Counterfactual Mugging Poker Game

Con­sider the fol­low­ing game:

Player A re­ceives a card at ran­dom that is ei­ther High or Low. He may re­veal his card if he wishes.

Player B then chooses a prob­a­bil­ity that Player A has a high card.

Player A always loses dol­lars. Player B loses dol­lars if the card is low and dol­lars if the card is high.

Note that Player B has been given a proper scor­ing rule, and so is in­cen­tivized to give his true prob­a­bil­ity (un­less he makes some deal with player A).

You are play­ing this game as player A. You only play one time. You are look­ing at a low card. Player B is not try­ing to make a deal with you, and will re­port his true prob­a­bil­ity. Player B is very good at rea­son­ing about you, but you are in a sep­a­rate room, so Player B can­not read any tells un­less you show the card. Do you show your card?

Since your card is low, if you show it to player B, you will lose noth­ing, and get the best pos­si­ble out­put. How­ever, if player B rea­sons that if you would show your card if it was low, then in the coun­ter­fac­tual world in which you got a high card, player B would know you had a high card be­cause you re­fused to show. Thus, you would lose a full dol­lar in those coun­ter­fac­tual wor­lds.

If you choose to not re­veal your card, player B would as­sign prob­a­bil­ity 12 and you would lose a quar­ter.

I like this var­i­ant of the coun­ter­fac­tual mug­ging be­cause it takes the agency out of the pre­dic­tor. In the stan­dard coun­ter­fac­tual mug­ging, you might re­ject the hy­po­thet­i­cal and think that the pre­dic­tor is try­ing to trick you. Here, there is a sense in which you are cre­at­ing the coun­ter­fac­tual mug­ging your­self by try­ing to be able to keep se­crets.

Also, think about this ex­am­ple the next time you are tempted to say that some­one would only Glo­ma­rize if they had an im­por­tant se­cret.

No nominations.
No reviews.