How do you determine the winner of a Texas Hold'em poker hand?

Question

In Texas Hold'em poker, there are some cases where a particular card, often called kicker, acts as a tiebreaker between players to determine who wins the pot, or if the pot has to be shared.

For some combinations such as three or four of a kind, there is no doubt: the kicker is the 5th card of the chosen combination, and determines the winner.

Example

Alice has 7 and 10
Bob has 7 and K
On the board are 7, 7, 2, 3, 4

In that case, both have a three of a kind. But because Bob has a king and Alice only a 10, he wins.

Most of poker websites explaining the rules clearly mention the role of the kicker applying to three of a kind and double pairs. But much fewer say something about colors, straights and full houses, and I found many contradicting answers. I also asked the question to some used online poker players and they also gave me contradicting answers.

I have tried to search for official tournament rules, but most of them only explain what happens with bad behaviors, bad deals, incorrect or confusing betting, showing cards when you shouldn't, acting when it's not your turn, etc. without mentioning subtleties about combinations at all.

To simplify my question, I will take three examples; I think it's better to start with examples before getting to the general answer if one exists. So, what's happening in the following 3 examples? Could you give a more generalized answer?

Example 1 - Flushes

Alice has 2♣ and 3♣
Bob has 4♣ and 5♣
On the board are 6♣, 8♣, 10♣, 2♥, 5♦

Contradicting answer 1: it's a tie, because the highest card included in the flush is the 10, which everybody chooses to include in their 5 showdown cards.
Contradicting answer 2: Bob wins, because he has the greatest private card that is part of the flush

Example 2 - Straight

Alice has 6 and K
Bob has 6 and J
On the board are 4, 5, 7, 8, 10

Contradicting answer 1: it's a tie, because the greatest card in the straight is the 8 for both Alice and Bob
Contradicting answer 2: Alice wins, because she owns an extra king, compared to the jack of Bob

Example 3 - Full house

Alice has 3 and 7
Bob has 3 and 6
On the board are 3, 3, 2, 2, 5

Contradicting answer 1: it's a tie, because one is supposed to choose only five cards to make a combination, and a full house is already five cards. There couldn't be any kicker, and thus their showdowns are strictly equals.
Contradicting answer 2: Alice wins because of her extra 7, compared to the 6 of Bob

Thank you very much for your answers.

Note: I'm unable to post next to you, so I edit my own post; strange not be able to answer to an answer.

Ok, So if I summarize what you are saying :

• In the flush case, Bob wins because at some point, their hand differs. Technically, I can continue comparing the highest private card to decide who wins. I had it correct.
• In the straight case, if the highest card of the straight is public, then it's technically always a tie, no matter what the players had as second private card (asuming that only one of the two cards was part of the straight).
• In a full house if both players have the same triplet and the same pair, it's always a tie, no matter what the players had as second private card (assuming again that only one of the two was part of the full house).

Please confirm me that what I summarized is correct. Thank you for your answer.

Answer 1

The first thing to mention is definitely that there are no extra cards. Poker hands are evaluated with exactly five cards. Sometimes you use all five community cards as your best hand, in which case your pocket is useless (bluffing aside, of course). So strike that right away: if you can't beat your opponent with five cards, you've lost (or tied).

This is true even if all players are using the community cards. In the extreme example, say a Royal Flush comes down in the community cards. Everyone gets a Royal Flush (in which case, obviously, you go all in hoping your opponent makes a mistake.) You're not forced to say "Well, I had an ace in my hand, so I break the tie between the two of us." No, no, and no, you all have the community cards, you tie, split the pot and move on to the next hand.

The next step is to evaluate the hands. It starts like this:

1. Does any single player have a straight flush? A straight flush is having both a straight and flush at the same time. A flush is all 5 cards being the same suit. A straight is having 5 consecutive cards, consecutive being defined with the order A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A. Ace is in the list twice, but you cannot overlap A. Thus, A, 2, 3, 4, 5 is a straight, 10, J, Q, K, A is a straight, but Q, K, A, 2, 3 is not. A Royal Flush (10, J, Q, K, A of the same suit) is sometimes considered a special straight flush, but in reality it's just the best straight flush.
   • If yes, that player is the winner.
2. Do multiple players have a straight flush?
   • If yes, the winner is the one with the highest card. (A-2-3-4-5 is the lowest straight, while 10-J-Q-K-A is the highest straight.)
   • If multiple people share the highest card (either in a different suit or because there is a straight flush in the community cards) they split the pot.
3. Does any single player have 4 of a kind?
   • If yes, that player is the winner.
4. Do multiple players have 4 of a kind?
   • If yes, the one with the highest 'set of 4' is the winner.
   • If multiple players have the highest set of 4 (which is not achievable with a standard poker deck, but is with a double deck or community cards), the one with the highest kicker (highest card not in the set of 4) is the winner.
   • If this card is the same, they split the pot.
5. Does any single player have a full house?
   • If yes, that player is the winner.
6. Do multiple players have full houses?
   • If yes, then keeping in mind that a full house is a 3-set and a 2-set, the player with the highest 3-set wins the pot.
   • If multiple players share the highest 3-set (which isn't possible without community cards like in hold 'em, or a double deck) then the player with the highest 2-set is the winner.
   • If the 2-set and 3-set is the same, those players split the pot.
7. Does any single player have a flush? A flush is defined as all 5 cards of the same suit.
   • If yes, that player is the winner.
8. Do multiple players have a flush?
   • If yes, the player with a flush with the highest unique card is the winner.
   • This hand is similar to 'high card' resolution, where each card is effectively a kicker.
   • Note that a flush requires the same suit, not just color. While the colors used on the suit are red and black, two each, there's nothing to that connection. A club is no more similar to a spade than it is to a heart - only suit matters. The colors are red and black for historical purposes and so the same deck can be played for other games where that might matter.
9. Does any single player have a straight? A straight is 5 consecutive cards that do not wrap around the Ace. See full explanation in the Straight Flush description.
   • If yes, that player wins the pot.
10. Do multiple players have straights?
    • If so, the player with the highest straight wins. (A-2-3-4-5 is the lowest straight, while 10-J-Q-K-A is the highest straight.)
    • If multiple players share the highest straight, they split the pot.
11. Does any single player have a 3 of a kind?
    • If yes, that player wins the pot.
12. Do multiple players have 3 of a kind?
    • If yes, the player with the highest 3-set wins the pot.
    • If multiple players have the highest 3-set, the player with the highest kicker wins the pot.
    • If multiple players tie for highest 3-set and highest kicker, the player with the highest "second kicker" wins the pot. (For example, A A A K Q beats A A A K J - here, A A A are the 3-set, K is the kicker, and Q and J are the second-kickers.)
    • If the players tie for the highest 3-set, highest kicker, and highest second kicker, the players split the pot.
13. Does any single player have 2-pair?
    • If yes, that player wins the pot.
14. Do multiple players have 2-pair?
    • If yes, the player with the highest pair wins the pot.
    • If multiple players tie for the highest pair, the player with the second highest pair wins the pot.
    • If multiple players tie for both pairs, the player with the highest kicker wins the pot.
    • If multiple players tie for both pairs and the kicker, the players split the pot.
15. Does any single player have a pair?
    • If yes, that player wins the pot.
16. Do multiple players have a pair?
    • If yes, the player with the highest pair win.
    • If multiple players have the highest pair, the player with the highest kicker wins.
    • Compare second and third kickers as expected to resolve conflicts, or split if all three kickers tie.
17. At this point, all cards are kickers, so compare the first, second, third, fourth, and if necessary, fifth highest cards in order until a winner is resolved, or split the pot if the hands are identical.

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Note that when comparing two hands, all suits are equal in Poker - the ranking of suits from games such as Bridge and Five Hundred have no bearing on evaluating Poker hands. Bridge order is, however, used for certain "bring in" tie breakers, such as 7 card stud.

Answer 2

First of all: in your examples 2 and 3, the 'extra cards' (Alice's king and Bob's jack in example 2, and Alice's 7 and Bob's 6 in example 3) effectively don't exist: for comparison purposes, you use precisely each player's best 5-card hand. Those hands are 45678 in example 2 and 33322 in example 3; any additional cards in the player's hands are entirely moot.

Example 1, on the other hand, follows exactly the rules for kicker cards: Alice's flush is 2, 3, 6, 8, ten (of hearts), while Bob's is 4, 5, 6, 8, ten (of hearts). Since Bob's cards are 'higher' than Alice's, Bob wins the hand. More generically, what's used for flushes (and technically for straights) is lexicographic order : compare the highest cards (in the player's 5-card hands, remember!); if they're the same, compare the second-highest cards; and so on down to the end. As soon as you find one card higher than the other, that player wins; if you never find any cards higher than the other, then it's a tie. This is the idea that covers examples 2 and 3. (Technically it covers example 1, too - it's just that that example falls into the tie case.)

The same concept works for the other classes of hands too, but you have to be careful about comparison order; for pairs, two pairs, and three of a kinds, compare the 'feature' cards first (this is where kickers come into play, when the feature cards are the same: ten, ten, king, three, two beats ten, ten, queen, jack, nine, but king, king, 4, 3, 2 would beat queen, queen, jack, ten, 9). For full houses, always compare the three-of-a-kinds first (so a hand of 5, 5, 5, 4, 4 would still beat a hand of 3, 3, 3, King, King).

Answer 3

@corsica has given an excellent answer, which is extremely thorough, but isn't at the right level of abstraction (in my opinion). Here is a different way of conceptualizing the rules for people looking for a more rule-based understanding of how the hand ranking system works.

Rules for poker hands

There are 6 rules, applied in the order they are listed. While this list is specifically for Texas Hold'em, all but Rule #2 are used by most poker variants.

Rule #1: Anyone who has folded the hand is ineligible to win or split the pot.

It doesn't matter what someone's hand "would have been" had he/she stayed in. If you fold, you lose. If all but one player folds before the end of the hand, the remaining player wins without the hand being played to completion. Because betting happens sequentially, it is not possible for all players to fold during a hand, so there will always be (at least one) winner.

Rule #2: You make the best 5 card hand you can of your cards and the community cards. Cards that are not part of that hand are irrelevant.

In particular, if the best 5 card hand you can make is the community cards, you have no particular card advantage over other players and your best possible outcome is to split the pot. This is the only rule that's specific to Texas Hold'em. Variants of poker with just 5 cards do not need this rule. Other variants of poker with more than five cards (like 7 card stud or Omaha) have a variation on this rule.

Rule #3: The player with the highest category of hand wins.

The categories are (from best to worst):

0. Five of a kind [only possible when playing with wilds]: all 5 cards of the same number (ex. 7-7-7-7-Joker)
1. Straight flush: all 5 cards are consecutive* and of the same suit** (ex. 5♠-6♠-7♠-8♠-9♠)
2. Four of a kind: four cards with the same number (ex. 5-5-5-5-3)
3. Full house: three cards with the same number as each other and the other two cards are the same number as each other (ex. Q-Q-Q-9-9)
4. Flush: all 5 cards are the same suit** (ex. Q♠-10♠-5♠-4♠-2♠)
5. Straight: all 5 cards are consecutive* (ex. 5-6-7-8-9)
6. Three of a kind: three cards with the same number as each other (ex. J-J-J-K-2)
7. Two pair: two cards with the same number as each other and two other cards with the same number as each other (ex. 10-10-5-5-K)
8. Pair: two cards with the same number as each other (ex. K-K-A-4-2)
9. Junk: none of the above (ex. K-10-6-5-3)

*Aces can count as low (effectively a 1) or high (effectively a 14) (ex A-2-3-4-5 or 10-J-Q-K-A) when determining consecutiveness for a straight or straight flush.

**Same suit means they are all ♣, ♠, ♥, or ♦. Color is irrelevant.

Rule #4: If players have the same category of hand, the player with the highest unique participating card wins. For hands with multiple groups of the same number (i.e. the full house), the larger group is more important.

For a straight, a flush, or a straight flush, all cards are participating. For the other hands, it's the cards involved in the sets of the same number. The order of cards from highest to lowest is A (14), K (13), Q (12), J (11), 10, 9, 8, 7, 6, 5, 4, 3, 2. Exception: when an ace is used in a low straight (A-2-3-4-5), it is below a 2 in value (effectively a 1). Example 1: Q♠-J♠-8♠-5♠-4♠ beats Q♠-J♠-6♠-5♠-4♠ because the 8 is higher than the 6. Example 2: A-K-Q-J-10 beats K-Q-J-10-9 because A is higher than K. Example 3: a 3-3-3-2-2 beats a 2-2-2-K-K because the triplet of a full house is more important than the pair.

Rule #5: If players are tied on category of hand and are also tied on all participating cards, the player with the highest unique non-participating card wins.

These cards are known colloquially as a kicker, second kicker, etc. Hands that use all 5 cards (the straight, flush, full house, and straight-flush) don't have kickers. A junk hand is all kickers, and a face-off between junk hands is simply a competition of highest card. Example: 8-8-8-A-K beats 8-8-8-A-J on the second kicker (King beats Jack).

Rule #6: If the players are tied on category of hand and are also tied on the value of all cards, the hand is a draw between those players and all players tied for best hand split the pot.

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The examples in the question

Example 1 - Flush

Alice has 2♣ and 3♣
Bob has 4♣ and 5♣
On the board are 6♣, 8♣, 10♣, 2♥, 5♦

Alice and Bob both have a flush of clubs. Bob wins due to having a higher participating card (5 vs 3). See rule #4.

Example 2 - Straight

Alice has 6 and K
Bob has 6 and J
On the board are 4, 5, 7, 8, 10

Alice and Bob both have a straight (4-5-6-7-8). The hand is a tie because their participating cards are the same and there are no non-participating cards. See rules #2 and #4.

Example 3 - Full house

Alice has 3 and 7
Bob has 3 and 6
On the board are 3, 3, 2, 2, 5

Alice and Bob both have a full house (3-3-3-2-2). The hand is a tie because their participating cards are the same and there are no non-participating cards. See rules #2 and #4.