Abstract
In a time-spanning competitive environment, at each time a player competes by investing some of her budgets or resources in a battle to collect a value or prize if winning the battle. There are multiple battles to fight, and the budgets get consumed over time. The final winner is the one that collects the largest amount of total value. Examples of such competition include real-world campaigns for elections, and some computer or board games. A player needs to make adequate sequential decisions to accumulate small winnings to dominate against dynamic competition over time from the others possibly along with external factors. We are interested in how much budgets the players would need and what actions they should take over time in order to perform well. We model and study such dynamic budget-constrained competition where each battle is a first-price or all-pay auction. We focus on analyzing the 2-player budget ratio that guarantees a player's winning, or falling behind in just a bounded amount of collected value, against the other omnipotent player. In the settings considered, we give efficient dynamic programs to find the optimal budget ratios and the corresponding series of bidding strategies. Our definition of game, budget constraints, and focuses on budget analyses have not been observed in the related context.