TITLE: Improving Von Neumann's Unfair Coin Solution
QUESTION [12 upvotes]: If a cheat has altered a coin to prefer one side over another (a
  biased coin), the coin can still be used for fair results by changing
  the game slightly. John von Neumann gave the following
  procedure: 
Toss the coin twice. If the results match, start over, forgetting both
  results. If the results differ, use the first result, forgetting the
  second.

Are there ways of doing this (can we tweak this procedure) to reduce the number of expected flips needed (for a realization of heads or tails)?

REPLY [2 votes]: Yes.
If the coin has a bias $0<p<1$ then the information you get from tossing the coin is
$$
\operatorname{E}(p):=-p\log_2(p) - (1-p)\log_2(1-p).
$$
For each coin toss as input you receive $\operatorname{E}(p)$ entropy but only return on average $p(1-p)$ entropy in the output so you must be losing something.