There is a flavor of non-sequential learning games in a well-known saying of Confucius: “Consistency is the virtue of fools and wise people change their minds as they grow wiser.” The formulation of common knowledge is not obvious, but commonly believed to be due to Allmann (1976). However one can also sense the notion of common knowledge in Confucius’ dialogue with Ming, which runs as follows: “I know that you know, you know that I know, I know that you know that I know, and so on” (see Last Emperor of China).
Economists began to realize the importance of limitation on the information possessed by individuals in understanding economic behavior because such limitation induces agents to change their behavior. The standard assumptions of perfect competition, that individuals are mere price takers, is no longer relevant. Rather, the strategic interactions have potentially profound implications on the behavior of agents in the decision-making process by altering behavior in the rest of the market. Game theory is well suited to modeling takeovers because of the importance of the information and its ability to include a number of sharply delineated sequences of moves and events. Precommitment and information transformation are the two pillars of modern game theory. Thus, the stylized facts and rationality of game theory may be more appropriate for markets in corporate control than for vegetable markets in developing countries.
In the business world, the power of game theory as a management tool rests on reasonably comprehensive assumptions that are imbedded in the rules of the game. Players can experiment with different solutions and concepts to problems that are intrinsically insoluble. In other words, there are no unique solutions to the problems. The analysis of the results can be used for greater insights into the real problems the game simulates. In a game involving a large number of players using a wide range of strategies, it is possible to identify strategies that do better than others even if there is no unique correct strategy at all. Allmann (1987) defines game theory as a sort of umbrella or “unified field” theory for the rational side of the social science, where “social” is interpreted broadly to include human as well as nonhuman players (computers, animals, plants, etc.).
In game theory, the prisoner’s dilemma is commonly used to describe certain real-world problems. The central characteristics of a prisoner’s dilemma are an array of benefits and detriments associated with alternative course s of action so that the dominant individual strategy is not to cooperate even though, if the parties do not cooperate, pursuit of individual self-interest yields less than optimal results.
There are a wide range of applications of game theory in finance. Typical examples of models are signaling through information transmission in corporate takeovers, capital structure as a commitment, and incentive design for financial intermediation. Game theory has been applied in other literature in finance, for example, market microstructure, executive compensation, dividends and stock repurchases , external financing, debt signaling, etc.
Application 1—The Theory of Corporate Takeover Bids
Grossman and Hart (1980) explain a particular free-rider problem using a game-theoretic model with a continuum of players. Suppose that under status quo management, a corporation has value v and if a raider can improve the target’s value by x, then its potential value is v + x. If the takeover bid is conditional and v <p < v + x i.e. pricep is below the potential value, no shareholders will sell, even though shareholders and management would jointly profit. The shareholders are in the prisoner’s dilemma and if the takeover bid is to be successful, then a holdout is better and the shareholders no worse off if it fails. Hence, tendering is not a dominant strategy. So every shareholder hold s out and in Nash equilibrium, takeover will never occur. Grossman and Hart strongly argued in favor of exclusionary devices by suggesting that the raider be allowed to dilute the value of the minority shareholder if the raid is successful.
Shleifer and Vishny (1986) point out that if the raider is a large shareholder and, if permitted to profit from secretly purchasing a proportion of shares prior to the tender offer, the free-rider problem can be solved even without dilution. The tender offer can be profitable because the raider can profit on his own share s even if he offers p > v + x and loses on the tendered shares.
Hirshleifer and Titman (1990) relax the assumptions of Shleifer and Vishny and present a model of tender offers in which the bid perfectly reveals the bidder’s private information about the size of the value improvement that can be generated by a takeover. They argue that bidders with greater improvements will offer higher premiums to insure that sufficient shares are tendered for majority control. They explain why offers succeed sometimes, but not always. Following Milgrom and Roberts (1982), nature moves first and chooses the raider’s “type” to be x s (0, 11 (CI’ *1) and the raider offers a premium of x for each of a proportion of shares. Each of the continuum of shareholders decides whether to sell or not to sell his shares. If over (0.5 – a) shareholders accept the tender offer, the payoffs are p for those who accept and v + x for those that refuse. Otherwise, all payoffs are zero.
Bradley and Kim’s (1985) analysis of the free-rider problem demonstrated that a necessary condition for a tender offer to be successful is that it should be front-end loaded and this condition should hold regardless of whether the tender offer is a partial or two tier. This is another application of the prisoner’s dilemma. Suppose a corporation is equally owned by two shareholders and its underlying value is US$80. Let the raider make a tender offer in which 51 percent shares will be purchased at a price of US$50 and the remaining 49 percent offered a lower price of US$25 on the condition that 51 percent shareholders tender. If both tender, the share will be purchased pro rata. Under the se conditions, tendering is a dominant strategy, even though all the shareholders would be better off refusing to sell. It is argued that twotier tender offers must be outlawed because of their coercive nature. However, Bradley and Kim see no reason to outlaw two-tier offers because it helps reallocate corporate resources to their highest valued use. This allows for greater flexibility in financing takeover activity by reducing the amount of cash that a potential raider must accumulate to pursue an acquisition. They further suggested that the potential for competition among raiders and a dominating intra-firm tender offer can solve the prisoner’s dilemma.
Deman (1991, 1994) re-examines Grossman and Hart’s (1980) paper and shows under complete and imperfect information that the prisoner’s dilemma can be solved. The existence of the mixed strategy symmetric equilibria with or without the dilution shows that we do not really need assumptions of a continuum of play ers. Deman explores possibilities of two kinds of equilibria: one is “separating equilibria” in mixed strategies in which each type of raider behaves differently and the shareholders randomize their payoffs. The raider of a high type will not offer a low price because such an offer would more than likely not succeed and the raider would lose the potential gain on his initial shares. A less plausible class of equilibria are “pooling equilibria” in which different type s of raiders behave in the same way. However, pooling equilibria are ruled out by the reasonable “out-of-equilibrium belief” that price offers will signal the raider’s type. In that case, a low-type raider could profitably differentiate himself from the pooling equilibrium by offering a low price and the shareholder would accept his offer. In fact, a model of finitely many players under potentially confusing signals gives the same results as the continuum-of-players model in which the decision of any individual player does not affect the success o f the tender offer. Deman applies a corporate finance-game theoretic model to real-estate takeovers. For example, when considering the problem of the developer negotiating with landowners, a model of finitely many owners appears to be much more realistic. It is well known that takeovers do occur with positive probabilities in models with finitely many players. This result holds independently whether or not these many finitely owners believe that they have an impact on the success of the sale, as pointed out by Shleifer and Vishny (1986), Bagnoli and Lipman (1988), Bebchuk (1989), and Deman (1991).
Kyle and Vila (1991) investigated a model of takeovers in which “noise trading” provides camouflage and makes it possible for a large corporate outsider to purchase enough shares at favorable prices for a takeover to become profitable. Although the model accommodates the possibility of dilution (Grossman and Hart, 1980) and a large incumbent shareholder (Shleifer and Vishny, 1986), neither dilution nor a large incumbent shareholder is necessary for costly takeovers to be profitable. Noise trading tends to encourage costly takeovers that otherwise would not occur, and discourage beneficial takeovers that otherwise would occur.
Application 2—Capital Structure as Precommitment
Unlike the first example, this is a game under the assumption of symmetric information. The main focus of the game is on commitment rather than on information transmission. In the game, each firm purposely risks bankruptcy to create a conflict of interest between debt and equity that increases its aggressiveness in seeking market shares. The outcome is worse for the firms if they jointly avoid debt, because de bt lowers firms’ profits while helping the firm that uses it as a commitment tool.
Harris and Raviv (1988) focus on capital structure as an anti-takeover device because common stock carries voting rights while debt does not. The debt-equity decision may effect the outcome of corporate votes and may partly determine the corporate resources. Thus, incumbent management can use short-term financial restructuring as a tactic to influence the form of the takeover attempts and their outcom e, assuming that managerial ability to identify good projects is unknown. In a subgame, perfect reputational equilibrium, managers may choose too much safety compared with the shareholders’ optimum. If the firm issues debt, then this incentive aligns the manager’s interest with the interests of the shareholders and thus reduces their agency costs of debt. This implies higher optimal leverage when the manager is motivated by his personal reputation than when he is not. This result is different from that of Harris and Raviv.
Application 3—Financial Intermediation — An Incentive Design
In most models, the players begin with symmetric information, but they know that some players will later acquire an informational advantage over the others. The model that I am going to use here is an example of theory-based institutional economics. The purpose of this is to show that: (1) an intermediary is useful only if there are many investors and many entrepreneurs; and (2) incentive contracts have economies of scales compared to monitoring.
Diamond (1984) provides a model of financial intermediaries, so that M risk-neutral investors wish to finance N risk-neutral firms. Each entrepreneur has a project that requires 1 unit in capital and yields Q level of output, where Q is initially unknown to anyone. If Q < 1, the entrepreneur genuinely cannot repay the investors, but the problem is that only he, not the investors, will observe Q, so they cannot validate his claim Q < 1. The investors must rely on one of two things to ensure the truth: namely, monitoring or incentive contract. Under a monitoring scheme, each investor incurs a cost C to observe Q, which makes it a contractible variable, on which payment can be made contingent. The entrepreneur suffers a dissipative punishment S(x) under the incentive contract i f he repays x. The cost of monitoring is MC, while the expected cost of an incentive contract is ES. In the absence of an intermediary, if ES < MC, the incentive contract is preferred. The underlying idea behind the financial intermediary is to eliminate redundancy by replacing M individual monitors with a single monitoring agency. The intermediary itself re quires an incentive contract, at cost E S. To justify its existence, it should spread this cost over many entrepreneurs. If N = 1, the intermediary incurs a cost of C for monitoring and ES for its own incentive, whereas a direct investor-entrepreneur contract would cost only ES. In the above scheme, while information is still symmetric, the institution assumes a particular form to avoid information problems by contracting.
The main driving force behind the existence of financial intermediaries is the asymmetric information which opens doors for a much wider application of game theory. Reputational issues on the part of borrowers become very important and were first analyzed by John and Nachman (1985) in a two-period model. They depicted, in sequential equilibrium, a problem in which agency debt can be decreased when compared with a single-period model. Diamond (1989) uses a somewhat similar model in which borrowers deal with banks over more than one period and have an incentive to build a reputation for repaying loans. This provides a partial improvement of the agency problem in one-shot games in which the borrower prefers riskier investments than the lender would like.
Conclusions
Game theory has emerged as one of the most powerful techniques of analysis because, in the game, both players are actively trying to promote their own welfare in opposition to that of the opponent. It develops a rational criterion for selecting a strategy in which each player will uncompromisingly attempt to do as well as possible in relation to his opponent by giving the best response. However, game theory is often criticized on the grounds that it is sensitive to minor changes in assumptions and lacks empirical verification. The existence of various equilibria depends on what information is available to players or who moves first. Deman (1987) basically identifies three criteria for a theory to be considered useful: (1) it is consistent with known facts; (2) provides greater insights and understanding than earlier theories; and (3) it can be used for forecasting future trends, particularly under conditions that differ from the past. The underlying assumption is that both theorists and empiricists have common objectives to describe, explain, relate, anticipate, and evaluate phenomenona, events, and relationships crucial to decision making through theory construction and data collection. Unfortunately, crucial variables are hard to measure, but that does not diminish their importance. As Rasmussen (1989) pointed out, the economist’s empirical work has dominated case-by-case verification replacing the traditional regression running. A theory’s sensitivity to assumptions is not a shortcoming. Rather, it is a contribution of the theory, pointing out the important role of what were once thought to be insignificant details of reality in the world. To blame game theory for any failure to predict or for selfishness is like blaming cardiology for heart disease. The failure of macroeconomic forecasts and the growing importance of the microeconomic theory of the firm have brought game theory to the forefront of economic decision making.
