Dividing And Shedding Light

Posted in Serious Fun on July 12, 2005

By Anthony Alongi

Anthony continues his game theory series, with a colorful twist.

Ed. Note: While House of Cards is on hiatus we'll be moving Anthony's column back to Tuesdays. The new column will appear on Wednesdays, starting tomorrow. – Scott Johns

First, I'd like to welcome myself back to Tuesdays. Wednesdays were nice, but ultimately too hard to spell.

As we continue "Rainbow Week" here at magicthegathering.com, I wanted to come at the theme indirectly. (When we did theme week assignments, I heard Bennie Smith would cover Rainbow Stairwell, an increasingly popular casual format in Magic Online. Enthusiasts of this format can look forward to that! And this leaves me free to go in another direction.)

Rainbows are essentially "divided light" – that is, the sun's ordinary white light split by water droplets into different colors. They are a reminder that every time we see something "ordinary" like sunlight (kids, don't stare at the sun!), we are in fact seeing a ton of hidden information. All the little wavelengths in there we don't really recognize from day to day – we suddenly see the complexity, with the help of a little rain.

Last week I introduced game theory to readers. I did this not because I expect game theory to explain every situation you'll face in Magic, but because I think it's helpful to have the tools game theory provides in your "mental toolbox". There are at least three benefits you'll gain from this series:

  1. Making the effort to understand game theory simply exercises your mental muscles – as you get better at analysis, you get better at Magic (and other things) because you're just plain smarter, even if you never see a direct application.
  2. You will, at some point, actually face a decision where recognizing game theory will help you make a better decision.
  3. As you recognize different bits of game theory in Magic, you'll discover more complexity in the game, which can be just plain rewarding on its own – and you may appreciate some cards, formats, and strategies that may have appeared confusing or "dumb" before.

This week, I want to look at a single game situation (the sunlight), and then throw some rain around so that readers can see the different game theory concepts (the colors) inside.

My disclaimer, for easy copying and pasting on the message boards – my primary goal is not complete mathematical or technical precision. My goal is broad understanding of general concepts, and having a bit of fun doing it. If you want to enjoy a college-level course in game theory, attend one.

A Sunny Situation

Okay, I don't often ask readers to pay lots of attention to a game board, but I kinda need you to do that now, because we're going to use it all article long. Don't worry – it's a simple board.

We're in a three-player free-for-all game – Andy, Bob, and Christina. Any player can target or attack anyone else. The game has progressed to the point where all players are sitting at two life. Winning is important to all of them – and coming in second is better than coming in third.

Every player has no cards in hand, only one card left in library which they'll draw and play on their turn, and no cards in graveyard. A carrot-loving dog upset the card table just a few seconds ago in an attempt to snag a tasty, rooty treat, so everyone knows what their opponents' last draw will be. (We're setting up some unusual conditions, but it's only to help settle issues of imperfect information.)

In turn order:

The first critical point I want readers to notice: no one at the table is playing white or black. Remember the rainbow! Cripes, I can't believe some of you forgot that already. You have such short attention spans!

The second point: If nothing happens, Christina will win, because Andy and Bob will each lose on their next draw step.

The third point – well, to understand the third point, we have to understand dominant strategy. So that's the first part of the spectrum we'll look at.

The First Color: Dominant Blue Doesn't Dominate

If you recall last week, we talked about rational decisions. It's a bedrock principle of game theory, and here's why: if someone is rational, they'll always look for the "dominant strategy". A dominant strategy is the path you'll take when no matter what your opponent does (or opponents do), you'll do better for yourself than you would have doing anything else.

In a simple, non-Magic example: my lovely wife MaryJanice and I value time with our children, even though we loathe each other. (This is not true, of course! We actually prefer not to be with the children, har har.) On a given night, we can each either go out alone, or go with our children to their favorite movie. If I go out alone, I'm sad no matter what my beautiful but devious wife does. If I go to the movies with the kids, I'm happy. I'm slightly less happy if MaryJanice comes along – but still happier than being alone.

Game theorists love 2x2 grids. (I simply don't have the heart to tell them that they're actually 3x3 grids, with titles in the first row and first column.)

  Anthony goes out with kids Anthony stays home.
MaryJanice goes with kids. Anthony is happy, albeit annoyed at his beautiful but devious wife's ability to understand game theory. Anthony is sad – and bitter at his wife's superior understanding of game theory.
MaryJanice stays home. Anthony is happy, and gloats over his superior understanding of game theory. Anthony is sad, trapped at home with his beautiful but devious wife, and guilty of letting his preteen children out in public without proper adult supervision.

As the grid shows, no matter what my wife does, I'm better off in the column that has me going out with the kids. (Though look at how happy I am if my wife stays home!) Grids – they just don't lie.

Okay, now look at the Magic example we set up with Andy, Bob, and Christina. What is Andy's dominant strategy?

First, Andy could do nothing. If he does, he puts the onus of the game on Bob to finish off Christina – which is a mean thing to do to poor Bob. But even if Bob succeeds, Andy still loses first – because that draw phase is hanging over his head. This strategy is weak (and, we'll soon see, easily dominated by another).

Second, Andy could split his energy – sending the Seal at one player or creature, and the Sandstalker in the other direction. This is tantamount to doing nothing – Andy will almost certainly end up losing first, unless his opponents make horrific mistakes.

Third, Andy could try to kill Bob. With enough effort, he'll succeed – the Seal can finish off the Wellwisher (if Christina cooperates), and then the Sandstalker can overwhelm whatever Bob's life total will be. This effectively hands the game to Christina, who can then pass the turn and watch Andy die.

Fourth, Andy could try to kill Christina. Success depends on what happens when he attacks with the Sandstalker. Christina has a tough choice to make – she can either block with the Willbender and leave herself open to the Seal of Fire (which Andy was smart enough to keep around), or she can let it through and die.

The decision trees get pretty branchy at this point, but it boils down to this: if Andy attacks Christina, she has a bad choice to make. Either choice she makes (let the Sandstalker through and die, block the Sandstalker and let the Seal go where it may) is bad for her. It is most likely that she will block the Sandstalker. Her morph dies, and the Seal is now free to go to either Bob or Christina. A small mistake (or a simple kingmaking preference) on Bob's part gives Andy the game.

Here is a grossly simplified chart for Andy, which abuses notions of probability and risk for the sake of simplicity and poetic license. Deal with it.

  Do Nothing Attack Bob Attack Christina
Opponents make no mistakes Andy loses and comes in third. Andy loses, but comes in second. Andy guarantees himself second place, with a very faint chance of winning.
Opponents make mistakes Andy loses and probably still comes in third. Andy loses, but likely comes in second. Andy guarantees himself second place, and has a fair shot of winning.

Attacking Christina is the dominant strategy – even though chances of actually winning are slim if he does. The far right column simply looks better than the other choices, no matter what row you're in.

Each player has a similar grid – though in fact, the players who come later have different grids, depending on what Andy does. This gets really griddy, really fast. Assuming Christina blocks the Sandstalker and Andy passes the turn, Bob's chart for his own turn looks something like this – again, with gross simplifications and several impractical options left out:

  Do Nothing Attack Andy Attack Christina
Andy sends the Seal at Bob Wellwisher available to tap for life; survive Seal and take second (to Christina). Wellwisher cannot gain life. Bob comes in third. Wellwisher cannot gain life. Bob comes in third.
Andy sends the Seal at Christina Win during Andy's draw step. Pump the Wellwisher with Predator's Strike and take second (to Christina). Win during Andy's draw step.

Does Bob have a dominant strategy? Technically, no: if Andy sends the Seal at Christina, Bob would do equally well either doing nothing or attacking Christina. A dominant strategy always does better than the other options.

But that's a technical quirk. It's obvious Bob does the best he can if he just sits and does nothing. Keeping the Wellwisher untapped puts the red mage in a box – Andy has to send it at Christina, since we've already asserted that coming in second is "better" than coming in third.

So What Do You Do with This? For each of these concepts, I'm going to try to sum up what you should keep in mind and why it's important. For dominant strategy, here's what you need to remember: dominant strategies come up all the time. They appear most often (or I should say they're most fruitfully analyzed) toward the end of a game, when the payoffs and costs are clearer than they are at the beginning – and when players tend to have fewer pieces of hidden information (cards in hand). Whenever you make a decision during such times, you should put some effort into analyzing the paths you could take, and the most likely paths your opponents would take.

Don't cop out by saying "it's all too complex." Real life is always more complicated than a theory. Put some mental effort into working out your best chances, and go with the path that seems to let you come out on top most often.

Okay, that's dominant strategy. Now let's go somewhere else over the rainbow.

The Second Color: When It's Not Easy Being Hunted Green

There's a game called the "Stag Hunt", which a French philosopher named Jean Jacques Rousseau came up with. Two hunters can either jointly hunt a stag, or individually hunt a rabbit. The stag offers more meat, but is impossible to catch when the hunter is alone (and hard enough when it's two hunters!). Rabbits offer less meat, but are easy to catch when the hunter is alone.

It may seem as though hunting stag is the easy decision – but the twist is, hunting a stag together doesn't mean you walk side by side. You have to come at the stag from different parts of the forest – and one hunter may decide it's just easier to go get a rabbit, leaving the other one fruitlessly chasing a stag all day long.

My wife grew up hunting. I did not. During our courtship, is there any doubt I experienced the harsher corners of this 2x2 grid?

  Anthony chases the stag, per pretty girlfriend's request. Anthony gives up and looks for rabbit.
MaryJanice keeps her promise to help Anthony with the stag. They catch the stag together and have memories of bloody venison that last a lifetime. Anthony eats well but misses out on a second date.
MaryJanice laughs over rabbit stew later on with a smarter, more handsome hunter. Anthony doesn't even have a pet rabbit to console him. Both parties at least have enough energy from rabbit stew to seek out other romantic interests.

For each of us, there's a path dominated by payoff (the rabbits, which are a more sure thing) and a path dominated by risk (the stag, which gives more reward if we crazy kids can stick together).

So how does this all work out for Andy, Bob, and Christina?

Let's change a fundamental assumption of our Magic game. Let's say at this point, winning is still important, but perhaps not as important for Andy and Christina. They've all analyzed dominant strategy, see Bob probably come out on top – and darnit, they wanna do something about that. They wanna hunt Bob. This could be easy – after all, one of Andy's "bad" options above was to just kill Bob with combat damage from the Sandstalker.

So let's add a couple of cards to the table too – Bob has a Traproot Kami next to his Wellwisher, and Christina has a Seal of Removal on the table.

Now Andy clearly needs Christina's help to get through Andy's defenses – one seal for each creature, and then the Sandstalker punches through.

Notice that to kill Bob, Andy and Christina must cooperate. They can cooperate on Andy's turn as noted above – or they can cooperate on Christina's turn (Andy spends his turn eliminating a blocker with the Sandstalker, then Christina eliminates the other one with her blue seal, rendering Bob's Predator's Strike useless, swings for two…and Andy finishes Bob off with a seal.)

The wheels are a bit wobbly on this example because the Stag Hunt assumes simultaneous actions – the hunters are making their decisions at the same time. So we don't see technical perfection here. But what does carry through is the idea of cooperation to get the highest payoff – a dead Bob. If either party reneges (Christina watches Andy's seal hit the Wellwisher, but then doesn't sack her own seal to get rid of the Kami; or Andy decides to go after Christina instead), the payoff is less rewarding – maybe each party has increased their own chances of winning or done something interesting, but winning doesn't feel as good as getting rid of poor Bob.

  Andy sticks to the anti-Bob plan Andy attacks Christina
Christina sticks to the anti-Bob plan Bob dies first. Andy and Christina are so wild with joy, they don't even remember who won. Andy ambushes Christina with the Seal of Removal gone. Christina likely dies to Bob. Andy enjoys a glimmer of interesting strategy. Christina sulks.
Christina leaves Andy in the cold Bob still dies, but only after Andy goes. Andy sulks. Christina probably wins and Andy got to do something interesting, but Bob didn't die first. Meh.

So What Do You Do with This? In your own Magic games, you probably see Stag Hunts all the time – two players who implicitly (or explicitly) cooperate to get rid of a third. The third player is powerful enough to cripple one attacker, but not both. Maintaining cooperation under these circumstances can be pretty tough.

Whether you cooperate or not at all – well, that depends on your assumptions, doesn't it? Sometimes collaboration is about dealing with a common threat. Sometimes collaboration is about "getting even" for something else that happened earlier, or in another game. Sometimes collaboration is just a whim.

One of the reasons I prefer implicit collaboration – collaboration based on true threat assessment, and not wheeling and dealing and whining – is because it's more stable. For those of you who noticed how tenuous the Stag Hunt analogy is above – there's a reason for that. Alliances and shared interests never work out as nicely as you think they will. Keep your eyes open. Chase the stag if you like, but be ready for the hunt to fail.

This is as complicated as we'll get today. The last part will be nice and easy.

The Third Color: Seeing Red At Auctions

A lot of Magic games have elements of auctions in them, but very few are exact replicas. Perhaps the most common type of auction to see in a Magic game is a Dutch auction.

A Dutch auction starts with a really high price for an item – a price so high, it's unlikely anyone will bid for it. Then, using a clock or similar device, the price ratchets down bit by bit, until someone "buzzes in". That first bid wins the item.

(This approach was traditionally used for selling flowers in Holland. I have no idea if they still do this. Frankly, when I buy flowers for my beautiful but devious wife, I do not need the added nerve-jangling tension from a bidding process that's likely to end abruptly, loudly, and expensively. But the Dutch may do as they like.)

The ingenuity of this approach is that the seller is guaranteed the highest possible price for a given item. In fact, it works so well, department stores use a similar approach – whenever they start a suit jacket off at $500, don't sell too many, then reduce the price to $420, sell a few, reduce the price to $350, sell a few more, and so on. The Dutch auction process figures heavily into this sort of "price discrimination" – an economic term which is perfectly legal and more ethical than it sounds. Sellers want to get the highest price consumers are willing to pay, and only come down if no one's willing to pay. Dutch auctions and clearance racks are two ways of reflecting this.

So what does this have to do with multiplayer Magic?

Go back to the original Andy, Bob, and Christina. Look at the basic assumptions we made – empty libraries, life totals at two, no cards in hand.

Each player's survival is a cheap commodity. Any one player can look at the other two and say, "you know, it wouldn't take more than a bit of concerted effort on my part, and a good stiff wind, to take you out." It makes each more willing to take a serious, hungry look at the other two.

Compare that situation to this: with the exact same permanents on the board, rewind the clock so that every player is at 20 life, with 50 cards in their libraries and five cards in hand.

Should Andy be thinking of blowing that Seal of Fire up? Does it matter so much where the Viashino Sandstalker goes? Should he even show the board that he has a Sandstalker?

He's not going to die next turn. There's not much pressure on him to push hard (though the Wellwisher and the Willbender can certainly both get troublesome, if left untouched). Blowing a ton of cards and resources just to punch through for two or four damage – well, the price is just too high. The clock hasn't ratcheted down to the point where he wants to make a critical move.

So What Do You Do with This? Multiplayer games do go through a sort of rhythm that feels very much like a Dutch auction – there is some establishment of position in the early game. (This doesn't make first- or second-turn plays inconsequential – in fact, I argued the reverse just a few weeks ago. On the contrary, it makes them more important, since you are helping set the price opponents must pay to attack you early on.)

During the mid-game and late game, different players start to snap. Something makes them change – they shift from defense to offense, or they send more attackers than they did before, or they put two or three pieces of what may be a combo out, or they blow up permanents that may or may not appear to affect them. All of this is in preparation for victory. The price on their opponents' heads – your head – is coming down.

It's important to get a feel for when the price on your head is coming down. When have you lost too much life to be a viable threat? When should you let an attack through, so you can preserve a blocker for a possibly worse assault? When would a "finisher" like Overrun or Inferno be likely to show up?

Knowing your group and the personalities within helps a lot. You learn how people act when they're tasting victory. You see when they attack the wounded, and when they leave them alone. You see the "tells" involved when someone draws the fifth piece of their combo.

It's important to get a feel for when the price on your head is coming down.

And from this, you plot your own strategy. You bait attacks into spells like Warpath and Spinal Embrace. You occasionally forgo the Counterpell or Captain's Maneuver in your hand, letting a minor spell resolve and appearing weaker than you actually are. You alpha strike faster than an opponent expects, using Reverent Mantra when you're tapped out.

In essence, you learn how important instant-speed spells and abilities are. The clock is always ticking – not just on your turn, but on opponents' turns as well. Someone is always looking to bid. Someone always sees red. You have to be ready to respond.

Chasing The Game Theory Rainbow Away

There are, of course, times when you shouldn't worry so much about game theory or analysis. You're not playing this game to write really cool equations or sketch 2x2 matrices. (Not always, anyway.) Can't this game just be about fun?

Sure. I'll return to game theory in later weeks – while I usually do series in "chunks", I prefer to keep this as a random return topic. But for now, relax, close your eyes, chase all the little rainbows away…and play some Magic.

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