A two-player game is a tool for paying attention to someone and expressing yourself to them. With more players, group dynamics creep in; there's more to pay attention to and it's harder to give your focused attention to an individual. On the other hand with a larger audience self-expression is amplified - games that ask you to act, make jokes, and generally be extroverted work much better with larger groups. I value the focused attention-giving of one-on-one games over larger group interactions (though I do value both); similarly to how I perform better one-on-one conversations than in group discussions.
Good two-player games present many opportunities to impress and admire your adversary. What can these opportunities look like? When you're figuring out a game, learning together, you can show off by figuring out how to do something before your opponent does. Strategy games like Chess let you spot moves your opponent hadn't, or think ahead to come up with a cunning plan, and then surprise them with it. These surprises don't feel strictly delightful in the context of the game - they hurt your position - but there's an intellectual delight in being shown something you weren't expecting, and in being presented with a new challenge to think about. Chess also lets you demonstrate your memory and experience by learning openings and endgames, though personally I'm less interested in this.
Many different skills can be expressed through games - creativity, reasoning, reading your opponent ("yomi"), attention, humour, strength, and so on. It is unfortunately difficult to express moral virtues like generosity, selflessness, courage, love, honesty, faithfulness in many games, because either these actively cost you through inefficiency, or they're actively rewarded and feel less meaningful because they don't cost you; perhaps this sense that morality should cost something excludes it from being fully expressed in games because all in-game costs are imaginary.
Expressing your virtues in a game is not about trying to be the smartest kid in the room, to figure out who's smartest, to lift yourself up by putting your opponent down. These are attitudes of insecurity. Let's try to help each other feel more secure, to lift each other up by appreciating each other's abilities - whether it's the grudging respect of nemeses as portrayed by Holmes and Moriarty, or more likely in the context of a friendly relationship. Maybe one of us is smarter, though this is impossible to measure absolutely, but even if so we can still admire each other.
When designing a two-player game, ask these questions:
* What virtues does it allow the players to display to each other?
* How can I direct the players' attention onto each other rather than just onto the game?
* How can I avoid the feeling of "multiplayer solitaire", playing together alone?
* How can the game continually present new opportunities for players to express themselves to each other?
* How can it give scope for moves that are particularly notable in their cleverness or idiosyncrasy?
* Can it provide high-pressure moments requesting quick/clever thinking followed by low-pressure moments to reflect on the previous?
* Are there hidden depths that players can impress each other by revealing and share the joy of exploring?
* Are there unexpected combinations requiring creativity to uncover?
* Does the mastery curve exclude players low on it from having anything interesting to say to those higher?
* Are one player's clever actions sufficiently clear for their opponent to see why they're clever?
* How will actions in the game provoke admiration?