Game Theory: A Long-Term Look at Selling Young Players

My last post looked at the calculations involved in whether a team battling relegation should sell strong young players, and showed that in a one-shot game it basically comes down to your perceptions of probability of remaining in the league if you sell the young star vs. keeping the young star. For those who didn’t read (and don’t want to click the link above) here’s a quick refresher on the calcuation:

EV(Keep Grealish) = Pr(Staying in the EPL)* (Total Revenue from being in the EPL)  + Pr(Relegated)* (Total Revenue from the Championship)  – (Money spent improving the squad)

EV(Sell Grealish) = Pr(Staying in the EPL) * (Total Revenue from being in the EPL) + Pr(Relegated)* (Total Revenue from the Championship) + (Money gained from Selling Grealish) – (Money spent replacing him)

If EV(Keep) > EV(Sell), then Villa should keep Grealish. If EV(Sell) > EV(Keep), then Villa should sell him.

I left the last post as/is in the interest of simplicity and parsimony. However, the full math is a little more complicated than this and I wanted to walk through it a little more fully for interested readers here.

First, there is a psychic cost/benefit to selling young players. Fans may become upset at selling a club’s young star, there may be a decrease in locker room morale, or some other effects I can’t anticipate along those lines.  So add that to the “sell” side of the equation.1

Second, this isn’t really a one shot game. Aston Villa (or any team) retaining their Premier League status is a yearly task. 2 So they’d have to factor in the probability of retaining Premier League status this year (“t”), next year (“t+1”), the year after (“t+2”), etc. Presumably, if Grealish improves like he is expected to over the next 5-10 years, he will grow into a valuable member of Aston Villa’s squad, decreasing their probability of relegation in t+1, t+2, etc.  So with this in mind, here’s the new equation (some of the names are abbreviated so it doesn’t get too unruly.

Pr(Stay)* (EPL Revenue)  + Pr(Relegated)* (Championship Revenue)  – (Net Transfer spend) 

EV(Keep) = t + Pr(Stay)(t+1))* (EPL Revenue(t+1))  + Pr(Relegated(t+1))* (Championship Revenue(t+1))  

EV(Sell) = t + Pr(Stay)(t+1))* (EPL Revenue(t+1))  + Pr(Relegated(t+1))* (Championship Revenue(t+1)) + (Money gained from Selling Grealish(t+1)) – (Money spent replacing him(t+1))

In this formula, Pr(Stay)(t+1)  for EV(keep) presumably is higher based on Grealish’s improvement over a year. He will be a better player, and will be able to make a higher contribution to the team. How much higher, and how much will that improve their likelihood of staying in the Premier League?

Similarly, his value will increase over the year, so Money gained from Selling Grealish(t+1) will be higher, increasing EV(sell). So this changes the calculation, but both sides probably increase proportionally (his affect on probability of staying in the league will go up as his value goes up).

I’m not going to go through year two, but the process is the same, just adding the EV of t and t+1 to the formula for t+2, although Grealish’s value will continue to go up.

The final wrinkle to the process is that the game presumably ends if Aston Villa gets relegated. A prospect like Jack Grealish wouldn’t want to play for a team in the Championship, and would be expected to leave in the summer instead of sticking around another year. So our new formula would be3:

EV(Keep(t+1)|EPL(t)) = t + Pr(Stay)(t+1))* (EPL Revenue(t+2))  + Pr(Relegated(t))* (Championship Revenue(t+2))  

EV(Keep(t)|Championship(t+1)) = Championship Revenue(t+1) + Sale price for Grealish(t+1 Championship) 

EV(Sell(t+1)) = t + Pr(Stay)(t+1))* (EPL Revenue(t+2))  + Pr(Relegated(t+1))* (Championship Revenue(t+2)) + (Money gained from Selling Grealish(t+1 EPL)) – (Money spent replacing him(t+1))

This version reflects Grealish’s presumably dramatically changing value based on Aston Villa’s success in year t. If Aston Villa keeps him and stays up, his value increases ever year. But if Aston Villa keeps him and goes down, his value presumably decreases because teams know he won’t want to stay at a Championship level team which changes the expected value calculations yet again. There’s a risk/reward involved in keeping young stars around, and so much uncertainty that what a team chooses depends on assigning the correct probabilities to all the events and determining your risk aversion.

Finally, we add the psychic benefit/cost in, weighting for relegation and staying in the league to get our final, overly complicated looking equation.

EV(Keep(t+1)|EPL(t)) = t + Pr(Stay)(t+1))* (EPL Revenue(t+2))  + Pr(Stay)(t+1) *(Psychic Benefit Stay (t+2))  * Pr(Relegated(t))* (Championship Revenue(t+2))  – Pr(Relegated)(t+1) *(Psychic Cost(Relegated) (t+2)) – Psychic Cost(Sell)

EV(Keep(t)|Championship(t+1)) = Championship Revenue(t+1) + Sale price for Grealish(t+1 Championship) – Pr(Relegated)(t) *(Psychic Cost(Relegated) (t+1))

EV(Sell(t+1)) = t + Pr(Stay)(t+1))* (EPL Revenue(t+2))  + Pr(Stay)(t+1) *(Psychic Benefit Stay (t+2)) + Pr(Relegated(t+1))* (Championship Revenue(t+2)) – Pr(Relegated)(t+1) *(Psychic Cost(Relegated) (t+2)) + (Money gained from Selling Grealish(t+1 EPL)) – (Money spent replacing him(t+1)) – Psychic Cost(Sell)

All of this being said, the money earned for keeping/selling a young star is small compared to the money earned from staying in the EPL vs. the Championship. A couple million pounds difference in selling/keeping a young star would have only matter if the odds of relegation changed marginally based on selling him. The driving force here is the first part of the equation:

Pr(Staying in the EPL) * (Total Revenue from being in the EPL) + Pr(Relegated)* (Total Revenue from the Championship)

How much better would Aston Villa be by selling Jack Grealish and buying more experienced players with the money they made? Would it be enough to offset the psychic cost, increased likelihood of remaining in the EPL the subsequent year, and profit they would make by keeping him an extra year? Given that Aston Villa, like all EPL teams, are single-minded forsakers of relegation, how risk-tolerant are they? If it were up to me, looking at the numbers, I’d sell now, but I likely estimate their probability of finding replacements who can improve their chances of staying up more highly than most do. Regardless of the conclusion, the goal here was to formalize the thought process of any team in the decision to sell a young player. It’s a complicated process with a lot of uncertainty in many different places, which is why you’ll see people argue both sides so passionately.

  1. In political science, William Riker asserted there was a psychic benefit to voting, e.g. wearing your “I voted today” sticker makes you feel good about yourself, and this is really the only reason why people vote.
  2. One could even apply this to the 7 or 8 teams who are perennially “safe” – change “relegated” to “Champions League” or “Winning the Title” or whatever your goal is.
  3. EV(Keep(t)|EPL(t+1)) is a statement of conditional probability that should read “The Expected Value of keeping Grealish in year t+1 given that they remained in the EPL after year t is…”

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