Ntuple Indemnity

As some of you know, a week and a half ago I played in the Settlers of Catan Canadian National Championship (Western): 60 players, 4 round-robin games, and a break to a 16-player semi-final based on wins, victory points and victory point percentage. In short, I bombed; you can see the results for yourself, and observe that I did not manage to finish in the top third.

On a game-by-game basis, it was far from a disaster by any standard, as I won one game decisively (6 VPs on the table, 4 for Longest Road and Largest Army, and another one just to make sure) and tied for a close second in two others in the face of unfavourable dice. But I needed two wins to advance to the semis. As with most games that involve a tenuous balance of skill and chance, consistency in the face of misfortune is the mark of a skilled player. It looks like I have some ways to go.

We will speak little of Round 3, in which I did not show up, nor did the number 5, and I finished dead last with only 4 VPs. That debacle was almost entirely due to tactical and strategic miscalculation on my part, as I drew first position on the snake (placing my initial settlements first and last), grabbed a 5/9/10 and went for an expansionist wood/brick strategy, which I almost never do. Anyone who has heard me mope about Scrabble knows that I have a remarkable tendency to get overly experimental in untimely competitive situations. All three of the other players went for an ore-heavy game, and rolled the numbers to match, erecting early cities and grabbing piles of development cards while I sat on my hands with nothing of significant trade value. With no cities, no soldiers to keep the robber off my property and no 5s, my exit from contention was terrible, swift and entirely my fault.

But it happens. And while the knowledge that you played poorly is hardly a happy thought, it gives you the opportunity to shoulder the responsibility and think about how you might act differently in the future. In spite of the dice, you retain the impression that you have control over your own destiny.

However, three key elements of Settlers of Catan elevate its strategic complexity to a whole other dimension that I am only beginning to grasp. It’s non-zero-sum, it’s for four players, and it’s built on trade.

Settlers is non-zero-sum in that you can check a player, or you can block him from victory to some extent (ask Jake Troughton about the game he had elsewhere in the tournament where the other players held him at 9 VPs for well over an hour), but aside from snatching away the Longest Road or Largest Army, you can’t roll him back. Once you have points on the table, they stay on the table. And when someone coasting to victory settles on the same numbers as you, the rolls that permit you to develop confer no relative advantage. The dice may well push them over the edge, and there isn’t a damnable thing you can do about it.

And more to the point, in a four-player game, someone can hand someone else the game with a series of bad decisions, and there isn’t a thing you can do about it except sit there and watch. If the initial settlement placement pans out so one opponent is left with a disproportionate tract of uncontested land open for expansion, because the other two decided to crowd you instead, it sucks to be you and that’s that. Similarly, if another player decides to go through with an uneven trade, or break a three-way embargo, there’s little to be done apart from taking a bullet and proposing an even more ridiculous counter-offer. I’ve been on both sides of this situation. On the winning side, you try to get yourself into a position where someone else has to cooperate with you just to stay in the game.

I’m not an Xbox 360 owner, so I haven’t had a chance to poke and prod the AI behind the Settlers of Catan video game, but I’d be very interested in examining it for its shortcomings. Like the human element behind games like poker or Diplomacy, it seems to me that the trading element that fundamentally drives the course of proceedings in Settlers would be extremely difficult to formalize and refine. One of the biggest challenges when it comes to artificial intelligence is getting over the assumption that you’re playing against someone rational. Game theory was designed to fight rational fire with rational fire, which is why you end up with suboptimal equilibria like the Prisoner’s Dilemma. In Settlers, you have to account for deeply suboptimal opponents – the ones who, in the Prisoner’s Dilemma, could actually be hoodwinked into cooperating instead of defecting; the Kevin Lowes who will actually give away their Chris Prongers for less than nothing (to your divisional rivals, no less). What you do at a table of championship-level thinkers is and must be different from how you behave when someone is just there for the burgers and door prizes.

And I wouldn’t have complained about a little help from the dice.

Belated congratulations to my compadres Chris Samuel, who finished 13th with two wins and contended in the semis, and Steve Smith, who finished 19th with one win (Round 4 against me, knocking me out), and sat (but did not play) as an alternate.

Hockey anthem addendum: I have a lot of respect for Holly Cole for being one of Canada’s leading jazz ambassadors, but here’s a three-letter memo about her performance of “O Canada” at the Duck Pond in Game 2: WTF?

Never seen me interpret video game music? YouTube to the rescue!

If you’ve had a Wii for any amount of time, you’ve probably spent a couple hours in the Mii Channel making caricatures of friends and celebrities alike to fill up your ragtag baseball team. (Dan Lazin, for one, sent me a most excellent Lieutenant Worf.) And if you have, then it’s almost a certainty that the Mii-making music has been stuck in your head at least once. Naturally, I set about figuring it out on the keyboard, only to discover that I couldn’t quite get all the chord progressions right by ear.

So the other night, I turned on my Wii, set it to the Mii Channel, and did a rough transcription of the music as it played, mostly to figure out what was going on harmonically. Like most of the repetitive but catchy incidental music that comes out of Nintendo, there’s a great deal of complexity under those unassuming bleeps and bloops. So I switched up the rhythmic feel from Latin to a medium swing, jotted down some fancy chord substitutions, and decided to see where I could take the tune. Here’s the result:

I’m not all that happy with my solo, but I almost never am, and given that most listeners are absurdly easy to impress, I doubt a lot of people will complain. I gave myself a fairly challenging set of chord changes to play over, so the take I recorded was more about surviving four choruses and staying in time than actually taking risks and coming up with lovely melodic architectures. It’s easy to stretch out and aim for the pretty notes when you’re just jamming, but recording a complete take creates considerably more room for error. Apart from cutting back on the arpeggiation and going for longer melodic lines, there are two other things I’d change should I do this again. First, it swings a bit hard for a two-beat feel, and probably isn’t as laid back as it should be. Second, my left hand is mostly preoccupied with spelling out the bass line here, so the chord voicings are quite sparse; if I were to do a bassless recording, it would free up the left hand to highlight some of the more interesting substitutions I found.

As I said earlier, this is a surprisingly deep tune, compositionally speaking. I’ll go into some specific analysis for the benefit of the musically literate.

The biggest wrench in the whole affair is the oddball 25-bar form. There’s a straightforward 16-bar A-section that modulates to the subdominant (in the original, from A to D major; in my version, from B-flat to E-flat), followed by a 1-bar break and an 8-bar B-section (the only part in the original that really casts a melodic line into the foreground). In the video, I chose to keep the break at bar 17 in the solo choruses just to keep the tune quirky, and encountered all of the expected difficulties. I may do another take at some point that keeps bar 17 when playing through the head, but removes it for a more predictable (and playable) 24-bar solo form.

Harmonically, the most interesting part is probably how the B-section modulates back to the original key (again, A in the original, and B-flat in my version). The tonal centre moves up a major third: there’s a II-V-I in E-flat followed by a II-V-I that resolves to G major, which then drops to a minor and proceeds down the circle of fifths until we’ve returned to the key of B-flat. Jazz musicians will recognize the major-third jump as one of Coltrane’s “giant steps,” which most obviously predates Coltrane in Rodgers and Hart’s “Have You Met Miss Jones?” and becomes commonplace in a lot of post-Coltrane compositions by Wayne Shorter, Herbie Hancock and the like. Here, we don’t take the tonal centres all the way through a circle of thirds (which is more of a triangle), but the modulation from E-flat to G suffices to make the tune particularly susceptible to jazz improvisation with a modern sound, and generally fun to play.

I hope Nintendo lets the cat out of the bag regarding the composer of all the Wii’s onboard music, as I’d really like to give credit where credit is due. I’m sure some people have speculated that it’s the work of Nintendo legend Koji Kondo, but I’m inclined to put my money on Kazumi Totaka, or Totakeke to his legion of Animal Crossing devotees. No sign of Totaka’s Song yet, though.

Surely, by now, we’ve all seen the commercials where an elite and well-dressed tag team shows up at someone’s door with the mantra “Wii would like to play,” leads an attentive crowd in an exhilirating and playful experiment, and drives off into a satisfying sunset. Now imagine this scenario with science projects instead of Nintendo consoles, and elementary schools instead of households equipped with high-definition TVs. This is my job.

From a sixth-grade workshop about flight: We discuss air resistance and drag and demonstrate how a piece of paper travels farther when thrown if it is crumpled up. Then we guide the class in the construction of paper airplanes, which they were free to decorate with felt markers, and hold a contest to design the plane that would fly the farthest. The resident keener, who knew the answer to every question we asked in our presentation, crumples up a sheet of paper and asks me if it counts. No, I tell him, it doesn’t count. He promptly affixes a tiny pair of wings to his paper ball, and then he chucks it. Another child – no more than eleven years of age, remember – walks up to me with a paper glider covered in dollar-signs and assorted, uh, bling, and asks: “Can we have a contest for the most pimped-up plane?”

From a second-grade workshop about insects: We give the kids assorted goods like marshmallows, pipe cleaners and coloured tissue paper in order for them to construct their own bugs. They finish early, and we have some time at the end of the class to ask them to tell stories about each of their bugs and what makes them special. One boy (about seven years old, I’m guessing) raises his hand and shows off his rather elongated bumblebee. “My insect is the Lord of the Bees,” he declares. I ask him to tell the class its name; I’d heard it earlier, and thought it was well worth sharing. “His name,” he says with total seriousness, “is Montflyington the Third.”

From a fourth-grade workshop about light: At the end of the class, a girl comes up to me with her agenda and asks for an autograph. I sign her notebook with my name and my trademark blowfish insignia; if you’ve ever received a letter or postcard from me, you’ve probably seen some variation of the latter. But remembering proper autograph protocol from all the books I got signed this year, I asked the girl for her name, so I could leave a personalized message. She was roughly nine years old, so it shouldn’t take a genius to figure out wherefore her name was Xena. I don’t know about you, but I was both amused and depressed.

I’m behind enough on my novel that the additional negative harm of dedicating a few words to the blog instead of my little green notebook is negligible. Ergo, let’s talk about board games.

But first, as I have been tending to do with increasing regularity, I am going to precede the main feature with some ancillary featurettes.

I don’t write that much about hockey (proportionate to my other interests) because other, more prolific minds know a lot more about the ins and outs of the subject than I do. But I do know music, and I’m a bit of a stickler when it comes to the singing of the national anthems. I remember, with astounding clarity, that the most execrable part of watching Tampa’s home games in the Tampa-Calgary series in 2004 (aside from seeing the Lightning hoist the Cup) was Brooke Hogan’s consistently lamentable butchering of “O Canada” – and I’m sad to report that since then, Hulk’s baby girl hasn’t faded into deserved obscurity, but has instead released an album and graduated to B-list pop-stardom.

So, for all the press about how epic the Buffalo-Ottawa series will be when it comes to the action on the ice, I think it falls to somebody to mention two of the lesser-recognized reasons why it will be (and already is) so fun to watch: Doug Allen and Lyndon Slewidge. I haven’t seen every regular anthem singer currently making the rounds in the NHL, of course, but of the ones I have, these two are the very best. (And being from Calgary, where we get the excellent Heather Liscano, I do have standards.) What does the trick is their clarity, charisma, genuine quality of musicianship, and unflinching respect for what they are singing. Oh, and they stay in time. I’ve lost count of the wannabe divas who trill left and right and demonstrate nothing but a lack of education in the basics of metrical tempo (to say nothing of tone control). We’ll have none of that.

Next: Its run at the Varscona Theatre closes on Sunday, but Shadow Theatre’s production of David Belke’s Dreamland Saturday Nights is an absolute delight. I’m not as well-versed in appreciating the subtleties of the stage as I am with cinema or music, but this was my best experience with live theatre in some time. I do wonder, however, if it plays nearly as well to people who aren’t major cinephiles, as so much of its magic is referential.

And now, as promised, a bit about board games.

Last week, I went out into the wilderness with my new colleagues (pamperedness level: heated cabins), and one of them had the presence of mind to bring along the familiar diversion that we all know as Connect Four. On the last evening of our trip, when I was busy nursing a pot of cheap strawberry tea and making absolutely no progress on my cute little novel-in-progress about space colonies and married women, he challenged me to a friendly match. One friendly match quickly ignited into thirteen. We stopped when I hit ten wins. For those of you in the audience who can’t add, that’s a 10-3 record.

On paper, it looks like I demolished the poor guy; he certainly thought so, and it wasn’t until the next morning (when he beat up on someone else) that he regained some elementary semblance of confidence. To be fair, I started with six straight wins, and finished with a considerably weaker 4-3 thanks to careless errors in observation, unnecessary experimentation, and (in my opinion) noticeable improvement on the part of my opponent.

Now, I hadn’t played Connect Four in an absurdly long time: I only vaguely remember playing round after round on my Macintosh SE back in the black-and-white halcyon days of System 6, and I may have faced the occasional human opponent, though I’ve never owned a physical set myself. But the rules are simple enough that it’s not like you’re liable to forget how to play. The name of the game and the design of the board (i.e. it’s vertical, and pieces dropped in the slots tend to fall) make the mechanics about as self-evident as is possibly conceivable. That said, it would be foolish to understate its strategic complexity.

I can’t say I’ve ever considered myself any more than a casual player, and by purely casual standards, I don’t think my opponent was exceptionally bad. How, then, did I manage to be so dominant? What has really changed since the age of nine? Board game skills aren’t like fine wines: they don’t improve with age alone.

Two things come to mind: 1) serious exposure to other games of strategy, and 2) an education in the little-understood field of computing science.

The first is probably obvious. In the case of Connect Four – elegant, turn-based (but otherwise symmetrical) and fundamentally geometric – it’s easy to see how some half-decent Scrabble skills might help: looking for hotspots, settling into a rhythm of risk and reward, differentiating high-risk offensive positions from conservative and defensive ones. More pertinent would be a deterministic game like chess, where you learn to recognize certain patterns as opportunities for setting up zugzwangs and forced mates. I kind of wonder how well someone with a background in grid-based video games like Tetris Attack would fare.

There’s also the psychological element of the discipline it takes to focus on a game, exercise the patience to find the best move, intuit what your opponent is trying to do, and avoid stupid mistakes, but I don’t have that.

Less obvious to the layman is the value of a computing science education, which a lot of people out there think of as the study of computers, when in many cases it would actually be more accurate to say it is the study of problem-solving. (That’s a crude generalization as well, but still more accurate.) It’s not too hard to see how excellence at games of strategy is so often conceived as the pinnacle of whatever it is computing scientists do.

I am not saying that in a given turn in Connect Four, I’m culling a game tree in my head or performing a systematic limited depth-first search. But just as a basic level of comfort with calculus provides you with a substantially better intuitive grasp of infinities and convergences even if you don’t remember the ins and outs of solving triple integrals by substitution, I wager that an understanding of rational, computational processes provides one with an intuitive basis for pattern recognition and logical reasoning.

On top of that, it provides you with the tools to easily pursue further study. The reason I am writing this post is in part because tonight, I thought about how I might design a Connect Four AI, and I started looking at what had already been done. As a mechanically straightforward (and fully deterministic) game, it did not come as a surprise that Connect Four had already been solved. (The relevant thesis, which you can download as a 91-page PDF, is Victor Allis’s “A Knowledge-Based Approach of Connect-Four,” published in 1987. I’m not through much of it, but so far, it’s a terrific read.)

It turns out that with perfect play, the first player to move (“White” in the standard literature, as I don’t think many people with the standard Milton Bradley set care if red or yellow go first) always wins if the first move is down the centre column, just as how X can’t lose in tic-tac-toe if the first move is in the centre square. It seems intuitive, but there’s no comfort like the warm cushion of a mathematical proof.

Apart from questions of the “how did I win” variety, I think there’s another issue in play: why did I have fun? (This might not occur to those of you on another planet who play games solely for amusement and diversion and not necessarily to win, but I do, on occasion, actually have fun.)

Let’s proceed by example and analogy, shall we?

Anybody who has known me for any period of time probably has at least a cursory knowledge of where I stand when it comes to board games. There are three games in the top tier. I play Scrabble (far and away my favourite game of any sort) seriously enough to compete but not obsessively enough to truly excel; I’m fascinated by Diplomacy, but inexperienced; I adore Settlers of Catan, but I’m not nearly as good as the statement “I finished second at the University of Alberta’s POGOB! tournament in October and I’m playing in the Western Canadian championship next week” might suggest. (With games like Diplomacy and Settlers, a good reputation is a huge liability. People start ganging up on you.)

I have the deepest respect for chess, checkers and Go, but there’s such a gulf between my level of skill and what I would consider an acceptable competitive standard that I view them much like I view movies that are obviously imbued with the finest craftsmanship, but do not stir any passions on a personal level, and cannot be said to be among my favourites.

Among the other popular entertainments: I used to enjoy Risk, but I abandoned it after I found Diplomacy, and I haven’t played it since. Back when I was an only child, my family played Monopoly, but we would fight over the game and inevitably leave it unfinished; later, I discovered that we weren’t following the rules correctly anyway. I’ll play Battleship, but it’s just not the same without the “Sploosh!” and “Kaboom!” sound effects from the version in The Legend of Zelda: The Wind Waker. Then there’s the whole class of games like Cranium and Trivial Pursuit that I’m not even sure I should count: they suit social settings well, but you can’t “study” the game without feeling like you’re cheating.

I hate the Game of Life. (Not Conway’s cellular automaton, but that thinly-veiled wheel-spinning instrument of social engineering where you drive around in a little car and add pins to it when you get married, have kids, and coast towards your inconsequential demise in the thorny brambles of the American Dream.) Life sucks. No strategic depth whatsoever.

Now, let’s play Spot the Correlates. Is there anything I consistently like or dislike about board games?

If you look at the top tier – Scrabble, Diplomacy, Settlers – all three of them have an element of nondeterminism (stochastic in Scrabble, qualitative in Diplomacy, both in Settlers), but success or failure is primarily and repeatably dependent on conscious strategic decisions on the part of the player. In addition, a significant number (but not all) of these decisions are independent of who the opponent is, because they involve tactical knowledge of the game mechanics. It is possible to get progressively better as a player, but impossible to win all the time.

Connect Four initially falls into the same qualitative category as chess – although Go would be a better comparison, as it is based on placing pieces, not moving them, and there is only one type of piece. Connect Four doesn’t belong to nearly the same order of complexity, of course, but it feels like a faster-paced incarnation of the same deterministic experience. There is at least the persistent illusion of a fair fight, whereas in chess, I will almost always lose to someone who has a library of openings down pat, and almost always beat someone who can’t find his way around basic pins and skewers.

In a casual or novice-level game, you will almost never see a stalemate in chess except by accident, because the players aren’t skilled enough to set one up to avoid a loss, but they are skilled enough to finish off a familiar pattern like having a king and a rook to the opponent’s king. In expert play, draws are virtually the norm, and the result is a phenomenon called draw death: the prevalence of stalemates between players of sufficient skill exposes a serious air of futility that hangs over the game as a whole. A more familiar example of draw death is in tic-tac-toe. It doesn’t take very long for even a small child to figure out how to play every game of tic-tac-toe to a draw, at which point there is no real incentive to play anymore.

To be fair, this also happens in expert-level Diplomacy, where it is possible and not at all uncommon for the last two remaining players to entrench themselves in a stalemate line that cuts across the map, preventing each other from reaching the 18 supply centres required for victory. And a game like chess is sufficiently complex that unlike tic-tac-toe, it’s still possible to hold a championship. Draw death isn’t a reason to not play chess unless you are outstanding, in which case your name is Bobby Fischer and you remedied the problem by inventing a chess variant (Fischerandom/Chess960) where the initial placement of pieces is randomized within certain constraints.

I’m sure that against sufficiently skilled players, draw death presents a huge problem for Connect Four. But the real barrier to enjoying games of strategy is when improvement relies on the memorization of positions instead of principles; I also see it occur in Scrabble, where you need to dedicate a tremendous effort to memorizing words just to have a chance to apply strategic ideas. I like words enough that I don’t mind so much, but that obviously isn’t the case for everybody.

The elegance of Connect Four is how much complexity it manages to preserve in order to remain interesting, while its simplicity entails that it is never an exhausting game to play; it goes by so quickly that one can play round after round with little trouble. It owes at least some of its speed to a certain lack of choice, with only seven or fewer possible moves per turn. It also does not seem as susceptible to massive disparities in skill as many other games of its ilk. On the other hand, it’s entirely possible that I just don’t know the game very well yet, and that once I do, the number of players with whom I feel evenly matched (in either direction) will plummet.

And that’s why I’m behind on my novel.

Nobody known to me who reads this space was there to see it, but I guarantee you that my jaunt to the Yardbird tonight led to what is unquestionably the worst I have played in years. For some reason, I just flat-out forgot how to work a piano (in a profoundly public situation, no less). It may have been because the B-flat feedback on my monitor sent me into a timid corner wherein I performed with an impotent absence of confidence and conviction, or because I hadn’t so much as touched a keyboard in three days and had committed my fingers to more rudimentary motor functions like the inspection and sorting of resistors, or because the hundred-some tunes I’d taken to the woodshed in the past half a year curiously did not include much in the way of rhythm changes (let alone a head as tricky as “Oleo”), or because I’ve fully diverted the attention of my Creative Processing Unit to the writing of fast-food prose (which is going quite smoothly, thank you), or because of that Cabernet Sauvignon from Chile (I remain convinced I play better after a drink or two, but maybe that only applies when I’ve had dinner), but no matter the cause, that was an embarrassment on the order of a slaughter on the sandlot with Charlie Brown on the pitcher’s mound.

Well, let us make the best of this debacle and not smother the furious passion of disappointment, but stoke it into a phoenix of a bonfire. I have a 50,000-word trek ahead of me, and I need fuel.

But first, I’ll get lost in the woods.