Rule of Law – OBP and The Mulligan Decision


What is Optimization By Proxy (“OBP”)?

Optimization by proxy sounds like a playtesting method by which you write on cards with a Sharpie. It isn’t.

I was just recently introduced to this concept by an article on the rationality blog “Less Wrong.” That blog post can be found here.

The Less Wrong post discusses a fascinating study about what makes herring gull hatchlings peck their mother’s beak to obtain food:

The first thing a newly-hatched herring gull does after breaking out of its shell is to peck on its mother’s beak, which causes her to give it its first feeding. Puzzled by this apparent automatic recognition of its mother, Dutch ethologist and ornithologist Nikolaas Tinbergen conducted a sequence of experiments designed to determine what precisely it was that the newborn herring gull was attracted to. After experimenting with facsimiles of adult female herring gulls, he realized that the beak alone, without the bird, would elicit the response. Through multiple further iterations he found that the characteristics that the newborns were attracted to were thinness, elongation, redness and an area with high contrast. Thus, the birds would react much more intensely to a long red stick-like beak with painted stripes on the tip than they would to a real female herring gull. It turns out that the chicks don’t have an ingrained definition of ‘motherness’ but rather determine their initial actions by obeying very simple rules, and are liable to radically miss the mark in the presence of objects that are explicitly designed to the specification of these rules.


Generalising the above example, we can say that Optimization by Proxy occurs when an algorithm substitutes the problem of measuring a hard to quantify attribute, with a usually co-occurring a proxy that is computationally efficient to measure.

I started thinking about the many ways this applies to Magic: The Gathering strategy, and the most salient one for me was the mulligan decision. Conley Woods and PV recently wrote about the mulligan decision (with the articles here and here), and their examples and analysis were a great read. I do, however, feel that those articles lacked an ability to generalize across formats, situations, and even other decisions besides mulliganing. Walking through examples is helpful, but what do people use when they encounter a novel situation (besides ask it where Snookie is at)? It turns out people encounter novel situations all the time, and each one is kind of like hatching out of a herring gull egg in a way. In each situation, we have heuristics, or shortcuts, with which we try to optimize our decision. We can’t do a complete analysis of a Magic game decision any more than a baby gull can do a maternity test. What we can do is understand the shortcomings of our heuristics, which then helps us identify and avoid traps.


Whether or not to mulligan is such a complex problem that it is no surprise we all use heuristics to help us solve it. To correctly determine whether or not to mulligan a given hand, we would need to determine the likelihood of winning if we mulligan, and compare that to the likelihood of winning if we don’t mulligan (for the statisticians among you, p(win|mulligan) vs. p(win|~mulligan)). Breaking that down a bit, we’ll need to know the probability of every possible game that could occur given our present hand, as well as the win/loss outcome for each of those games. We’ll also need the probability of every possible post-mulligan game and corresponding outcome, and then we can compare the probability of winning given a mulligan to the probability of winning given a keep. It should be obvious that we cannot complete this task, and we certainly can’t even come close to figuring all this out in the minute or so we have to make a decision. Remember though, that the baby herring gull isn’t equipped to definitively identify its mother moments after hatching. To do that it would need to perform an analysis far too complicated given its ability and the time within which it must make a decision and act. The gull instead uses an algorithm that identifies thinness, elongation, redness and an area with high contrast as a proxy for a process which would identify its mother. What are the algorithms we use to determine the mulligan decision in place of the complex analysis given above? What are our “long red sticks” that we look for since we can’t look for “what really matters”?

Below is a list of things we often look for in our hand, and a discussion of when optimization by proxy with regard to this particular stimulus can lead us astray. If you haven’t yet learned to look for these things, this list is a good primer on mulligan strategy. If you’re just learning how to mulligan more effectively, be particularly wary of the “all my colors” and “particular important cards” methods, for now, since those are common traps for beginners. Everyone should be looking to avoid the situations where our OBP can do more harm than good, discussed below each method. I proceed (roughly) starting with simple heuristics and working my way up to the more nuanced, as a beginning player might in his Magic career.

Land-to-spell ratio

We’ll start with perhaps the most common heuristic, “what is my land-to-spell ratio?” A beginner using this method might look at whether he/she has a “good” or “bad” ratio of lands-to-spells, with 1:6 being bad, for example, and 3:4 being “good.” A slightly more experienced player might view all the ratios on a continuum from 0:7 to 7:0, rather than binary categories, but this player is still using the land-to-spell ratio as a proxy for the strength of the hand.

OBP based on land-to-spell ratio can be problematic in a number of ways. The most obvious is that it disregards which lands and which spells have been drawn. Taking the Standard Naya deck as an example:


We wouldn’t want to keep this, but the land-to-spell ratio doesn’t tell us that. That’s the obvious case, so without much analysis let’s look next at a hand like so:


I imagine this is an easy mulligan for top Naya players like The Boss or LSV, yet I suspect many beginning Naya players will keep this hand. I urge them not to. Card quality matters. Even if you have enough land to cast all of your spells, and you’ve got three spells to cast, this is too little information to reliably arrive at a correct decision by proxy.

“All my colors”

In a multicolor deck, such as a 3- or even 2-color Limited deck, there is a temptation to use the following algorithm: If the hand can produce all my colors of mana, then do not mulligan the hand. Players might add a level of sophistication by running the hand through the land-to-spell algorithm as well before deciding to keep, resulting in: If all my colors of mana can be produced AND the ratio of lands to spells isn’t extreme, then do not mulligan the hand. The above example of Mountain, Forest, Plains, Sejiri Steppe, Stoneforge Mystic, Basilisk Collar, Behemoth Sledge again is illuminating here as well. I mentioned above that I suspect many beginners would keep this hand, but I didn’t say why. It is the commonly used algorithm I just described that explains it. The “If all my colors of mana can be produced AND the ratio of lands to spells isn’t extreme, then do not mulligan the hand” algorithm has all the makings of a great OBP trap. It very often accurately approximates the optimal result; lands that can produce all your colors and have 3-4 spells in them tend to be keepable hands. Furthermore, in the situations in which the algorithm leads us astray, it is often non-obvious that our mulligan decision was flawed. Since the hand does allow us to play lands and some spells, by definition, we won’t lose due to mana screw very often. How can we tell which of our decisions was flawed when we look back on the game? Many of us use whether we got mana screwed as a proxy for whether our decision to keep was good or bad! Thus, in another example of OBP, we don’t even examine our poor mulligan decision because this “wasn’t one of those games where I kept a 2 lander and never drew a land” or the like. If you’re one of the players who uses this kind of mulligan strategy or a similar variant, take a close look at whether you’re considering more than just colors of mana and land-to-spell ratio.

Productive early turns

Some people realize that “how the game is likely to play out” is an important element in predicting the outcome of the game. Since no one can fully predict how the game is going to play out (and if you could, that would be your answer), one proxy for this is to answer the much simpler version, “How are the early turns of this game likely to play out.” This is a productive exercise, but as with any OBP, there is a downside. The downside here should be fairly obvious: the early game isn’t the entire game. This error shows up as a sort of “overcompensation.” I say overcompensation because the natural tendency for Magic players seems to be to mulligan too infrequently. Well, some players have figured this out, and they use methods like looking for “productive early turns” to make sure they aren’t keeping poor hands. In doing this they can develop algorithms that instruct them to mulligan too much. Fortunately for me, Rise of the Eldrazi is the perfect context to illustrate such an error. In ROE sealed deck it could be perfectly acceptable to keep a hand with 5 land, a 4 drop such as Kozilek’s Predator or Ondu Giant, and an 8 mana spell, such as Ulamog’s Crusher. In Zendikar x6 sealed deck, you would be crazy to keep such a hand, but that’s the nature of OBP: contextual differences are what make your heuristic fail.

Particular important cards

This example of OBP can trick the beginner and the expert alike. Sometimes a card is perceived to be so important that its presence in the hand is used as a proxy for whether the hand should be kept or mulliganed. Of course, you wouldn’t keep a 7-spell hand, so we can assume you’re running the hand through one or more other algorithms, like “all my colors” or “land-to-spell ratio” or both. Still, some draws won’t be good enough to allow you to survive to cast the bomb you’ve drawn. Even more subtly, sometimes the card you think is of supreme importance isn’t as important as you think! The opponent may have sideboarded 5 cards that kill your Baneslayer or counter your Ajani, so what was critical game 1 might be a trap game 2. Looking for important cards is something to be aware of, but we need to be careful to only let it be a proxy for strength of hand in very limited circumstances, such as when your only way to have a reasonable chance at victory is to begin the game with a particular card (think Leyline of the Void in constructed or a bomb rare in limited in a very, very lop-sided matchup).

“Is a 6 card hand likely better?”

This heuristic is so broad that is almost no help at all. It is perhaps more question than answer (it even contains punctuation to that effect). However, this can still be OBP since it isn’t the same thing as probability of winning given mulligan versus probability of winning given no mulligan. There are several ways “If a 6-card hand is likely better than these 7 cards, then mulligan” is a flawed operation. First, it doesn’t account for how much better the typical 6-card hand is. Imagine hand A has a 40% chance of winning. 51% of my 6-card hands will have a 45% chance of winning, while the other 49% will have a 10% chance of winning. This example is too simplified to be a real scenario obviously, but we can see the problem nonetheless. (.51 x .45) + (.49 x .1) = .3145, so in other words our chances of winning when we mulligan are only around 31%, and we gave up 40% to get there. This is true even though it is likely that a 6-card hand is better than our current 7-card hand. You can adjust the heuristic to account for this deficiency, but you’ll find that either a) there is still something you aren’t accounting for, or b) your heuristic isn’t a shortcut at all, it is a restatement of the original problem “should I mulligan this hand?” Perhaps surprisingly, I still find this kind of heuristic to be helpful. Framing the problem in terms of your “likely” or “average” hand of 1 less card that you presently hold at least means you aren’t using only the more seriously flawed heuristics listed above, and it means that you are taking a big-picture approach rather than focusing on certain details by proxy.

“If I draw a land, will I win? If yes, the odds of drawing land determines whether I keep.”

This is a popular shortcut for the land-light hand. The probability of drawing a land (or two straight lands, or whatever the case may be) becomes a proxy for the odds of winning. This has at least two problems. 1) the odds of winning if you keep are only half the equation, it doesn’t say anything about the odds of winning if you mulligan, and you need both sides of the coin to do a comparison, and 2) it is almost never the case that if you draw your land or two land, your chances of winning rise to 100%, which is what is implied in the stated shortcut. As with all the other heuristics, knowing the exceptions allows you to better utilize the rule. There are plenty of times you’ll want to go ahead and use the odds of drawing a third land as a proxy for the odds of winning, but remember to view the 6-card hand’s chances of winning as the other side of the coin, and that you need to back off the chance of winning by some amount that reflects the chances of drawing a land and still losing the game.


So how do I advocate approaching the mulligan decision? What we need to keep in mind is that all we can hope to do is find the correct OBP, we can’t hope to avoid OBP altogether. What works for me is a combination of intuition, “is the average 6-card hand better than these 7,” and trying to figure out how the game will play out, as far out as I can reliably see. I’ve written previously about the value of intuition and of listening to and refining your “gut” instinct. I stand behind that sentiment, and once again add that intuition isn’t the only thing to consider; you need to check your intuition against rational thinking, and check your rational thinking against your intuition. If they agree, great, act. If they disagree, you must be prepared to decide which tool seems most likely to fail in the given context, and act accordingly.

I usually start with a “feeling” or gut reaction about the hand, but I then make sure to be cognizant of how my deck desires to have the game go. Do I want early plays? Do I need early plays? Is finding my one [card]Sphinx of Jwar Isle[/card] more important than having a turn 2 play? Does my opponent’s deck put me on a fast clock? Over time you learn which questions to be asking, and I recommend these 2 recent articles from Paulo and Conley to begin to think about what questions to ask. I’m more concerned here with the process, and those articles largely only talk about one step, the rational explanations for keeping a hand. I wouldn’t even use that as a starting point, rather, I get a feel for the hand first, then do some analysis about whether a 6 card (or 5 card if I’m already at 6, etc.) presents a better chance of winning, given all the heuristics I have available, while being cognizant of their shortcomings. If I feel that the “all of my colors” aspect of my hand is one of the reasons it “feels” like a keep, I’ll bring to my conscious mind the ways I can lose a game even when I have all my colors of mana. You could say I consider these things: intuition, applicable heuristics, and contexts where those heuristics may fail.

-Matt Sperling
@mtg_law_etc on twitter


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