player is making. In any game, a good move is indistinguishable from luck to the observer unless the observer knows what the alternatives were.

The second way is that a stronger player is able to advise a weaker player. However, I recommend giving children only advice when they solicit it. Let them learn on their own, and lose if necessary. Losing too often is unfair and upsetting, but handicaps address this better than lowering the opponent's level of play or artificially boosting the child's.

Under our rules, a typical game only consumes about half of the draw deck, so card counting is thankfully not a strong advantage.

Our rules do not scale to multiple players. The lack of hidden information in a multiplayer game can make it very slow, and the way that we've modified skipping and drawing rules causes a random walk in hand size rather than a consistently decreasing one.

The Rules

Like base Uno, this variant can be played with two standard decks of playing cards. I recommend purchasing Uno cards. They aren't much more expensive than generic cards, the bold art style matches gameplay, and they speed play because they are easier to recognize. Our deck reinterprets the 108 card Uno deck as:

  • Four color suits: red, green, yellow, and blue; each containing the following ranks:
    • One zero card
    • Two of each number card, 1 - 9
    • Two Skip cards, which skip the opponent's turn
    • Two Draw Two cards, which force the opponent to draw two cards immediately but do not skip the opponent's turn
    • Two Color Wild Cards labelled with two arrows ("R" = Reverse in older decks), which allow the player to name a new color suit (including the current one)
  • Four Wild cards, which can be played on any suit and allow the player to name a new color suit
  • Four Wild Draw Four cards, which can be played on any suit when the player does not have another card in that suit. These force allow the player to name a new color suit and then force the opponent to draw four cards immediately but do not skip the opponent's turn
You can always play a card with the same rank on another, for example, a red Skip on a green Skip or a blue Color Wild Card on a Yellow Color Wild Card.

Begin the game by shuffling the deck thoroughly and then dealing each player seven cards, face up. Stronger players can be handicapped by dealing them extra cards at the start of the game. The remaining cards become the draw deck. Flip the top card from the draw deck to form the face-up discard pile. The starting player, who lost the previous game, plays first. Players then take turns, except when a Skip card skips a player.

On a turn, a player may optionally discard from his or her hand a card that satisfies one of the following conditions:

  • Match the suit of the top of the discard pile
  • Match the rank of the top of the discard pile
  • A Wild or Wild Draw Four, if the conditions are satisfied

If the player does not discard, then he or she must draw one card from the top of the draw deck. If it is legal to play this card, then he or she may optionally play it immediately. Otherwise the card is placed face up in the player's hand.

The first player to discard his or her entire hand wins. Players are not required to shout "Uno!" when reaching a single-card hand.

Handicap a strong player by giving him or her extra cards at the start of the game and removing cards from the weaker player, Do not go below 6 cards or the game degenerates to luck for the weaker player's starting draw. When playing with my six-year old, I start with 8 cards and he starts with 6.

Observations

There is a slight first-player advantage. Because the game is short, we prefer to address this by playing multiple games rather than completing rounds. Rarely does one player win by a single turn; we observed over about 50 games that the losing player has on average five cards in hand.

Because players can see each other's cards, they plan ahead at least one move and typically an entire suit run of moves. This obviously creates more opportunities for strategy, and strategic play is often slower than random play. However, in this case it is much faster than blind play because players quickly execute the pre-planned moves. Pauses occur mainly when a player must evaluate the odds of drawing a particular new card, or consider the best tactic for a card that has just been drawn.

Players do not tend to race towards a minimal hand. Instead, a better strategy is to reduce hand size while building an unassailable run of cards that will trigger forced moves. This resembles the gameplay of Flash Duel, an excellent newer (and more complicated) dueling game by David Sirlin. It is typical to play a series of Skip and Wild cards, mixed numbers in a suit that the opponent cannot match, in the final run towards victory. We've identified several kinds of end run that allow forcing a win, which as ending on a string of Skip cards followed by a number card in the color suit, ending on a Wild card, and ending on a number pair that the opponent cannot interrupt. Rarely does good play end by luck--both players know about four turns from the end who is going to win and the player in the weaker position is desperately trying to draw a card to interrupt the run. I consider this an ideal ending for a game.

Players often will choose to not play cards in hand that would have been legally allowed on a given turn. That is because reducing hand size is often secondary to building a viable victory run, or to attempting to gain Draw or Skip cards to interrupt the opponent's victory run mid-execution. This creates a dueling feel of lunches, parries, and blocks. There is still enough randomness that one can't plan too far ahead and can enjoy the thrill of occasional unlikely upsets, however, strong players will consistently beat weak ones.

I do not recommend Mattel's new Uno Attack variant
Shuffling an Uno deck is hard. It is twice the size of a single standard card deck. Play naturally sorts cards into runs within a suit, so a perfect (Faro) shuffle will merely deal one player an ideal hand. Shuffles don't randomize a deck. They permute it in a predictable way. The semi-random offset caused by an imperfect shuffle and imperfect cut introduce some randomness, but anything close to a set of perfect shuffles is often close to a specific permutation. Depending on the number of shuffles, that can be a step in a sorting algorithm or a cyclic shift, which may actually maintain or increase ordering in the deck instead of decreasing it. We address this with two methods. One is to split the deck into four small decks and shuffle these together sequentially, and then repeat the process a few times. The second is to leverage a card shuffling machine to shuffle the deck many times. These (surprisingly) tend to shuffle so much worse than a good human shuffler that they generate more randomness over a large number of shuffles. They are really fast and can help poor shufflers (which both children and players with arthritis may be) to quickly shuffle the deck or sub-decks many times. A shuffling machine of course makes the game less portable, but also is fun to operate.

Mattel would prefer that, instead of playing our strategic variant, you take the already too-random version that they sell and add more randomness by purchasing Uno Attack for $24. This version replaces drawing with a machine that ejects pseudo-random numbers of cards at a button press. I prefer to teach young players strategy and planning instead of gambling, so I don't recommend Mattel's version.


Morgan McGuire (@morgan3d) is a professor of Computer Science at Williams College, visiting professor at NVIDIA Research, and a professional game developer. He is the author of the Graphics Codex, an essential reference for computer graphics now available in iOS and Web Editions.