If many parties are mining to try to establish a block, and one succeeds, does that mean the efforts of the other miners (whose blocks have at least some common transactions) are lost; the time and energy spent to create their blocks must be discarded?
[I have read some of the links, and some of the links of the links, and am still a bit confused.]
Here is an example using a simplified coin. Imagine there are only 1000 possible choices to complete the block. And only one miner, Alice. Since she has to guess which one of the 1000 choices will complete the block, she will make, on average, 500 tries to complete the block. It could be more or less, but 500 would be the average.
Now add another miner, Bill. Alice and Bill start mining at the same time and at the same rate. Since there are now two miners, and only one needs to succeed to complete the block, together, they need to each try only about 293 times before, on average, the block will be complete. That is quite a bit faster, because only 293 tries-times have elapsed, but the total number of tries to complete the block is larger, 586 instead of 500. The 86 tries seem wasted because a single miner would only need, on average, 500 tries.
I now realize that each miner (or at least each mining group) is working on their own, unique block. The block is unique if only because of the miner designation (who gets the bitcoin reward) that is part of the block. So, in general, no two miners, outside of a mining group, are ever working on the same block.