Having a restricted 4-way or 8-way digital joystick for an arcade game is fine if the joystick is built into a game cabinet that plays only one game — 4-way for Pacman and 8-way for Super Cobra. But [Tinker_On_Steroids] wanted a joystick that could be restricted as either 4-way or 8-way for a cabinet that could play a multitude of games, and it had to switch from one type of restriction to the other automatically based on the selected game.
His digital joystick already came with a plate that can be mounted for either 4-way or 8-way restriction, but it has to be manually screwed in place for one or the other. He removed it and designed two 3D-printable parts, one that is to be mounted firmly to the bottom of the joystick and the other that rotates within the first one. Rotated in one orientation gives 4-way restriction and in the other orientation gives 8-way. That left only attaching a servo to do the rotation. The first video below shows mounting this all to the joystick and demoing the servo using a Teensy. The STL files for the parts are on his Thingiverse project page.
He also shows a simple circuit board he made that has two buttons and two LEDs on it for connecting to the Teensy and controlling the servo. And as an added option he shows how to talk to the Teensy from his desktop computer through USB and control the servo that way. In the second video below he details all that and also does a walk-through of the code he wrote for the Teensy. On the Thingiverse page he provides only the hex file but it’s likely you’d write your own software for interfacing with a game anyway.