If Arsenal keep Sanchez and this results in their reaching 4th in the EPL, Arsenal gets:
£32.3m for 4th in the EPL and
€12.7m for reaching C/L qualification (2017/18 figure)
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A Euro is now worth slightly more than a pound but say they're equal.
So 4th in the EPL is worth about 32.3+12.7 = £45.0m
Now suppose that Arsenal sell Sanchez this window. And let's say that drops them from 4th to 5th in the EPL. Then, Arsenal get:
£30.4m for 5th in the EPL and
€2.6m for reaching the Europa League (2017/18 figure)
So 5th in the EPL is approximately worth 30.4+2.6 = £33.0
The difference between 4th and 5th in the EPL is approximately 45-33 = £12 (approximate due to £ to € conversion and 2017/18 C/L, E/L qualification figures used).
We've offered £20m for Sanchez - so, prima facie, Arsenal would be better off accepting our offer than keeping Sanchez.
This rough analysis is obviously incomplete in that it doesn't consider the possibilities of Arsenal finishing above 4th or below 5th following the sale/retention of Sanchez and b/c 2017/18 figures were used for C/L and E/L qualification awards.
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TL/DR - It looks like Arsenal gains somewhat by selling Sanchez to us and take the £20m we have on offer.
As-is, it's unlikely that Arsenal will finish 4th or higher no matter what they do in the Winter window and it's virtually certain that they'll lose Sanchez on a free if they keep him until next season.
So... If Arsenal's board/owners are in it for the money (I suspect that they are) I expect that Arsenal will take our offer.
Arsenal have sold to us in the past under - I think? - the same board, so this deal won't probably be scuppered for some sort of hate versus Manchester City or policy of never selling to rivals.
On the other hand, £12 isn't a huge sum. So maybe Arsenal will hold out - hoping that the short term expected loss is offset by gains in future deals where clubs intentionally way-overbid for desired players.