Dynasty, in Theory: Hyperbolic Discounting

There's a lot of really strong dynasty analysis out there, especially when compared to five or ten years ago. But most of it is so dang practical-- Player X is undervalued, Player Y's workload is troubling, the market at this position is irrational, take this specific action to win your league. Dynasty, in Theory is meant as a corrective, offering insights and takeaways into the strategic and structural nature of the game that might not lead to an immediate benefit but which should help us become better players over time. (Additionally, it serves as a vehicle for me to make jokes like "theoretically, this column will help you out".)

The Rise and Fall of Draft Picks

Anyone who has been in dynasty leagues for a while has probably observed certain fluctuations in the price of rookie draft picks throughout the season. These fluctuations are sinusoidal. They are also predictable and reliable. We are now reaching the point of the year where future pick values are hitting their nadir (at least as far as late firsts are concerned), and if this column was devoted to giving you actionable advice, I would probably be suggesting you run to buy some right about now.

Instead, I want to look at some theories about why these fluctuations happen in the first place.

From an individual standpoint, it makes perfect sense that the amount of value I place on individual draft picks will change over time. If I open the season 4-0, I will naturally place much less value on my own 2023 picks than I would if I opened 0-4 because the picks I am in line for will be intrinsically less valuable.

From a league-wide standpoint, these individual changes shouldn’t impact the total market. Wins and losses are a zero-sum game, so if I get 4 wins, that means others in my league have accrued 4 matching losses. As my expectations for my draft picks fall, their expectations for their picks should rise commensurately.

And yet we know that the market for picks rises and falls all the same. Why is this the case?

The Unexaggerated Truth about Hyperbolic Discounting

The “hyperbolic” in “hyperbolic discounting” refers, not to hyperbole or other gross exaggeration, but to the humble hyperbola. So what is hyperbolic discounting?

Imagine I was offered a choice between receiving $100 today or $250 ten days from now. Imagine I also had access to an amazing investment opportunity that guaranteed me a 10% return compounded daily. If I opt for the $100 today, I could invest it, and after ten days it would be worth $259. $259 is greater than $250, so my strictly rational self should prefer the immediate payoff.

Let’s say my investment opportunity wasn’t expected to pay off quite as well, and I was only anticipating an 8% return each day, (still one heck of an investment opportunity!) In that case, if I invested the $100 today, I would only expect to have $216 at the end of the ten days. This is less than the promised $250, so I would prefer the future payoff.

So far, so good. These are perfect examples of “exponential discounting”. The problem is that, in real-world situations, we often find that our discounting models do not behave exponentially. Instead, we find that the initial discount is too steep- or, rather, that it is “hyperbolic” instead of “exponential”.

Consider the following: a man comes up to you and tells you he will either give you $50 today or he will give you $100 one year from now. In this situation, most people opt for the immediate $50 reward. Now imagine the same man tells you he will either give you $50 in five years or $100 in six years. This is the exact same scenario (wait a year to receive double the reward), but in this case, most people opt for the $100 payoff.

Exponential discounting cannot explain this discrepancy, but hyperbolic discounting can. The initial discount from right now to a point a short time in the future is too steep. Later discounts from a point sometime in the future to another point even further in the future become even shallower in response.

In fact, this is the key characteristic distinguishing (rational) exponential discounting from (irrational) hyperbolic discounting. Exponential discounting models produce consistent preferences across any range of time spans. Hyperbolic discounting models, on the other hand, are characterized by preference reversals. If presented with a choice, the hyperbolic discounter would prefer one side. If presented with an identical choice over a different time scale, the hyperbolic discounter would prefer the other side.

These preference reversals provide plenty of opportunities for profit. A trade that would be laughed at in May becomes a trade that is eagerly accepted in October. In the most extreme examples, these preference reversals can be exploited on both sides for even greater returns. One could buy first-round picks in October when their value is lowest, then sell them right back again in May when their value is highest— often to the same owner who sold them in the first place!

Explaining the Ebb and Flow

“But Adam!”, some among you might interject, “hyperbolic discounting might explain why future picks are undervalued, but it still doesn't explain why the amount they are undervalued by fluctuates throughout the year!”

I’m thankful that I have such astute readers (who are totally not just figments of my imagination I invented to serve as rhetorical tools designed to provide transitions and advance my thesis). You see, the value of future picks changes throughout the year because how various assets are perceived changes throughout the year.

In May of 2022, it would have been hard to convince a manager to trade a 2023 first-rounder for DeAndre Hopkins. There are a number of reasons for this (we weren't sure how good Hopkins would be, we can't be confident how late our 2023 first will be, etc), but a big part of it is that DeAndre Hopkins' looming suspension meant he was a "future asset". He wasn't someone you could plug straight into your lineup, he's someone you'd have to wait for. And the future first was also a "future asset", something you'd have to wait for. Such a proposed trade wouldn't trigger that sweet hyperbolic "distant cost for immediate benefit" brain chemistry.

Today that's not the case. If you trade for DeAndre Hopkins today, you can start him immediately. (Or at least once Arizona is past its bye). And all of a sudden, teams are much more willing to start discounting their future assets if necessary to acquire him.

Given the specifics around Hopkins, maybe he's not the clearest example. Imagine instead approaching one of your leaguemates and trying to trade a 2033 first-rounder for a 2034 first-rounder. How much extra do you think you would be able to get in such a trade? If your leaguemate is like most, he or she would be unwilling to give much at all to move his gains from eleven years in the future to ten years in the future.

Contrast this with the regular trades involving a team trading a current pick for a pick one year in the future. Typically, these trades command quite the premium, indeed.

Indeed, to take advantage of hyperbolic discounting, one needs to be able to trade assets that are immediately valuable-- the more immediate, the better-- for assets that are immediately worthless, but which hold future value. And that particular mix occurs most prominently during the season itself (or, for picks, during the rookie draft when specific players are on the clock).

Key Points to Remember

Valuing current assets more than future assets is not, in itself, irrational. Current assets truly are more valuable. A 2023 first-rounder is absolutely worth more than a 2033 first-rounder.

The irrational part is when those preferences become “time inconsistent”. Time inconsistency means that a decision made today might be different from one that would have been made a month ago. In other words, the decisions are subject to “preference reversal”.

Few owners would trade a 2033 first and a 2032 second for a 2032 first. At the same time, plenty of owners will be happy to make that trade… in 2032. That is an example of a preference reversal. Their preference is not consistent over time.

Savvy owners are able to identify when those preference reversals occur and exploit them for a profit. Much ink is devoted in fantasy football to the concept of “buying low” and “selling high”, but in practice, it’s very difficult to determine when the low point and high point have been reached.

Hyperbolic discounting and other related time-inconsistent biases solve that problem for us, offering a clear and predictable low and high point. The only question left is what we choose to do with that knowledge.

On the other end, if you're worried that you are falling prey to hyperbolic discounting yourself, consider whether you would have made the same trade six months ago if you knew then what you know now. Consider whether six months from now you'll still believe this was a reasonable and defensible trade.

One Final Thought That Likely Offers Little Comfort

Hyperbolic discounting is not strictly a human problem, either. These same results have been replicated in studies of monkeys, rats, and pigeons, too. So when you get right down to it, we’re really not any more illogical than pigeons.

So that’s something to hang our hats on.