This dissertation consists of three essays studying the theory of dynamic contest. This analysis mainly focuses on how the outcome and the optimal design in a dynamic contest varies on contest technology, heterogeneous players, contest architecture, and bias instruments. The first chapter outlines the dissertation by briefly discussing the motivations, methods, and main findings in the following chapters. Chapter 2 considers a situation in which two groups compete in a series of battles with complete information. Each group has multiple heterogeneous players. The group who first wins a predetermined number of battles wins a prize which is a public good for the winning group. A discriminatory state-dependent contest success function will be employed in each battle. We found that in the subgame perfect Nash equilibrium (equilibria), the lower valuation players can only exert effort in earlier battles, while the higher valuation players may exert effort throughout the entire series of battles. The typical discouragement effect in a multi-battle contest is mitigated when players compete as a group. We also provide two types of optimal contest designs that can fully resolve the free-rider problem in group contests. Chapter 3 investigates optimal contest design with multiple heterogeneous players. We allow the contest designer to have one or multiple/mixed objectives, which includes the following parts: the total effort; the winner's effort; the maximal effort; and the winning probability of the strongest player. We provide a one-size-fits-all contest design that is optimal given any objective function. In the optimal contest, the designer will have one of the weaker players exhaust the strongest in the contest with infinite battles. We obtain the required conditions on different contest frameworks (e.g., all-pay auctions and lottery contests) and bias instruments (e.g., head starts and multiplicative bias). This means the contest designer has multiple alternatives to design the optimal contest. The last chapter investigates a situation where two players compete in a series of sequential battles to win a prize. A player can obtain certain points by winning a single battle, and the available points may vary across the battles. The player who first obtains predetermined points wins the prize. We fully characterize the subgame perfect Nash equilibrium by describing the indifference continuation value interval. We found that when two players are symmetric, they only compete in the separating battle. In the general case, we found that winning a battle may not create any momentum when the weight of the battle is small. A small enough adjustment of a battle's weight will not change both players' incentive to win the battle. Increasing (or decreasing) a battle's weight weakly increases (or weakly decreases) both players' incentive to win. / Doctor of Philosophy / A contest in economics is defined as a situation in which players exert positive effort to win a prize. The effort can be money, time, energy, or any resource that is used in a competition. The prize can be monetary or other perks from winning a competition. In this dissertation, we explore dynamic multi-battle contests where the winner is not decided by one single competition but by a series of sequential competitions. For example, the US presidential primary begins sometime in January or February and ends about mid-June and candidates will compete in different states during the time. In NBA finals, the winner is decided by a best-of-seven contest. The team that first wins four games becomes the champion. In the second chapter, we explore multi-battle group contest in which each group has multiple heterogeneous players. The group who first wins a certain number of battles wins a prize. The prize is a public good within the winning group so players in the winning group can enjoy the prize regardless their effort. We found that players with high prize valuation will be discouraged in earlier battles due to high expected effort in later battles. This may make high-value players only exert effort in later and more decisive battles. The low-value players will exert effort in earlier battles and will free rider on high-value players in later battles. We also provide the optimal contest design that can fully resolve the free-rider problem. In the optimal contest design, the designer should completely balance two groups in every battle. In the third chapter, we explore the optimal contest design in the multi-battle contests with multiple heterogeneous players. The contest designer can have one or multiple/mixed objectives. We found a "one size fits all" multi-battle contest design that is optimal for various objective functions. In the optimal contest design, the designer should give different advantages to the strongest player and one of the weaker players. More specifically, the weaker player is easier to win each battle, while the strongest player needs to win fewer battles. This overturns the conventional wisdom that the advantage should be only given to the weaker players. In the fourth quarter, we explore the multi-battle contest that in which each battle has a different weight, that is, some battles may more or less important than others. We found that when a battle's weight is small, players may feel indifference between winning or losing the battle. Therefore, winning such battles will not create any momentum, and players tend to give up those battles by exerting no effort. We also found that when we increase or decrease a battle's weight, if the adjustment is small, it will not change players' incentive to win a battle. However, if the adjustment is large enough, it will increase or decrease players' incentive to win in the same direction.

In this series of essays, we study the influence of weight and group size in the sequential multi-battle contest with laboratory experiences (Chapter 2 and Chapter 3). We then develop an empirical method to model perceptual present and time inconsistency (Chapter 4). Chapter 2 examines how the weight and the ordered weights in battles affect the behavior in sequential multi-battle contests with an experiment. We find robustly that the weight of the current battle consistently influences contestants' efforts. Additionally, we discover the math-point-oriented behavior despite differences in history. In other words, the weight effect is expressed in two ways: influencing the effort of the current battle and transferring a contest to the next battle with a designated intensity. Chapter 3 explores the group size effect and how the contest success functions influence the group size effect in sequential multi-battle contests with an experiment. We capture the negative group size effect on the leaders' efforts, participation and dropout rates; contrarily, the positive effect on the non-leaders' efforts. Compared to the Tullock lottery, the all-pay auction intensifies the group size effect of the high effort in the initial battle. It also enlarges the observed group size effects of the effort gaps between the leaders and the non-leaders. Chapter 4 develops the quasi-hyperbolic discounting model into the general beta-delta model to parametrically detect and measure the inconsistency in revealed time preference. This method empirically classifies time preference into four categories, i.e., time consistent, present bias, future bias, and mixed inconsistent. Then we applied this method to the convex time budget data of seven experiments, including 3670 subjects. We discover empirical evidence supporting perceptual differences in the present-future threshold. Traditional present bias models may interpret the time preference imprecisely. / Doctor of Philosophy / Competition and Time are two essential aspects of life. Many decisions are made in a competitive environment. Some other decisions are made when time serves as a critical factor. We divide this dissertation into two parts. In the first part, we study strategic behavior in competitions. Specifically, we examine how (1) the importance of each round (weight), (2) the number of competitors, and (3) the ambiguity of the rule affect the result of a multi-round competition. In the second part, we study people's subjective understanding of time, generally the personal beliefs and preferences of the present and future. In part one (Chapter 2 and Chapter 3), first, we find people are very responsive about the importance of a round in a multi-round competition. When a round is more important, people make more effort in such round. People are also sensible about the competition's current status (leading, behind, or tied) rather than the history. Second, at the beginning of a competition, we find an increase in the participation rate when fewer competitors exist. Suppose there are more competitors; the leading position players compete more brutally; on the contrary, the non-leading players are discouraged more. Third, people spend more energy when the rule is less ambiguous in a multi-round competition. In part two (Chapter 4), We find a very diversified subjective belief in the word "present." The concept of "now" lasts longer than we conventionally thought. When the subjective "present'' is captured at the individual level, we find the immediate now is not necessarily the best way to represent the "average present'' for the population.