As I am writing this, I have a Skype interview early in the morning for the job that, out of all those I’ve applied for, I want the most. There are bills to pay, and with my graduation in May comes the need for a gig other than an internship through the university. As you are reading this, however, the interview will have come and gone, and I will have driven two hours to Bowling Green, joined in playing Friday Night Magic, and tried to put thoughts of the real world out of my mind for the weekend.
And playing card games is just the remedy for that.
Vincenzo De Florio, whose paper on fractals I reviewed months ago and whose workshop on antifragility I am on the Programme Committee of, had another paper published in the Complex Systems journal. This work describes an Italian card game, “strappacuore,” which is popular with young children and is much like the game War here in the US.
The rules of strappacuore are fairly simple. Kings, Queens, and Jacks are removed from a standard deck of playing cards, the deck is shuffled, and the cards are dealt evenly between two players. During the game, the players engage in two modes of play, which I will call “digging” and “challenging.”
The game starts in digging mode, in which players take turns flipping the top card of their face-down decks onto a shared pile of face-up cards. Once an Ace, Two, or Three has been flipped, a challenge begins.
The player who flipped the Ace, Two, or Three, known as “good cards,” challenges the other to flip a good card in a certain number of flips: Aces allow one flip, Twos allow two flips, and Threes three flips. If the challenged player fails to flip a good card this way, he or she fails the challenge. But, if he or she succeeds, a new challenge begins with challenger/challenged roles swapped and the newly revealed good card determining how many flips are allowed.
Once a player fails a challenge, the challenger turns the pile face-down, adds it to the bottom of his or her deck, and resumes digging by flipping the top card of that deck. Once a player’s deck becomes empty and he or she in unable to flip a new card, the game immediately ends and the player still with cards wins.
A multiplayer variant exists, in which proportionately more cards are used, the last player standing wins, and challengers always challenge the next player in turn order. In a nondeterministic variant, players choose which opponent to challenge.
However, De Florio considers only the original version in his paper, since it suffices to describe a startling complex system.
Strappacuore is, after the initial shuffle and turn order has been decided, completely deterministic. That is, if the order of the cards is known, along with who’s turn it is, the outcome of the game is precisely determinable by simply simulating the rest of the game—assuming, of course, that the game will end and won’t enter some sort of infinite loop.
But being deterministic does not imply that the outcome will be easy to predict. Because there’s no way to tell how long the game will last, there’s also no way to tell how long the simulation will take. So, is it possible to predict who will win some other way?
De Florio’s analysis suggests that, in most cases, there isn’t.
He depicts strappacuore as a chaotic system. That is, it’s a game where an estimate of initial conditions does not permit outcomes to be estimated as well, since slight variations missed by the intial estimate are magnified every turn, leading to vast differences down the road.
For example, consider a game between Jack and Neal. As the game progresses, we chart the size of Jack’s deck over time. During a digging mode, his deck will decrease by one each time he flips a card, producing a slowly descending line. Once a round of challenges begin, both decks might momentarily drop in size as challenges are issued back and forth. Afterwards, Jack’s deck may be left crippled, or it may have almost doubled by winning cards away from Neal. Also possible, though, is that Neal flips an Ace, Jack losses the challenge, and is down only one card. De Florio claims games are possible where a player can be reduced to only a few cards, yet comes back for a landslide victory.
The behavior of the good cards, coupled with the randomness of the pregame shuffle, leads to wild behavior in the long run. An Ace challenge is more difficult to win, but typically costs the loser a single card. In contrast, a Three challenge is far easier to win, but a loss can be more devastating. Additionally, the rules of the game require one player to, essentially, steal all cards from the opponent. To do this, however, he or she must steal the opponent’s good cards—which, in turn, require a challenge to be met with a challenge to be met with another challenge, and so on.
Victory in this game, therefore, is nearly dependant on large piles that can go either way, to Jack or to Neal. ∎
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