fixed income - Pricing Treasury futures

I've recently learned that at the delivery of Treasury futures the short side can decide which of the $n$ Treasury bonds (with relevant maturities) to deliver. If the short side chooses to deliver the $i$-th bond, the the long side has to pay $c_i\cdot F$ for delivery, where $c_i$ is the conversion factor for the $i$-th bond, which is the value everybody knows. If the bond price at delivery is $B_i$ then the short side delivers the cheapest-to-deliver (CDT) $\hat i$-th bond where

$$\tag{1} \hat i = \operatorname{argmin}\limits_{1\leq i\leq n}\left(B_i - c_i\cdot F\right).$$ As a result, CDT bonds depend on the $F$ whereas $F$ itself shall be some kind of expected value of the delivered bond, that is $$\tag{2} F = \frac{\Bbb E B_{\hat i}}{\Bbb E c_\hat i}.$$ From these arguments it seems that to find $F$ one has to solve a fixpoint problem $(1)-(2)$, and it does not seem to be trivial even to prove existance there. Am I missing something?

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In case I don't miss anything, I think the problem can be simplified as follows. To avoid all the discounting/margin account issues, let's talk about the forward contract the expires on one of the stocks $S_i$ where $1\leq i\leq n$. To each of these stocks we assign some constant adjustment weighting $c_i$, and at the maturity of the contract the short side has to choose $i$ and deliver the cash difference $S_i - c_i\cdot F$, where $F$ is the forward price we agree upon now, and $S_i$ is the stock price at the delivery. I assume that neither of stocks pays dividends, and interest rates are $0$. Of course, since it's a cash delivery, the short side will deliver $S_{\hat i}$

$$\hat i = \operatorname{argmin}\limits_{1\leq i\leq n}\left(S_i - c_i\cdot F\right).$$ Under these conditions it may be ok to assume that the forward price satisfies $$\Bbb E[S_\hat i - c_\hat i\cdot F] = 0$$ which of course translates into $(2)$.