I'm reading this paper about building a combat simulator for 8 unit vs 8 unit mini combats in StarCraft: Brood War. The basic idea is to build a search tree simulating these small combats in order to determine next best move in combat scenarios in StarCraft game play. Section 3.2 (which probably is necessary to understand my question) is the part I am having trouble with, where he talks about approximating a combat (where simultaneous moves occur) with a version where alternating move occur. This allows him to use minmax trees and alpha-beta search.
The part I don't understand is when he describes how building the minmax tree gives an advantage to one player, while a maxmin tree gives an advantage to the other player. For one, his diagram labels the tree where "max" goes first as "maxmin", but his text (and wikipedia :P) describe the tree where max goes first as "minmax". Perhaps that is just a mislabelling.
The main part I am confused about is when he goes on to say:
Proposition 1: For stacked matrix games G, we have $$mini(G) \leq Nash(G) \leq maxi(G)$$
My understanding of this is that $Nash(G)$ is the real final game state taking into account simultaneous moves. Then $mini(G)$ is the final game state if we approximate with MAX moving first, and $maxi(G)$ is the final game state if we approximate with MIN moving first. Besides the Nash equilibrium stuff, which is a bit beyond my education so far, I don't understand how the inequality $$mini(G) \leq maxi(G)$$ can be true. Below are 4 examples. The leaf node values come, I believe, from an evaluation function of that game state's value to MAX:
The two on the left are maxmin, meaning MIN goes first. The two on the right are minmax, with MAX going first. The top pair contains one set of leaf node values that lead to $$5=maxi(G) \lt mini(G)=8$$ This goes against the proposition above. However, the bottom pair contain a different set of leaf node values that lead to $$4=maxi(G) \gt mini(G)=3$$ Here the proposition seems to hold. I believe I could also construct a scenario where $mini(G) = maxi(G)$.
What am I missing here? Is there really a hard relation between $mini(G)$ and $maxi(G)$ or doesn't it just depend on the leaf node values?