Chapter 1　Introduction

One ordinary morning in 356 BC, standing aside a 7-meter-long slender wood at the south gate of the capital of the State of Qin, Shang Yang, the then chief advisor of the state, announced to a crowd of citizens that the government would give 3.8 kg brass as a reward to the first man who managed to move the wood from the current position to the north gate of the city. The task was of course not so easy that everybody could finish it without any difficulty, nor was it too difficult to achieve for any well-trained manual worker. The reward, on the other hand, was tremendous, considering that brass was the most precious hard currency at that time. However, no volunteers appeared: Although the reward seemed compelling, the crowd believed that the announcement was a prank played by some noble to make fun of them, and the incredibly huge reward could never be realized.

Widely regarded as the most successful reformer in Chinese history, Shang Yang did not play the game for fun. Prior to the scene above, with support from the Duke Xiao of Qin, Shang Yang proposed a reform plan, which consisted of a package of new laws that encouraged agricultural production, provided promotions and rewards based on contributions rather than kinship, reduced privilege items of the nobility, allowed for change of social class, opened the market for land transactions, etc. The main goal of the reform was to stimulate ordinary citizens to contribute in both internal developments and external wars, through devolving power and vested interests from the nobility. This kind of re-form usually would be opposed by most of the nobility, which explained why some much milder reforms carried out previously (for instance, Wu Qi's reform in the State of Chu) eventually failed. (1) Knowing this, Shang Yang realized that he needed support from the ordinary citizens to help his reform survive the attack by the nobility. To achieve this goal, the first thing Shang Yang had to do was to build the trust between the reformers and citizens. By rewarding any man who finished such a non-difficult task, his plan was to make the citizens believe that the government would always commit itself to its own promises. Observing that nobody would like to go for the reward, Shang Yang reannounced that the government was serious about this rewarding policy and the reward increased to 19 kg brass. Finally, a volunteer came out and successfully moved the wood to the north gate. Shang Yang rewarded him the 19 kg brass immediately. The package of reform plans was then enacted and announced to the public. Because of the previously established trust in the government, nobody doubted its resolve in implementing the new laws. Finally, although the reform was a very thorough and extreme one, it was successfully carried out and indeed built the foundation that enabled the State of Qin to eliminate all other states of China, and to unite the whole country for the first time as the Qin dynasty.

The story above illustrates the two themes of the present works: ①In scenarios where a significant change happens if and only if a critical mass of individuals agree on it, there usually exist multiple subgroups whose interests are in conflict with each other. ②In such environments, public signals matter. Chapter 2 is mainly devoted to the first theme. It compares the efficacies of different intervention policies aiming to stabilize a regime in which there are both stakeholder and speculator groups. Chapter 3 deals with both themes. We study a modelwith the coexistence of investors, a speculator, and an authority, with a focus on the signaling effects of the authority's and the speculator's actions. Chapter 4 investigates a game with features belonging to the second theme. It checks whether or not the publicly observed actions taken by the winners of previous stages push players' equilibrium actions to converge in a repeated game.

Although the specific economies investigated in the subsequent chapters are different from each other, they do have a remarkable feature in common: the strategic complementarities between individuals. In the regime-changing context, the larger the mass of attacking individuals, the safer it is for an individual to attack. In the context of financial crises, the more holding investors there are, the more attractive it is for any investor to hold; and given more others withdrawing, it is less attractive for an investor to hold. In the beauty-contest scenario, e. g., in a stock market, it is rational for an investor to take her action based on some noisy but publicly observed signals (act as a momentum investor) if others are mainly momentum investors, and to behave according to her understanding of the fundamental of the corresponding company (act as a fundamentalist) if others are mainly fundamentalists.

For games with strategic complementarity under complete information, Topkis (1979) and Vives (1990) show the existence of pure strategy equilibria and that the equilibrium set is indeed a lattice with a maximum element and a minimum element, which are often different. As a result, there usually are multiple equilibria, and the outcome of the game is crucially determined by which equilibrium is chosen by individuals to coordinate on. Diamond and Dybvig (1983) use such a model to describe the scenario that a bank run happens just because depositors coordinate on a socially inferior equilibrium, rather than because of a deterioration in the fundamental.

By absorbing the techniques and treatments used in the global games investigated by Carlsson and van Damme (1993), Morris and Shin (1998, 2000, 2002) introduce incomplete information to games with strategic complementarities, and construct their forms of global games suitable for studying scenarios such as regime changes, financial crises, price bubbles, and so on. Instead of making the fundamental variable (e. g., the strength of the authority in the regime-changing model, the profitability of the investment portfolio in the bank-run model, the "true beauty" in the beauty-contest model) perfectly observable to all players, their models assume that all players receive noisy and private signals about the fundamental variable. They further argue that their settings have several merits. Firstly, it is more realistic to introduce the incomplete information to the model, since individuals often cannot observe perfectly the underlying fundamental, but instead have to deduce it based on their private information. Secondly, after introducing the incomplete information, under parsimonious conditions, there exists a unique equilibrium. What is more, the result of the game is determined by the underlying fundamental, e. g., a financial crisis happens if and only if the fundamental deteriorates to some threshold. This thus provides a more reasonable explanation for the occurrence of financial crises, and enables the researchers to investigate effects of other parameters and policies through comparative statics analysis. These models are further discussed and summarized in Frankel et al. (2003) and Morris and Shin (2003), and are widely adopted in the literature to explain regime changes, bank runs, financial crises, etc. The examples include Morris and Shin (2004); Rochet and Vives (2004); Goldstein and Pauzner (2005); Corsetti et al. (2006); Edmond (2013).

Global games have also been extended to contexts with dynamic settings. These works can mainly be split into two categories. In some of them, e. g., Allen et al. (2006) and Amador and Weill (2006), the authors want to depict the equilibrium dynamics of some variables. As a result, games admitting a unique equilibrium are used in these works. The work in Chapter 4 belongs to this category. In the other papers, the main objective of study is to investigate the informational roles of the endogenously formed public signals. Many of them regain the multiplicity in the equilibrium structure, and thus re-explain the occurrence of crises and bank runs as mis-coordinations between individuals. Examples in this strand include Atkeson (2000); Angeletos et al. (2006, 2007); Angeletos and Werning (2006); Costain (2007).The dynamic games investigated in Chapter 2 belong to the latter category. The work in Chapter 3 shares some common features with many papers in the second category. For instance, we focus on the speculator's and the authority's signaling effects on the investors'strategies, and obtain the multiplicity property for some cases and use it to explain the inefficacy of certain intervention policies designed to prevent a financial crisis from happening. However, it also differs from those works, in that we are not merely interested in whether or not introducing endogenous public signals helps regain the multiplicity. Actually, we refine the initial perfect Bayesian equilibrium set using some selection rules and focus on solutions that are the most likely to occur and interest us the most.

In Chapter 2, we investigate global games for regime changes. Observing that different social classes are often treated differently by the distribution rule embedded in a given regime, we divide the traditional continuum of players in the benchmark model into a stakeholder group and a speculator group. Compared to the speculators, stakeholders receive extra payoffs, the vested interests, if the regime survives the attack in the end. It is found that a unique equilibrium exists in the revised game. Comparative statics analyses are then adopted to investigate how changes in the fraction of stakeholders, variances of the noises, and the potential punishment for attacking stakeholders affect the vulnerability of the regime.

The static game is then extended to two-stage ones in which the authority has the chance to stabilize the regime through intervention policies that allow for changes of players' labels. Specifically, we consider two forms of label changing mechanisms: one allowing for downward social mobility and the other one that allows for upward social mobility. It is shown that although the former mechanism helps stabilize the regime in the first stage, it may increase the vulnerability of the regime in the second stage because of the multiplicity problem. In contrast, the latter mechanism generally decreases the vulnerability compared to the benchmark model.

Chapter 3 investigates global games in a bank-run context. The speculators have long been criticized for making the financial system more vulnerable, and many suggestions about what should be done to deal with it have been put forward in both the literature and the mass media. However, little effort has been given to formalizing theses statements and concerns in a game-theoretical model. The work in this chapter tries to address this issue. We first add a speculator to a benchmark model. It is found that in the resulting two-stage game, the existence of the speculator increases the vulnerability of the financial system, especially when the attacking cost is relatively low.

To counteract this effect, we compare three different intervention policies imposed by the authority: deterring the speculator, rewarding holding investors, and eliminating the preemption motives among investors. We argue that the first method may not work because of the multiplicity problem; the second one is almost useless when a crisis is about to happen; the last tool works given enough effort. We then use a discussion of different intervention polices employed by governments during the 1997 Asian Financial Crisis to illustrate the theoretical results. Specifically, we explain why the capital regulation policies did not work, why interest rate defense policies could not stop capital flights, how Hong Kong's intervention worked, and how the IMF programs failed to rescue the participating countries by impeding them from eliminating the preemption motives.

In Chapter 4, we build a dynamic game consisting of a continuum of players to investigate the effect of previous winners' actions on the spread of subsequent players' actions. In each stage, besides the private signal, each player also observes actions taken by the winners of all previous stages as public signals. A unique equilibrium of the game is found and characterized. We then define variances of three forms of gap: variance of the gap between the average play and the underlying fundamental value, variance of the gap between a generic player's action and the average play, and the variance of the gap between a generic player's action and the winner's play. By checking their dynamics in the equilibrium, it is shown that the accumulation of private signals always reduces the first variance and the accumulation of the public signals always reduces the second variance. However, the accumulation of public signals reduces the first variance if and only if the public signals are sufficiently precise compared to the private ones, and the accumulation of private signals reduces the second variance if and only if the private signals are sufficiently precise compared to the public ones. We also show that the third variance is a weighted sum of the other two, which turns out to be useful in applications where the fundamental is unobservable.

Based on the theoretical results, we conduct an empirical study on the last twenty editions of the Miss Korea contest. We find a descending trend in the variance of the gap between the average face and the underlying "true beauty" face over these years. Moreover, this process is accompanied by ascending trends in the other two variances, indicating that contestants' faces have been converging to the "true beauty" overall but diverging from each other over the two decades.

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(1)See Sima (2011) for the details of Wu Qi's reform as well as Shang Yang's.