Screw Your Neighbor

Introduction

Screw Your Neighbor is a relatively straightforward card game, often associated with dealer's choice poker games. It is known by several nicknames, such as Ranter-Go-Round, Le Her, and other names that are too inappropriate to mention. Based on my observations, this game tends to be played when participants are either too fatigued or intoxicated to engage in a more traditional poker game that requires careful thought.

Rules

Game rules can differ significantly depending on the location. For the purpose of this explanation, I’ll present the rules as outlined by Stewart N. Ethier in his book. In 'The Doctrine of Chances: Probabilistic Aspects of Gambling', the game is conducted using a standard deck of 52 cards. In this game, kings hold the highest value while aces are treated as the lowest. . In the book, the game is called Le Her.

1. At the beginning, each participant contributes a specified amount of money, typically three times a predetermined unit, such as $1, leading to a total risk of $3.
2. To determine the initial dealer, the deck is cut.
3. The dealer distributes one card to each player, including himself.
4. Players will then inspect their cards; if someone has a king, they will reveal it immediately; otherwise, they keep their card face down.
5. Starting with the player sitting to the dealer’s left, each participant has the option to either keep their card or exchange it with the person next to them. However, if the player to the left reveals a king, that player loses the ability to swap cards, as the king serves as a blocker.
6. After all players have taken their turns, the dealer acts last. Should the dealer choose to swap, which is often the case, he can exchange his card with the dummy card located at the top of the deck. However, if that dummy card is a king, it also acts as a blocker, and the dealer cannot choose it.
7. The player holding the lowest-valued card loses and must contribute to the pot. In the event of a tie, the player closest to the dealer's left in the tie loses.
8. If a player runs out of units to contribute to the pot, they are eliminated from the game.
9. This process continues, repeating the rules above, until just one player remains. The last person with units in play claims the entire pot.
10. Let’s denote the first player to take action as player 1 and the second as player 2. If player 1 decides to swap, player 2 should logically swap with the dummy card if they have a worse card. However, if player 1 holds steady, player 2 must then consider their options. Here are four potential outcomes based on strong players’ strategies.

Strategy

Two-Player Case

Should player 1 switch with a value of 6 or less, while player 2 swaps with 7 or less, player 1's chances of winning stand at 51.1855%, which translates to an expected value of 3144/132,600.

• If player 1 exchanges with a value of 6 or lower while player 2 opts to change with 8 or lower, player 1’s odds of victory slightly increase to 51.2941%, giving them an expected value of 3432/132,600.
• In a scenario where player 1 switches with a value of 7 or less, and player 2 does the same, player 1's likelihood of winning is further elevated to 51.3665%, with an expected value of 3624/132,600.
• Conversely, if player 1 swaps with a value of 7 or less, and player 2 exchanges with 8 or lower, player 1's winning probability reverts back to 51.1855%, maintaining the expected value at 3144/132,600.
• Thus, when player 1 switches with a value of 6 or lower, player 2 should respond by switching with a value of 7 or less. If player 2 switches with 7 or less, player 1 should ideally switch with 7 or less as well. Similarly, if player 1 switches with 7 or lower, player 2 should consider switching with 8 or lower. This back-and-forth decision-making continues and takes on a resemblance to a game of rock-paper-scissors.

After some analysis of the game theory involved, realistically, player 1 should always switch when holding 6 or less, stay with 8 or higher, and switch with a 7 60% of the time. Player 2 should opt to switch with 7 or less, hold steady on 9 or above, and switch with 8 30% of the time.

If at least one player adheres to this strategic approach, player 1’s chance of winning will be approximately 51.2534%, yielding an expected value of 2.5068%. Should either player deviate from this strategy, the other player may recognize the change and take advantage of it in future rounds.

I present a resolution to a similar game theory dilemma in my work.

Utilizing the same Stewart Ethier rules mentioned earlier, here are the insights I gained from a three-player game. If a player receives a more favorable card than one they initially passed, it is clear they should hold that card rather than return it; otherwise, they must pass along the lesser card to the next player unless obstructed by a king. MathProblems.info site, problem 192.

Three-Player Case

If Player 1 decides to hold, Player 2 should consider switching with a card of 6 or less.

If nobody has a king then:

1. Player 1 should switch with 6 or less.
2. If both Players 1 and 2 decide to hold, Player 3 should exchange with a card of 7 or less.
3. In the event that Player 1 has switched and Player 2 has stayed, Player 3 should then opt to switch with a card of 4 or lower.
4. If Player 1 is holding a king, Player 3 should switch with the dummy card holding 6 or less.

Conversely, if Player 3 possesses a king, then Player 1 should exchange with Player 2 using 6 or lower.

In the way that I played Screw Your Neighbor while in Seal Beach, California, the rules mirrored those of Stewart Ethier, with the exceptions being that (1) all players who matched the lowest card had to contribute to the pot, and (2) a king did not act as a blocker if the dummy held that card. Apologies, but I have yet to formulate a strategy for these specific rules, which would indeed present a greater challenge.

Seal Beach Rules

In the video that follows, I illustrate the process of playing Screw Your Neighbor. We adhere to the rule allowing players with kings to keep their cards hidden unless they opt to switch. Additionally, we employed the rule where aces are considered high, which typically is not the norm.

Video

by Stewart N. Ethier. This college textbook includes extensive discussions on the mathematical aspects of the two-player version of Screw Your Neighbor, which it refers to as 'Le Her.'

Links

• In 'The Doctrine of Chances: Probabilistic Aspects of Gambling', the game is conducted using a standard deck of 52 cards. In this game, kings hold the highest value while aces are treated as the lowest. Strategies grounded in mathematics and insights for casino games such as blackjack, craps, roulette, and countless others are available for those interested.
• Ranter-Go-Round , as the game is titled at Wikipedia.
• cahillfamily.com has a good explanation of the rules

Written by: Michael Shackleford