A Commentary on the Work of Saks and Wigderson 1986: Correlated Distributions on an Unbalanced Binary AND-OR Tree (Proof theory and related topics)

We discuss the game-theoretical equilibrium of an AND-OR tree. Here, correlated distributions on the truth assignments are taken into consideration. In the seminal paper of Saks and Wigderson (1986), they proposed a recurrence formula method to calculate the equilibrium value. They provided a proof only for the case of complete binary trees. As to unbalanced binary trees, without a proof, they wrote that the main result for the complete binary trees holds for nearly balanced trees. However, their definition of a nearly balanced tree is vague. This note is a commentary on the paper of Saks and Wigderson with a particular focus on unbalanced binary trees. We propose the concept of a weakly balanced tree as an alternative to the concept of a nearly balanced tree. We demonstrate that the recurrence formula method of Saks and Wigderson works well for weakly balanced trees