How to Play Dundee

TL;DR

This paper analyzes the game of Dundee, determining optimal strategies for different variants, including adaptive and pre-planned bidding, and provides solutions for specific deck configurations and game lengths.

Contribution

It offers the first comprehensive analysis of optimal strategies in Dundee, solving both adaptive and fixed-sequence variants for certain deck structures.

Findings

Optimal strategies are characterized for adaptive bidding.

Explicit solutions are provided for fixed-sequence bidding when deck sizes are equal.

The problem is solved for arbitrary game lengths with equal deck sizes.

Abstract

We consider the following one-player game called Dundee. We are given a deck consisting of s_i cards of Value i, where i=1,...,v, and an integer m\le s_1+...+s_v. There are m rounds. In each round, the player names a number between 1 and v and draws a random card from the deck. The player loses if the named number coincides with the drawn value in at least one round. The famous Problem of Thirteen, proposed by Monmort in 1708, asks for the probability of winning in the case when v=13, s_1=...=s_{13}=4, m=13, and the player names the sequence 1,...,13. This problem and its various generalizations were studied by numerous mathematicians, including J. and N. Bernoulli, De Moivre, Euler, Catalan, and others. However, it seems that nobody has considered which strategies of the player maximize the probability of winning. We study two variants of this problem. In the first variant, the player's bid in Round i may depend on the values of the random cards drawn in the previous rounds. We completely solve this version. In the second variant, the player has to specify the whole sequence of m bids in advance, before turning any cards. We are able to solve this problem when s_1=...=s_v and m is arbitrary.