The GalCom hy­po­thet­i­cal is a thought ex­per­i­ment where send­ing a bit of in­for­ma­tion is a clearly defined and very ex­pen­sive ac­tion, which makes it use­ful for un­der­stand­ing var­i­ous con­cepts in in­for­ma­tion the­ory and prob­a­bil­ity the­ory.

The year is 21026. Hu­man­ity has be­come a flour­ish­ing in­ter-stel­lar civ­i­liza­tion. Cap­i­tal­ism never died, and stock mar­kets are thriv­ing in var­i­ous star sys­tems. Due to the light-speed limi­ta­tions, the mar­kets are out of sync, and there’s a lot of money to be made by set­ting up an au­to­mated trader in one star sys­tem, mov­ing to an­other star sys­tem, and send­ing in­for­ma­tion about the stock mar­ket to your au­to­mated trader the in­stant that that in­for­ma­tion be­comes available.

You set up your own au­to­mated trader in the Vega star sys­tem, and you cur­rently make a liv­ing by trans­mit­ting mar­ket data from the Deneb star sys­tem. To trans­mit that data, you use the deep-space Galac­tic Com­mu­ni­ca­tions net­work known as “GalCom.”

You have a lot of data to trans­mit, and send­ing in­for­ma­tion on GalCom isn’t cheap. For­tu­nately, it is effi­cient: GalCom is a highly op­ti­mized net­work used by trillions of cit­i­zens to send and re­ceive pulses of light across the cos­mos. To send in­for­ma­tion via GalCom, you pur­chase bits. For each bit you pur­chase, you are al­lowed to con­trol a sin­gle pulse in the GalCom sig­nal, mak­ing it ei­ther be pre­sent (rep­re­sent­ing a 1) or ab­sent (rep­re­sent­ing a 0).

It’s rel­a­tively cheap to buy bits on non-peak hours (to, e.g., re­pro­gram your ma­chine on Vega), but it’s very ex­pen­sive to re­serve bits dur­ing peak hours (i.e., just af­ter a juicy earn­ings re­port is pub­lished). You make your money by be­ing among the very first peo­ple to send in­for­ma­tion about the stock mar­ket, so you have to re­serve those very ex­pen­sive pre­cisely-timed bits in ad­vance. This means that you have to know what mes­sages you might want to send in ad­vance, and re­serve enough bits for the longest pos­si­ble mes­sage that you might send. For­tu­nately, there’s a sec­ondary mar­ket on peak-hour bits, so if you end up re­serv­ing 5 bits and then only us­ing 3 of them, you can re­sell the last two bits.

For ex­am­ple, say you know that you’re go­ing to send one of the fol­low­ing mes­sages: “buy”, “sell”, or “hold”. GalCom would re­quire you to pur­chase 4 bytes of in­for­ma­tion (enough to trans­mit four let­ters), and then if you ac­tu­ally end up trans­mit­ting “buy” you can sell the last byte back (be­cause you only used 3 of the 4).

Of course, you can send that mes­sage us­ing much less than 4 bytes, if you’re clever, both by de­vel­op­ing effi­cient en­cod­ing schemes and by ac­count­ing for the fact that some out­comes are more likely than oth­ers. Your profit mar­gins de­pend on you find­ing ways to trans­mit in­for­ma­tion as effi­ciently as pos­si­ble. To do that, you can use in­for­ma­tion the­ory.