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I've been generally wondering about what sort of winrate good players can expect vs. worse players, and thought of an interesting limit to this winrate. In NLHE, if the worse player just blindly shoves all-in before the flop every turn, what kind of winrate can an optimal player hope to have against them? Their strategy probably amounts to just waiting for a good enough hand to call; at best you get AA and have a 85% chance to with the hand, but in practice you probably end up having to call with worse hands. Has anyone done the math and figured this out?

EDIT: After some discussions with answerers, I want to clarify some points in this question:

  • Each player starts with 50BB, and the shover starts as BB
  • Each player stays until one player runs out of chips, at which point the game ends
  • The goal of the pro player is to win this single game with as high a probability as possible, knowing their opponent's strategy is to shove pre-flop every hand
  • What's the practical use of this question? Beside maniacs, players on tilt and players that want to lose their last dollars in their bankrolls, nobody does that. – Bogdan Doicin Jul 24 at 5:32
  • @BogdanDoicin There are actually situations where you should be shoving 100% in theory. That's assuming villain is playing correctly and mostly only in high ICM situations such as on the bubble of a tournament and close to the money in satellites. – Jonast92 Jul 24 at 13:30
  • It's mostly just curiousity, though I imagine that if you're playing against a player who you know is much better than you, this might be your best strategy. If my friend who has never played poker before was about to play heads-up vs. Phil Ivey, and I had 1 minute to coach them, I'd tell them to never look at their cards and shove every hand, and they'd have a better than 1/3 chance of winning. – user68945 Jul 24 at 17:25
  • Better than 1/3 vs what range? – Bogdan Doicin Jul 24 at 19:02
  • vs every range: if the pro played optimally (i.e. used the most optimal calling range in every hand), they would have at best a 64% chance of winning. The specific optimal range for the pro when both players have 50BB is 66 or better (i.e. any hand with at least a 64% winrate vs. a random hand heads-up), but it gets larger when either player has a chip advantage. – user68945 Jul 24 at 19:59
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Assuming good depth (50bb+) if someone calls with the top 50% of hand i.e. everything that is +ev to call, he would have 57% chance to win

However a good player realizing he has 57% chance to win and is against a very good client, could call everything that has 57% chance to win or better, i.e the top 20% of hands, giving up some per-hand ev in exchange for higher winrate. His chances to win the all-in are 63% but he would give up 4/5th of the blinds, depending of depth his overall winrate would be prob. between 58 to 63%.

To put this in perspective, HU SNG players consider 54% to be a good winrate, so the strategy is at best mediocre - this is because NLHE is a game where hands are strongly dominated right at preflop.

Note that effective strategies to reduce opponent edge (GTO stategies) are the one used against unknowns by pro-level players : in a zero-sum game, there is no distinction between reducing your opponent chances to win and increasing yours.

  • I definitely agree this is a bad strategy, and a great situation for the better player. But I'm curious about what exactly the best overall winrate is that a player could achieve vs. the shover. i.e. if I told my friend who has never played poker to just shove his chips in whenever the dealer gives him cards, what are the chances of him beating a top player? – user68945 Jul 22 at 18:20
  • Sorry, I just realized you sort of addressed this in the second paragraph. Just to make things unambiguous, let's say each player starts with 50bb. I'm curious what the GTO strategy is for this game, and especially in how likely it is to win. It seems like a natural thing to consider but I haven't been able to find anything online where someone actually computed it – user68945 Jul 22 at 19:55
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    @ user68945 It depends on what you define as beating someone. Is this a single buy-in game? Is he rebuying in nonstop? For a single buy-in HU game, the shover probably can win about 30-35% of the time. – sakon Jul 23 at 6:32
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    Single buy-in; the goal is to win the 1 game with as high a probability as possible. And yeah, I actually wrote code to calculate it, and it looks like it's about a 36% winrate for the shover if their opponent plays optimally – user68945 Jul 23 at 18:44
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It would depend on the blinds:stack ratio, but in general you can call when you have >50% equity against a random hand. Which is something like, 22+, any A, K or suited Q, Q6o+, J6s+, J8o+, T7s+, T9o+, 98s+. This is 52% of hands. (I individually keyed in hands against a random range on an EV calculator).

When the stacks get shorter, then you may need to loosen up and call a wider range, but this shud be fairly ok call range for >30BB or so. If its multi-handed then you probably should tighten up unless you're closing the action.

You'll have a much wider variance than lets say someone who is playing properly, but id take this cash cow any day.

EDIT (scenario given)

Youre dealt 22 on the SB (50c), and BB shoves you $50 without looking at his hand. You have ~50.4% equity against a random range. If you call, youre putting in $49.50 to win $50.40 (50.4% of a $100 pot). You are +EV (0.90c).

It doesn't matter what you do with other hands. You can call with just AA, or top 20% of hands or whatever, but the moment you fold 22, you lose 0.90c. End of story. Thats 0.90c off your winrate. You'll never get this 0.90c back by waiting for a better hand.

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    calling with 52% of hands seems way to loose. Your opponnet is shoving with every hand, you can do better by waiting for a better hand. – Cohensius Jul 22 at 12:58
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    True, let's say SB is 1% of each player's starting stack. And I agree with Cohensius; if the opponent shoves every hand, I imagine you can do better by just giving them a few blinds and waiting for an extremely good hand to call them with. – user68945 Jul 22 at 18:23
  • @Cohensius If you have math to back up your opinion, i'll be happy to hear it. Your definition of "better" means less variance, not more equity. If youre afraid of being felted, play lower stakes. – sakon Jul 23 at 6:30
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    It's important to remember this is a weird hypothetical situation, where we know that the opponent will continue to shove every hand. In a normal game you would be happy to call a >50% equity all-in, but I think the difference here is that you know that if you wait a few hands, you're likely to find an even higher equity call later. – user68945 Jul 23 at 18:47
  • In response to your edit: in this scenario our goal is to win the single heads-up match. If we play optimally, we have about a 63% winrate (see my accepted answer); calling with 22 gives us only about a 51% chance. Even though calling with 22 is +EV, it's suboptimal in this scenario since it stops us from getting an even higher EV call later. From the book I cited: "Anyhow, this is definitely a situation where we want to maximize our win percentage as opposed to our chip-EV." – user68945 Jul 24 at 17:41
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I found an answer to this question in Will Tipton's Expert Heads Up No Limit Hold'Em, Volume 2, Section 16.2.1:

at the starting stacks of 50BB, Hero actually expects to win over 63% of the prize pool.

Where the Hero is the pro player and the villain is the shover; so the winrate of playing this "always shove" strategy is about 37%.

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