Bidding problem From Peter Winkler's 'Mathematical puzzles' > You can make a bid on a widget whose value to the owner, as far as you know, is uniformly randomly distributed between 0 and 100 dollars. > > However its value to you is 80% greater than its value to the owner. > > If you offer the same or more than the widget is worth to the owner S/he will sell. > > How much should you bid? I do not understand the answer: > Do not bid at all. > > If you bid $x$ than the value to the owner given that S/he sells will be $ x/2$ and therefore the value to you $1.8 * x/2 = 0.9x < x$ Why the $x/2$ ?

If a random variable $Y$ is a priori uniformly distributed in $[0,100]$, and you learn later that its value is in fact $\leq x$, then conditioned on this additional information it is uniformly distributed in $[0,x]$. It follows that the conditional expectation of $Y$ then is ${x\over2}$.