We are opponents. Neither of us has any cards in hand and the only card on the battlefield other than basic lands is your Ulamog, the Ceaseless Hunger. We both know I have exactly 26 cards in my library because we counted last turn when you played Ulamog. I am at 20 life and you are at 7 life. You know for certain that the only relevant card remaining in either of our libraries is a lethal Rolling Thunder in my library that you have no way to stop if I draw it. I draw and play Fertile Thicket, looking at the top five cards of my library. I rearrange them and put them all on the bottom of my library, leaving me with 25 cards total remaining. You untap and draw a basic land. Do you attack with Ulamog? Why or why not? If not, how many draw steps do you give me before attacking with Ulamog and why?
This may be the most challenging Magic strategy question you've ever been asked! It illustrates the deeper strategic implications of multilevel thinking in Magic. If you haven't already, I would suggest thinking long and hard about the question before reading on, treating it as a complex puzzle to solve. That way you will get the most out of this article.
Before providing my own answer I decided to pose this exact question to several of the game's top pro players in order to find out whether their reasoning matched up with my own. Here were some of their responses.
Pro Player Responses
Pat Cox: "I would skip one attack. Most people would make the obvious play, assuming I'll attack with Ulamog."
Luis Scott Vargas: "I would probably skip at least one so I don't just lose to the obvious play, but I would not skip five because that would basically be the same as guessing wrong. I would at least take a guess."
Josh Utter-Leyton: "If I make the same play in that spot every time, my strategy would be exploitable, so I would give you something like: zero draw steps 25% of the time, 1 draw step 50%, and 2 draw steps 25% of the time."
Jon Finkel: "I would skip one attack against most opponents, but I would skip two against you, Craig."
William 'Huey' Jensen: "I would give you zero draw steps, but against most opponents I would give them one draw because most likely they put it on top."
Ben Stark: "If I know that you know what's going on, then I just attack. Some amount of the time you're going to level yourself and lose by not putting it on top, so there's no point in giving you any free draw steps."
Eric Severson: "I would skip two attacks. You might make the obvious play and put it on top or you might think I'll play around the obvious play by skipping an attack. So I think it's worth giving you two draw steps to play around both cases. There's no way I'm giving you more than two draws though."
Now that we've gotten a few different pro player perspectives on the Ulamog Gambit, here is my own reasoning:
My Detailed Response
With 25 cards in my library I am 20% to find the Rolling Thunder off Fertile Thicket (5/25 = 20%) and 4% to draw it naturally off my first draw step (1/25 = 4%).
If I find it off Fertile Thicket, the Level 0 strategy is for me to put it 5th from bottom so that after attacking me with Ulamog and exiling my top 20 cards, I will draw it to kill you on my turn.
If I put the Rolling Thunder anywhere else (4th from bottom, 3rd from bottom, 2nd from bottom, or bottom), then I won't survive long enough to draw it, assuming you attack me with Ulamog on your turn. I can only survive one Ulamog attack.
Given that my Level 0 play is to put the Rolling Thunder fifth from the bottom, the Level 1 strategy would be for you to skip one attack so that I have 4% to hit instead of 20%. If you somehow know for certain that I'll make the Level 0 play and put the Rolling Thunder fifth from the bottom, then the correct mathematical play is definitely to give me exactly one draw step.
But what if I know that you know this?
If I know that skipping one attack is the optimal Level 1 strategy, then I could put it 4th from the bottom to play around Level 1. Then when you attack the following turn after giving me one draw step, I will draw the lethal Rolling Thunder. This would be the Level 2 strategy.
Optimal Level 3 strategy would then be for you to attack since attacking would give me no free draw steps and would result in my drawing the card fifth from the bottom, which is not Rolling Thunder since I put the Rolling Thunder fourth from the bottom qua Level 2 strategy.
Notice that Level 3 is identical to Level 0 (i.e. attacking immediately).
So would it make any sense to give me more than one draw step?
The answer is yes, if you believe my range is such that I could very likely be operating on Level 0 or Level 2 and you want to hedge against both.
If you want to play around Level 0 and Level 2, you could operate on Level 3A by skipping two attacks. This gives me two draw steps to find Rolling Thunder (or 8%), which is still less than 20%. If you were somehow certain that I was operating on Level 0 or Level 2 but you did not know which, then it is mathematically optimal to give me exactly two draw steps since giving me zero draw steps would be 10% and one draw step would be 14% (10% + 4% for one draw step), assuming it is equally likely that I operate from Level 0 or Level 2. And 8% odds of losing are mathematically favorable to 10% or 14%.
Based on the feedback received by the pro players interviewed, it seems as though this is a very realistic strategy to employ since no one wants to go deeper than Level 2, yet there is quite a bit of variation between Level 0 and Level 2.
But it doesn't necessarily stop here.
If I somehow put you on Level 3A, then I could go to Level 4A by putting the Rolling Thunder third from the bottom. Then after you give me two draw steps and attack me, you mill me right into the lethal Rolling Thunder that I put third from the bottom and I kill you with it.
If you believe I am definitely operating on Level 4A (i.e. putting the Rolling Thunder 3rd from the bottom), then the correct strategy is to just attack me since that would give me no free draw steps and would result in me drawing the card I put fifth from the bottom, which is not the Rolling Thunder since I put that third from the bottom qua Level 4A strategy. I am then 0% to win if I operate on Level 4A and you operate on Level 0.
At his point it should be clear that attacking is optimal (i.e. employing Level 0 strategy) if you know I am operating on any level other than Level 0 since it minimizes my draw steps without resulting in me drawing the Rolling Thunder off the five cards seen with Fertile Thicket. But of course you don't know I won't be on Level 0, which is what makes this puzzle so complicated.
So what if, instead of knowing for certain I am operating on Level 4A strategy, you believe Level 0, Level 2, and Level 4A are all within my range. If you somehow know for certain (or are reasonably sure) that those are the only three levels within my range, then it is still not mathematically optimal to operate from Level 5A and give me three draw steps because this would make it 12% likely that I draw Rolling Thunder. While 12% is still less than 20%, it would be better to randomly select one of the three levels within my range and assume I will operate from that level. If I'm equally likely to operate from Level 0, Level 2, or Level 4A, then each is on average 11% likely to win me the game ([20/3 = 7%] + [(0%+4%+8%)/3 = 4%] = 11%), which is less than the 12% likelihood if you give me three draw steps. Hence Level 5B would be to randomly determine whether to give me zero, one, or two draw steps.
If I somehow knew for certain you would operate from Level 5B, then putting it anywhere other than the bottom two cards would be a mathematically co-optimal strategy since my odds of winning are on average 11% whether I put it fifth from the bottom, fourth from the bottom, or third from the bottom. Level 5B is an unexploitable strategy that insures you are exactly 11% to win (plus 100% bonus equity if I choose to put Rolling Thunder on the bottom or second from the bottom).
Since Level 5B cannot be exploited, there is not a Level 6B.
Given that 5B is unexploitable, would a Level 3B be superior to Level 5B?
The short answer is NO because giving me zero draw steps would be 10% and one draw step would be 14% (10% + 4% for one draw step), assuming it is equally likely that I operate from Level 0 or Level 2. Hence Level 3B strategy would allow me on average to find Rolling Thunder 12% of the time (10% and 14% averaged) whereas Level 5B strategy would only allow me to find it on average 11% of the time. Hence, since neither is an exploitable strategy, 5B is mathematically superior to 3B by 1%.
Given that level 5B (randomly giving me 0, 1, or 2 draws) is a mathematically favorably strategy to level 5A (i.e. giving me 3 draw steps) and also favorable to level 7A (4 draw steps = 16%) or 9A (5 draw steps = 20%), it would be indefensible for me to put the Rolling Thunder on the bottom or second from the bottom and it would likewise be indefensible for you to give me 3 or more draws to find the Rolling Thunder.
So this leads us back to the original question: which strategy is optimal?
Now that we laid out all the reasoning and showed the math that supports each strategy, we can definitively narrow down the range of possibly correct strategies to:
Level 0: Attack. (0-20% chance to lose, only losing to me finding Rolling Thunder with Fertile Thicket and putting it fifth from the bottom)
Level 1: Skip 1 attack. (4-20% chance to lose, only losing if Rolling Thunder is on top of my library or if I find it with Fertile Thicket and put it fourth from the bottom)
Level 3A: Skip 2 attacks. (8-20% chance to lose, only losing if Rolling Thunder is on top or second from the top of my library or if I find it with Fertile Thicket and put it third from the bottom).
Level 5B: Randomly decide whether to give me 0, 1, or 2 draw steps. (11% chance to lose ([20/3 = 7%] + [(0%+4%+8%)/3 = 4%] = 11%)
Based on this math, it would be optimal to operate from Level 5B unless you had reason to believe the opponent is more likely to operate from a particular level than any other level contained within the 5B range (i.e. Level 0, Level 2, or Level 4A). For instance, as Austin Bursavich put it, "in real time it will be very obvious if they hit the Rolling Thunder off the Fertile Thicket. They will repeatedly count their deck and do the calculations and get excited." Or as Pat Cox put it, "Most people would make the obvious play, assuming I'll attack with Ulamog." Level 5B would mostly come into play in the situation where you not only know that the opponent knows about the mind game going on but also that he/she knows that you know about it.
The Final Answer
As it turns out, the math is such that three possible answers can be eliminated as incorrect (giving me three draws, four draws, or five draws) while the remaining three (zero draws, one draw, or two draws) are each defensible and depend on whether you have reason to believe you can improve your odds from 11% by leveling the opponent. The risk of course is that you open yourself up to getting leveled. Level 5B is the safe, unexploitable strategy, but 11% chance of losing is worse odds than 0%, 4%, or 8% if you correctly guess the level the opponent is on (though better than the 20%, 24%, or 28% if you're wrong).
For further reading on Multilevel Thinking, here are my previous two articles on the topic:
Multilevel Thinking 2
Now that you have all the tools at your disposal for making the decision, I have one final question for you:
If faced with the Ulamog Gambit, when do you attack?
@Nacatls4Life on twitter
Thanks to everyone who responded to my question! Also thanks to Dustin Faeder for posting the original formulation of the question on my Facebook wall and to Kai Budde, Austin Bursavich, and everyone else who commented on the post for helping to refine the question.