Can a King be beaten in one card poker?

Question

This question is about the following theoretical model of poker, which is a greatly simplified version of my earlier question. Suppose that from one deck of cards, we remove all the Kings, and give one of them to one of the players, whom we'll call the King. The other player, whom we'll call Challenger, gets one card randomly from the remaining deck. Only Challenger can see this card, so it's not known to the King, but King's card is known to the Challenger. Thus, they both know that the probability of Challenger having a better hand, i.e., possessing an Ace, is 1/12.

Now they start bidding in the usual way. If the blinds are big compared to the stacks, King will have an advantage in this game. But what happens if the blinds are small?

Also, instead of the King, which card would make the above game fair?

I would also like to formulate a general conjecture: King should never raise, so the game looks like Challenger makes a raise, then King folds or calls.

Update: J would be an almost perfectly fair card if the blinds are small enough, see here: https://mathoverflow.net/a/270877/955

Answer 1

The king still has a big edge. Chances of him being beat are 4/48. It is impossible for the challenger to play an unexploitable strategy that is profitable in the long term.

The challenger cannot profitably bluff all of his hands, since most of the time he is beat and the king can just call him down. The challenger cannot only bet his value, since the king can just always fold to his bets. A mixture of both strategies will not work, since the challenger, if he is bluffing balanced say with only fives and sixes, will still have way too many hands that are beat by the king. The king can just fold to the bets and still show profit. If the challenger starts bluffing more, the king can start to call everything down and also show profit.

Bluffing all your hands doesn't work. Bluffing no hands doesn't work. Bluffing a mixture of good and bad hands doesn't work.

Bet sizing will not matter either. Betting exclusivly small with bluffs and value will get the same results as mentioned above. Betting exclusivly big will as well. Betting small with bluffs and big with value is exploitable and the king can only call the challenger's small bets and vice versa. Anything between "small" and "big" will result in the same outcome.

Conclusion: the king will beat the game no mather how big blinds are.

What card would make this game fair? For this to be the case the challenger has to 1. have the same number of value bets and bluffs in his betting range and 2. he should bet 50% of the time. This way the king will be breaking even whether he calls or folds. Given that the blinds are infinitesimally small compared to the bets the game will be break even if these conditions are met.

There are a total of 48 combinations possible for the challenger. The challenger thus must bet 24 combinations, of which 12 combinations need to be value. If the 'fair card' is a Jack, the challenger can exactly have 12 combinations of value (Aces, Kings and Queens). The challenger should look to bluff with 12 combinations and his range is complete.

Conclusion: the 'fair card' is a Jack.

Answer 2

In GTO the the theoretical maximum bluff rate is 1/2 but with blinds it is less

Put villain in worse reasonable spot. They are in the SB and get raised 10BB.
How often can villain be bluffed and have no value in calling.

0 = -19 + bluff(40)
bluff = 19 / 40 < 1/2

Hero needs to pick of 7 of 12 binds at as minimum
If the maximum buff rate is < 1/2 then need 4 good hands AKQJ
The best card you can give the villain is T

Pretend the deck is 12 card (villain has T) and in a round get all 12

EV for villian

-19 A
-18 K
-19 Q
-18 J
-19 + 40 bluff
-19 + 40 bluff
-19 + 40 bluff

if villain folds then lose blinds -9
if villain calls then -11

2 pick up blind hero folds
1 pick up blind hero folds
1 pick up blind hero folds
1 pick up blind hero folds
1 pick up blind hero folds

villain +6

villain cannot win

villain should fold to every bet and still losing 3 a round of 12

get tricky and change bet size just makes it more complex without changing the math

If villain gets a J then cannot beat them unless they play just plain stupid

Answer 3

Even the king can be beaten.

All the models you had rely on having a single betting round. But in poker, you often play multiple rounds. What would happen if we had 6 betting rounds instead of a single ?

Considering big enough stack sizes, the challenger can apply the following pot-sized bets strategy:

    1  3  9  27 81 243
Ax  B  B  B  B  B  B | 
4%  B  B  B  B  B  B | -- Core 1/8 range that will bet to the end
6%  B  B  B  B  B  -  <- Extra range that will bet on the 5th street, but not on 6th 
9%  B  B  B  B  -  - |
14% B  B  B  -  -  - |
20% B  B  -  -  -  - |-- And so on, growing the range by 50% each street
30% B  -  -  -  -  - |
10% -  -  -  -  - -  |

On the 6th street, calling and folding to a bet is EV equivalent (33% chance to win a 2:1 pot odds), so King effectively looses the pot when he faces a bet.

Since he will face 66% of the time a bet he will looses pot to, it is equivalent for him to call or fold on the 5th street, because calling a pot size bet with 33% equity is equivalent to folding to it.

Recursively, when facing a bet on a Nth street, since on N+1th street King will face 66% of the time a bet he will "fold" to, it is equivalent for him to fold in the Nth one. Even on first street, where Challenger bets 90% of his hands.

So at each street, there is no point calling or folding, it's all the same, all of this because of the possibility he might as well be building an enormous pot for aces.

This is an unexploitable strategy for Challenger. And we proved that it unexploitably pockets 90% of pot equity, beating the King.

The more betting rounds, the more uncomfortable it is to be face up

The important difference between a single betting round and a multiple one is that information of how much will be bet is partially removed from the caller.

This is what makes it possible to dynamically adjust the range, and be on the 1st street way above the bluffing frequency of a single street example.

Real life poker lessons

This is part of the explanation why things like continuation betting are so popular, and why stack depth matters : on a hand where the aggressor could hold the nuts, calling down on several streets is hardly profitable. Aggressive poker player always threaten to bet on several streets, yet not doing it all the time, in order to obtain the exact same effect than with this theoretical example.

Another lesson is to avoid face-up situations altogether by playing strong hands in different ways. Being able to have a nuts hand even only 1/100th of the time as the King player would be enough for him to call down all bets and actually come out ahead of the game. Protecting a weak range with nuts is the reason why Cepheus never 4-bets even if this might look profitable.