Interaction and You
One of the main things that draws me to tabletop games is the promise of interacting with other human beings in a real space. I work from home, and a considerable amount of my life is spent in front of one screen or another and so, the promise of spend time with actual humans is a big draw. Now, a person with better developed social skills would actually, you know, spend time with people in a less-structured way, but because I am who I am, interacting in a highly-structured, rule-heavy environment is extremely attractive.
However, despite the promise of human interaction being a draw, a lot of (even relatively popular games) make player interaction something that happens on the fringes of the game, rather than being a core element. For example, consider the player interaction in something like Wingspan. Players can take food from the shared food tray or birds from the shared bird display that other players were interested in. That's the extent of the interaction available. Of course, I think the very limited interaction was a deliberate choice by the designer, because a lot of player interaction that people are familiar with can have very negative player experience.
Before I do a deeper dive into these forms of player interaction, I'm going to discuss an idea from game theory, the "Sum". Here, the sum refers to the state of the players before and after the interaction. A positive-sum interaction is one in which both players are better off after than before. A zero-sum interaction is one in which for every gain of one player, the other player experiences a loss, and a negative-sum interaction is one in which both players are worse off.
Positive Sum Interactions
A positive-sum interaction is one in which both players benefit. The classic example is something like trade in Settlers of Catan. If I have extra sheep and you have extra wheat, we can trade and we'll both be able to build something we couldn't before. Another positive-sum interaction (even though it might not feel like it) is taking resources from a common pool. If I take a shared resource, I have gained the value of the shared resource, and you have lost the opportunity to take the shared resource. However, for most circumstances, the opportunity to take a resource is worth less than the resource (because taking a resource isn't free), so we (evaluated as a pair) have had a net gain of value.
You might think that positive-sum interactions are positive player experiences, and, generally, you'd be right. However, some kinds of positive-sum interactions are very un-fun for uninvolved players. If my trade isn't well-balanced, or if a player is very close to victory, even if I'm benefitting from the trade, I'm "giving the game away" to the other player in a way that feels like king-making. For this reason, a lot of positive-sum interactions in games are channeled into common pool resource taking, since I'm only ever benefitting myself, and the state of the table is always advancing, so the game is moving towards a conclusion.
Zero Sum Interactions
In a zero-sum interaction, every gain for Player A is a loss for Player B (or a summed loss for players B-X). Zero-sum interactions are common in two-player games. Consider a card-battler (like Magic or Star Wars: Unlimited) in which you play a unit and I play a card to defeat your unit. The net sum of these two interactions is a state that's exactly (for a given value of exactly) the same as it was before.
With that said, the net of effect of zero-sum interactions is not to move the game towards the conclusion. Games with lots of zero-sum interactions (or near-zero sum interactions) tend to drift along until someone finds a way to do something powerful enough to break the tit-for-tat dynamic.
A modern development in card-battlers is to reduce the amount of "pure zero-sum" interaction in favor of positive-sum interaction so that the game moves towards a conclusion speedily. For example, rather than just a kill-effect, instead, we have a kill-effect with a unit attached, so when I remove your unit, I have a unit of my own, and the game is moving towards an end.
Negative Sum Interactions
The classic negative-sum board game interaction is attacking in something like Risk. One player is spending their resources to hurt another player, in hopes that it will pay off later. In a game with 3 equal players, if Player A attacks Player B, Player A is hurting their position and the position of Player B right now in hopes that, in the future, they will be in a better position. However, Player C is unaffected and can (and likely will) just attack whichever player comes off worse from the attack, spending fewer resources for whatever long-term benefit is gained by attacking.
Another example of negative-sum interactions is in take-that games like Munchkin or even Magic in multiplayer matches. A player spends their resources to stop another player from winning or getting ahead, causing the two players involved to be down in value relative to the rest of the table. Negative-sum interactions naturally tend to feel bad for everyone involved. However, they are a very common occurrence in classic multi-player free-for-all battle games, which is one of the reasons multi-player free-for-all battle games are not something you see as often in modern board games.
Consider the shift in something like Kemet or even Small World from something like Risk. In these games, conquering a territory gives immediate rewards that cannot be taken away (victory points). In Risk conquering a territory gives the promise of future troops and a card, which can also be used to generate future troops. However, these rewards aren't immediate, and can be lost or unable to be used.
With as much as I've been talking down on them, I do believe that negative-sum interactions have a place in games, because of the concept of Altruistic Punishment. Altruistic Punishment is a negative-sum interaction in which a player spends resources to harm another player for norm violation or other bad behavior. Altruistic Punishment allows for players to punish "greedy" or "rude" behavior while still allowing players to indulge a bit in a way that I find interesting. Another aspect of negative-sum interaction is that hittin' people is fun! I can be fun to be able to say "no" to something (in moderation), and I think that some games can really benefit from letting people bump into each other, even in negative ways.
Conclusion
In general, each player (and game) is going to have a different tolerance for interaction, but the net overall trend is towards more restrictive, positive-sum interaction, with good reason! That said, the carcinization of interaction is towards common pool resource taking, which has a very limited range of expression - "I got the thing I wanted :) " "You took the thing I wanted :(" and "I'm gonna take this thing so you can't have it :P" (with the third category being discouraged). There is room for games which are kind or sneaky or even cruel, and you, as a designer, need to figure out what your game is trying to evoke, and to make sure that you're leveraging the most important part of the game, the other players.