Screw Your Neighbor

Last Update: Aug 11, 2010

Introduction

Screw Your Neighbor is a fairly simple card game, sometimes called in dealer's choice poker games. The game goes by other names including Ranter-Go-Round, Le Her, and something too indecent to put in writing. In my experience, it tends to be called when everybody is too tired or drunk to call a real poker game requiring serious thought.

Rules

The rules can vary from place to place. For purpose of this page, I'll explain and analyze what I'll call the Stewart N. Ethier rules, which he explains in his book The Doctrine of Chances: Probabilistic Aspects of Gambling. In the book, the game is called Le Her.

1. The game is played with a single 52-card deck. Kings are high and aces are low.
2. Each player puts up a certain amount of money. The amount will usually be three units of a specified amount, for example $1, for a total of $3 at risk.
3. The deck is cut to determine who is the first dealer.
4. The dealer will deal each player one card, including himself.
5. Each player should look at his card. If it is a king, then he will immediately turn it over, otherwise he will keep it face down.
6. Starting on the dealer's left, each player may choose to keep his card or exchange it with the player to his left. However, the king is a "blocker" and can't be switched. So, if the player to your left flipped over a king, then you lose your choice to switch.
7. After everyone else has taken his turn, the dealer is the last to act. If the dealer wishes to switch, and often he will, then he can switch with the dummy on the top card in the deck. However, if the dummy card is a king, it acts as a blocker as well, and the dealer can't have it.
8. The player with the lowest card loses and must put a unit into the pot. If there is a tie, the tying player closest to the dealer's left loses.
9. If a player is eliminated because he runs out of units to put into the pot, he no longer plays.
10. Keep repeating rules 1 to 9 until only one player is left. The last player with any units wins the entire pot.

Strategy

Two-Player Case

Let's call the first player to act player 1 and the second player to act player 2. If player 1 swtiches, player 2 should obviously switch with the dummy if he got a worse card. However, if player 1 stood, then player 2 has a decision to make. The following are the four plausible outcomes of good players.

• If player 1 switches with 6 or less, and player 2 switches with 7 or less, then player 1 will have a probability of winning of 51.1855%, or an expected value of 3144/132,600.
• If player 1 switches with 6 or less, and player 2 switches with 8 or less, then player 1 will have a probability of winning of 51.2941%, or an expected value of 3432/132,600.
• If player 1 switches with 7 or less, and player 2 switches with 7 or less, then player 1 will have a probability of winning of 51.3665%, or an expected value of 3624/132,600.
• If player 1 switches with 7 or less, and player 2 switches with 8 or less, then player 1 will have a probability of winning of 51.1855%, or an expected value of 3144/132,600.

So, if player 1 switches with 6 or less, player 2 should switch with 7 or less. However, if player 2 switches with 7 or less, player 1 should switch with 7 or less. However, if player 1 switches with 7 or less, player 2 should switch with 8 or less. If player 2 switches with 8 or less, player 1 should switch with 7 or less. If player 2 switches with 8 or less, then player 1 should switch with 6 or less. And around and around it goes. It becomes similar to a game of rock paper scissors.

After some game theory, which I won't get into, player 1 should always switch with 6 or less, stand on 8 or more, and switch with a 7 with probability 5/8. Player 2 should always switch with 7 or less, stand on 9 or more, and switch with 8 with probability 3/8.

If at least one player follows this strategy, then player 1 will have a probability of winning of 51.2534%, for an expected value of 2.5068%. If either player were to deviate from this strategy, the other player might pick up on it and exploit it in the future.

I show a solution to a similar game theory problem in my MathProblems.info site, problem 192.

Three-Player Case

Under the same Stewart Ethier rules explained above, the following is what I get in a three player game. If at any time a player is passed a better card than what he passed back, he should obviously stand, otherwise pass the lower card to the next player, if not blocked by a king.

If nobody has a king then:

1. Player 1 should switch with 6 or less.
2. If Player 1 stood, then player 2 should switch with 6 or less.
3. If Player 1 and 2 stood, then player 3 should switch with 7 or less.
4. If Player 1 switched and player 2 stood, then player 3 should switch with 4 or less.

If player 1 has a king, then player 3 should switch with the dummy with 6 or less.

If player 3 has a king, then player 1 should switch with the player 2 with 6 or less.

Seal Beach Rules

The way I played Screw Your Neighbor in Seal Beach, California, is the same as the Stewart Ethier rules, except (1) all players who tied for the low card had to put a unit into the pot, and (2) a king did not serve as a blocker if the dummy had it. Sorry, I have not worked out a strategy for these rules, which would be a lot harder.

Links

• The Doctrine of Chances: Probabilistic Aspects of Gambling by Stewart N. Ethier. This college text has several pages devoted to the mathematics of two-player version of Screw Your Neighbor, which it titles "Le Her."
• Ranter-Go-Round, as the game is titled at Wikipedia.
• cahillfamily.com has a good explanation of the rules