Could you provide some specific examples of Western and Eastern games?

If I'm to guess, it comes down to, in a broad generalization, a difference in the predominant roles of cavalry between Europe and East Asia. Western armies tended to be partial to shock cavalry, where mounted soldiers would often be used as an armored fist, designed to break through enemy lines, which also oftentimes makes cavalry a good counter to some sword-armed infantry in some western strategy games such as Total War. Of course the best counter to an armored man charging home with a lance on an armored horse is a spear, which a horse would refuse to go near. However, the predominant Chinese, Japanese, and Korean experiences with cavalry come from horse archers. China gets this from its experience with steppe peoples, which oftentimes served in Chinese armies as well as fought against them. Japan and Korea both get their respective experiences from a long cultural tradition of archery. Mounted archers are effective against infantry because infantry can't catch them, but archers can just shoot them. TLDR: Western and Eastern games have different understandings of "cavalry"

I can't speak for eastern games, but in Total War (a series I feel can represent western grand strategy) there is a distinct difference in spear vs sword/melee units to where they can be distinguished in the "rock paper scissors" perspective. Melee beats Spears beats cavalry beats Melee.

This is for direct combat. Once you throw in Archers and artillery a "rock paper scissors" outlook no longer applies. For example, anything beats archers in hand to hand decisively, but conversely archers can pelt anything to death given time.

Bah! I had book marked this weeks ago, mentally sketched out a response and then forgot to post it. Ah well, might as well do it now and hope OP gets his question answered.

I assume you know the historical context about the western mode of warfare games like Total War emulate, so just to give you a bit of historical context about the Eastern mode of warfare which gave rise to this very simplistic rock-paper-scissors idea.

A lot of this stems from the Chinese confrontations against various Steppe nations in the Warring States/Qin/Early Han Dynasties. In the early Han Dynasty China's infantry & chariot based armies were at a severe disadvantage in steppe terrains against the Xiongnu confederacy's (the dominant Steppe power of the time) horse archers. China was forced to adapt and switch to cavalry armies to fight the Xiongnu confederacy. I should note that these cavalry armies employed by both sides are not really the kind of heavy shock cavalry of Europe. The Chinese cavalry has mixes of horse archers and really something akin to dragoons (eg. mounted infantry, archers, crossbowmen who can dismount and fight). So this is where the cavalry -> infantry/chariots came from. And in this mode of warfare you can see how this logic would arise, and indeed the Mongol horse archer method of warfare proved very powerful against standard infantry army as we saw during Subutai's raids/invasions of Eastern Europe.

The archer -> cavalry logic arose from observations that the Han armies' coordinated firepower could decimate steppe cavalry based armies, which were lightly armed, compared to European knights etc.

We're talking about fairly large focuses on firepower/maneuverability here for professional ancient Chinese armies outside of civil war eras (when mass untrained levies were raised and seemed to for all intents and purposes act as rather ineffective line infantry). For example, the Eastern Han Dynasty's elite Northern Army, tasked with guarding the Capital, was broken into five regiments of about 800 men each. The five regiments were:

-Agile Cavalry
-Garrison Cavalry
-Chang River Regiment (Steppe cavalry auxiliaries)
-Archers
-Infantry

That's 60% cavalry & horse archers, 20% infantry, 20% archers. You can see, compared to say, a Roman army of the time, these armies skewed very heavily towards cavalry/archers.

Sources:

Cosmo, Nicola Di (2002). Ancient China and its Enemies: The Rise of Nomadic Power in East Asian History.

de Crespigny, Rafe. Later Han Military Organisation: An Outline of the Military Organisation of the Later Han Empire