What Are the Odds to “Draw 2?” Part 1

Part 1.  The Question

The principal purpose of this blog is to bring you living debates from game design and our solutions to them.  An interesting one occurred recently which drove me on a bit of a mathematical quest.  Having taken the computational plunge, it’s now time to share with you fine folks my results.

Draw 2So it happened that we were playtesting a card game.  Luther on his turn played a card we’ve seen in a huge number of games; he played the special “draw 2 cards” card.  So far, so good, right?  We’ve seen this card many times in many different games.  Our postgame discussion led to a debate over whether playing this special meant that he was more likely or less likely to be holding another.

We began to explore the possibilities.  Luther holds more cards, that increases the odds but he also played one of these specials and that reduces the odds. It was hard to tell from the outset.

We did the calculations.  This particular game has a 99 card deck and 9 of these cards are the “draw 2 cards” special.  Players are dealt an 11 card hand.

The easiest way to determine the odds that Luther is holding a special is to calculate the odds that he is not and then subtract this result from 1.  The probability that he is not holding a special in his first eleven cards is

Probability Part 1 Equations 1

In the interest of time and space, I elected to assume that you are already familiar with the mathematics of permutations. If you find yourself wanting a refresher, there is a solid description at Math is Fun.

Subtracting this from 1 gives us

Probability Part 1 Equations 2

So there was roughly a 67% chance that he received a special in his opening deal.

Now he plays a special, which reduces his hand by one card and draws two cards.  He is now holding twelve cards but there is one fewer special in the available supply since he just played one of them.  The odds that he is not holding a special in these twelve cards is

Probability Part 1 Equations 3

Subtracting this from 1 gives us

Probability Part 1 Equations 4

There is now roughly a 71% chance that he holds a special.  We have our result.  Not only did the odds go up, the odds rose by nearly 4%.

But this is not the end of the story.

It’s nice to know that the odds go up by 4% in this particular game but what about all the other games with similar conditions?  What about later in this same game, once a few cards have been seen?  Is there some point at which the odds slip?  When is that?  These questions propelled me to dig deeper.  Our next installment will seek to answer these questions.

What do you think of “draw 2 cards” cards?  Have you incorporated them into your designs?  What did you learned from the experience? Do they improve a design or weaken it?  Share with your fellow readers in the comments below.  And if you’re enjoying what you’re reading, create an account with WordPress and follow this blog.  If you keep reading, I’ll keep writing.


Tricky, Tricky, Part 2

The Story So Far…

Trick taking games are defined by the following criteria:

(A)  Each player has a hand of cards.

(B)  These cards are played in a series of rounds (tricks).

(C)  Each player in turn must play to the trick.

(D)  Each player plays to the trick exactly once.

Trick-taking games were in a slump they’re coming back.  Let’s look at how to write one that stands out from the crowd.

Variations On The Basics

Even with the four criteria listed above, there’s a great deal of room for any designer who’s willing to explore the design space.  Each of these criteria carries assumptions with it.  Let’s set these assumptions aside and ask “which traditional rules am I willing to break?”

(A) Each Player Has a Hand of Cards

The vast majority of trick taking games deal the same number of cards to each player.  There’s no particular reason for this, however.

Each player in the traditional game Euchre receives a five-card hand.  Then a sixth card is offered to the dealer.  If any player accepts it, that card goes into the dealer’s hand and the dealer then discards one card.  While this still leaves the dealer holding five cards, these five are chosen from six.

What if we allowed the dealer in Euchre to keep all six cards, simply discarding the last card when all the other players have run out their five?  This would tend to make the dealer’s position stronger and reduce the need for hand-evaluation skills.  But what if your card game had a significant dealer disadvantage?  Then offering that player a larger pool of cards from which to play would go a long way toward addressing that weakness.

A recent prototype in my group similarly gives the dealer one extra card but then requires a discard.  In this case, the discard dictates trump but can never be played, thereby forcing the dealer to give up a trump card in order to promote its suit-mates.  In this case, the extra card forces the player to exercise her hand-evaluation skills even more carefully.

By offering significantly different hands of cards to different positions at the table, we can give our games a type of texture which same-hand card games traditionally lack.

A Sample Game With Different Hand Sizes

What if in addition to varying the hand size, we also made different positions receive their cards from different sources?  Here’s an outline for such a game.  Feel free to run with it as far as you wish:

Imagine a three-handed game.  This game uses four suits with cards numbered 1-9 in each suit.  Before dealing, these cards are divided into three decks.  Deck A contains the 1, 2, 3, 4 of each suit (16 cards).  Deck B contains the 5, 6, 7 of each suit (12 cards).  Deck C contains the 8, 9 of each suit (8 cards).

Each deck is shuffled. and dealt out.  The start player receives a fifteen-card hand–eleven cards from deck A and four cards from deck B.  The middle player receives a twelve-card hand–five cards from deck A, four cards from deck B, and three cards from deck C.  The dealer (last player) receives a nine-card hand–four cards from deck B and five cards from deck C.

These cards are then played over 12 tricks.  The dealer will play out his hand in the first nine tricks but has the best cards.  The middle player will play the entirety of her hand but has a medium-strength hand and three tricks in which the dealer does not loom over her.  The start player has the weakest hand but gets to choose three cards to leave unplayed.

(B)  Cards Are Played In a Series of Rounds (Tricks)

Most trick taking games start with the assumption that only one trick will be going on at a time.

Hattrick allows two tricks.  After the first player plays a card, any player who wishes may begin a second trick in a second suit.  Because of the possibility of this second trick, Hattrick is at its best when it is played by its full compliment of 6 players.  When 4 players participate, Hattrick is simply too forgiving.

Victory & Honor is a game for exactly four players and has exactly three tricks going on at all times–the left, the center, and the right.  None of these tricks are evaluated until all three have been completed.

(C)  Each Player In Turn Must Play To the Trick

In addition to its three simultaneous tricks, Victory & Honor also allows players some control over the order of play. If you play a card in your left area, your left-hand opponent must play next.  Similarly, play to your right area makes your left-hand opponent play next.  If you play a card in your center, your partner plays next.  This variable player order gives Victory & Honor a texture unlike any other card game I’ve seen.

Most trick taking games also assume that each player will play exactly one card in each trick.  What if your game allowed cards to be played in combinations?  In games of that type, there might not be any need at all to keep hand sizes even.

For example, your game might allow players to play sets and add their value.  Thus, when my opponent opens with the 8 of spades, I might respond with a pair of 5s–the 5 of spades and 5 of diamonds–and count this play as the 10 of spades  (the rank of my cards–5–multiplied by the number of cards in the set–2).

Alternately, you might allow your game might allow players to play runs.  When my opponent opens with the 8 of spades,  I might respond with a run of three cards–the 4, 5, 7 of spades–and count the play as 12 spades (the lowest card–the 4–multiplied by the number of cards in the run–3).

A Sample Game that Allows Multi-Card Plays

I particularly like set-playing in games which feature unevenly distributed ranks.  Here’s another game outline for you to run with:

Imagine a four-handed game.  Cards in this game are ranked 2 – 9 but high-ranked cards are rarer.  There are twelve 2s, eleven 3s, ten 4s, nine 5s, sis 6s, five 7s, four 8s, and three 9s.  A player may follow a trick by playing sets as described above–when your opponent opens with the 8 of triangles and the 8 of spirals, played as a pair, he declares the value to be 16 and must choose either triangles or spirals as the suit.

(D)  Each player plays to the trick exactly once.

If you were to ask most any card game player who takes their turn, they’d stare at you with the sort of look generally saved for the simple-minded.  Of course, I take my turn.  Who else could it be?

But Twilight/Dr. Jekyll & Mr. Hyde threw this assumption right off a cliff.  This game features two decks–one white, one black–which are shuffled together and dealt out.  These colors correspond to the two teams in the game.  Because these decks have different card backs, it is clear to everyone who holds how many of which team’s cards.  When it is your turn to play, you may name any player holding cards belonging to your team to play for you.  This leads to some engaging decisions.  Can you force an opponent to take a trick for you?  Can you find opportunities to waste the opposition’s cards?

What variations do you like to see in a trick-taking game?  What makes them so special to you?  Which ones do you dislike?  Why?  Share with your fellow readers in the comments below.  And if you’re enjoying what you’re reading, create an account with WordPress and follow this blog.  If you keep reading, I’ll keep writing.

Asymmetrical Suits in Card Games, Part 1

You must walk–or so they say–before you can run.  You must similarly design from the familiar before you can find your own voice.  Freshman designers tend to begin by working with familiar things.  This means reaching for games and components already lying around.  The consequence of this is that most of us have written several card games using a regular deck of cards.  I certainly have.  In the 1990s, living the life of the cash-poor graduate student, I frequently gave these games away as birthday presents, frequently naming them after their intended recipient.

Starting with a regular deck is great.  The components are well known, readily available, and cheap.

There is a pitfall to using a standard deck, however.  This issue is subtle but important.  Mainstream card decks are completely symmetric.  Each suit is identical to every other suit, having the same number of cards and the same card ranks.  This is also true of the Scopa, Tarot, and Uno decks.  These were the decks I’d grown up with.  They were all symmetrical give or take a few special cards. It did not occur to me that decks should be any other way.

It was the flood of European games card game designs in the 1990s which revealed the power of asymmetry.

Asymmetry Serves Player Engagement

Asymmetric decks are not automatically well-known to the players.  They offer your players a puzzle.  How do these suits interact?  Which cards are now better or worse by suit? In his book A Theory of Fun for Game Designers (http://www.theoryoffun.com/), Ralph Koster notes how much our brains love to search for patterns.  Asymmetry challenges our brains to discover new patterns.


Asymmetry Serves Game Balance

Asymmetric decks allow the designer to balance rarity against utility.  Richard Garfield’s climbing game Dilbert: Corporate Shuffle, has one 1, two 2s, three 3s and so on up to ten 10s and a few special cards.  Low numbers beat high numbers so the most powerful cards in the game are extremely rare.  You must play the same quantity to follow another player’s lead so weak cards can defeat strong cards by outnumbering them.

Many designers prefer to make powerful cards rare.  I often find the opposite works better.  By inserting several copies of high-utility cards, you give each player a better chance of getting it.  Weaker cards and situational cards then add texture and encourage players to think laterally rather than being the majority of the hand while one lucky player draws the best card and runs away with the win.


Asymmetry Serves Story

Asymmetric decks are give you as the designer another tool for representing how one group is stronger, weaker, or simply different than another.

Doris and Frank’s wonderful card game Frank’s Zoo has circular card ranking.  Every card can defeat at least one other card and is in turn defeated by at least one other card.  No one card is the best.  No one card is the worst.  No player can put a card forward with impunity.  There is always a risk of it being defeated. Add that one suit contains only 4 cards while the other suits contain 11 and we see how asymmetry reinforces the game’s story.


Asymmetry Serves Sales

Game reviewers and the gameratti are quick to identify a game which can be played with a standard deck of cards.  Once the word spreads–and be certain that word will spread with the internet being what the internet is–many of them will throw together a set from a deck they have lying around and go straight to it rather than purchasing your game.

If you simply wish to create games for the enjoyment of others and have no interest in monetization then this is perfect.  This was certainly my intent when I created games as gifts.

You need to offer players more if you wish to sell your game.  You are asking gamers for their money.  They want to see that you earned it.  Offering an asymmetrical card deck shows them that you put some of yourself into your game.  Making it a little bit more difficult to make that home set gives them another reason to pay for your effort.


The best card game in the world–Tichu–uses nothing more than a standard deck of cards with 4 differentiated jokers (I games of Tichu being played in just this way at a few conventions).  That Tichu can be so excellent with a symmetric deck shows that symmetric decks still have plenty of value.  I’ve come to feel however that asymmetry should be the assumed condition over symmetry rather than the other way around.

What are your favorite asymmetric card games?  Share them with your fellow designers in the comments below.  And if you’re enjoying what you’re reading, subscribe to this blog.  It makes a big difference.

Our Friday installment will examine specific examples of asymmetry. I will attempt to pick apart compare designs with a symmetric deck against identical rule sets with asymmetrical decks.  Come back Friday and see how I do!