Earlier this week Sam Black wrote a thought-provoking article refuting the long-held notion that there is always a "correct" play. His point was, given that we are all imperfect players, we are better off making a play that utilizes our strengths rather than an abstractly better play that runs contrary to our strengths. For example, even if a control deck is the best deck, if you are much better at aggro decks and there is a somewhat-weaker-but-still-good aggro deck, the decision that will maximize your chances of winning is to play the aggro deck instead of the abstractly better control deck. He goes even further to claim that if we have lots of success drafting monoblack and little success drafting white decks that we should take Tormented Hero over Elspeth, Sun's Champion.

While the takeaway message of playing to your strengths is certainly worthwhile, I disagree with his assertion that there is fundamentally no such thing as a "correct" play. Over the years I have given quite a bit of thought to understanding what constitutes a "correct play." Whether I am right or Sam is right, I would like to take this opportunity to discuss many of the thoughts I've had on this subject in hopes that they will be helpful to you as a player just as Sam's article was very instructive regardless of whether his theory ultimately proves to be right or wrong.


What does it mean to make the "correct play"?

In order to avoid talking past each other, it's important to clearly define what exactly is meant by the term "correct play." As most magic pros use the term (in my estimation), the correct play is the one that maximizes a player's win rate if always made from the current game state.

To elaborate on this definition, "if always made from the current game state" means if a thousand identical game states were simulated and a decision is called to be made, the decision that leads to the most number of wins is the "correct" decision. For instance, assume the decision is whether to attack with Watchwolf on the second turn or to hold it back to potentially play around an opposing Boggart Ram-Gang that might be in the opponent's hand. If in 500 of those 1000 games you attack with Watchwolf and in the other 500 games you do not, assuming all other things are equal, whichever decision results in more wins is the "correct" play. I say this in a purely statistical sense, as should be clear from the first part of my definition as "maximizing a player's win rate."

Grounding the "correct play" theory in probability is the only useful way I can conceive of grounding it and I don't think this is really a point of contention by anyone. I mention it only because Sam asked an interesting question about hindsight confirmation. He asked, "Sometimes when the right play is a losing play, there is another play that would be a winning play. Can the only winning play ever really be the wrong play?" In order to definitively answer "yes," we must appeal to the thousand identical game states. When faced with a decision between option A and option B, if option A results in 50 wins and option B results in 950 wins, clearly option B is the better play even for the 50 identical board states in which it results in a loss when its opposite would have resulted in a win.

The difficulty arises when we try to apply the theory to a concrete game state without access to a thousand identical game states or a program to simulate the results of each decision. We have to rely on rules of thumb or to rationale from relevant variables. If the opponent is at three life with no cards in hand and no way to survive a Lightning Bolt, clearly it is "correct" to target the opponent with your Lightning Bolt instead of a creature. But usually decisions are not so cut and dry. Most game states, at least the interesting ones up for debate, are complex and involve a variety of variables to consider. So when it comes to applying the correct play theory to a concrete game state, the most pressing questions tend to be:

1. What are the relevant variables?
2. What are the probabilities associated with each variable?
3. How do we compute the correct play based on these values?

Let's consider each of these questions as we go through a few example exercises, in order to understand how to arrive at the correct play. In order to get the most out of these examples, read the scenario description and then try to answer the above three questions and arrive at what you take to be the correct play. Then read through my answers to the questions to see how I arrived at what I take to be the correct play. Do this for each of the examples, or at least try to think of the correct play and why before reading my answers.


Game State 1: Keep or mulligan?

You're on the draw (post-board) playing Selesnya against Esper Control (in RtR Block Constructed) and you draw the following opening hand:

Forest, Forest, Experiment One, Experiment One, Call of the Conclave, Call of the Conclave, Selesnya Charm.

You have full knowledge of both deck lists and you sideboarded zero cards:

DECKID=1178630

DECKID=1178631

Do you keep this hand? Why or why not?

1. What are the relevant variables?

- The hand is very good if we draw a white source quickly but gets much worse if we don't.

- He has inevitability. We need a fast start to win.

2. What are the probabilities associated with each variable?

- We have 14 white sources in the deck (nine Plains, four Temple Garden, one Selesnya Guildgate). Since we are on the draw, we have three draw steps to find a white land for turn three, except the Guildgate would be too slow on the third draw step. I will have 53, 52, and 51 cards in my library during my next three draw steps respectively. So the math for drawing a white source on time is: 1-[(39/53)*(38/52)*(38/51)] = 60%

3. How do we compute the correct play based on these values?

- The question then is whether an average six-card hand gives us a better chance to win than this hand, which has a 60% chance of being a good seven card hand. Since 60% chance of being a good seven card hand makes it better than the average seven card hand (which is always 50%), then it can logically be deduced that this better-than-average seven card hand is better than the average six card hand we would be mulliganing into. Remember we are basing our decision on the results of a thousand identical game states, so assuming we mulligan into an average six card hand is a valid assumption in this case.

Hence the "correct play" is to keep the hand. It does not boil down to preference, play style, whether you are comfortable keeping one-landers, or whatever else. If you mulligan, you are decreasing your chances of winning the game, plain and simple.


Game State 2: Doom Blade or Hero's Downfall?

Oftentimes it is correct to conserve one particular removal spell over another. In general you want to save the more versatile one and exhaust the less versatile one. Given this rule of thumb, what is the correct play in the following scenario:

It is a Standard Monoblack Devotion mirror. You are holding Doom Blade, Swamp, Swamp, and Hero's Downfall as your four cards in hand. The opponent has a Swamp and a Mutavault untapped and no other untapped lands and no creatures on the battlefield. It is your pre-combat main phase. You have six untapped lands (all Swamps), two Pack Rats in play, and the opponent is at nine life. You attack with the two Pack Rats and the opponent activates Mutavault. Which removal spell do you use on it and why?

1. What are the relevant variables?

- There are none.

2. What are the probabilities associated with each variable?

- You win 100% of the time if you use either removal spell on the Mutavault and discard any other card in hand to activate either Pack Rat to pump each to a 3/3.

3. How do we compute the correct play based on these values?

- This scenario was chosen in order to illustrate that sometimes there are multiple equally correct plays. These are sometimes called co-optimal strategies. Be careful not to assume just because two plays are close that they are equally correct, however.


Game State 3: Lightning Bolt the Opponent or Firedrinker Satyr?

For this example I am going to intentionally leave out some details in order to walk you through the appropriate thought process for identifying the relevant variables.

You're playing a Boros mirror in Modern. Your opponent is at three life with one (unknown) card in hand. He has a Firedrinker Satyr as his only creature in play and four lands. You have no cards in hand, are at four life, and draw Lightning Bolt. Do you target the opponent with it or the Firedrinker Satyr? (note: Firedrinker Satyr would deal three damage to its controller if you target it and it resolves.)

1. What are the relevant variables?

- The opponent's card in hand is unknown.

- If the card in hand is Lightning Helix and we target the player, he can target us in response and win the game next turn by attacking with the Satyr.

- If the card in hand is Lightning Helix and we target the Satyr, he has to Lightning Helix us in response to survive, but he will no longer have a Satyr to kill us with.

- If the card in hand is a Path to Exile and we target the Satyr, he can path his Satyr in response and survive.

- If the card in hand is a Path to Exile and we target the player, we win.

- If the card in hand is anything other than these two cards, we win by targeting the player or the Satyr. Hence we can ignore all other possibilities as long as we target the opponent or the Satyr.

2. What are the probabilities associated with each variable?

- In order to compute the probabilities of him having Path to Exile or Lightning Helix, the first thing to consider is how many Paths and Helixes he has already exhausted. If he used two Paths and only one Helix, it is more likely he has a Helix than a Path (assuming it is reasonable for him to have four of each, or at least the same number of each). Also you should consider how long the card in his hand could have been either card. Did he have no cards in hand at the beginning of his last turn and just drew this one or could he have been holding onto it for a while. If the former, then the math is the same. If the latter, then ask yourself, "Has he had a reasonable opportunity to cast Helix or Path recently?" If he had two cards in hand last turn and used a Path to Exile on your Firedrinker Satyr while you were tapped out, then it is reasonable for his other card to be another Path but unreasonable for it to be a Helix because clearly he would have used the Helix in that spot over the Path. He still could have drawn either one in the subsequent draw step though.

- What are your chances of winning if he has Lightning Helix and we target the Firedrinker Satyr? We would be at one life with no cards in hand and he would be at three with no cards in hand and (more importantly) he has the first draw step to draw a winning card.

- What are your chances of winning if he has Path to Exile and we target the Firedrinker Satyr? We would be at four life with no cards in hand and he would be at three life with no cards in hand and (more importantly) he has the first draw step to draw a winning card.

3. How do we compute the correct play based on these values?

- The math gets very complicated when we ask the latter two questions. Clearly our chances of winning are higher if we are at four life instead of one life, but it is unclear by just how much. Fortunately we don't need to know the exact win percentage in order to compute the correct play. Instead we only need to know the win percentage relative to the win percentage of a competing scenario, and that can be figured out conclusively without doing exact math, using the following shorthand method:

- There are four possible scenarios relevant to our decision:

1. We Bolt opponent and he has Helix in hand.
2. We Bolt opponent and he has Path in hand.
3. We Bolt Satyr and he has Helix in hand.
4. We Bolt Satyr and he has Path in hand.

- Assuming each of these scenarios is equally likely, bolting the opponent gives us a 50% chance of winning (since we are 0% to win in scenario one and 100% to win in scenario two).

- Computing the correct play then boils down to whether scenarios three and four average a combined win rate greater than 50%.

- In scenario three we have a pretty low chance of winning since we are behind three-to-one in life totals and the opponent gets the first draw step.

- In scenario four it is much closer since we are ahead in life totals four-to-three, but he still has the first draw step, making us probably not much better than 50% to win, if even that.

- I am more certain that we are behind in scenario three than I am that we're ahead in scenario four and therefore the average win rate between scenario three and four is less than 50%.

- Since the two scenarios involving Bolting the Satyr combine to average a lower win rate than the two scenarios involving Bolting the opponent, it is therefore correct to Bolt the opponent (assuming Path and Helix are equally likely, which you should see question two to determine). The two plays are not co-optimal, nor does it come down to anything other than math. There is a correct play and it is Bolting the opponent.

You don't always need exact math to figure out the exact correct play. Oftentimes relative math is sufficient for arriving at the exact correct play.


Conclusions

While it's uncertain from a deeply philosophical standpoint whether there really is a "correct play," I take Sam and myself to each be speaking from a practical standpoint. In other words, is it more useful to strive to make the abstractly "correct" play or should some alternate paradigm be used for decision-making? What works for Sam may just be different than what works for me, but I'm clearly of the mindset that striving to deduce the objectively "correct play" is the most useful paradigm by which to make decisions concerning Magic game states.

Given the "correct play theory," the way I have defined it, these are some of the takeaways outlined in this article:

- There is always a correct play, a play that is mathematically superior to all other plays, excepting co-optimal plays. From a practical standpoint, competing co-optimal plays are inconsequential since arriving at any of them would amount to arriving at the correct play and hence result in maximizing your chances of winning.

- Sometimes the correct play will lose you a game that an inferior play would have won you, but that does not make the "correct play" any less correct. From a practical standpoint, however, make sure you are not overlooking an important variable that may have made the other play actually better.

- In order to arrive at the correct play, you first want to identify the relevant variables and then compute the probability of winning based on these variables to determine the correct play.

- Sometimes complex math can be broken down into simpler math, especially when comparing probabilities relative to other probabilities. Determining "greater than" is much easier than computing "greater than by X%," and determining the correct play often only requires the former.

- Regardless of whether the correct play theory Withstands all criticisms and proves correct in some deep philosophical sense, it is at least highly useful. If you're able to master it, you'll be amongst the game's best. Until a different theory proves more effective, I'll stick with this one.

Craig Wescoe
@Nacatls4Life on twitter
*Thanks to Chris Mascioli for help coming up with the formula for Game State 1.