### Running it Twice: Analysis

Many poker spectators were recently introduced to the concept of "Running it twice" on the television show High Stakes Poker on GSN. When two players agree to "run it twice", they are agreeing to complete the hand with two sets of outcomes. If there is one card left to be dealt, then two cards will be dealt, each card acting as a the final community card for two separate outcomes. When a hand is run twice, the unknown cards are not shuffled between the two outcomes.

If you win both times, you win the entire pot. If you lose both times, you lose the entire pot. If you win one and lose the other, then the pot is divided in half and distributed to the two players. Players who opt to run the hand twice are trying to lower the variance of the outcome. When you run a board twice, you are giving up some of your odds of winning in exchange for tieing a good percent of the time. This is an option that is only available in a cash game (as opposed to a tournament). In a tournament, players are not allowed to make any deals that affect their chip stacks.

It is easy to see that your variance will go down. There are many opportunities to tie and your chances of winning or losing both decrease. What many people do not immediately understand is that your expected value (EV) does not change.

The easiest non-trivial situation to analyze is a one outer situation with one card to be dealt. An example of this would be

Player 1: 2♥ 3♥

Player 2: A♥ K♥

Board: 4♥ 5♥ 9♣ Q♥

Both players have flushes, but player 2 has the higher flush. The only way player 1 can win is if the river is a 6♥, making a him a straight flush. There is a 2.27% chance that he will hit his one outer (1/44 chance). The total EV of this situation is 0.0227.

If the players run the board twice, he has absolutely NO chance of winning. Since cards are not shuffled between the deals, it is impossible to hit his one out both times. He does have a 4.55% chance of tieing though.

This is easy to calculate because it is 1 - the probability of losing. The probability of losing is (43/44) x (42/43). When calculating the EV for the 2x board, it is just the probability of winning plus the (probability of tieing)/2.

EV = P

So

EV

EV

I created an Excel spreadsheet with all of the possible outcomes for every situation with 1 or 2 cards to come (assuming there is no possibility of tying or redrawing). In every situation, the EV is identical.

Download Spreadsheet

If you win both times, you win the entire pot. If you lose both times, you lose the entire pot. If you win one and lose the other, then the pot is divided in half and distributed to the two players. Players who opt to run the hand twice are trying to lower the variance of the outcome. When you run a board twice, you are giving up some of your odds of winning in exchange for tieing a good percent of the time. This is an option that is only available in a cash game (as opposed to a tournament). In a tournament, players are not allowed to make any deals that affect their chip stacks.

It is easy to see that your variance will go down. There are many opportunities to tie and your chances of winning or losing both decrease. What many people do not immediately understand is that your expected value (EV) does not change.

The easiest non-trivial situation to analyze is a one outer situation with one card to be dealt. An example of this would be

Player 1: 2♥ 3♥

Player 2: A♥ K♥

Board: 4♥ 5♥ 9♣ Q♥

Both players have flushes, but player 2 has the higher flush. The only way player 1 can win is if the river is a 6♥, making a him a straight flush. There is a 2.27% chance that he will hit his one outer (1/44 chance). The total EV of this situation is 0.0227.

If the players run the board twice, he has absolutely NO chance of winning. Since cards are not shuffled between the deals, it is impossible to hit his one out both times. He does have a 4.55% chance of tieing though.

This is easy to calculate because it is 1 - the probability of losing. The probability of losing is (43/44) x (42/43). When calculating the EV for the 2x board, it is just the probability of winning plus the (probability of tieing)/2.

EV = P

_{win}+ 0.5 P_{tie}So

EV

_{run it 1x}= 0.0227 + 0.500 x 0.000 = 0.227EV

_{run it 2x}= 0.000 + 0.500 x 0.0455 = 0.227I created an Excel spreadsheet with all of the possible outcomes for every situation with 1 or 2 cards to come (assuming there is no possibility of tying or redrawing). In every situation, the EV is identical.

Download Spreadsheet