Guaranteed To Win: Optimal Strategies For Discrete Bidding Games

Abstract

Many of us are familiar with two player games, such as Tic-Tac-Toe or chess, where each player alternates taking turns. Players compete against each other, strategically making a move once it’s their turn. The goal of the game is simply to “win”, depending on the rules of the game. We can add an extra layer to these games that creates some mathematical questions. Instead of alternating turns, players are now “bidding” to make a move. Not only does this add more competition, strategy, and excitement to the game, but it also adds mathematical intricacies. We call these Richman games, studied by David Richman in the 1980s. In Richman games, players make a bid (or auction)[1] of a nonnegative number of chips to make a move. The player that bids the most plays their turn, and then “pays” their chips to the other player. By studying Richman games, this paper will explore the optimal bidding strategies to maximize game play. The goal of each player is to win the game - not have the most amount of chips. In order to win the game, players need to have bidding strategies to ensure they are making moves. The proportion of chips a player has in their possession at a certain point, or critical threshold, is crucial within bidding games. We will explore how to find the critical threshold for games, and how it optimizes a player’s chance of winning (also referred to as winning strategies). We will also dissect the use of the tie-breaking advantage when two players bid the game amount of chips. Through these strategies, we will explore a game of bidding Tug O’ War and applications to more extensive games, such as bidding Tic-Tac-Toe.