michmarc
Well-Known Member
I was tweaking my spire solver to fix a minor bug and came across an unexpected result. It turns out that the optimal way to play the spire depends on exactly how much you value the goods being used (relative to the other goods asked for).
As a simple example, you're playing a round with 4 different goods to choose from. You guess AABCD as your original guess and get back the answer CCSSS (first two correct and the last three 'someone else'). Great -- you are guaranteed success as the only two possibilities are AACDB and AADBC. Now what?
We assume that the value of the goods to you is A<=B<=C<=D. And let's put a comparative value on the goods.
Let me describe three strategies:
1) Guess AACDB. If that's wrong, then guess AADBC.
2) Guess AAAAB. If B was right, guess AACDB; otherwise guess AADBC.
3) Guess AAABB. If the first B was right, guess AADBC; otherwise guess AACDB.
If all goods cost the same (10, 10, 10, 10): Strategy 1 costs either 30 or 60 for an average cost of 45. Strategy 2 costs 50 or 60 for an average cost of 55. Strategy 3 always costs 50. #1 is best.
If you think A is much cheaper than B, (10, 30, 30, 30): Strategy 1 costs 90 or 180 for an average cost of 135. Strategy 2 costs 50 for the first guess and either 60 or 90 for the second guess for a total average of 125. Strategy 3 costs 70 for the first guess and 60 for the second for a total cost of 130. #2 is best.
If you think B is cheap enough relative to A but expensive enough relative to C and D (10, 15, 30, 30): Strategy 1 costs 75 or 150 for an average cost of 112.5. Strategy 2 costs 35 for the first guess and either 60 or 75 for the second for an average cost of 102.5. Strategy 3 costs 40 for the first guess and 60 for the second for a total cost of 100. #3 is best.
Deep.
This also directly affects whether or not you should give up halfway through and start over or press on hoping to get lucky.
As a simple example, you're playing a round with 4 different goods to choose from. You guess AABCD as your original guess and get back the answer CCSSS (first two correct and the last three 'someone else'). Great -- you are guaranteed success as the only two possibilities are AACDB and AADBC. Now what?
We assume that the value of the goods to you is A<=B<=C<=D. And let's put a comparative value on the goods.
Let me describe three strategies:
1) Guess AACDB. If that's wrong, then guess AADBC.
2) Guess AAAAB. If B was right, guess AACDB; otherwise guess AADBC.
3) Guess AAABB. If the first B was right, guess AADBC; otherwise guess AACDB.
If all goods cost the same (10, 10, 10, 10): Strategy 1 costs either 30 or 60 for an average cost of 45. Strategy 2 costs 50 or 60 for an average cost of 55. Strategy 3 always costs 50. #1 is best.
If you think A is much cheaper than B, (10, 30, 30, 30): Strategy 1 costs 90 or 180 for an average cost of 135. Strategy 2 costs 50 for the first guess and either 60 or 90 for the second guess for a total average of 125. Strategy 3 costs 70 for the first guess and 60 for the second for a total cost of 130. #2 is best.
If you think B is cheap enough relative to A but expensive enough relative to C and D (10, 15, 30, 30): Strategy 1 costs 75 or 150 for an average cost of 112.5. Strategy 2 costs 35 for the first guess and either 60 or 75 for the second for an average cost of 102.5. Strategy 3 costs 40 for the first guess and 60 for the second for a total cost of 100. #3 is best.
Deep.
This also directly affects whether or not you should give up halfway through and start over or press on hoping to get lucky.