The argument needs to look at the wider situation. How did the burger shop decide on their restocking algorithm? By looking at demand. They will continue to look at demand and review their algorithm from time to time. Buying one burger contributes to that, so the situation is that one more customer may result in them changing the numbers from 100 and 120 to 110 and 130. Ten extra burgers a day until they review the numbers again.

Incorporating this into the toy model shows this isn't enough to guarantee proportionality either.

My hypothetical burger shop is now in Sphericalcowland, a land where every month is 30 days. It also has a new burger-buying policy. On day 1 of each month, it buys 100 burgers for that day, then uses a meta-decision rule to decide the burger-buying decision rule for the month's remaining 29 days. Let Y be the number of customers in the previous month. If Y was no more than, say, 2500, the shop uses my earlier 100/120 decision rule for the remaining 29 days. But if Y > 2500, it uses your upgraded decision rule (buy 130 burgers if there were >100 customers the previous day, otherwise buy 110). X ~ Po(80) as before, so Y ~ Po(2400). (I've deliberately held constant the burgers bought for day 1 of each month to avoid applying a previous day-based decision rule for day 1 and causing inter-month dependencies.)

With the 100/120 decision rule, the shop buys an average of 3007.639 burgers a month. So with the 110/130 decision rule, it buys an average of 3297.639 a month.

If I don't buy a burger, E(burgers bought next month) = (3007.639 × P(Y ≤ 2500)) + (3297.639 × P(Y > 2500)) = 3013.624 burgers, by my working.

If I buy a burger, E(burgers bought next month) = (3007.639 × P(Y+1 ≤ 2500)) + (3297.639 × P(Y+1 > 2500)) = 3013.920 burgers.

Hence in this example, the upstream marginal effect of my buying 1 burger is only 0.296 burgers. The presence of feedback doesn't suffice to guarantee proportionality.

A crucial detail: "most passengers did not eat the olives in their salads". At least, in that telling of the story. So there was reason to think that the olives weren't earning their place in terms of passenger satisfaction.

For all I know, neither do sesame seeds on buns! In any case, the American Airlines story might be apocryphal in itself. I just bring it up to illustrate that there's a countervailing anecdote to the parable.

You knew the timetable and caught the same train?

Exactly.