10 Aug 2024 EV equity probabilities variance No one likes bad beats in poker, or in mathematical terms, variance. For many players, including professionals, this is one of the most significant downsides of the game, often leading to negative emotions such as anger, despair, and even the desire to quit poker altogether. Importance of Variance in Poker Due to variance, it’s impossible for a professional poker player to accurately predict their "salary" for the near future, whether it’s for a week, a month, or even a year! This is especially uncomfortable if poker is your only source of income and you need to pay bills or support a family. Nevertheless, any long-term winning player must understand that their profitability is, in fact, due to poker variance. If there were no element of short-term luck, all weak players (aka "fish") – whom we profit from – would quit the game, leaving only the regs. Eventually, even the regs would quit, realizing their inability to beat the rake. It would turn poker into a pure game of skill, like chess. Variance and these occasional streaks of luck encourage fish to make new deposits, hoping that the poker gods will soon favor them. "I just need to wait out the bad luck", they often reassure themselves. Without bad beats, there would be no fish, and without fish, there would be no poker. Variance keeps poker afloat! Moreover, variance allows not only poker but the entire gambling industry — including roulette, slots, lotteries, and more — to live and thrive. The difference between poker and casinos or lotteries is that in poker you play against people, not the casino, so you can create a positive expected value (+EV) for yourself, unlike roulette and lotteries where the rules inherently create -EV for players and +EV for the casino. What is "Run It Twice"? Many of you are likely familiar with the poker feature "Run It Twice," first introduced online in 2009 at Full Tilt Poker, quickly gaining popularity among players and later adopted by most other poker rooms. It was introduced as a response to players' constant complaints about bad beats, intended to slightly reduce the swings of short-term variance in poker. For those unfamiliar, 'Run It Twice' is a feature offered voluntarily to two players who go all-in before the river. Instead of dealing just one board, two boards are dealt. The pot is then split in half, with each half awarded to the winner of each run-out. In some poker rooms, more run-outs are offered, such as three ("Run It Three Times"), where the pot is divided into three equal parts and follows the same scheme. However, the introduction of this feature sparked heated debates within the poker community about its advisability. Yes, it does reduce the swings of variance, which is welcomed by everyone. However, the main debate centers on how it affects EV. Today, we will address this once and for all, leaving no questions unanswered." Does "Run It Twice" Affect EV? The main argument of opponents of this feature is that "Run It Twice" reduces the EV of the player who is currently the favorite. This implies that to reduce variance, we sacrifice our EV, which is unacceptable for winning players. If a player is a long-term winner with a proper bankroll capable of withstanding any variance, they should not sacrifice their EV for the sake of comfort. That's the absolute truth! But is this really the case? Does the number of run-outs affect our EV? At first glance, everything seems logical. For example, in a showdown of KK vs AA, it’s obvious that with two boards, the probability of a king appearing increases. As a result, AA will win the entire pot less often and will split it more frequently. Yes, that’s true. The number of splits will increase, but will this really affect the long-term EV? To answer this question, let’s start with a simplified example. Matchstick Example We have 5 matchsticks: 3 long and 2 short. The rules are as follows: draw one matchstick; drawing a short one means we lose $1, and drawing a long one means we win $1. Let’s calculate the probabilities and EV for this scenario with 1 and 2 attempts. 1 attempt: The probabilities of drawing a short and long matchstick are 2/5 and 3/5, respectively. Therefore, we calculate EV: EV = (+1) * (3/5) + (-1) * (2/5) = 1/5 Thus, our EV for one run-out in this case is +$0.2. 2 attempts: To calculate EV in this case, we need to break down the situation into all possible outcomes: Outcome 1: We win both run-outs. Probability: (3/5 * 2/4 = 0.3 = 30%). Outcome 2: We lose both run-outs. Probability: (2/5 * 1/4 = 0.1 = 10%). Outcome 3: We split the pot. Probability: (1 - (0.3 + 0.1) = 0.6 = 60%). Now let’s calculate our EV: EV = (+1) * (0.3) + (-1) * (0.1) + (0) * (0.6) = +0.2 As you can see, the EV did not change. Yes, the probabilities did. Now, we will win less often in the short term—30% instead of 60%. We will split more often—60% instead of 0%. However, we will lose less often—10% instead of 40%. The redistribution of probabilities is the effect that "Run It Twice" provides, influencing the reduction of variance. However, the long-term EV remains unaffected by the number of run-outs. EV remains unaffected by the number of run-outs. Poker Example From matchsticks to poker! For simplicity and time-saving calculations, let’s consider a spot on the turn, where only one river card remains to be dealt, and we have 15 outs to win. 1 Board: The probability of hitting one of our outs on the river is 15/44. EV = (+1) * (15/44) + (-1) * (29/44) = -14/44 = -0.318 2 Boards: Outcome 1: We win both run-outs. Probability: (15/44 * 14/43 = 210/1892 = 11.1%). Outcome 2: We lose both run-outs. Probability: (29/44 * 28/43 = 812/1892 = 42.9%). Outcome 3: We split the pot. Probability: (1 - (210/1892 + 812/1892) = 870/1892 = 46%). EV = (+1) * (210/1892) + (-1) * (812/1892) + (0) * (1022/1892) = -0.318 In both cases, the EV remains the same! Meta-Factors in Using "Run It Twice" As you now understand, from a purely mathematical perspective, and assuming other conditions remain unchanged, 'Run It Twice' does not affect our expected value (EV). Therefore, at first glance, it seems that whether to use this option is a personal choice for each player—it doesn't affect anything! However, there are additional non-mathematical factors that should also be considered. Let’s call them "meta-factors". Meta-factors are off-game factors not related to mathematics, covering broader aspects of the game such as psychology, emotions, etc. In the paragraph above, I emphasized the phrase "other conditions remain unchanged". This is the key hint towards these meta-factors. So, let’s get to the point! Impact on Tilt By using 'Run It Twice,' you reduce variance, which decreases the likelihood of bad beats for both yourself and your opponents. As we know, bad beats can severely affect players' emotional states and put them on tilt. Therefore, we can formulate the following rule for using the Run It Twice: If you have a robust bankroll and consider yourself a mentally resilient person — someone who is difficult to put on tilt, unlike your opponent — then it’s better not to use Run It Twice. This way, you will more quickly knock your opponent off their emotional balance and A-game, gaining a psychological edge. In this case, it doesn’t matter whether you are ahead or behind – AA vs KK or KK vs AA. Your EV will be the same regardless of the number of run-outs, but winning with KK against AA, or AQ against AK will be very painful for your opponent, potentially causing them to go on tilt. "Run It Twice" Against Fish Meta-factors also include the specifics of playing against fish. As we know, fish are the main source of money in poker. That’s why we strive to play as many pots with them as possible, looking for various profitable situations to take their stack. So, when you’re all-in against a fish while ahead, avoid using 'Run It Twice,' as we want to strip the fish as quickly as possible before other hungry regulars do. With 'Run It Twice,' the probability of splits increases, which is not advantageous for us when playing against fish. We want to strip the fish right here and right now, not in the long term! The fish's bankroll is usually limited, so the longer we allow them to stay at the table, the higher the likelihood that someone else will take their stack instead of us. Eventually, the fish will leave, and we won’t be able to capitalize on them in the long term — as is theoretically expected. Final Thoughts Now you know that 'Run It Twice' does not affect our long-term expected value (EV) when meta-factors are not considered. However, when taking meta-factors into account, it’s generally better not to use this feature, as reducing variance can diminish the profitability of the game. Instead, focus on training your discipline and self-control! Finally, I’d like to warn you about some truly losing features in poker rooms, such as "All-in Insurance". This is when a player, who is the favorite, pays a fee to insure their all-in, receiving a payout if they lose. What’s wrong with this? While "Run It Twice" simply redistributes probabilities between two players and the room gets its fixed rake, "All-in Insurance" adds another element for the room to profit from! The room doesn’t care who wins each hand; their main goal is to maximize their earnings from the game. By preying on their plyers’ fears, the room offers this feature. Clearly, purchasing such comfort will cost you EV from a mathematical standpoint, unlike 'Run It Twice!'" I hope this was a useful and comprehensive article. Have a great grinding weekend, and good luck!