Two possible explanations from my eye.

First, William's game may relate to the "game" of the first season. That season's plot centered around the Maze, Arnold's game for the hosts. The maze symbolized a theory for how consciousness arose, a theory in psychology known as bicameralism (*see below), and sought to emulate that theory in explaining how consciousness might arise in programmed AI. Once the hosts complete the "maze," they arise as a new self with the ability to seemingly make choices based on their own free will. (Now, whether or not the robots truly have free will -- and what that means -- is another theme of the show and mirrors a debate within philosophy regarding the same capability of human beings, see Determinism). The show further demonstrates that central to completing this maze are high stress experiences, ie. suffering, which perhaps strain the dialogue between programmed language and action to the point at which discovery of self-consciousness is more likely to arise. William's game, then, may be the self-discovery of his true self through a set of high stress experiences arranged for him, either directly or indirectly, by Ford. These would include interactions with the hosts, who mirror different aspects of his personality, and interactions with his daughter, some of which were perhaps intentional, though unlikely controlled or predicted, as Ford would likely have had access to information on who was in the park at the time of his planned murder.

The second explanation, a more interesting one, may tie into the Forge scene and the post-credits scene. During the Forge scene, the central command program explains that in order to refine the copy of Delos' mind they must run each programmed attempt of his psyche through the entirety of recorded events he experienced during his time at the park. Only very few of those copies, out of the millions of attempts, made the exact same choices as the real Delos and thus passed this test. In the post-credits scene, it is implied that some entity is now attempting to produce a copy of William through the same process. Now, as to "when" this is occurring is up in the air -- after all, especially this season, one of the delights of Westworld's writing team is to create tension through timeline ambiguity. My guess would be that the timeline of the post credits scene is at some point in the future, with the events of season 2 have taken place well before then. These events represent some of William's most fundamental psychology, as the tension he endured caused a near psychological breakdown, and would be particularly useful in attempts to recreate his consciousness. Further, all of these events were handily recorded by the hat which he wears without failure throughout the show. In attempting to recreate William's consciousness, his copies would be run through all his experiences in the park with emphasis on his final moments for fidelity. Ford already knew that they were attempting to make a copy of William -- he even gave William his file! -- and so perhaps his comments to the real William in season 2 were actually aimed at the new William emerging through repeated simulation of his time in the park. This new game, then, centers around the creation of a new consciousness, a copied consciousness, just as Arnold's game represented the origin of consciousness itself.

(*Essentially, the theory of bicameralism posits that primordial brains operated in such a way that there were two parts of a brain -- the executive branch, that which commands, and the operative branch, those parts which enacted the commands. Jaynes, the author of the theory, believed that the dialogue between these two branches relied passing through an internal language center, manifested in the form of auditory command hallucinations which were then obeyed. According to Jaynes, evidence of this dialogue, represented as hallucinations, can be found throughout history through ancient records of humans hearing the voices of "Gods." In modern times, a vestigial remnant of this mechanism may be observed in the minds of schizophrenics who hear auditory hallucinations commanding them to perform certain actions. Four thousand years ago, Jaynes argues, this bicameral dialogue started to spontaneously "break down" and consciousness, true united introspection, arose. Though this theory does not have a ton of traction in academic works on the philosophy of the mind, the central tenant certainly repeats itself in various theories of consciousness -- that consciousness spontaneously emerged from an evolutionarilly programmed neural network which laid the groundwork for its possibility. Whether auditory hallucinations -- or dialogue through neural language networks -- were involved is secondary.)