I've talked about how much I enjoy treachery in games, so it should be no surprise that I'm also a big fan of lies. Games involving deception have been around a long time and even games that are not supposed to involve it can do. I remember a multiplayer game of scrabble in which one player was worried about exposing a triple word before my turn and a casual "I don't have the tiles to make much use of it." was enough to persuade them to open it up. I could have played a higher scoring word, but sometimes an option in irresistible.
While deception is common in most games, some games have it built into their core. Broadly I'd divide the sorts of deception used in games into four categories, based on two dimensions; Lies are either implicit or explicit and deceptions either concern the game state or the player's intentions.
Explicit lies about game state
Cheat asks players to explicitly lie about the game state. They place cards facedown and identify them for other players. The game mandates that players identify every card placed and in some situations correctly identifying the card will lead to a certain penalty that is no greater than the penalty for getting caught - this encourages players to lie about the current state of the game on at least some occasions. Other players may call out a statement as a lie, penalising the lying player if correct but being penalised themselves if they are wrong.
Implicit lies about game state
In liar's dice each player rolls a handful of dice, observes their own but not the other players and makes guesses concerning the total number of dice showing each value. Guesses must increase in value as a round goes on, so while it is initially possible to name combinations of values that you are aware of due to the dice you can see it is eventually necessary to makes guesses which may be untrue. In the context of the game other players can call you a liar and you lose the round if successfully identified.
However this isn't the sort of lie that the game is really asking you to make. It's not even a lie, more of an incorrect guess. The lies that make this game function are those concerning the current value of your dice. If you say "There are at least three sixes" on the table early on, other players can be lead to conclude that you have rolled a six or two. Later this might cause them to guess that there are a high number of sixes and you can call them on this, if you in fact had none. The real lie in this game is not the statement of what is in the game (which you cannot know) but the implicit statement about what dice you can see that is contained within your guesses. If you are able to deceive another player about your dice you will be able to catch them out later and win the round, if you fail they may call you on a set that you cannot contribute towards and will lose.
Explicit lies about future intentions
Diplomacy is a good example here, but merits a quick rant. This game is a personal bugbear for me. Everything about this game implies that I should be good at it. I'm individually great at all of the skills that it involves when they appear in other games, but whenever I play Diplomacy I lose and lose badly. I'm still not sure why this happens. I'd love to say people gang up on me because how I've done in other games makes me look like a threat, but it happens with total strangers too.
Anyway, in this game players have access to equal amounts of force and combat is deterministic. To effectively make attacks players need to engage their enemies with support from other players. Orders are written simultaneously and executed once everyone has completed recording them. Telling a player that you intend to support them when you intend to attack them can lead to large gains, but if they predict your attack and share this information with other players you may be attacked from a direction that you've had to remove troops from. There's also a penalty for incorrectly distrusting someone, as the troops positioned to defend against them would have been more valuable elsewhere. Arguably nothing in the rules mandates explicit statements of future intent, but the mechanics encourage every game to end up this way.
Implicit lies about future intentions
It was really hard to pick a game for this slot, I think I like to imply falsehoods in almost every game I play, but can't think of a single prototypical example of the behaviour. I settled on Junta as our games involve a lot of implicit deception, but I'm not sure how much of this is related to how we play as opposed to what the rules of the game call for. In this game one player is the president and receives the nations budget each round, which must be distributed among the players. Everyone is trying to grab as much cash as possible and sooner or later there isn't enough to go around. When that happens, players can try to remove the president through various means, most of which will require the support of other players.
In our games there are plenty of explicit and implicit deceptions. It's practically tradition for the president to complain about how poor the budget is every round. The implicit lies tend to be the most important; When someone supports the president in a small event they're implicitly stating that they'll keep supporting him in a coup. When the president pays someone a big share of the budget they're implicitly stating that they will continue to pay them for their support. In both cases a successful deception is helpful, a loyal subject who turns almost guarantees the removal of a president and it can be good to get a lot of support and then not have to pay someone for it because you can have them killed.
Bringing it together
Pulling these examples together, deception games require a few elements to work. Players need to have something to gain for getting away with a lie and something to lose by getting caught in one. Other players need motivation to overturn a lie and motivation not to mistake a truth for a lie. Sometimes these things can serve each other, for instance a player's motivation to overturn a lie may simply be to see the lying player punished as this improves their chances of winning. The most games that seem the most elegant to me work in these consequences as an automatic consequence of gameplay rather than needing explicitly punishing rules.