In Texas Hold'em poker, is the ranking according to chance of beating 1 opponent's hand the same as according to beating multiple opponents?

Question

In Texas Hold'em poker, you can rank hands according to the probability of beating one randomly generated opponent's hand. However, no one can compute the exact probabilities of beating $8$ random opponents' hands (a full game is usually $9$ players). Regardless, if you rank hands according to either probability metric, do you get the same ranking or can you get a paradox where one hand is better than another against one random opponent, but not if you are against between $2$ and $8$ inclusive random opponents?

Answer 1

That "paradox" is absolutely possible. Holding a pair of deuces, you are a favorite against any single non-paired hands (coin flips against some suited hands), hence a favorite to beat a random hand. But if several people are in play, then there are so many ways for one of them to outflop you that you are a serious underdog. There are many articles on the web about such calculations.

Answer 2

Another way to answer this question is to compare this same "paradox" using a simpler game, rolling a single fair die and the person with the highest number wins. Ask yourself do the odds for player A change if (s)he is up against multiple opponents vs. only a single opponent? Each die roll is analogous to a poker hand. Suppose player A rolls a $5$ and there is only one opponent (B). What are the chances of A outrolling B in that case? How about in the general case? Now imagine if A rolls the same $5$ but there are $8$ other opponents. Did that $5$ become not as good?

In the first case (only A and B playing and A rolls a $5$), there is only a $1$ in $6$ chance of B rolling a winning $6$ (B rolling a $5$ would be a tie with A's $5$). However, look what happens when you have $8$ opponents to A rather than just $1$ and A rolls a $5$. In that scenario, any of the other opponents has a $1$ in $6$ chance in beating A so the probability of nobody beating A goes down a lot. It would be $(\frac 5 6)^8$ which is about $23$% which means A's chances of winning went from $5/6$ = about $83$%, down to only about $23$% but with the same $5$ die roll.

A slight "flaw" in this analogy is with poker hands, the cards in hand are "tied up" for that round so that nobody else can get those same cards whereas in the this simple die roll game, no matter what A gets, the other players can get the same exact roll. I think the concept of more competition requires a better "hand" is accurate though in both games.

Answer 3

So I found a website that gives odds for starting hands winning in terms of the number of opponents 1 to 8. It turns out that ace-2 is better than 2-2 against 1 random opponent, but the opposite is true for 7 or 8 random opponents. In retrospect I guess that makes sense, because there is a 20ish percent chance that 2-2 will have a 3rd 2 come out in the middle to make 3 of a kind or a full house if the board pairs which would be very hard to beat even with 8 opponents. On the other hand, it's hard for ace-2 to make anything better than a pair (less than 10 percent chance), and even a pair of aces or 2 pair or 3 aces could get beaten by another player having an ace with a higher kicker. All the more likely as the number of opponents goes up.

Answer 4

Most answers seem to indicate that a strong hand with few players is not ncecessarily a strong hand with many players. Of course that makes sense, but more interestingly, is there a certain hand A that you would rather have than hand B with two players, whereas you'd rather have hand B than A with nine players? This appears to be the case. My monte carlo simulation tells me that with two players, you'd rather have 77 offsuit (78%) than AK suited (73%), whereas with nine players 77 offsuit is 21% and AK suited is 26%

Answer 5

But you can compute versus multiple opponents

Let me give a golf analogy. Tournament and only 1st gets paid and you do not know your opponents scores. If I am favored and against a field of 5 I am going to lay up and play a conservative game. If I am against a field of 100 I am going to take more chances because I have to. I need a better score to win against 100 versus 5.

In poker the same thing. Against a single opponent JJ is a monster. Against a field of 10 then JJ is not as likely to hold up. Against a single opponent 78 suited is weak but against a 10 opponents it goes up in values. Your chance of 78s hitting does not go up but if you hit the straight or flush you likely have the best hand even against a multiple opponents. Against multiple opponents you need a straight or flush to win more often so the value of suited connectors goes up.