Sometimes, however, players will want or need to participate in the weird activity. This is true of dancing, where knowing the right moves can impress a space princess, and it's definitely true of sports and games, where the plot might turn on the outcome of a game.
How to Win at Sports and Games
In both cases, the easiest way to represent a sport or a game is a quick contest between certain skills. That's it. The biggest difference will be between what skills. Most sports represent a contest between common skills. For example, Fencing is likely a contest between Fencing (Sport), while track is a contest of Running, Archery is a contest of the Archery skill and so on. Some sports become so codified that they don't really explore a "real" skill: What skill covers Cricket? Throwing and Broadsword? It's better to simply use the Sports skill in these cases, though certain skills (like Throwing or Broadsword for cricket) might offer a complentary skill roll: being a master swordsman will not let you win cricket, but it might help you play cricket better than someone who lacked the skill at all. The Games skill is used for understanding the rules of your sport, and nothing more. It's the skill of referees, or someone who wants to defeat their opponent on a technicality (such as noting that one can "foul" an opponent up to three times without being kicked out of the game).
Games, on the other hand, are the exclusive domain of the Games skill. Chess isn't really a contest of Strategy and Poker isn't really a contest of Acting and Body Language, but rather ones skill at the game itself. That said, like with sports, certain skills might offer complementary skill rolls.
Detailed Rules for Sports and Games
Sometimes, a quick contest isn't enough. Sometimes you want more detail than that, particularly if the game turns on the outcome of a particular match as happens more often than one might expect in the Action (or Planetary Romance!) genre. If you need extra rules, let me offer some thoughts below.
Slow Match: A quick contest is done all at once, which suits our purposes well, but if we want to slowly ramp up the tension and narrate what's happening, we could use a series of Regular Contests instead of a Quick Contest, or even a series of Quick Contests representing quality of play, with someone trying to accumulate a sufficient margin of success, similar to the Chase rules. GURPS Martial Arts has some additional thought starting on page 134 if you want even more detailed fight scenarios.
Different Modes of Play: Most contests will exist between a set of specific skills, like chess might be a contest of IQ-based Games, with Tactics as a complimentary roll. However, some games might have multiple modes of play, allowing different skills to come to the fore. For example, American Football can be about contests of Running and ST, but it can just as much be about throwing and catching, or even kicking. A team that excels at ST will want to make the game about tackling while a team that excels at DX will want to make it about throwing and catching. Other games change over time, or can be changed by the player themselves. For example, Go or Chess have different optimal strategies depending on the point of the game in which the player is, and games like Poker can vary depending on what the character has drawn.
This can be handled in several ways. First, in games where each participant can choose his own approach to the game, offer a variety of modes for the player to choose from, similar to how Martial Arts will let you choose between ST-, DX- and IQ-based "Feints", giving each a different name. If changing modes can be affected by the other player (that is, you can force the entire game out of DX-based Games to IQ-based Games for both sides, or you can force your opponent to abandon his Fencing (Sport) skill for Sleight of Hand, etc) then that generally requires sufficient mastery of gameplay to perform. This generally requires a quick contest of IQ-based Sports or Games.
The Gambit: Sometimes a game doesn't come down to skill, but to luck. In that case, the characters involved roll against a static value (say, 10 for a 50/50 chance of win or loss) to see if he won. This is typical of lotteries,or certain games where luck matters far more than skill, like Baccarat. Here, the Gambling skill is more useful as it gives you better insight into what your chances are (but high levels of gambling won't let you win, it'll just give you a heads up as to what your chances of winning are). Luck definitely applies, and you may use Serendipity to automatically win such a contest.
This can interact with "Different Modes" above in a few interesting ways. First, the gambit might be one of the modes of play. It can be useful to force the game into a gambit if your opponent has far superior skill than you. For example, if a novice faces a master in a Poker-like game, but has the option of turning the game into "whoever draws the highest card next wins" is well served by doing so, because the chances of him winning a straight game are fairly low. On the other hand, the gambit can shape modes. Perhaps a game has a variety of possible modes and which mode you play is determined randomly (perhaps by draw).
Playing for Keeps: What would an Action movie be without at least one scene in a casino? Here, Games or Sports might matter, but Gambling is necessary to navigate the amounts on offer. The specifics of what is being won or lost isn't important (generally, the higher the BAD, the higher the stakes), because the GM shouldn't allow the PCs to keep any major winnings from adventure to adventure, instead favoring the standard budget (though a major win at a casino could justify a character spending saved up CPs on improved wealth). If the game can turn on how much one bets, such as poker, then treat Gambling as an alternate mode of play.
Cheating: Many games are played openly, like chess or tic-tac-toe, but some games can be played deceptively, like poker. In such a game, it becomes hard to know for sure what's going on behind the scenes. Not all games should allow cheating, or be equally easy to cheat, but other games should definitely support it. GURPS basic already discusses how to cheat, but for our purposes, a successful cheating roll bypasses the normal results of the game. That is, in a way, it forces the game into a new mode, one between the cheater's Sleight of Hand vs the opponent's Gambling or Observation. This covers classic cheating, but more subtle cheating might be harder to spot, such as fixing a match, or taking performance enhancing drugs before a match. Rules for these are left up to the GM.
Sports in Psi-Wars
We could divide up sports into a few broad categories, to get some additional ideas for how we might allow a sport to play out.
Substitute for War: Many games represent combat training that has turned into a sport for the sake of sport. This includes most Track and Field events, Archery, Fencing and Wrestling. The rules began as a way to keep players focused on learning important skills, but eventually the drama created by the sport itself overtook the importance of the skills themselves. This is an ideal sort of sport for Psi-Wars, as it allows our dramatic heroes to take their already useful skills and apply them in an attempt to win the day, albeit at a minor penalty.
These sports generally focus on a Sports-version of a combat skill, though characters might try to use Games to play for technical points or obscure victory conditions. Psi-Wars already includes two sports like this, namely Pit Fighting and Pain-Blade Dueling.
Performance Sports: Some "sports" double as beauty contests, where the athletes look for a reason to master a particular skill that is entirely subjective, but still useful. This includes dancing, ice skating and gymnastics. Here, again, a normal skill is used, though usually with a Sport or Art specialization (if necessary, such as a contest between whose kung fu kata are "prettier"), with Games reflecting an attempt to game the scoring system, or to understand the judging involved.
Community Sports: Many sports start as childhood activities. We might take baseball or soccer very seriously, but the vast majority of participants in these sports are under the age of 18 and lurk on playgrounds and in empty urban lots. Such sports often have minimal equipment, little more than a ball and some points marked out as a goal, to play. These sports are usually best covered with a unqiue DX-based Sports skill, plus a few complementary skills, usually physical skills like Running, Throwing, Climbing, Acrobatics, or a few team-oriented skills, like Leadership or Tactics. The benefit of learning such a sport is often interacting with a local culture.
Ceremonial Sports: A surprising number of sports, like dances, have ceremonial origins, with the sport representing something fundamental about the culture. Knowing the Games skill for the sport is key, as it might reveal some of the inherent symbolism behind the sport, which means learning it does more than just connect you with the culture in question, but also into some of their mythology as well. A good real world example of this is Ullamaliztli and Sumo-Wrestling.
Games in Psi-Wars
Like sports, we can divide up games into a few categories useful for inspiration. Generally, games will have less mechanical diversity than sports (that is, they're mostly just contests between Games), but they often serve very different social rolls
Conversational Pastimes: These games matter less for whether or not the players can win, and more that they present a momentary diversion for the participants and an excuse to talk. Knowing the favorite game of some local crime boss or a valuable scientist you want to kidnap can bring you closer and help you gain their trust. Such games might act as complementary rolls for Savoir-Faire or Carousing, or grant a +1 reaction from the right culture or target. Games like this are numerous and include pinochle, scrabble, and certain forms of mah jong or dominos.
Strategy Games: Just as some sports act as substitutes for war, so too do some games. The game usually gains a reputation for a high-level contest of minds, and generally allows Strategy or Tactics as a complimentary roll. Often, because the game relies on out thinking your opponent, they tend to have a reputation for allowing you to better understand your opponent. If so, a Games roll might substitute for Psychology in a limited manner, or act as a complementary roll for Psychology. Strategy games are often high status because of their refinement and deep knowledge might act as a marker of high status. Examples of strategy games include chess, go, many wargames and, to some extend, poker.
Games of Chance: The game of chance turns on the gambit rather than on skill, or it might involve an element of both. Mastery of the game requires some level of Gambling to guess your chances of victory, and when to apply your Games skill and when to simply walk away from the game. Games of Chance often have a reputation for daring and high drama. As such, they're often associated with bets, and might act as complementary rolls for (or replacements for) Intimidation, Sex-Appeal or Streetwise. Daredevil should definitely apply for taking high risks. Real world examples include most casino games, like craps, roulette or blackjack, or daring games like Russian roulette.
Massively Social Games: More advanced civilizations have the tools necessary for very sweeping games played by hundreds of players at once, and even less advanced civilizations can play such games and often do, for example at royal courts where courtiers have a lot of time on their hands and want to show off their flair for the dramatic. Such games might be unofficial, like a courtly "whisper" or rumor-game, or it might be a "real" game, like Pokemon Go or the game of Orson Scott Card's "Breaking the Game." They tend not to require a board, and can be played in such a way that "everyone is playing, all the time." The key distinction here is that many people play the game and that your standing in the game is apparent to everyone. This means a well-played game is definitely worth a bonus reaction to those who also plays the game, and the game might be wide-spread enough to earn you a reputation. For courtly games, it definitely provides a complimentary roll on Savoir-Faire.
Richly Complex Games: The modern world sees an explosion of extremely rich and complex games, like collecting games, or extensive wargames, or roleplaying games themselves. The distinction here is the really deep ruleset and the amount of knowledge necessary to master the game and make use of it. This typically manifests as an increased penalty (treat it as a familiarity penalty) for the uninitiated to join the game. The advantage of such a game is that it acts a a sorting mechanism between "one of us" and "outsiders." If the game is build deeply into a culture or a class, knowing the game to a decent degree is definitely necessary to get past any cultural familiarity penalty.
Ceremonial Games: Like ceremonial sports, certain games can be embedded with hidden meaning, whether intentional or otherwise. Those who master the game can often apply their game knowledge as a complementary bonus on their Philosophy or Theology rolls, or vice versa. Real world examples might include snakes and ladders (virtues and vices and their impact on your life), monopoly (the evils of capitalism) and some games played with Tarot cards (though this is questionable).
The Game of Fans: A courtly world has a traditional pastime associated with what color/pattern of fan chosen at the beginning of the day. Players score points by challenging one another, and then accruing support from other fan-players who carry the same color or pattern of fan. Thus, selection of fan determines who your allies and enemies will be for the day. The entire game is played with gestures with the fans themselves, so the aristocrats will happily discuss political matters while battling one another with minor, almost unseen gestures. The game doubles as a way of checking political acumen, as one needs to be able to astutely guess if someone is willing to support you or not, or what fan someone will choose the next day. Thus, the game has an interesting relationship with Politics and, of course, is massively social, so a well played move will impress most members at the court. (A massively social strategy game).
Ten Kings: A very ancient board game played with wide variety of traditional pieces. Each player must choose his ten pieces from a great assortment of possible pieces, and each piece has its own rules and its own way of playing the game. The player gets a certain amount of choice and strategy, mainly in the choice of pieces, but what matters most is an understanding of the rules and the symbolism of the pieces, which tie into a the mythology of the culture that plays the game. The culture will often offer unique or rare pieces to others as a gift, or a statement. Deep understanding of those pieces will not only help you understand the culture, put the outcome of a game is often seen as a sort of omen (A Richly Complex, Ceremonial Game of Chance).
Sport is too many varied to get into without specifics, though it might be applicable to certain backgrounds. Games are much more likely, especially given that it governs the rules of sports, and especially when paired with Gambling. These will generally be background skills, however.