New topic: Mindgames vs Random Guesses
I'm sure you're all familiar with Rock-Paper-Scissors, where you without knowledge of your opponent's decision have to pick the option that counters it. Often times this will lead to players trying to get into each other's head after repeat games "he just picked scissors so he probably won't pick it again". This is mindgames in it's purest form, where there are no other methods to make the right choice than the psychological one. The most famous game with a strong psychological foundation is probably poker (*). To win one usually has to guess what cards the other player has in his hand. The big difference with RPS however is that there are factors besides the psychological one. Because certain cards have an advantage over others, they influence the player's playstyle, which makes it possible to determine a player's cards by analysing their playstyle. What makes poker fun for most people is that this results in the mathematically optimal playstyle not being the actual optimal one. If you always play in the most mathematically optimal way your opponent can predict your moves and cards and exploit this to win, even if he had worse cards (ie. bluffing)! Players will still generally have a thought process and play according to risk/rewards probabilities though, so the game becomes about discerning your opponent's thought progress and applying it to the 'situation' which the cards that you can see create, to predict their moves and cards. What's important here is that each round the cards create a different 'situation' which provides clues to your opponent's thought process in contrast to RPS where there are no extenuating factors and every round is the same 'situation' which holds no clues.
The point I want to make is the difference between the mindgames in RPS and poker. In poker you have clues in the form of the cards and the variety of playstyles with down- and upsides, where as in RPS you basically have to be a mindreader and guess between three equal choices (unless your opponent for some reason follows a really predictable pattern like alternating rock and scissor). Also worth noting is that it becomes more plausible to make an accurate prediction after multiple rounds.
Now let's look at some examples in actual videogames. For starters, something relatively simple:
Team Aerial Combos in UMvC. You have three choices with different advantages. Your opponent has to predict which choice you make to prevent you from gaining those advantages. This is a good implementation of mindgames as the optimal choice depends on the situation (eg. stealing meter from Morrigan), thus there are clues upon which to base your predictions. However, there is a
glitch that basically changes the situation into two choices for the attacking player, neither with specific advantages or disadvantages. Thus the defending player doesn't have any real clues to base his prediction off and has to randomly guess, which would be a bad implementation of a mindgame.
Finally, a more complex example. Opening moves (or build orders) in Starcraft 2. In SC2 the optimal strategy is highly dependant on your opponents' strategy. However determining that requires you to scout him first, which always has a cost (lost mining time+unit cost). Thus not only is your opening move 'blind', how long you stay 'blind' is also part of your opening move (how long do you build up economy before sending a scout). The following factors provide clues to determine opening moves however: the map, metagame, viability of the strategy itself, your opponent's playstyle (most starcraft players have at least distinct preferences in strategy), what your opponent thinks your playstyle is, previous matches (it's less desirable to repeat the same opening too much since your opponent will have had time to analyse it), situation in a tournament (do you really dare to pull off a risky opening when it's the deciding match for a 100$k prize?) and your experience with executing a certain opening. Also interesting is that it's possible to mislead your opponent by showing him misleading information (cancelling or hiding buildings). All these factors contribute to making SC2 openings fascinating mindgames. As you might have noted some of these factors aren't a direct consequence of game design, but the game design does in a way allow them to be important.
In conclusion I'd posit that mindgames in videogames are better when there are more clues to add depth to the decisionmaking, and that variety to the decisionmaking can be introduced due to having a multitude of situations that affect the mindgame. Mindgames are also a way to make it harder for games to be 'solved'.
(*):I'll be using 2-card texas hold'em no limit for my example to avoid misunderstandings