[hardy263]

As a curiosity, how many bits were gained (rather than lost) when the second and third characters obtaining a death note were entered? (female and male respectively)

Do the bits of entropy add on, or does it not matter at all?

[gwern]

Depends on the assumptions you make, I think. If we make the assumption that #2 and #3 obtained Death Notes at a known time, then any observations before then still pin down #1 - if there are kills at morning Japan-time, that serves to pin him down to Japan, etc.

But once #2 and #3 become equally active, now any evidence like that serves to narrow down the propositions 'any of #1, #2, #3 are in Japan', so if someone asked you what's the odds that #2 lives in Japan, you would only have 1/3 the evidence you did before - because any kill linked to Japan would only have 1/3 chance of having been #2. If a kill is made using information from a rural Iowan newspaper with circulation of 100, well, all you know is that any of 1/2/3 had access to it (and maybe all 3!). And so on.

(In bit terms, if there were 4 Death Note users, then any observation has 1/4 the power it did, or 1/2^2; similarly, if there were 16 users, or 32 users... Once you established someone was a Death Note user you would still need to do that many more bits of work to figure out which Death Note user you want.)