Sup­pose no one, given steps of com­pu­ta­tion, is able to com­pute any in­for­ma­tion about the par­ity of the th digit of , and ev­ery­one, given steps of com­pu­ta­tion, is able to com­pute the th digit of . Sup­pose that at time , ev­ery­one has steps of com­pu­ta­tion, and at a later time , ev­ery­one has steps of com­pu­ta­tion. At the ini­tial time , Omega se­lects a prob­a­bil­ity equal to the con­di­tional prob­a­bil­ity Omega as­signs to the agent pay­ing $1 at time con­di­tional on the digit be­ing odd. (This could be be­cause Omega is a log­i­cal in­duc­tor, or be­cause Omega is a CDT agent whose util­ity func­tion is such that se­lect­ing this value of is op­ti­mal). At time , if the digit is even, a fair coin with prob­a­bil­ity of com­ing up heads is flipped, and if it comes up heads, Omega pays the agent$10. If in­stead the digit is odd, then the agent has the op­tion of pay­ing Omega \$1.