Information: Perfect, Imperfect, Virtual, and False

Magic is a game of information. The better your knowledge of the game state, the better your decisions can be. The greater your information advantage over your opponent, the more likely you are to be able to outplay him.

But what is information? Let Chad explain all!

Magic is a game of information. The better your knowledge of the game state, the better your decisions can be. The greater your information advantage over your opponent, the more likely you are to be able to outplay him.

But what is information?

According to Wikipedia:

The mathematical theory of information is based on probability theory and statistics. The most important quantities of information are entropy, the information in a random variable, and mutual information, the amount of information in common between two random variables. The former quantity indicates how easily message data can be compressed while the latter can be used to find the communication rate across a channel.

The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. The most common unit of information currently in use is the bit, based on the binary logarithm. Thus base is usually assumed to be 2. In addition, due to limit behavior, the usually undefined “0 log 0” is considered to be 0.

Just kidding. Come on, I’m a geek… but a StarCityGames.com article should link to Shannon Elizabeth or Shannon Elizabeth, depending on whether you want traditional hot or gamer-hot. Claude Shannon, although sexy in his own way, doesn’t quite measure up.

For the purposes of this article I’m going to focus on three information concepts — degree of information perfection, virtual information, and false information.

Perfect versus Imperfect information

You have perfect information if you know every relevant fact for your decision; the more missing relevant facts, the less perfect your information. (The key is relevant facts; this becomes a key determinant in forming virtual information.)

Chess is a game of nearly perfect information. The only thing you don’t know in a game of chess is what your opponent is thinking. Everything else is right there on the board — that Rook isn’t suddenly going to turn into a Knight, and while you might miss one of your opponent’s possibilities you’re theoretically aware of all of them.

Poker, by contrast, is a game based almost entirely on imperfect information. Even if you know you have the nuts you don’t know what your opponent’s cards are and thus don’t know what bet is most likely to extract the most money — moreover, he (critically!) doesn’t know that you’ve got the nuts, so you want at least not to give that away and ideally to get him to think that you’re weak.

Magic plays in a broad range between Chess and Poker. Most of the time you play with imperfect information, with a varying degree of missing significant facts, and a key part of the struggle is reducing imperfection in your own information while maintaining or increasing it for your opponent.

Consider the following situation:

You have four Bears in play. Your opponent has two Spiders. You’re both at four life. Your opponent has ample mana untapped and three cards in hand. You also have plenty of lands in play and two more lands in your hand (no spells). You’re about to draw for your turn.

Obviously there are some premium cards you could draw here (e.g. an X-spell) but in the current situation Peek is a potentially game-winning card! You’re currently in a dicey situation — an alpha-strike could win you the game but it could also lose it for you, if your opponent has a single removal or damage prevention spell. Peek probably* elevates you to perfect information for the decision — once you know your opponent’s hand, you can likely determine for sure who wins the game if you alpha-strike.

Question. If you had to choose between your opponent getting a game-winning spell for free and your opponent not getting a game-winning spell for free, what would you choose? The answer seems obvious, but player after player gives up the equivalent of that game-winning Peek by creating situations in which their opponent has perfect information, turning a risky play into certain victory.

Players do this by playing out the last card in their hand or by tapping out in a situation where they are dead on the board and where there are no free spells that might save them. If you do this you are putting all of your hopes for that game on your opponent being unable to assess the board position once all of your options are known. You can sometimes get away with that and sometimes you even have to try, but it’s an incredible long shot against any but the weakest opponents. You should never do it unnecessarily.

Virtual Information

Imagine the following scenario. You’re in an odd casino in which the only game consists of Roulette and where the only bets allowed are Red or Black; each pays 1:1. You have $1,000 in your hand and are considering betting, although you know that the odds are slightly in favor of the house because of the two Green spots on the wheel.

Now imagine that someone has a gun to your head and is going to shoot you (and take your money) unless the next spin pays off Red.

Suddenly you can, and should, eliminate the possibility of any non-Red outcome on the next spin. You’re dead if it isn’t Red and you can’t even leave the $1,000 to your next-of-kin, so the optimal play is to bet it all on Red, as indicated in the following payout table:

. Wheel Pays Red Wheel Pays Black Wheel Pays Green
No Bet
Dead and Broke
Dead and Broke
Bet X on Red
$1000 + X
Dead and Broke
Dead and Broke
Bet Y on non-Red
$1000 – Y
Dead and Broke
Dead and Broke

Essentially you eliminate the Black and Green columns from consideration because you can’t affect the outcome if they occur; instead you focus on what happens if Red pays off, in which case you’re clearly better off betting on Red and the more the better.

As awful as the starting situation is, it’s one in which you actually have perfect information. You may not know which color will be paid but that only seems like a relevant detail; in reality it isn’t.

Beg pardon? That bit of information decides whether I live or die… that’s not relevant? Nope… it’s important, sure, but it’s not relevant to your decision.

The same thing happens in Magic quite often. You’re in a desperate situation and realize that the only way you can win is if you draw a particular card on your next turn — say, Wrath of God to get rid of your opponent’s army. Anything else and you’re sure to lose. At that point you must play as though Wrath of God is on the top of your deck and do whatever will maximize your chances to win if that happens.

The converse can happen as well. Your position may be so good that your opponent is down to one draw to find his Wrath. In that case you have virtual information that his next card is a Wrath and should play accordingly, i.e. don’t add another creature to the board!

False Information

Darwin Kastle has graciously invited me to team with him at the 2HG Pro Tour, so we decided to get some practice at MA States. We went 5-1, couldn’t draw in and lost a really tight and exciting game in the final round… but my story begins a few rounds earlier.

We kept hands that looked pretty solid (I had mana issues but Greenseeker, while Darwin had Looter Il-Kor and a generally good hand) that got a bit more complicated when they led with Icatian Javelineers. Boourns.

We decided to bait the javelin with Greenseeker so Darwin’s Merfolk would have a shot, which in retrospect may have been a mistake since the Merfolk is clearly worth a removal spell (it got one) while I could have played Forest and Plains plus a turn 2 Greenseeker that might have been viewed as bait and ignored. As it was I began drawing Red card after Red card but no Mountains, forcing Darwin to do most of the heavy work of keeping us alive. They, meanwhile, played a few mediocre cards but also an uncomfortable number of evasion creatures, putting our life total under pressure.

Darwin’s draw had gone fairly well and he was actually more than holding our own, along with help from the Green creatures I’d drawn, when they played Damnation and began establishing board superiority with cards they’d held back. All would be well if I could draw a Mountain, but…

I finally hit five mana (still no Red) and could at least start the long process of getting Stormfront Riders into play. Cast them, get a token… cast them again, bounce them and the token to get two… finally cast them and have them stick. Whee. Darwin had been forced to play out most of his resources prior to the Damnation and was now low on gas — in particular he’d had to kill their evasion creatures and now had no answers in hand to their Mire Boa. Two, two, two… while I sat there looking at the Sudden Shock in my hand.

Six life. Come on, Mountain. At least the rest of the board is now stable and even slightly in our favor, with Chronozoa giving us long-term inevitability.

Four life. Come on, Mountain.

Two life. Now the Mountain I finally drew is probably too late anyway, since any random Red spell can just kill us.

Oh, wait… that’s not a Mountain. It’s just the last Red card in my deck**.

Fortunately Darwin drew an answer to the Boa. Naw.

So Darwin, as he describes it, is feeling incredibly sad because we’re not only getting knocked out of the tournament but are getting knocked out by people who don’t seem terribly good, and in large part because I’ve been almost unable to play the game.

Time to start playing. I lean over to Darwin to plan out our turn. Pointing to a card in my hand I say, “okay, now if they attack with the Boa next turn I can do this.” Darwin manages not to ask what on Earth my Red card is going to do and asks me if I think it’s time to start counter-attacking. We think about it for a while and I finally say that I think we should hold off for another turn or two just to make sure they can’t punch through with Strength in Numbers; instead we should just add another guy to the board.

Out of the corner of my eye I could see their faces fall slightly.

Did I bluff well? I think so. I conveyed 100% confidence that the attacking Boa would not make it in, and Darwin and I constructed a vision of the game two to three turns out in which we were the ones finally starting to attack. Clearly that vision wasn’t compatible with their Boa making it through, and given their relative inexperience I suspect that they thought there was a card in the format like Condemn that would punish the Boa for attacking.

That’s not to say that it should have worked, but we took advantage of the fact that 2HG involves communication (i.e. the supply of information) that is unusual for Magic and that opponents are naturally trying to siphon off. We just gave them the information we wanted them to siphon and then hoped for the best.

What happened instead is that they cast Armageddon, presumably to prevent me from doing anything nasty to their Boa. That, of course, turned off Swampwalk and our inability to cast new spells didn’t hurt so much when our overall board position was superior and Darwin’s flying Hill Giant was about to split. We were soon attacking for massive damage in the air and took the game in a few turns.

Again, that sort of bluff shouldn’t work often, but who cares? We were dead unless they decided they had to do something other than attack. Even if they decided to hold off for one turn that could be enough — and in fact would have been, as my next draw was a Mountain.

While the example in question is a bit out there, 2HG players should be aware of — and indeed, should practice — giving false information to opponents as they share information with each other. Suppose you know full well that you’re not going to attack… but your opponent has a tapper. Saying, “First, we should bash,” before casting your main phase spells may nudge them to use the tapper rather than waiting and perhaps tapping something better EOT. If you don’t have a drop and ask your partner, “Should I play this or hold off for this?” may cause your opponents to play around a spell you don’t have. Meanwhile, by engaging in this sort of patter regularly you force your opponents to wonder whether you’re giving real or false information out, improving the odds that they won’t gain reliable information from you and will thus fail to anticipate your actions.

With that in mind, let me close with an apropos section from Raphael Levy Singapore Fling article

I learned something quite interesting in the last round…

I’m paired against Erick Purnama, playing TEPS: one of Zoo’s bad matchups. The problem is that your clock is usually one turn too slow on the draw, and that you have very few disruption spells, only Meddling Mage after sideboard. We are in game 2. I lost game 1, and I’m on the play. Erick lays a Lotus Bloom on turn 1. My hand served me a turn 4 kill, but I definitely needed to figure out my options. With the multitude of lands I could search for, the different creatures I could cast, and the burn / pump spells I was holding, I really needed some time, and some paper, to see how I could win the game.

After a moment, I managed to figure out the best play, allowing me to kill him on turn 4, before he could play his Lotus Bloom. I acted confident in my ability to pull out the play, and my opponent got the information. On his turn 3, the turn before the Lotus hit play, feeling he was very likely to die on next turn, he sacrificed one of his lands, drawing from a Chromatic Sphere, just in case. He was not ready to go off, and passed the turn. I proceeded to kill him on my turn 4.

What’s the deal with this situation?

I usually keep a blank poker face when I’m playing, revealing as little info as possible. In this case, I needed to write everything down for my maths, and it was quite obvious that my results were satisfying, giving away that I would kill him before he could go off on turn 4. So he tried to go off on turn 3, just in case he would succeed… but he probably compromised his chances, at least a little bit, to secure his kill on turn 4 had he got another turn. If he had actually pulled a turn 3 kill, that would have entirely been my fault, as I don’t think he would have even tried if my math proved that I wasn’t able to kill him on my turn 4.

What I should have done was to try to show him that my math didn’t work, and that I would have to draw into something really good to win. This way, he would have felt more confident in the fact that he was winning on his turn 4, and wouldn’t have tried a turn 3 kill.

Had I not had the turn 4 kill, I should have clearly showed that my math were correct, forcing him to try to go off on turn 3, and hopefully compromising — a little bit — his chances to go off on turn 4.

But then again, what I did could have been a double-bluff. Trying to make him think that I could kill him, so he could think I was bluffing, but that in fact I wasn’t bluffing…. but would he have thought that far ahead?

Anyway, the point is, when you’re playing against TEPS and you can’t put a clock fast enough to win before he goes off, try to make him think you have the perfect math that would kill him in just one or two turns (when you can’t). The more convincing you are, the more risks he will be willing to take to win. In the best-case scenario, he will sacrifice all his resources and fizzle, and you’ll win what seemed to be, on paper, an unwinnable game.

Levy gave Purnama some real information — that Levy would win on turn 4. That, in turn, gave Purnama critical virtual information — that by burning a Chromatic Sphere he would be able to go off on turn 3. The virtual information turned out to be false — it’s funny like that — but by giving his opponent information Levy gave him a chance to win, and Purnama took it.

Meanwhile, as Levy points out, it is entirely possible to give your opponent (especially one playing a deck that involves a do-or-die “go for it” moment) false information about when they are going to die that might cause them to pass up their one chance to win (i.e. when you’ve got the kill but convince them you don’t) or else to take a big gamble on going off early when they could actually wait for a much more favorable opportunity.

Hugs ‘til next time,

* Peek only probably achieves perfect information for the alpha-strike because your opponent could have no answer in hand but a card like Think Twice that might result in an answer.

** It may not actually have been my last Red card but it certainly felt that way.