probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

and

game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

, the Waldegrave problem refers to a problem first described in the second edition of

Pierre Raymond de Montmort Pierre Remond de Montmort was a French mathematician. He was born in Paris on 27 October 1678 and died there on 7 October 1719. His name was originally just Pierre Remond. His father pressured him to study law, but he rebelled and travelled to E ...

`s '' Essay d'analyse sur les jeux de hazard''. This problem is remarkable in that it is the first appearance to a mixed strategy solution in game theory. Montmort originally called Waldegrave's Problem the ''Problème de la Poulle'' or the Problem of the Pool. He provides a

minimax Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When de ...

mixed strategy In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...

solution to a two-person version of the card game

le Her Le Her (or ''le Hère'') is a French card game that dates back to the 16th century. It is quoted by the French poet Marc Papillon de Lasphrise in 1597. Under the name ''coucou'' it is mentioned in Rabelais' long list of games (in Gargantua, 1534). ...

. It was Isaac Todhunter who called it Waldegrave's Problem. The general description of the problem is as follows: Suppose there are n+1 players with each player putting one unit into the pot or pool. The first two players play each other and the winner plays the third player. The loser of each game puts one unit into the pot. Play continues in like fashion through all the players until one of the players has beaten all the others in succession. The original problem, stated in a letter dated 10 April 1711, from Montmort to Nicholas Bernoulli is for n = 2 and is attributed to ''M. de Waldegrave''. The problem, according to Montmort, is to find the expectation of each player and the probability that the pool will be won within a specified number of games.

References

Sources

* Game theory Probability problems {{mathematics-lit-stub