you said that I said that you said that…

A riddle from The Riddler on limited information decision making, which I tought I failed to understand:

Two players, Martina and Olivia, are each secretly given realisations, m and u. Starting with Martina, they must state to the other player whom they think probably has the greater number until they agree. They are playing as a team, hoping to maximize the chances they correctly predict who has the greater number. For a given round, what is the probability that the person they agree on really does have the bigger number?

A logical strategy is as follows: If m>.5, P(U>m)<.5, hence Martina should state her number is probably bigger, which conveys to Olivia that M>.5. If u<.5, Olivia can agree for certain, else, if u>.75, P(M>u)<.5 and she can state a probably larger number, while if 0.5<u<.75, Olivia can state (truthfully) that her number us probably smaller, although there is a ½ probability she is wrong. As detailed in the solution, the probability of finishing on a false statement is ¼²+¼³+…, equal to 1/12.

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