ISSN 0032-9460, Problems of Information Transmission, 2013, Vol. 49, No. 2, pp. 163–166.
Pleiades Publishing, Inc., 2013.
Original Russian Text
E.L. Lakshtanov, A.I. Aleksenko, 2013, published in Problemy Peredachi Informatsii, 2013, Vol. 49, No. 2, pp. 73–77.
Finiteness in the Beggar-My-Neighbor Card Game
E. L. Lakshtanov and A. I. Aleksenko
Department of Mathematics, Aveiro University, Portugal
Received August 26, 2011; in ﬁnal form, February 19, 2013
Abstract—For card games of the Beggar-My-Neighbor type, we prove ﬁniteness of the math-
ematical expectation of the game duration under the conditions that a player to play the ﬁrst
card is chosen randomly and that cards in a pile are shuﬄed before being placed to the deck.
The result is also valid for general-type modiﬁcations of the game rules. In other words, we
show that the graph of the Markov chain for the Beggar-My-Neighbor game is absorbing; i.e.,
from any vertex there is at least one path leading to the end of the game.
In , ﬁniteness of the mathematical expectation of the number of moves in the card game of
War was proved for various modiﬁcations of the game. Recall the simplest version of this game:
a deck consists of cards of a single suit, and at the beginning of the game the deck is divided evenly
among two players. Each player keeps his cards as a stack; all cards are kept face down. Players
reveal the top card of their stack, and the player with the higher card takes both cards played
and moves them to the bottom of his stack. For versions of this game, see . Mostly, variations
concern the number of cards in the deck, card ranking (in a Russian version, the lowest card wins
against the ace), and also the players’ actions if the two cards played are of equal value. The game
of War can naturally be considered as a Markov chain, since the order in which cards of a pile are
returned to the deck is not speciﬁed by game rules.
The Beggar-My-Neighbor game (there is an economic term of the same name , which originates
from the name of the card game) was mentioned as early as in Charles Dickens’s 1860 novel .
The rules of this game are deterministic: At the beginning of the game, the deck is divided equally
between two players, and the two stacks of cards are placed on the table face down. Cards are not
shuﬄed during the play; each player can play only the top card of his stack. The players alternately
lay down their top cards face up, until an honor card (ace, King, Queen, or Jack) appears; in this
case, the other player has to pay a “penalty” by laying down a certain number of his cards: four
cards for an ace, three for a King, two for a Queen, or one for a Jack. However, if a player turns up
another honor card in the course of paying to the original penalty card, his payment ceases and the
other player must pay to this new card. If a penalty is paid completely, the player of the penalty
card wins the hand, takes all the cards in the pile and places them under his pack. As is seen from
the description, it is essential who started the game, i.e., played the ﬁrst card. A player who takes
a pile is always the one to start the next pile! If a player runs out of cards, he loses.
One can consider other games where rules determining the process of the play are changed: the
number of cards required to pay the penalty can be changed, as well as the set of penalty cards
Supported by the European Union Funds FEDER/COMPETE, project no. FCOMP-01-0124-FEDER-
022690, coﬁnanced by the Portuguese Foundation for Science and Technology (FCT, Fund¸c˜ao para a
Ciˆencia e a Tecnologia), project no. PEst-C/MAT/UI4106/2011.