Wobbleknock is the latest fad in DS City. It is played with two teams of four players each who aim to score points against each other. The City Shakes have solicited your help in determining their lineup for the upcoming game against their main rival. Can you help?
The Shakes have two players for each position. Using DS, each player was rated for offense and defense strength in their respective position.
From our lineup, we can compute the offense and defense scores:
OS = 3 * P4 * (P2 + P3)
DS = P1 * (Min(P2, P3) + P4 / 2)
whereby Px is the offense or defense strength we assign to position x, respectively.
From these scores we can then compute our team's strength as
TS = OS / 100 - 100 / DS
and the respective chances of winning
Odds = 1 / (1 + EXP(1.5 - TS))
These formulas were found using extensive data analysis. Your task is to find a lineup that maximizes the Shakes' team strength.
Extra Credit: Consider the situation where Stella and Esme are a bit inconsistent in their performance. Assume their respective strengths vary normally around the listed values with a standard deviation of 20%, while all other players have no variation in performance. Should this affect your lineup at all when our objective is to maximize the expected odds of winning?
Carefully consider the 'min' operator used for calculating the defensive score. Is it enough to compute with expected strength values?
Since the player in position 1 has no bearing on the team's offensive strength, the best option here is always Brian. When Stella and Esme play with consistent defensive strengths, together they make the best midfield. Their defensive performance is then so strong that they are best complemented with Dash in position 4.
The situation changes fundamentally when Stella and Esme vary in their defensive performance. As Scott Nestler pointed out correctly, the 'min' operator in the calculation of the defensive score models a 'weak link' system where the weakest-performing player limits the strength of the system. Within such systems, consistency matters a great deal. If Stella and Esme, who both have a defense score of 7, have good and bad days independently from one another, we would expect the strength of the system to perform under 7 75% of the time. Therefore, the expected performance of the weak-link system is less than the minimum of the expected strengths of each player.
In this case, the best lineup is Brian, Kai, Esme, and Zoey. Jax' extraordinary defensive strength makes no difference because he will have to play alongside Stella or Kai anyway. Therefore, we still land on Esme, who has a better offense score than Jax. We trade Kai for Stella because Stella's higher average defense score matters less when Esme's performance is inconsistent, but Kai gives us an extra point in offensive strength. Finally, we put Zoeye into the team, as her defensive strength can help make up the loss in defensive performance.
What do we learn?
- We cannot aggregate player performance into a single score. Each player's strength consists of different dimensions, a performance profile, and the composition as a team matters more than aggregate values.
- If you look at Dash and Zoey, you see that whether or not you make it on the team may not depend on your aggregate score but on how well your performance profile fits with the rest of the team.
Finally, for the optimization enthusiasts: This model shows how we can use Seeker to find the optimal team composition. Set variable_esme_and_stella to True to solve the extra credit question.