Previously, I had been using my goal predictions to determine which teams were most likely to score or have clean sheets. Then I would try to find the best way of packing the best players from those teams into a lineup.
This approach has one particular problem. When deciding whether to start Tevez or van Persie, for example, I would know only the expected number of goals for each team and the average number of points per game for each of these players. It's not obvious how you would use those two numbers, though, to estimate the expected number of points for each player because I don't know whether this week's expected number of goals is more or less than usual for that player. I also don't know, if the expected number of goals is different this week, how many more points that should translate into.
However, I now have all the information necessary to do this the right way. What I really want to know is the expected number of points for a given player. This is just 2 plus [probability of a clean sheet] times [clean sheet points for that player] plus [expected number of goals] times [points per team goal for that player]. (All other sources of points amount to less than half a point per match, on average, so it's fairly safe to ignore.) I computed that last part, which tells me how many points the player should get for each goal scored by the team, from the official statistics of each player over the last 12 weeks.
Putting that all together, I get a list of all the players and their expected number of fantasy points this week.
All that remains is to figure out the best way to put them into a lineup. This is a simple exercise in search. For those interested, I used a branch-and-bound approach where the upper-bound comes from ignoring the team constraint (only 2 players per team) and simply taking the best players at each position.
Surprisingly, the resulting algorithm takes very little time to run, just a few seconds. I computed the best lineups in each of the allowed formations.
This week, I get the following 4-3-3:
- Forwards: R. van Persie (ARS), W. Rooney (MAU), N. Zigic (BIR)
- Midfielders: F. Lampard (CHE), Nani (MAU), K. Nolan (NEW)
- Defenders: J. Terry (CHE), L. Baines (EVE), P. Bardsley (SUN), K. Richardson (SUN)
- Keeper: B. Foster (BIR)
Not too many surprises there, I suppose, except for the players from Birmingham. Still, I expect that this more analytical approach will pay off in the coming weeks. At least, it will save me some time.
Update: It turns out that Kevin Nolan is suspended, so I needed to do this again.
I added more data from some teams less likely to score goals and was surprised to see some were picked. In particular, Charlie Adam is a fantasy team gem. Even though Blackpool aren't as likely to score as some other teams, Adam's share of every goal is so high that it cancels that out.
The new team put Adam in for Nolan and DJ Campbell in for Zigic. It also picked Seamus Coleman. However, I remain worried that he will not be available for tomorrow's match. (He is currently recovering from injury.)
Once I listed him as still injured, a different formation gave the best results. In particular, I get the following 3-5-2:
- Forwards: R. van Persie (ARS), W. Rooney (MAU),
- Midfielders: F. Lampard (CHE), Nani (MAU), C. Adam (BPL), A. Young (AST), T. Cleverley (WIG)
- Defenders: J. Terry (CHE), L. Baines (EVE), K. Richardson (SUN)
- Keeper: S. Mignolet (SUN)
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