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In game design, multiple emotions often turn up woven together in complex stories, but for our purposes we'll attempt to address them in isolation, as part of the game experience itself, using the simplest, most pure examples possible.
Suspense (or Thriller) is perhaps the most common emotion in games, and turns up in any game with an uncertain outcome. An entire genre of kids' games (that I like to call "Russian Roulette games") is based entirely on suspense and includes the Water-balloon Toss, Crocodile Dentist, and Don't Wake Daddy.
But more than these, any games that build on anxiety contain suspense, such as children playing hide and seek in a dark house, the card game Slap-Jack, and the aspect of survival horror games in which monsters spring on you from unexpected directions.
Romance has a few representatives that often share duty with suspense; Mystery Date is a weak example, as are likely those dating simulators sold in Japan.
Less vicarious examples include spin the bottle and strip poker; some would argue actual flirting and dating are part of a ritualistic mating game. I would also add that situations where sexual tension might develop in online social games should count (I certainly saw plenty of flirting when I was working on Zynga's poker.)
Comedy is clearly a driving objective of many creativity-based party games like Taboo, Once Upon a Time, Balderdash, and Apples to Apples. The latter two even directly reward humor via voting mechanics.
Yet you probably won't find more than a passing mention of comedy in these game's rules; comedy instead seems to be an emotion just waiting to happen anywhere players are given the context to express themselves in a free format. As evidence, you probably don't have to look any further than the activity of your friends on your Facebook wall to find a series of quips and one-liners.
Horror as entertainment requires a certain terrifying visceral experience that aims to instill disgust and build a sense of dread. While gladiatorial combat was probably quite the spectacle in its time, in the modern era horror as entertainment is largely limited to pure narrative fantasy. Horror games are usually clearly packaged, including the likes of FEAR, Resident Evil, BioShock and Dead Space.
Adventure covers just about every video game featuring a protagonist ever made. The player overcomes adversaries, cheats death, and races against time. In a lot of ways, adventure sits at the intersection of many of the other emotional genres; a little bit thriller, horror, comedy, suspense, and even romance. It covers none of these emotions too deeply, and perhaps this balance is what makes an adventure so universally appealing (and so unlikely to be obtained in simple non-game experiences).
Drama is something of an umbrella term. Dramas tend to cover a range of emotions not already mentioned; things like jealousy, suspicion, honor, guilt, greed, ambition, ennui, and repression. Collectively, they seem to describe interpersonal relations taken to dysfunctional extremes.
Game stories certainly take advantage of drama regularly (any JRPG), but in actual gameplay, games tend to rely on the more competitive aspects like suspicion and ambition. Multiplayer games that require balancing competition and cooperation, like Avalon Hill's Diplomacy, represent the rare few that accurately model the emotional strain of shifting trust, honor and guilt of real drama.
Tragedy seems to be the least-represented emotional genre, with only the rare game dedicated to it (Shadow of the Colossus and Sword and Sworcery are two examples and both play tragedy with subtlety).
But this should be expected: to constitute a tragedy, the experience must ultimately be a failure, and failure is not a satisfactory outcome in most game designs. Although it is worth observing that, in most multiplayer games, everyone but the winner ultimately fails, and that in itself is something of a controlled tragedy.
Emotion is a difficult element to instill into games. It tends to be highly contextual and not usually pursued as an explicit objective beyond the scope of traditional story-telling. When considering emotion as a gameplay objective, I think it's beneficial to view the available emotions in three groups of utility:
Those that integrate well with traditional game mechanics: Suspense
Those that can be worked into a design, if the proper considerations are taken: Romance, Comedy, Drama
Those that really only work in the context of a story: Horror, Adventure, Tragedy