How about a full auction in which the teams alternate bids, each one (a positive integer) less than the previous bid, until one team concedes, and the winning bid gets the ball on their own N yard line where N is the winning bid. Coin flip for who gets first bid. Less complex strategy involved this way which is good, but the averaging approach takes into account the different valuations that different teams would place on being on offense or defense.
The practical reality is that no bidding system would ever be used-- too complicated. What might work is an agreement that starting at, say, the 12 yard line, or whatever yard line is neutral with respect to who will (on average over all games and teams) score first -- to be computed by looking at past data and fixed once and for all for every game. Then flip a coin to in which the winner decides whether to start on offense or defense. Quick and easy, if not as interesting as the bidding methods.
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