What makes a board game fun?

Carcassonne bothers me.

It bothers me because it is a lot of fun to play… but I feel like it should not be fun.

For those not familiar with the game, Carcassonne is in my opinion one of the better so-called German-style board games.  I think this way for several reasons.  The game doesn’t suffer from the bloat of equipment and rulebook pages that plagues many other games in the genre; it has the “minutes to learn, years to master” feel.  (Sort of, anyway– more on this later.)  The game is relatively short, and it also has the rare quality among German-style games of allowing just two players to play.  My wife and I play the game quite frequently during evenings after work.

And yet I have a sneaking suspicion that the game lacks something critical that a “good game” should have.  Sometimes after finishing a game (and yes, after winning as well as losing), I have the impression that we just completed a half-hour coin toss.  That is, the outcome of the game was more a function of the random draw of the tiles than of any strategy of mine or my opponent’s.

At other times during a game, sometimes just once or maybe twice throughout an entire game, I seem to recognize that the next move is crucial somehow, and perhaps that that next move should be one of only two or three possibilities… but I have very little insight into which move is best.

Despite these impressions, I really enjoy playing the game.  Why?  What makes this game fun?  Why do you love (or hate) to play chess, or backgammon, or Settlers of Catan, or Carcassonne?

Before presenting my attempt at answering these questions, I recommend J. Mark Thompson’s Defining the Abstract.  This is an interesting short article that I think does a great job of describing four key characteristics that distinguish good games from not-so-good games.  (The article focuses on “abstract” strategy games, and there are a few specific claims with which I don’t quite agree, but it certainly does a good job of provoking thought.)

In discussing these four properties, it is useful to consider the idea that every two-player game of perfect information may be viewed as a tree.  The start of the game corresponds to the root of the tree, and each branch corresponds to a possible move by one of the players– or by a possible random outcome such as a die roll or drawing of a tile in the case of games involving elements of chance.  The leaves of the tree correspond to end game states.  A game progresses from the root, down a particular path from one branch to the next, as players select a specific sequence of moves, draw particular tiles, etc.

(Note the progression “down” from the root.  Computer scientists always visualize/draw their trees upside down, with the root at the top, and branches extending downward, with leaves at the bottom.)

Thompson’s four characteristics are easily expressed in terms of a game’s tree structure.  A game’s “depth” is the simplest example: a game has depth if its tree is “deep.”  That is, even at some distance from the root, the tree still looks like a dense tree with many possible choices and outcomes, and better players are able to “see farther down” the tree than more novice players.

“Clarity” is also rather easily described in terms of a game tree… and this is one of the properties that I think Carcassonne may lack.  When evaluating which move to make– which branch to take– there should be a reasonably sharp difference between the estimated value of the “best” moves and the value of the “worst” moves, at least during the mid-to-end game.  (In more mathematical terms, the principal variation should stand out, so to speak.)  In other words, you should not frequently find yourself just guessing, not knowing whether this move is much better than that move.  I wonder if the random element in Carcassonne (i.e., the remaining shuffled tiles) diminishes too much the strategic value of one placement of a tile over another.

I’ll let you read the article for details about the other two properties, drama and decisiveness.  They also have simple descriptions in terms of the game tree.  For example, I think Settlers of Catan is an example of a game that suffers greatly from lack of drama– you can lose a game very early on, with no hope of regaining any ground.  Check out the article and see what you think about games you like or don’t like to play.

(Postscript: my latest project is a computer version of Carcassonne.  There are several goals here: (a) keep me busy; (b) let my wife and I play the game with an automated aid that I think is missing from the board version, namely keeping track of not only the points from completed features, but also those from incomplete features and farms; (c) implement a computer AI player; and (d) use (c) to do some interesting analysis that might provide some definite answers to the questions raised in this post.

The game is coming along very nicely… and in the process is raising some very interesting questions about intellectual property, which will probably be the subject of a post in the near future.)

Statistics of Deadly Quarrels

I started this post with the title “Death and Taxes.”  (I know– I will move on to lighter subject matter soon.)  But my references already both have this same new title above, and it sounds more interesting to me anyway, so I continue the theft.

This is a bit of a hodge-podge of ideas stemming from a convoluted train of thought this past weekend, but there is at least some common ground here.  It began with the common saying about death and taxes, and led from there in two interesting directions.  The first direction was taxation, specifically an argument about a flat tax, and an interesting counter-argument for progressive tax (i.e., higher tax rates for higher incomes) based on the idea of logarithmic utility of wealth.

The second direction was death (!), where I was reminded of an interesting article in an interesting book, that also involved the useful application of logarithms, in this case to a study of humans’ capacity to kill each other.  This is the source of the title of the post; we’ll get to this in just a moment, but first let’s tackle progressive tax and log utility.

The idea is pretty straightforward: if you have $1,000,000 and your neighbor has $10,000, then adding– or taking away– $1,000 means a lot more to your neighbor than it does to you.  In other words, the marginal usefulness of income decreases as the absolute income increases.  This is often made mathematically precise by using the logarithm of some numeric representation of wealth as an indicator of the perceived utility of that wealth.  As applied to a discussion of taxation, the argument against a flat tax is that with a flat (i.e., constant percentage) income tax, the perceived loss in relative utility from income tax is less for the wealthier than for the poorer.

I am not sure I buy this as a defense of progressive tax.  But it’s an interesting idea.  And this is only the lesser, motivational half of the story anyway.  The more interesting half, to me, is an essay by Brian Hayes titled “Statistics of Deadly Quarrels,” discussing same-titled research by one Lewis Richardson on the history of “deaths… caused by a deliberate act of another person.”  This includes not just wars, but individual murders, covering the period from about 1820 to 1950.

As Hayes points out, this aggregation of small and large scale, of “abominable selfish crime” and “heroic and patriotic adventure,” was intentionally provocative: “One can find cases of homicide which one large group of people condemned as murder, while another large group condoned or praised them as legitimate war.  Such things went on in Ireland in 1921 and are going on now in Palestine.”  Note that this was written in 1960.

Ok, enough heat, now for some light.  Richardson took the very useful, and necessary, approach of classifying the various conflicts according to not their absolute number of deaths involved, but to the (base 10) logarithm of the number of deaths.  For example, there were (and still are) only two wars with a “magnitude” of 7, meaning that 10-100 million people died, namely the two World Wars.

This logarithmic scale is handy, since it allows for the difficulty in obtaining very precise data, particularly for small-scale conflicts such as murders involving only a few people, but even for the larger-scale wars where it is not always clear exactly who died and when, and whether they should be “counted,” perhaps dying months or years later from complications resulting from wounds received in battle.  Using a logarithmic scale allows comparison of sizes of conflicts without needing to be as precise about the exact size of any one conflict.

The essay is just one of several interesting reads in Hayes’ book referenced below.  I recommend checking it out; to tease without spoiling the fun, another interesting observation from the study was the list of magnitude-6 wars.  These are the wars involving between about 500,000 and 2 million deaths, the largest conflicts other than the two World Wars.  It turns out that there are “only” seven such wars, and presumably we should all know what they were.  I didn’t do so well, only being able to name three, and one of those was an uneducated guess.  Can you name all seven?

Reference: Hayes, Brian. Group Theory in the Bedroom, and Other Mathematical Diversions. New York: Hill and Wang, 2008.  This is a great collection of essays, one of which is titled as below.  The “group theory in the bedroom” refers to mattress flipping; it is also an interesting read, and I think would make a great introduction of students to some simple group theory.

Reference: Richardson, Lewis Fry. Statistics of Deadly Quarrels. Pittsburgh: Boxwood Press, 1960.

Matters of death and life

It is hard to lose a loved one.  I have had occasion recently to contemplate the impact that we feel by the death of those close to us, and our reactions to that loss.  (I realize that the subject of this blog is science, mathematics, and computing.  I don’t mean to be gloomy here, just hopefully to provoke thought.  But mostly I am just moved to write about what I think about, and right now what I think about is loss.)

A couple of thoughts occur to me, both of which are recollections of better words than mine.  The first is an excerpt from Euripides’ The Trojan Women, where Hecuba tells the chorus to prepare her son Hector for burial:

“Go, bury now in his poor tomb the dead, wreathed all duly as befits a corpse.  And yet I deem it makes but little difference to the dead, although they get a gorgeous funeral; for this is but a cause of idle pride to the living.”

It’s been over 15 years since I read this, but it struck me when I read it and has stuck with me since.  In short, the flowers, the preparation of the body, the ceremonies, and all of the solemn words, are our means of dealing with loss– the dead do not care.  For me, I think this is helpful rather than depressing.  If they did care, if the loudness of our laments indeed mattered to the dead, then we fall short of the mark; one perfect rose or a carefully selected Bible verse is insufficient expression of either the sadness of losing someone or the joy of having been a part of their life.

The second recollection is of a letter written by Richard Feyman to his wife Arline.  The letter is included in the book “Perfectly Reasonable Deviations from the Beaten Track,” edited by Feynman’s daughter Michelle.

I found this letter moving for several reasons.  First, I think it shows the heart of a man who to me exemplifies the rational, scientific approach to understanding our world.  (Witness the title of this blog.)  Second, consider the dates involved.  Arline died on 16 June 1945, just one month before the Trinity test of the atomic bomb, as part of the Manhattan Project on which Feynman had been working:

To Arline Feynman, October 17, 1946


I adore you, sweetheart… It is such a terribly long time since I last wrote to you — almost two years but I know you’ll excuse me because you understand how I am, stubborn and realistic; and I thought there was no sense to writing.  But now I know my darling wife that it is right to do what I have delayed in doing, and what I have done so much in the past.  I want to tell you I love you.

I find it hard to understand in my mind what it means to love you after you are dead — but I still want to comfort and take care of you — and I want you to love me and care for me.  I want to have problems to discuss with you — I want to do little projects with you.  I never thought until just now that we can do that.  What should we do.  We started to learn to make clothes together — or learn Chinese — or getting a movie projector.

Can’t I do something now?  No.  I am alone without you and you were the “idea-woman” and general instigator of all our wild adventures.  When you were sick you worried because you could not give me something that you wanted to and thought I needed.  You needn’t have worried.

Just as I told you then there was no real need because I loved you in so many ways so much.  And now it is clearly even more true — you can give me nothing now yet I love you so that you stand in my way of loving anyone else — but I want to stand there.

I’ll bet that you are surprised that I don’t even have a girlfriend after two years.  But you can’t help it, darling, nor can I — I don’t understand it, for I have met many girls… and I don’t want to remain alone — but in two or three meetings they all seem ashes.  You only are left to me.  You are real.

My darling wife, I do adore you. I love my wife. My wife is dead,


PS Please excuse my not mailing this — but I don’t know your new address.

One final comment: I think it is important, and rather uplifting, to observe that Feynman did remarry.  He and his wife Gweneth were married for nearly 30 years before his death in 1988.

I think our lives are like books; there is resolution at the end of every chapter… but there is also suspense, because the book isn’t over yet.  And it is so wonderful to keep turning the page.

Math for the 4th of July

The lead article in yesterday’s Post described some interesting analysis of one of Thomas Jefferson’s early drafts of the Declaration of Independence.  Although several words can be seen to have been crossed out and replaced with others, in one instance where Jefferson initially used the word subjects, he did his best to not just cross it out but actually erase it and write citizens in its place.  (I assume the part in question is the following final version: “He has constrained our fellow Citizens taken Captive on the high Seas to bear Arms against their Country…”)

In a similar spirit of this holiday weekend, I can’t resist mentioning another interesting application of mathematics, in this case to the problem of determining authorship of the so-called “disputed” Federalist Papers.  This isn’t new stuff; however, I like problems (and solutions) like this because of the relative simplicity with which the ideas may be explained… while at the same time there is some meaty mathematics under the surface.  It is the kind of problem that has the potential to excite and challenge students.

The Federalist Papers are a collection of essays written by (variously) Alexander Hamilton, James Madison, and a few by John Jay, in support of ratification of the U. S. Constitution.  Although authorship of most of these essays is relatively certain, there has been some debate about twelve in particular.  These “disputed papers” are today generally all thought to have been written by Madison.

My first exposure to this problem was a 1998 paper by Bosch and Smith.  (Unfortunately, this JSTOR link is not accessible without a journal subscription.)  In it, the authors describe the idea of using “separating hyperplanes” to identify the author(s) of the disputed papers.  They compute, for each of the essays, a point in 70-dimensional space, with each coordinate indicating the frequency of occurrence of a corresponding “function word.”  Think of these function words and their frequencies of use as a “fingerprint” that is unique to a particular author.

Now, considering only the 65 points corresponding to the undisputed papers, the authors compute a “separating hyperplane,” or a hyperplane such that all of the points corresponding to Hamilton’s essays are on one side, and Madison’s on the other.  (This is where the interesting mathematics comes in; how do you compute such a separating hyperplane?  Under what conditions does a separating hyperplane even exist?  In the likely case that there is an entire infinite family of possible separating hyperplanes, how much does it matter which one you choose?)

Anyway, given such a hyperplane separating the two authors of the undisputed papers, the authorship of the disputed papers may be determined by observing on which side of the hyperplane the corresponding points fall.  It turns out that this approach yields the same conclusion, that all 12 were written by Madison.