自1961年William Vickery提出拍賣理論之雛形，許多人相繼投入這片領域。無論現實活動與遊戲競賽，不乏憑藉固有資源累積小籌碼以獲得最後勝利的情形，如選舉、商業貿易之區域競爭，抑或桌遊「上流社會」、「鬼屋拍賣」，都符合這種概念。我們建立一種賽局模型，試圖模擬並找出此情況下的最佳策略：玩家利用有限的初始預算，在有限局數中每局以第一價格暗標(first-price sealed-bid auction)競標，物品價值由給定的機率分佈函數所決定，得標者付出競標金額並得到該回合物品，最後取得總物品價值最高者獲勝。競標者必須審慎分配其預算至各局，才能在關鍵時刻拿下關鍵局數。
利用極小化極大演算法(Min-max algorithm)與動態規劃(dynamic programming)，藉由定義「勝利倒數值」(countdown value)，我們求得對任意局數T之賽局的理想預算比，以及相應的完整競標策略，亦即當初始預算是對手的幾倍時，能確保玩家獲勝。我們更進一步找到動態規劃矩陣的解析解，將演算法的時間複雜度降至O(T)。若允許玩家最多輸對手不超過給定的分差，經由平移原始矩陣，可以在僅改變初始條件的情況下，求得理想預算比。對全支付(all-pay)的拍賣形式，也有同樣成果。

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.

1 Introduction 3
2 Preliminaries 7
2.1 The Game Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Finding the Optimal Budget Ratio 9
3.1 OBR Countdown Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Getting OBR from the Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Further Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 All-Pay Auction 15
4.1 Closed Form of OBR in All-Pay Auction . . . . . . . . . . . . . . . . . . . . 16
5 Valuable Turn Fixed 18
6 Conclusions and Future Work 21
6.1 Analysis of Integer Bidding . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Reference 25
Appendix A 27
Appendix B 28

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