A poker player, a telepath and a clairvoyant

Question

There are three poker players, who are playing heads-up against each other. One of them is a usual poker player. Another - a telepath, who can read thoughts. So if you know your cards, he knows them too. And a third one - a clairvoyant who can't read thoughts, but knows which cards gonna come on flop, turn and river (but he doesn't know pocket cards). All three players are aware of poker maths and play correctly regardless of their skills.

So we have three match-ups.

1. A usual guy against a telepath.
2. A telepath against clairvoyant.
3. A clairvoyant against a usual guy. (he can bluff now, yeah?)

What are the best strategies of these players in their match-ups? Let's assume they should play even if they gonna lose (surely they can fold on any street), but want to minimize losses in the long run.

Let's assume they are playing limit holdem. (Being on the small or big blind each all the time)

Answer 1

Game 1

The telepath has a natural advantage, unless the usual guy does something to deal with it. There is no particular action the telepath can undertake beyond using their ability and otherwise playing poker.

The usual guy, on the other hand, has only one choice - play blind. Don't look at their cards, and play purely by watching the reactions of the telepath. Having skill at reading reactions, the regular poker player should be able to decide when to play or not, and thus should be able to gain a slight edge relative to any other plan. The best the telepath can do in this situation is to also play blind, in which case it basically becomes a 50% chance of winning for both sides.

Game 2

Here, the clairvoyant is at a disadvantage, because everything they can read, the telepath can learn from them. As such, the clairvoyant has to use their gift carefully. Under most situations, playing blindly as with the regular player is the only option. However, they can use their gift as a technique to encourage the telepath to reveal the strength of their hand, by seeing certain cards and observing how the telepath responds after reading it from their mind. This can give the clairvoyant a small benefit over the telepath, but a lot still comes down to pure luck.

Game 3

In this game, the clairvoyant has the advantage, and thus is likely to win. There is no particular strategy available to counter the advantage, as the regular player has to rely on their usual skills while having a lesser amount of knowledge of the hand. That said, some techniques can minimise the issue, and increase the potential for luck to go their way. The clairvoyant will, naturally, read every hand immediately.

If the clairvoyant has a tell, the regular player can use this to their advantage, as they can get some sense of the hand that is coming. By some effective bluffing in the first two rounds of betting, and then observing the reaction of the clairvoyant, they can get a sense of the kind of hand that is coming, and the clairvoyant's own hand. High bets on the first round suggests to the clairvoyant that the regular player has a strong hand, and thus will be confident if they've got a very strong hand or the cards are going to be bad for normally-strong hands, and thus will also be betting high. On the other hand, if they've got a weak hand and the cards are going to favour strong hands (lots of high cards, for instance), they'll bit minimally, or possibly even fold.

The second round of betting can then be used to determine further, in a similar way - observing the clairvoyant's reactions and tells should be enough to minimise the impact of their gift, but there's no way to completely neutralise it - that is, the clairvoyant will always have more information, and thus will always have a better than 50% chance of winning.

Answer 2

Game 1.

The telepath will never call on the river. She will either raise or fold. And she will only fold if she knows she's going to lose. She should bluff enough that nearly half her bluffs are called (assuming pot-limit raises). If half your bluffs are called on pot-limit raises, then the bluffs break even. The more your bluffs are called, the more often your real bets are called, and that's where the big money is.

If she has a tell, then she'll realize the other person is seeing the tell and she can work on covering/faking the tell.

The muggle should just accept that he's going to lose all his money. His only chance would be to bet big before the flop, but this is a risky strategy. When he starts with good cards, like a high pair, and the telepath starts with a higher pair, the muggle will lose big.

Game 2.

This is even worse: the clairvoyant knows what all the cards will be, so the telepath now knows this, too. The entire hand is a foregone conclusion in the telepath's mind. The clairvoyant has zero chance. The telepath will bet the maximum on half the hands (the one's she's winning) and often enough on the others to convince the clairvoyant to fold winning hands and bet on losing hands.

Game 3.

This game has some level of subtlety. The clairvoyant will know what she will have at the end of the game, but won't know what the other player will have. However, there are lots of hands where you know you almost certainly must have the highest hand: an inside straight, a high straight, a flush with a high kicker, full-house or four-of-a-kind. Even a high three-of-a-kind can often be an undisputable winner (eg. you have two aces, there's an ace on the table, and there are no other pairs or straight or flush candidates).

The clairvoyant requires skill with statistics and some skill at reading her opponent. But I suspect even a casual player with a strong statistics background would trounce most any professional.

Answer 3

Game 1

The telepath knows the normal player's cards at the start of the hand. The telepath can determine the probability of a person winning at any given point in time. The only way the normal person would make money would be if the odds were against him at the begginning, yet he bet (or called) substantial amount of money, and then the odds switched to his favour. This has a probability definitely less than 1/2.

Hence, he would prefer folding in every hand.

Game 2

As with game 1, the telepath is at a better position. Hence the clairvoyant will apply the same strategy as the normal guy in game 1.

Game 3

This game is also biased, but fair enough for both players to employ complex strategies. If they are playing optimal strategies, then the normal player will fold most of the time (not always). He has meek chance of winning. However, for human players, he still has some chance of winning, if he is better at bluffing and calling bluffs than the clairvoyant.

Note

You have not mentioned for how long they are required to play. If they are playing forever, the person at a disadvantage is going to go bankrupt after a finite number of games (due to the blinds), no matter what. If the number of games is fixed, the above rules apply.