# Variations on the Sleeping Beauty

This post won’t di­rectly ad­dress the Sleep­ing Beauty prob­lem so you may want to read the above link to un­der­stand what the sleep­ing beauty prob­lem is first.

Half*-Sleep­ing Beauty Problem

The as­ter­isk is be­cause it is only very similar to half of the sleep­ing beauty prob­lem, not ex­actly half.

A coin is flipped. If it is heads, you are wo­ken up with 50% chance and in­ter­ro­gated about the prob­a­bil­ity of the coin hav­ing come up heads. The other 50% of the time you are kil­led. If it is tails you are wo­ken up and similarly in­ter­ro­gated. Given that you are be­ing in­ter­ro­gated, what is the prob­a­bil­ity that the coin came up heads? And have you re­ceived any new in­for­ma­tion?

Dou­ble-Half*-Sleep­ing Beauty problem

A coin is flipped. If it is heads, a coin is flipped again. If this sec­ond coin is heads you are wo­ken up and in­ter­ro­gated on Mon­day, if it is tails you are wo­ken up and in­ter­ro­gated on Tues­day. If it is tails, then you are wo­ken up on Mon­day and Tues­day and in­ter­ro­gated both days (hav­ing no mem­ory of your pre­vi­ous in­ter­ro­ga­tion). If you are be­ing in­ter­ro­gated, what is the chance the coin came up heads? And have you re­ceived any new in­for­ma­tion?

Dou­ble-Half*-Sleep­ing Beauty prob­lem with Known Day Variation

EDIT: This prob­lem should have said: As above, but when­ever you are be­ing in­ter­ro­gated you are told the day. You may wish to con­sider this prob­lem be­fore the above one.

Sleep­ing Cou­ples Problem

A man and his iden­ti­cal-val­ued wife have lived to­gether for so many years that they have reached Au­mann agree­ment on all of their be­liefs, in­clud­ing core premises, so that they always make the same de­ci­sion in ev­ery situ­a­tion.

A coin is flipped. If it is heads, one of the cou­ple is ran­domly wo­ken up and in­ter­ro­gated about the prob­a­bil­ity of the coin hav­ing come up heads. The other is kil­led. If it is tales, both are wo­ken up sep­a­rately and similarly in­ter­ro­gated. If you are be­ing in­ter­ro­gated, what is the prob­a­bil­ity that the coin came up heads? And have you re­ceived any new in­for­ma­tion?

Sleep­ing Clones Problem

A coin is flipped. If it is heads, you are wo­ken up and in­ter­ro­gated about the prob­a­bil­ity of the coin hav­ing come up heads. If it is tails, then you are cloned and both copies are in­ter­ro­gated sep­a­rately with­out know­ing whether they are the clone or not. If you are be­ing in­ter­ro­gated, what is the prob­a­bil­ity that the coin came up heads? And have you re­ceived any new in­for­ma­tion?

My ex­pec­ta­tion is that the Dou­ble-Half Sleep­ing Beauty and Sleep­ing Clones will be con­tro­ver­sial, but I am op­ti­mistic that there will be a con­sen­sus on the other three.

Solu­tions (or at least what I be­lieve to be the solu­tions) will be forth­com­ing soon.

• This post caused me to type up some old, un­re­lated thoughts about Sleep­ing Beauty. I posted it as a com­ment to the stupid ques­tions thread at http://​​less­wrong.com/​​lw/​​n3v/​​stupid_ques­tions_2nd_half_of_de­cem­ber/​​d14z . I’d very much ap­pre­ci­ate any feed­back on this idea. This com­ment is just to catch the at­ten­tion of read­ers in­ter­ested in Sleep­ing Beauty who may not see the com­ment in the stupid ques­tions thread.

• The first one is straight­for­ward. Yes, you get ev­i­dence that not (coin heads && un­lucky 50%), so it’s definitely 33% heads. Would be the same to an out­side ob­server (re­place “you” with “the vic­tim” and sim­ply ob­serve whether the vic­tim is kil­led or not. The prob­lem with SB isn’t the prob­a­bil­ity, but the mul­ti­ple in­dis­t­in­guish­able ob­ser­va­tions.

The sec­ond one re-adds the mem­ory wipe, mak­ing it pretty much sleep­ing beauty. You have re­ceived no in­for­ma­tion—all paths lead to the same ob­ser­va­tion (you wake up). So the prior holds: prob­a­bil­ity is 5050, but in­ter­ro­ga­tion-in­stance-weighed op­por­tu­ni­ties for pre­dic­tion is 3366 in fa­vor of tails.

With known days, you do have in­for­ma­tion if wo­ken on Mon­day. You know it wasn’t heads-tails. On Tues­day, if you re­mem­ber Mon­day’s wake-up, you know it was heads-heads, and if you don’t re­mem­ber mon­day, you know it wasn’t.

Cou­ples: if part­ner1 is be­ing in­ter­ro­gated, they know it wasn’t heads-part­ner1kill. This is pretty much the same as prob­lem 1 for each of them. Without com­mu­ni­ca­tion, their ex­is­tence is ir­rele­vant.

Clones: no new in­for­ma­tion—out­side prior holds. Same as sleep­ing beauty.

• “Tues­day, if you re­mem­ber Mon­day’s wake-up”—You have no mem­ory of Mon­day’s in­ter­ro­ga­tion be­cause you are mem­ory wiped.

• Me­mory-wiped in the tails case, but not in the heads-heads case, right? If it’s heads-heads, you’re wo­ken on Mon­day and noth­ing hap­pens on Tues­day, so you pre­sum­ably re­mem­ber that you were wo­ken and in­ter­ro­gated yes­ter­day.

• You are in­ter­ro­gated on Mon­day, so what hap­pens on Tues­day is ir­rele­vant to the prob­lem for heads-heads. All that mat­ters is what you say when you are be­ing interogated

• Prob­a­bly doesn’t mat­ter how you phrase the han­dling of Mon­day night if there’s no Tues­day in­ter­ro­ga­tion. If you’re in­ter­ro­gated on Tues­day, you know it wasn’t heads-heads.