A conversation with Mischa led to the following question: what are the types of player interaction?
The short answer is: There are two types of player interaction: competitive and social.
All competitive interaction involves denying a limited resource to your opponents.
In all games, resources - points, transitory resources such as money, power cards, objectives - may be limited or (theoretically) unlimited.
For instance, a game may have 100 points (cards, countries) available for distribution, and whomever possesses the most at the end of the game wins. Or the money may be theoretically unlimited, e.g. players try to make the most money, but money earned does not diminish the available pool of money that can yet be earned.
Let's look at the competitive interaction in a few games:
Chess: The competitive interaction consists entirely of denying limited resources to your opponent, either by removing pieces, piece threats, or mobility.
100 meter dash: Players compete to acquire an unlimited resource before their opponent, namely distance traveled. Your distance traveled does not diminish the distance available for your opponent to travel. As a result, while the game is competitive, it is not interactive.
Tic Tac Toe: While both players can gain three in a row on the board, only one can do so first. Since the first player would always win if there were no interaction, players try to limit available safe moves of their opponent.
Scrabble: Points are theoretically infinite; taking points doesn't limit the number of points available to your opponent. However, point producing spaces are limited resources, and can be denied to your opponent.
Tug of War: Is a also a game of distance traveled, but distance gained is distance lost by the opponent, hence the resource is limited.
Puerto Rico: Here is another example of trying to deny your opponent resources by taking the roles, spaces, buildings, and goods most beneficial to you and leaving less beneficial ones to your opponent.
There are a number of forms for competitive interaction: auctions, bluffing, negotiation, trading, voting, and so on, and all devolve into the same competitive goal.
Games without competitive interaction may be called games or may be called competitions, but they are still interesting and enjoyable. They rely on internal rather than external challenges. A determination to outperform an opponent may be psychologically inspiring, but it's not truly interactive.
Negotiation, while a competitive interaction, is more than just one way to achieve a goal of gaining limited resources faster than your opponent. It is also a form of social interaction.
Some forms of social interaction in games are not at all competitive, such as much of what occurs during party games.
Video games provide avatars and chat for social interaction, and direct confrontation for competitive interaction. It's rare to see in them the integrated social/competitive interaction forms that occur in board and card games, such as negotiation or trading.
A game that fosters social interaction may be just as enjoyable for one crowd as one that fosters competitive interaction is for another. Some people want all games to be socially interactive, while others want all games to be competitively interactive.