What Are the Odds to “Draw 2?” Part 3

From last time we have that with

Probability Part 3 Equations 1

The change in odds when your opponent plays a special card and thereby draw two cards is

Probability Part 3 Equations 2

If this difference is positive, the odds go up.  If this difference is negative, the odds go down.  Let’s expand this formula and see where it can be simplified

Probability Part 3 Equations 3

Notice that the fraction shown in the first set of brackets must be positive.  We can set that aside for now since what we’re really interested in is when this formula is equal to zero.

Probability Part 3 Equations 4

Notice that the fraction in the left hand brackets must be positive.  Again we can set that aside for since what we’re really interested in is when this formula is equal to zero.

This leaves us with the formula and the information we need.  When a “draw 2 cards” special is played, the odds that this player holds another are…

When Probability Part 3 Equations 5 < 0, the odds decrease.

When Probability Part 3 Equations 5 = 0, the remain unchanged.

When Probability Part 3 Equations 5 > 0, the odds increase.

Hopefully, all this math will put a useful tool in your designer’s toolbox–and all arose from an off-handed comment I made about probability.

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.

What Are the Odds to “Draw 2?” Part 2

Part 2.  The Extension

The next questions was, how do these results vary from one game to the next?

With questions like these, it’s often useful to explore the extremes first.

Extreme #1.

Imagine a deck which only contains one special.  In this case, playing a special means that none remain and you have 0% probability of holding another.  Not very useful but worth noting.  We’ll bring this up again later.

Extreme #2.

Imagine a deck of seven cards which contains two specials in which you are dealt five cards.  If you play a special, you draw the rest of the deck and thereby guarantee that you’re holding the other one.  You have 100% probability of holding another.  This is actually more useful to note than Extreme #1.  Strategy in deckbuilding games like Dominion frequently revolves around building an engine that maximizes your chances of drawing all of your deck every turn.

So we have one extreme case in which the odds drop to 0%, and another extreme case in which the odds jump to 100%. What’s going on in the middle.  That’s what I set out to discover next.

The Restrictions.

To avoid ridiculous cases like Extreme #1 or Extreme #1, let’s set out a few assumptions.

First, we have a deck of size N and your starting hand size is n.  This deck contains a quantity k of special “draw 2 cards” cards.  To avoid the ridiculous extremes, let’s agree that the deck contains at least two specials and that the deck is large enough that after playing the special and drawing the extra two cards, there will be at least one card remaining.

Put mathematically, we have that

Probability Part 3 Equations 1

How much do the odds change when your opponent plays a special card and thereby draw two cards?  The quantity of specials which could be in her hand decreases by one.  The quantity of cards you have not seen decreases by one.  The quantity of cards in her hand increases by one.

The Formula.

The change in probability is given by these formulas:

Probability Part 2 Equations 2

Example:

The deck has 60 cards.  10 of these cards are “specials.”  Players are dealt 6 cards to begin.  When a player uses one of these specials, the odds drop by approximately 6%.

Probability Part 2 Equations 3

Example:

You are playing Dominion. Your deck has 25 cards.  10 of these cards have “Draw+2.”  Each round starts with 5 cards inn hand.  When you play one of these cards, the odds you are holding another drop by approximately 3%.

Probability Part 2 Equations 4

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.

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.

093 User-Friendly Games, Part 3

The Story So Far…

Deep games are cool.  Complex games can be awesome.  Many excellent games are excellent because they are complex and deep.  But when these elements aren’t well conveyed, good games get glanced over.  They get a single play but little more.  Rules complexity or obtuse play become an impediment, preventing the game from ever gaining the following it deserves.

This post takes a look at depth.  How depth be presented to our players and in accessible ways?  How can depth be user-friendly?

What Does Depth Mean

In his fantastic book A Theory of Fun for Game Design,  Ralph Koster opens with an examination of engagement.  Games need to be in the “Goldilocks zone,” in order to be engaging.  When a game is too easy, it bores us.  When a game is too hard, it confuses us.  To be Fun, a game needs to challenge us without overwhelming us.

Depth is not rules, nor is it card text, nor is it victory conditions.  Depth is in the ways rules, card text, and victory conditions interact.  In a deep game, each element you add brings new implications for your players.

DominicAdding game elements does not grant a game depth, however.  Adding elements and rules brings complexity but depth is as much about the elements you choose to leave out as the ones you choose to incorporate.  Designer Dominic Crapuchettes once remarked to me that “every rule must justify its existence.”  Dominic embodies the designer seeking depth.

Chess is the quintessential example of depth, of course.  The rules fit on the inside of the box top but thousands of books have been written on the subject.  Depth will keep players returning to your game long after its contemporaries have been “solved” and abandoned.

The Danger of Depth

Depth can be a trap for designers as well.  Inherent in designing for depth is the danger of unseen depth.  A Theory of Fun for Game Design cautions us about engagement–when a player cannot see the good ways to play, she will throw her hands up in frustration rather than keep looking for unseen depth.  You can hardly blame her; thousands of games come out every year.  Each of these begs for her attention.  If your game is inaccessible, she will quickly move on to the next one in line.

To the designer, who is intimate with each element of the game, this depth is transparent but working strictly from the rules, the depth is too great for many neophytes.  If the designer were included with the game to give new players guidance, its depth would become clear.  We call this designer in a box syndrome.

Designer in a Box: 1955, The War on Espionage

My first-hand experience with designer in a box syndrome happened in 2011 when Living Worlds Games released 1955: The War of Espionage.  Early reviews were solid.  Blind tests indicated that the play was clear.  We were hopeful that the game would be well-received.  Before going further in this story, it should be mentioned that 1955 is a card-driven  game and that each card can be played four different ways and that knowing which way to play each card is essential to victory. When we released to the public, we discovered that many players had trouble seeing through its depth.  I had stumbled into the land of designer in a box.

So how do we deal with this issue?  How can we put you–the designer–in the box?  I have identified three ways: Sidebars, Play Hints, and Strategy Guides.

Sidebars

SidebarRightSidebars are the margin notes in a rulebook.  Sidebars are principally used to remind players of key rules or to provide examples of play.  Sidebars can also be used to present play hints.  Use these play hints to give your players insight into your game’s depth.

To see examples of particularly good sidebars, take a look at rulebooks from Alea.  Their editors regularly use sidebars for all these reasons.

Play Hints

TipsThese are small tips to the players.  Play hints should be short and sweet–a quick sentence or two and nothing more.

Use your play hints to emphasize the play elements you believe should be obvious.  Remember that you are intimate with each element of your game.  Play hints are your opportunity to share your vision with your players.

Strategy Guides

When you feel verbose–as I often do–strategy guides are the place to exercise for that verbosity.  Play hints are terse.  Strategy guides can be exhaustive.  Use them to give players broad guides to play.  Use them to walk players through the major strategies you discovered during your game’s development.

Some players prefer to explore a game on their own.  Discovery is part of their core engagement.  These players plan to give your game multiple plays and discover their own strategies.  They may feel cheated when you show too much to them.  Because strategy guides may be seen as “spoilers” by these players, I recommend that they be saved for the end of the rulebook or put on the game’s website.

Deep Games on the T.A.B.L.E.

A few weeks ago, I had the pleasure of returning to the T.A.B.L.E.  convention.  Several attendees were aspiring game designers looking for feedback on their prototypes.  Talking with them was a pleasure.  Few things are more enjoyable to me than talking design with someone who’s passionate about their craft.

One game in particular caught my attention for its potential depth, Roller Derby Final Jams (image included with designer Steve Bissett’s permission).
RDFJ layout

RDFJ focuses on the closing minutes of play.  This is a wise decision on Steve’s part.  It allows him to focus gameplay on the most exciting parts of a match and dispense with the humdrum.

There were some areas that were still a bit clunky in this design and this was an opportunity for Steve and I to discuss his game at length, looking carefully for opportunities to streamline play, to increase its depth, and to convey depth through its components.  If you see a man dressed as a referee with “Tom Green” on his back at your local convention, I urge you to grab him for a play.

Closing Thoughts

I confess.  This entry kept me stumped for weeks.  Where the first two entries flowed out rather smoothly, the topic of depth keeps slipping out of my grasp.  Ironically, this is much like how game depth itself functions.  Depth in a game relies on a certain amount of subtlety and so too does describing it.  My instincts tell me that there’s quite a bit more to be said on the subject.  Hopefully, others can pick up where I am leaving off.

One good piece that helped me to clarify my thinking appears on Wikipedia. Unlike many other articles out there, it looks at the broader issue of usability, not just software but anything a human interacts with–books, tools, machines, processes.  I recommend checking it out.

What do you think of depth and its impact on user-friendliness?  How do you convey depth in your designs?  what have you learned from the experience? 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.