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It seems that it doesn't matter, whether you switch or not - you get the same prize on the average. An excel calculation using random numbers in the range [0,1) with 2 columns and 20 rows, where the smaller number in row was considered to be of value 1 and the bigger number was considered to be of value 2 showed it explicitly that you get the same amount whatever is your strategy (it was excatly 1,5 average for 20 pairs of numbers). The problem is thus equivalent to coin tossing.
Gleb Shiroki