Dominant Strategies in Egyptian Ratscrew
Egyptian Ratscrew (ERS) is a card game where you are rewarded mainly for your speed and reflexes, rather than your potential knowledge of other cards in play or in players’ hands. The deck is split among the players equally, and the goal of the game is to end up with the entire deck in your hand. The players place cards in the middle clockwise, grabbing from the top of their hand, never looking at their own cards or anyone else’s. There are multiple different rules, but the basics are that whenever a player places a face card, the player next in the rotation has a certain amount of attempts, where the cards they place must also be a face card, or the player that initially placed the face card is able to grab all the cards from the middle. For instance, if player 1 places a Jack, player 2 has one attempt to put a face card, and if the card that the player places in the middle is not a face card, player 1 grabs all the cards from the middle. In addition, any player is able to slap when specific, interesting combinations appear (for instance, a Queen right after a King or viceversa, or the same number placed twice in a row). Because of this, it was hard to imagine there is a dominant strategy, considering it’s a game mainly relying on reflexes and speed!
It was interesting to look at Diamond and Garcia’s paper, rooted on using computational game theory to find out a dominant strategy for ERS. Turns out, there is one! It still requires speed, since you still have to slap the deck of cards, but it’s much more mechanical and requires less thinking or observation. Basically, a player should “risk slap” whenever a card is played after a face card. Even though risk slapping and not being able to pick up cards will warrant a punishment, on average, the reward from risk slapping will be much greater than the punishment, as well as greater than the reward from “reflexive slapping”, which is trying to slap as fast as possible when a combination appears. Clearly, risk slapping is a dominant strategy, and has a much better win rate against reflexive slapping (90.689% in 2-player games, 73.118% in 4-player games against reflexive slappers). The researchers also explored different ways of risk slapping (e.g. slap when there are n cards in the deck, or only on certain face cards), but still concluded that risk slapping on face cards was the dominant strategy.
Source: https://scholarspace.library.gwu.edu/concern/gw_etds/5425kb44d?locale=en