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So there's this cool psychological effect whereby if you ask people "What are your odds of drawing a full house?" you get higher numbers on average than when you ask "What are the odds of drawing a full house?". People are almost universally aware that the cards drawn off a deck does not change depending on who is doing the drawing, but when asked - on average - will assign themselves higher odds. How do we use that?
I'd argue that there are two main ways to take advantage of this effect and that both are common in existing games, though are not often explicitly discussed in these terms. The first is to use it to create tension where the game state wouldn't otherwise justify it. The second is to use this sort of bias to generate emotionally tough decisions.
It's generally desirable to end a game once the winner is known. If a player gets into a position that they'll definitely win, but the game requires a long play time after that point it's generally frustrating for everyone to have to continue. The problem is that players are capable of very complicated reasoning that lets them predict outcomes. A written rule that was as good at spotting situations in which the outcome is effectively determined would be so unbelievably complicated that it'd be more frustrating to check every turn than it would be to just play on once the outcome is determined.
I believe that this is one of the reasons that hidden trackable information is so common. Certainly it's theoretically possible to know how many points every player is on, the vast majority won't bother and that creates a grey area. There'll be a range of situations in which an objective analysis might reveal "In all but the most perverse of worlds player one has definitely won" but the subjective analysis of the average human in second place would be "If I do really well and play the last turn just right I can still beat him."
The underlying structure of the game doesn't change at all - but by making players rely on estimates rather than definite numbers a wider stripe of "exciting closing game states" exists, which means that a more comprehensible "end game" rule can exist without players spending so much time frustrated at foregone conclusions.
The second use - generating a tough decision - works in the opposite way. Making difficult, but meaningful decisions is at the heart of game design. Ultimately if your players do not get the opportunity to meaningfully make decisions that alter the state of play then you've not made a game so much as a really inefficient movie.
Meaningful decisions depend upon a carefully managed level of ambiguity. If the best decision is too obvious, then the decision doesn't feel meaningful. Where every player can immediately see the "correct" option they'll take it, but the game will never engage them and the outcome will feel random because all players played equally well (by always being perfect). Conversely if the best decision is too obscure, such as where it is impossible for the players to understand the odds of different decisions paying off until after the decision is made, then the game is equally meaningless. Again all players will play equally well (by always getting random outcomes whatever their choices).
These sorts of biases help to make a choice more ambiguous, but in a way that most players experience as being "fair". The information that they need to make good decisions is available, it's just that figuring out what's important and arriving at a good strategy involves overcoming ones human limitations. There are plenty of times in games that a player is called upon to compare their odds to odds for the table. These tend to fall into a few categories:
Firstly, it's an intrinsic component of any straight up gambling scenario. Any time a player is looking to compare a straight up payout against taking some risky venture to improve their outcome they need to wrestle with the feeling that their odds are somehow better to make the wisest choice. Risk seems an aptly named example of a game in which a player thinking that they have above average odds might encourage them to attack where a strict mathematical analysis might call for them to remain stationary.
Secondly, it's inherent in a lot of cooperative games that ask players to make choices for themselves by comparison to the rest of the group. Consider something like Zombies Keep Out in which players must decide which of several bad things should happen each turn. Some options on the terrible things card might make a player ask "Can I deal with this issue?" where others lead more naturally to "Can someone else deal with this issue?". A lot of cooperative games use the difficulty inherent to seeing the difference between "The game state" and "The game state from my point of view" to generate tension. Particularly where they involve hidden traitors, as it provides them with some cover.
Thirdly, it allows actions to be weighted by personal stake. There is a difference between how "Roll a dice, on a 4+ you beat their armour" and "They roll a dice, on a 4+ their armour deflects your shot" are perceived and acted upon by players. Twilight Imperium has an optional rule about having a single fighter bring down a war sun (read death star) by shooting it with a torpedo in just the right place. The odds of success are somewhat less than if the fighter didn't use the optional rule and just decided to have it out with the war sun, but the option exists anyway and some players go for it because "I could have one of might fighters do an amazing bombing run" is a more powerful thought than "The odds of rolling a 10 to survive the initial attack and a 10 to hit on the run is literally a hundred to one."
These things are all interesting, some of them add more to their games than others, but they all have something to bring to the table. Handled well it's not hard to see how using this effect can enhance games, but there are two caveats to bear in mind:
The first is that this effect is inverted for some portion of the population. In particular people suffering from depression will tend to underestimate their chance of success compared to the general odds.They tend to underestimate by less than the ... this is a funny phrase ... "normal population" overestimate by, but it still means that over-reliance on this sort of effect is going to be wildly misplaced for some portion of gamers. I'm not sure how large this portion is, but anecdotally I'm inclined to believe it's not insignificant. Perhaps I should write something on depression and gaming some time, there's a lot going on there. In any event, it's worth bearing in mind, using these sorts of effects can make a game more challenging and interesting for most players, but you can't rely on it - use it as an enhancement, not as a basis.
The second is that there are other effects that operate in the opposite direction. In general most people are risk averse and will require a much higher level of certainty to come to a belief than you'd mathematically predict. There's this really cool collection of letters between a couple of psychologists in which one argues that being too certain of ambiguities demonstrated a manner in which schizophrenics were abnormal and the other points out that using the same statistical approaches the first used in his studies you could demonstrate that the schizophrenic population was closer to the "true" answer than the so called normal population. I digress, but the point is that there is more than one way in which the average human is terrible at perceiving reality. Relying on just one, without understanding any others, can lead to aberrant results. For the best outcome you'd want your situation in which you want a player to overestimate their personal odds to be for some sort of gain, rather than to avoid some sort of loss and for their prior experiences of the system to be generally positive and a whole bunch of other stuff. It's good to think about factors, but important to acknowledge that each one is one of many.
I hope that you've found this little detour into me vaguely remembering things from being a psychologist interesting. Every now and again I see things in game design that remind me of them and I think it's interesting to consider how they can be tempered as ways to create better games