Toy Games

2024-09-22 See all posts

I first learned what a toy game is by watching a poker strategy video from Ben Sulsky aka Sauce123 a decade or more ago. When there is a decision point in a game that you have not fully solved, you design a simplified version of it and solve that instead. Typically you cannot simply copy/paste the solution into the more complex game. But by revealing which factors are key and the magnitude of their importance in the toy game, you can better develop and execute on heuristics for the original game.

Today I was listening to my favorite fantasy football podcast host when he said something very wrong regarding best ball (Underdog Fantasy: Rules) strategy and did not correct himself. He said something like, “Diontae Johnson is the type of player I like in best ball much more than in re-draft because in best ball you want consistent production every week.”

Diontae Johnson was traded this offseason from the Pittsburgh Steelers to the Carolina Panthers. The Panthers have an unproven 2nd-year quarterback, run the ball more than they should, and are projected to score among the fewest touchdowns in the league this season. Diontae Johnson is known for getting open easily and catching a lot of passes. Because of his team and individual profiles we can describe him as a low variance fantasy scorer.

In best ball only your top scores at each position count towards your total for the week. For example, you might have 7 wide receivers on your team and throw away the 4 weakest scores each week.

For our toy game, consider two types of players: - Type 1 players score 10 points every single week. - Type 2 players score 11 points half the time and 9 points half the time.

To keep things simple, we will eliminate positions (QB, WR, etc.). We will have two teams go head to head each with two players. Team A has two Type 1 players and Team B has two Type 2 players. Let’s look at their outcome distributions:

Team A: Both players score 10 points. The highest score is 10 every week. Expected points = 10 * 1 = 10.

Team B: Four possible outcomes, each with the same probability: (9,9), (9,11), (11,9), (11,11). When we pick the highest score, three in four trials we get 11. One in four trials we get 9. Expected points = (0.75 * 11) + (0.25 * 9) = 10.5.

Because we get to throw out bad scores we gain expected points by pumping up the variance even though the player’s mean outcome is the same. We can also see that the effect gets larger the more variance is introduced; what if Type 2 players score 20 points half the time and 0 half the time?

Outcomes: (0,0), (0,20), (20,0), (20,20) Expected points = (0.75 * 20) + (0.25 * 0) = 15.0.

The closing lesson: players with higher variance in their weekly scores are more valuable in best ball.