I was at a wedding last week, and at some point in conversation with one of my uncles he posed the following question. What if you could play the following game?
The Rules:
1. The House will let one person play The Game. Assume The House can and will pay whatever it owes.
2. The House will auction off the right to play The Game. In the auction, you get one blind bid. If you win the auction, you pay whatever you bid and get to play The Game. If you don’t win the bid, you lose nothing.
3. Once you are in the game, you don’t risk anything more. The House puts $1 into the pot.
4. A coin is flipped. If it lands heads, you win all the money in the pot and the game is over. If it lands tails, The House doubles the money in the pot and you repeat this step.
Seems simple enough, right? So, how much would you be willing to pay to play?
I posed this very question to a number of great Magic players and Pro Poker players to get their perspective. Almost universally, their instinct was to try to calculate the EV (expected value, as opposed to exposure value) or ROI (return on investment).
As anyone who calculates their EV in this game comes to see, it is not as simple as just math.
What is your EV in The Game, as described above?
I will give you a minute to think about it.
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Seriously, work it out.
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Don’t know how? Hint, you calculate your equity for each way the flips could come. For instance, if it lands heads first, you win a dollar. Since that happens 50% of the time, you have 50 cents in equity from that possibility.
What are the chances of it lands tails, then heads?
Twenty-five percent. Since this would cause you to win $2, that is another 50 cents in equity.
Keep going.
What about tails, tails, heads? That would be 12.5% to win 4 dollars, another 50 cents.
See where this is going?
Infinite.
That’s right, your EV is actually infinite, under the crude, typical definition of EV. William “Baby Huey” Jensen summed it up well.
“You have a 50% of winning a dollar. That is 50 cents in equity. You have a 25% of winning two dollars. That is another 50 cents in equity. You have a 12.5% of winning four dollars. That is yet another 50 cents in equity. This continues infinitely. As a result, strictly speaking, your EV is infinite.”
Does this mean you should be willing to bid everything you own for the right to the game?
If you base your decisions strictly on what is the highest EV play, then you should, assuming of course that all you care about is getting the greatest chance to win the most money. However, as common sense shows you, this doesn’t mesh with real life.
Why does this calculation of EV break down?
The key is that there is an upper limit to how much more money is worth to you. What is the difference between 500 million and a billion? What about 500 billion versus a trillion?
To use an extreme example, would you rather have 100% chance at 100 million dollars or a 90% chance at 200 million dollars? See what I mean? Money has diminishing returns. Two dollars is probably worth twice as much to you as a dollar is, but 200 million won’t provide twice the “good” that 100 million would. This obviously assumes you aren’t able to arrange some crazy “hedge” such as a bet against yourself to win the flip.
In addition, there is the issue of an upward ceiling on how much money actually has any value at all. For instance, a billion dollars is a real amount of money, albeit one that may not have relevance in your life. However, what about a trillion? What would it even mean if a trillion dollars was printed to give to you?
Aside from side effects like inflation, or anything along those lines, there are issues like your own safety and what your relationship would be to other people. Once you have more money than almost any country, you are not going to be viewed as a person by a lot of people.
In addition, what value does money have to you at that point? What about a quadrillion dollars? That is more money than there is on Earth. So, let’s say they printed enough money so that you would get paid and now control most of the Earth’s money.
What would that even mean? What is it to ensure that everyone else agrees to value your money? Who is to say that people won’t just try to X you, one way or the other. They could go to war against you… or they could boycott dollars, now that you have them all.
The point is that a quadrillion dollars is not meaningful, at least not in a positive way. As a result, it or any amount of money more than it is not meaningful to win in The Game. This is relevant, because the calculation of EV is contingent on each flip doubling the “value” of the pot.
For the purpose of this exercise, I will neglect to factor in the diminishing returns of the doubling of the money and just focus on the fact that beyond a certain point, the money is essentially worthless.
The question is this: where do you draw the line?
I think that this is a personal choice that is completely contingent on how much money you would have use for. (I would describe this as the amount of money that you would keep, beyond which you would just give it away rather than be burdened with the problems inherent in the amount.)
So how much money would you have a use for in your life?
A million dollars?
A billion dollars?
Let’s calculate it with both of those numbers and get an idea of what our Real EV is, assuming that we don’t care substantially about money beyond a million, or beyond a billion dollars.
In order to evaluate the value of our equity in The Game, we use the method described above. We have a 50% chance to win a dollar, which is 50 cents in equity, a 25% chance of winning two dollars, which is another 50 cents, and so. However, this time we have an upward limit.
First, let’s look at a million dollars. With a million, our equity looks like this:
50% to win $1 = 0.50
25% to win $2 = 0.50
12.5% to win $4 = 0.50
6.25% to win $8 = 0.50
3.125% to win $16 = 0.50
And so on up to a 100/(2^20)% to win approx $1,000,000 = 0.50
Now here is the trick… if we are assuming that anything beyond a million would mean nothing to you, then the remaining 100/(2^20)% would no longer continue to cut in half over and over. Rather, instead the whole thing would be valued at just $1,000,000, since we are saying anything beyond that would be meaningless. That would add another fifty cents in equity for a total of $10.50.
Wow, so now our “infinite EV” has shrunk to $10.50? It is trivial at this point to demonstrate that even if a billion dollars is meaningful to you, there is still only $15.50 in equity. That would seem as good a way as any to determine how much to bid for the right to play, though again, you are obviously not factoring diminishing returns of large amounts of money.
Still, if all you are wagering is ten or fifteen dollars, it would seem that the opportunity to have the life experience of being the one to play this game would be worth it. That is a whole other line of discussion, regarding what is important.
I say all this to show that sometimes what is important is not just a straight measure of “EV” or how to get the most resources in “the big picture,” as it is not money that you really want… rather, that is just a means to an end.
The same is true in Magic with so very many resources. Extra cards tend to be good, as does mana, life, dealing damage to the opponent, creatures, etc. All of these things are good to have more of, in the abstract, but the goal is to win the game.
In life, you want to be happy, and money is just one tool of many to help shape the game. In Magic, you want to win, and there are a multitude of resources available to help shape the game.
Which would be better in Magic, drawing 5 cards 100% of the time or drawing 30 cards 25% of the time? Arguments can be made either way, but clearly it is not a matter of what gives you the greatest expected number of cards. The question is, does drawing 30 cards increase your win percentage at least four times as much as drawing 5 cards? I would say in general that it does not, however, now I am presupposing what I am trying to do is “win the game.”
Let’s say my goal was to set off some crazy combo and impress my friends. It may be that a chance at 30 cards would allow me to do this, where as drawing 5 would not. It makes more sense to make the riskier play there, since it is more likely to give me what I want.
In Magic, too often people get hung up on plays that are really big and really flashy or on powerful effects. The truth is, how powerful the effect needed to “typically” win the game is actually much lower than most players realize.
For instance, there are players who play with Aven Riftwatcher in their Reveillark deck because of the possibility of gaining an arbitrary amount of life, as opposed to Kitchen Finks, which will tend to only gain 4. It would appear that Aven Riftwatcher is expected to gain more life for you, but that is not what is important.
If your goal is to win tournaments, then you are not really concerned with how much life you gain, as opposed to how many games you win. Kitchen Finks is such a stronger card than Aven Riftwatcher, so it will give you a small but important advantage all the time. The Riftwatcher, on the other hand, appears to have an absurdly large high end, but what is the difference between gaining 1000 life and gaining a million life?
For instance, it is obvious to most tournament players that Tooth and Nail typically amounts to a nine-mana spell that wins the game. Most tournament players would agree that Enduring Ideal is a seven-mana spell that usually wins the game. However, spells of this power level are not the only ones that often translate into a win.
Look at a card like Firespout. Bitterblossom. Magus of the Moon. For sure, all of these cards can be beaten, and often are. However, they provide real edge so much more reliably, since they are so much cheaper and can be played so much easier.
A key to succeeding at designing new Magic decks, is to focus on gaining edge reliably. Also, keep in mind the diminishing returns of resources, whether they be cards, life, or creatures in play.
It is fine to look at strategies or combos that win the game in one fell swoop, but remember, for the most part the difference between 20 damage and 2 million damage is not that great. One Shivan Dragon may be more powerful than giving your opponent one poison counter, but ten Shivan Dragons is not as powerful as ten poison counters.
I guess I am saying that it is about scarcity, and about how valuable the resources are to you. Last week, Flores wrote a great piece related to economics in actual Magic game play. I recommend you check it out here. I myself took a convoluted route getting to my point of focusing on how life, cards, and creatures are all just resources, and to focus on what will allow you to win the most often, not what will give you the most life, cards, or creatures.
Honestly, I was gonna write a pretty off the wall piece that was more than a little… how shall we say this delicately… “crazy.” Then, this past week went by, and I felt this might be the wrong time for such craziness so went with something a little more reliable, albeit still unorthodox.
If you take nothing else away from this article, just remember to remind yourself that if your goal is to win the as many games of Magic that you can; resources are just resources, and not ends in and of themselves. It does not matter what will give you the most cards. What matters is what will make you win the most.
All resources have diminishing returns, so it is important to factor this in when making decisions. For instance, Mind Spring may lead to wins more than Oona’s Grace when you have ten land in play, but it is excessive. I mean, do you really need to draw that many cards. How big a difference is there between 4 cards and 8 cards?
Oona’s Grace is obviously much better earlier when you can just cycle it away (for later use, no less). However, the key is to identifying a card like Oona’s Grace is to realize that though it will correspond into “less card advantage” in those scenarios, such as when you flip tails 6 times in a row, it is the sure thing. It is the safe money that is always good, even if the payout is not as ludicrous when it hits.
Drawing a few extra cards over the course of the game is better than being stuck with a dead card sometimes and sometimes hitting the megamillions jackpot, because at the end of the day, we care about winning more than we care about what will draw us the most cards.
Try to approach new cards this way, as well as when you are going through old cards looking for new tech. Anyone can put together three-card combos that deal a million damage or gain a million life. It takes a real gift to discover role players that increase a stock deck’s win percentage against the field, or a new strategy that can actually stand up to an extreme metagame with Magus-Red, Faeries, and Five-Color Control.
See you guys next week.
Patrick Chapin
“The Innovator”