Photo: Evan Habeeb, USA Today Sports

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".)

Let's Get Blotto

Here's a game to play with a friend. Imagine that you two are generals, and you each have four units under your command. You are tasked with conquering three battlefields. If you send more units than your opponent, you win the field. If you send fewer, you lose. If you both send the same, the field remains neutral.

The catch is that it takes time for your army to travel, so you must give your marching orders in advance before you know what your opponent is going to do. You assign your armies, your friend assigns his or hers, and then you reveal your choices and figure the results. Whoever ends the game with more captured battlefields wins.

Let's think through the strategies of this game. There are only four ways to divide your units: 4/0/0, 3/1/0, 2/1/1, and 2/2/0. Obviously, if you and your opponent each select the same strategy, the net engagement does not favor one side or the other. But we can evaluate each strategy against each other one.

4/0/0 is obviously a terrible strategy, never capable of taking more than a single battlefield. There is literally no way for it to win; the best it can hope for is a draw. It can be safely discarded.

3/1/0 will lose 33% of the time against 2/2/0 (when your 3 faces off against their 0), win 33% of the time (when your 1 faces off against their 0), and draw 33% of the time (when your 0 faces off against their 0), so neither strategy dominates the other. It will win 0% of the time against 2/1/1, lose 33% of the time, and draw 66% of the time, so it can be said that 2/1/1 "dominates" 3/1/0 (meaning 2/1/1 is strictly better and cannot lose if your opponent selects a 3/1/0 strategy). Since 3/1/0 does not dominate any other strategy and is dominated by another strategy, there is no reason to ever use it. It can be safely discarded.

2/1/1, as has been mentioned, will dominate 3/1/0. Against 2/2/0, it will win 0% of the time, lose 33% of the time, and draw 66% of the time. So 2/1/1 dominates 3/1/0, but 3/1/0 is a failed strategy and that domination is pointless. Meanwhile, it will be dominated by 2/2/0. 2/1/1 never produces a winning result against a top strategy, so it can be safely discarded.

2/2/0 dominates 4/0/0 and 2/1/1, and it breaks even against 3/1/0, (and, of course, against another 2/2/0). At best, it's a win, and at worst, it's a tie. Under the terms of the game, it is the optimal strategy.

Congratulations, we've just walked through something called a "Blotto game", which is a 2-player, zero-sum "game" that game theorists use to entertain themselves at parties. (Game theorists are a wild bunch.)

If this Blotto game seems relatively simple, that's because it is. But Blotto games can get unbelievably complex. Both sides can have a different number of units at their disposal (if one side has six units and the other has four, there's a deployment strategy that guarantees victory; I'll leave it to the reader to puzzle it out). They can gain additional points for every enemy unit they defeat, or the different battlefields can be worth a different amount. If one of the battlefields was worth five points and the others were worth one, smart players would send four units to the important field every time and damn the rest.

Demonstrating how deceptive their simplicity is, after being invented in 1921, it took nearly 100 years for Blotto games to get fully "solved".

What Does This Have To Do With Fantasy Football?

They may seem pointless, but Blotto games have a shockingly large number of potential applications. Beyond the obvious troop deployment considerations in war, Blotto games have been used to estimate optimal research and development spending, allocation of marketing budgets, political campaigning tactics, salary cap spending, and more. Any time you have direct competitors operating in conditions of limited resources, Blotto games apply.

That includes fantasy football in general, and especially dynasty leagues in particular.