There are a few points I didn't make in my post on blame games because they seemed extraneous to the core point, which are still important enough to write down.
The Pecking Order game is a zero-sum game in which people closer to the center expropriate from people farther from the center, and use some of those resources to perpetuate the power imbalances that enable the expropriation. Players that fail to submit to expropriation by higher-level players are punished by those more-powerful players, often through intermediaries. Players that fail to help members of their class expropriate from those beneath them are excluded from their class, and often coordinated against more overtly.
This game isn't inherently majoritarian, - instead, it allows smaller groups to stably expropriate from larger ones, because every player in the middle has a short-run incentive to go along with the arrangement.
When pecking orders are overt and common knowledge, there can be an element of delegation that allows the pecking order to process some limited amount of information centrally, as a hierarchy. Feudalism is a simple example of the hierarchy game. Modern states almost always have some hierarchical arrangements, such as the police and military, and (less formally) economic class.
Around the time of the French revolution - a replacement of Feudal arrangements with Modern states - people started using terms like "left" and "right" to refer to political orientations. These terms are related to natural structural coalitions within a modern democratic state.
Political parties don't overtly promise to expropriate from 49% on behalf of an arbitrary 51%. This is probably in part because this would be correctly viewed as a proposal to massively increase expropriative activity relative to other activity, accelerating the rate of expropriation, which actually isn't in the majority's interests, and would quickly undermine the democratic paradigm without providing a replacement to enforce property claims. Instead, appropriation is opportunistic, and political coalitions seem to be oriented around natural power bases which could in principle replace deliberative democracy.
One natural sort of organization to orient around is the formal hierarchy with a monopoly on force - the military and police. The staffing needs of these organizations are substantial, especially in wartime (democracies perform well in wars in part because of their ability to mobilize a large share of the population without destabilizing their internal political arrangement) so they already form a natural constituency.
The obvious advantage of control over these organizations is in the event of a civil war, control over the army and police would be a massive advantage. So, building a group identity around those things is a pretty plausible way to expropriate the country from the other half.
The "right wing" is the part of the political spectrum that most resembles or is most naturally allied with this coordination strategy. Generally, if there's an identifiable majority group (ethnic, cultural, religious, etc.), the hierarchy of violence will perceive members of that group as more "central" and want to help them expropriate from minority groups more than vice versa, insofar as gaps in the rule of law allow this. People rewarded by the existing credit-allocation, the "upper classes," will also tend to favor and be favored by this arrangement.
The "left wing" is the natural complement to this strategy: a political "big tent" made up of all the noncentral groups. Such a coalition has a structural advantage as long as democratic institutions persist, since any new group (e.g. immigrants) that isn't part of the majority is a natural member of the "left wing" coalition. Such groups also tend to seek control of, and expropriate resources through, the parts of the state that are responsible for information processing, investment, and resource allocation rather than the administration of violence. In short, the bureaucracies and those who staff them.
As before, both sides are winning this civil war, at the expense of the people least interested in expropriation.
Related: Nightmare of the Perfectly Principled, Rentiers vs Apparatchiks, Arseholes, considered as a strategic resource, Pecking Order and Flight Leadership
Hierarchies wouldn't exist if they weren't better at defending themselves than other arrangements are, so there are non-zero-sum reasons for having them. But there's a zero-sum component that can be helpful to isolate and model. I was a little sloppy about this here, but Romeo Stevens pointed out the problem, at The Other Place.
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I don't think your definitions of Left and Right match up with how the terms have been used historically. We'd have to call the French revolutionaries, and the Jacobins in particular, right-wing, along with the Russian Revolutionaries. The original Nazis are a hard call, because they didn't build a power base in the police or military; they made their own police and military equivalents from the ground up. American Nazis today would definitely be left-wing, since they have little physical force and are not near the center.
Then we have to ask what "force" means, because humanities profs today think that social pressure and praxis count as force. In that case, universities like Berkeley, Yale, and Harvard would be right-wing, as one particular party has seized institutional power and uses that power to suppress other viewpoints, without any payoff system as in hierarchy.
I think you'd get a closer match if you said that the left wing was more ideological, by which I mean left-wing groups define the in-group in terms of what people believe. The right-wing organizes around some pre-existing group, based for instance on nation, race, or economic class. This classifies Marxists as right-wingers, while other types of communism are more likely left-wing. I think that might be a feature rather than a bug.
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