Massively Multiplayer Online Games (MMOGs) recently emerged as a popular class of applications with millions of users. To offer acceptable gaming experience, such applications need to render the virtual world surrounding the player with very low latency. However, current state-of-the-art MMOGs based on peer-to-peer overlays fail to satisfy these requirements. This happens because avatar mobility implies many data exchanges through the overlay. As state-of-the-art overlays do not anticipate this mobility, the needed data is not delivered on time, which leads to transient failures at the application level. To solve this problem, we propose Blue Banana, a mechanism that models and predicts avatar movement, allowing the overlay to adapt itself by anticipation to the MMOG needs. Our evaluation is based on large-scale traces derived from Second life. It shows that our anticipation mechanism decreases by 20% the number of transient failures with only a network overhead of 2%.