  In a multi-battle contest, each time a player competes by investing some of
her budgets or resources in a component battle to collect a value if winning
the battle. There are multiple battles to fight, and the budgets get consumed
over time. The final winner in the overall contest is the one who first reaches
some amount of total value. Examples include R & D races, sports competition,
elections, and many more. A player needs to make adequate sequential actions to
win the contest against dynamic competition over time from the others. We are
interested in how much budgets the players would need and what actions they
should take in order to perform well.
  We model and study such budget-constrained multi-battle contests where each
component 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 bidding strategies. Our definition
of game, budget constraints, and emphasis on budget analyses provide a new
perspective and analysis in the related context.
