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Here are the 102 strategies, written as length 100 strings. The 1 in each string represents the index of your first guess, then there are two 2's depending on whether your next guess is higher or lower, then four 3's, eight 4's and so on. Each strategy has an equal probability of being chosen.
There are 82 strategies that are optimal in the sense that expected value (if Ballmer picks randomly) is $0.20. All of these 82 strategies will score $0 if Ballmer chooses either 1 or 100, so you need to introduce a few other strategies just for these numbers. The other 20 strategies all have expected value $0.18, and each of them nets $1 when Ballmer chooses 1 or 100.
The ratio of 10:41 was chosen so that all numbers have equal expected value, so I believe 10/51 is the best possible mixed strategy. Adding more $0.20 strategies will increase the expected value in the middle and adding more $0.18 strategies will decrease the expected value. I searched for a little while but couldn't find a mixed strategy with 51 elements that was uniform, but I was able to find this solution with 102 strategies.
Here's a further improvement to 10/51 ≃ $0.1961.
Here are the 102 strategies, written as length 100 strings. The 1 in each string represents the index of your first guess, then there are two 2's depending on whether your next guess is higher or lower, then four 3's, eight 4's and so on. Each strategy has an equal probability of being chosen.
There are 82 strategies that are optimal in the sense that expected value (if Ballmer picks randomly) is $0.20. All of these 82 strategies will score $0 if Ballmer chooses either 1 or 100, so you need to introduce a few other strategies just for these numbers. The other 20 strategies all have expected value $0.18, and each of them nets $1 when Ballmer chooses 1 or 100.
The ratio of 10:41 was chosen so that all numbers have equal expected value, so I believe 10/51 is the best possible mixed strategy. Adding more $0.20 strategies will increase the expected value in the middle and adding more $0.18 strategies will decrease the expected value. I searched for a little while but couldn't find a mixed strategy with 51 elements that was uniform, but I was able to find this solution with 102 strategies.
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