Applying Game Theory in Finance

Table of Contents

List of Abbreviations

1 Introduction
1.1 Problem and Objective of the Paper
1.2 Organization of the Paper

2 Game Theoretic Foundations
2.1 Basic Definitions
2.2 Nash Equilibrium, Dominance, and Rollback
2.3 Asymmetric Information

3 Game Theory in the Context of Finance
3.1 First Game Theoretic Concepts in Finance
3.2 Enhancing Financial Theory with Game Theoretic Modeling

4 Selected Applications of Game Theory in Finance
4.1 Dividends as Signals of Future Cash Flows
4.2 Signaling and Agency Models of Capital Structure Decisions
4.3 Other Areas of Game Theory Application in Finance

5 Critique & Concluding Remarks

Appendix

References

List of Abbreviations

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1 Introduction

1.1 Problem and Objective of the Paper

The groundbreaking work of Modigliani & Miller (MM)^[1] introduced the rigors of economic analysis to financial research. This is “generally considered the beginning point of modern managerial finance.”^[2] Their first economic models were challenged by financial practitioners for being overly simplistic in their assumptions and therefore lacking real world application value.^[3] MM acknowledged and addressed this fact in their first paper.^[4] Later models relaxed some assumptions, such as symmetric information or complete contracts, while trying to retain an explanatory value in the spirit of the original MM papers.^[5]

This incorporation of more realistic elements, such as strategic interaction and asymmetric information, brought several problems to financial economists’ models: they required a lot of definitions, became even more complex, and were not easily comparable. Game theory provided a solution for those problems in its first applications to economics in the 70s and 80s: a set of common definitions and a basic language to guarantee comparability and empirical testability of financial models using game theoretic concepts.^[6] Nowadays, there are few issues in finance research which have not been modeled by applying game theoretic concepts^[7], and therefore it is crucial to be familiar with the basics of game theory and its application in finance.

The objective of this paper is to provide an intuitive approach to game theory in finance by first giving an overview of the basic foundations of game theory, and then providing a survey of some selected applications most relevant to the financial practitioner.

1.2 Organization of the Paper

The paper first develops the necessary foundations in chapter 2 to introduce the specific language and the most important theoretical concepts of game theory, and to demonstrate their application value in real world problems. The chapter serves to define game theory in the scope of this paper and explores the basic concepts needed to solve games of strategy. Special consideration is given to the incorporation of asymmetric information, as this area of research in game theory is most important to financial applications. Hence, it is central to the discussion in the following chapters.

This is shown in chapter 3, which explores how ideas of asymmetric information can be applied in finance. It aims to provide an intuitive approach to the subject by discussing early models of asymmetric information in finance. Building on those models, it demonstrates the usefulness of game theoretic concepts to enhance the economic modeling.

A survey of game theory in finance in chapter 4 integrates the prior chapters by showing the real application value of game theory with important historic and current examples. It provides a discussion of published papers using game theoretic concepts to enhance the understanding of two unresolved issues in finance: dividend and capital structure policy. To give a complete overview of game theoretic models in finance, it also provides selected examples from other areas, such as the market for corporate control, Initial Public Offerings (IPOs) and financial intermediation. However, taking the overall seminar topic into account this paper will mainly focus on corporate finance.

Finally, chapter 5 serves the purpose of closing the discussion with some critical remarks and drawing conclusions. Some current areas of research are addressed in order to indicate recent advances in game theory and the application possibilities in finance.

2 Game Theoretic Foundations

“Game theory comprises formal mathematical models of ‘games’ that are examined deductively.”^[8] These models provide three advantages: (1) a clear and precise language to transfer insights from one context to another, (2) the possibility to cross-check those insights for logical consistency, and (3) the ability to trace back conclusions directly to the assumptions.^[9] This chapter provides an introduction to the most important concepts of game theory and lays the ground for its applications in the context of finance.^[10]

2.1 Basic Definitions

Dixit & Skeath define game theory as “the science of rational behavior in interactive situations.”^[11] This definition serves as the basis of the following analysis to demonstrate the broad scope of situations game theory is able to encompass. Those interactive situations are called games of strategy, i.e. any interactive games which involve not only pure chance or skill, but also some amount of rational and strategic thinking.^[12] Games of strategy arise constantly in diverse areas, such as politics, business or sports. Game theory provides a methodology to analyze those games and to predict the outcomes.^[13]

In general, game theory can be divided into two branches. Co-operative game theory is concerned with games where “players can negotiate binding contracts that allow them to plan and implement joint strategies”.^[14] In non-cooperative game theory, such agreements are not enforceable; co-operation may arise only because such cooperative behavior is in the best interest of each individual on its own. As most games, in practice, do not provide adequate enforcement mechanisms, non-cooperative game theory is more useful in analyzing managerial decision making. Therefore, the analysis in this paper will focus on non-cooperative game theory.^[15]

A second important notion for classifying games is the distinction between simultaneous and sequential moves. In a game with simultaneous moves, also called static game, each player moves without being aware of the moves of the other players.^[16] The strategic thinking in such a game involves finding out what the opponent is going to do at that moment in time, and to act with the best response given this supposed action. On the contrary, in a game with sequential moves, also called dynamic game, players move in turns. In this type of game players are concerned with the calculation of future consequences, i.e. figuring out how other players might act at any later stage of the game and what the best response should be.^[17]

Simultaneous move games are normally analyzed in the normal or strategic form of the game, i.e. with a game table (also called a game matrix or payoff table) displaying the strategies available to each player and the associated payoffs in a format similar to a spreadsheet.^[18] Sequential move games on the other hand are normally analyzed in the extensive form, i.e. in game tree structures with nodes for the decision points and branches for the different actions available, leading to the final payoffs at the terminal node.^[19] In reality, combinations of both forms are necessary as both simultaneous and sequential moves appear at certain stages in most games.^[20]

Two words from the preceding paragraph warrant an explanation and definition in the context of game theory: payoffs and strategies. “A strategy is a complete plan of action”.^[21] This means that in a simultaneous move game, a strategy constitutes one of the choices available to the players. For a sequential move game it exactly specifies the player’s actions in every contingency, i.e. for all possible choices of all players.^[22] The results associated with the outcome of any combination of the players’ strategies are then called the players’ payoffs. Basically payoffs can be stated in any desired form under the condition that they capture everything the player cares about in the outcome of the game.^[23] Furthermore, the payoffs need to be stated as expected values if the player faces a random set of different outcomes in that particular situation.^[24]

One assumption mentioned in the general definition of game theory is central to the analysis: rational behavior. Game theory generally assumes that players are “perfect calculators and flawless followers of their best strategies […] Thus rationality has two essential ingredients: complete knowledge of one’s own interests, and flawless calculations of what actions will best serve those interests”.^[25] The rationality assumption is essential, otherwise the question about the degree of the irrationality of the players arises, and equilibrium analysis leads to circular reasoning and loses its predictive value.^[26]

2.2 Nash Equilibrium, Dominance, and Rollback

Equilibrium means “that each player is using the strategy that is the best response to the strategies of other players”.^[27] Equilibria can be found using different approaches depending on the specifics of the game. The two most important solution techniques for games with simultaneous moves are Nash equilibrium analysis and dominance.

A Nash equilibrium is “a configuration of strategies, one for each player, such that each player’s strategy is best for her, given that all the other players are playing their equilibrium strategies”.^[28] This definition needs to be narrowed down for the purpose of the paper. Only pure equilibrium strategies, i.e. strategies specifying a certain course of action without any uncertainty, will be considered in the following.^[29] There are several ways of identifying a pure strategy Nash equilibrium, a good one for finding all possible equilibria is inspecting the game table cell by cell and determining whether any player has an incentive to move given the supposed actions of the other players.^[30]

A more powerful method in determining the outcome of a simultaneous move game is dominance. A strategy is dominant if it “outperforms all other strategies, no matter what any opposing players do”.^[31] It is called strictly dominant if all payoffs are strictly greater than other strategies (> in mathematical terms) and weakly dominant if there are weak inequalities (≥ in mathematical terms). The strategies eliminated with this concept are called dominated strategies. From this argument it is obvious that the solution is very powerful: each player has clear incentives to play the dominant strategy if there is one.^[32] Nevertheless, as a lot of games do not have dominant strategies, Nash equilibrium analysis has to be used more often.^[33]

When moving from simultaneous move to sequential move games, the thinking involved in finding equilibria changes: given the chronological differences, solutions are found by looking ahead and reasoning back. This is done by examining the end of the game and determining which actions the last player will choose (equivalent to which ones will not be chosen). Reasoning back to the second-to-the-last player and determining the best strategy, given the choices of the last player, starts a series called rollback reasoning. It is used at each node until the initial node is reached and is therefore called rollback (or backward induction); the outcome is known as rollback equilibrium.^[34]

2.3 Asymmetric Information

One assumption underlying the equilibrium analysis described in the previous chapter is that of complete information about all moves, outcomes and probabilities, i.e. all possible states of the games. This chapter explores the concepts game theory offers to address situations with asymmetric information, i.e. situations where players are informed differently about certain aspects of the game.^[35]

There are two situations considered in the literature: imperfect and incomplete information. Imperfect information means that at least one player by the time of at least one action in the game does not know what outcome an action by a preceding player had brought about.^[36] If at the beginning of the game all players have symmetric information, but later one of them chooses an unobservable action influencing the outcome of the game, it is called a hidden action, otherwise known as moral hazard.^[37] Incomplete information means that a player is informed better about certain given aspects of the game, i.e. this player has private information. If the private information, often called the informed player’s type, is known to the informed player at the beginning of the game, it is called a situation of adverse selection.^[38]

The two most important strategic concepts used in games with adverse selections are signaling and screening. From the side of the uninformed player, “the strategy of making another player act as so to reveal his [private] information is called screening”.^[39] On the other hand, signaling is the strategy of the informed player to take actions which induce other players to believe the private information is advantageous. Nevertheless, not taking a signaling action is already a signal to the uninformed player, who can conclude that the information is not advantageous.^[40] Signaling theory is very important for the modeling of games with asymmetric information in finance, where most models see managers as insiders with private information who signal, through certain actions, to uninformed investors.

A field also based on issues of asymmetric information is the literature on agency conflicts, which can be interpreted as the game theoretic study of certain contractual structures under asymmetric information. The theory models contractual structures where a worse informed principal wants to induce a better informed agent to act in its interest.^[41] It was introduced to financial economic theory in the 70s, and laid the basis for contributions of game theoretic concepts to finance, as the analysis of strategic behavior becomes important in markets with information asymmetry.^[42]

3 Game Theory in the Context of Finance

3.1 First Game Theoretic Concepts in Finance

“The inability of standard finance theories to provide satisfactory explanations for observed phenomena lead to a search for theories using new methodologies”^[43] The agency literature is an example of a new strand in finance that searches for models explaining observed behavior with a managerial incentive based approach. Those “managerial theories [...] cut the umbilical cord that ties managers’ acts to stockholders interests”.^[44]

The literature on signaling^[45] and agency theory opened financial economics to the analytic tools of game theory. Suddenly, strategic behavior and information played a crucial role in determining important financial issues, such as dividend and capital structure policy. However, according to Thakor the early signaling models were not game-theoretic models in a strict sense. Granted, they used some game theoretic concepts as they implicitly studied games with strategic interactions among several players. On the other hand, they did not explicitly specify a formal structure for those games. This is a problem from a game theoretic standpoint, as concepts like Nash equilibrium require a specification of the outcomes off the equilibrium path of the game. So these theories needed a “set of ad hoc restrictions on reactions to out-of-equilibrium moves”.^[46] Whether those ad hoc restrictions could hold up to rigorous analysis using a completely specified, extensive form of the game remained an open question.^[47]

Those first non-game theoretic signaling models had another inherent problem: they required players “to abide ex post by strategies that were determined to be ex ante efficient but are not necessarily ex post efficient”.^[48] This is because an implicit assumption in those games is that the uninformed player moves first by pre-committing to a menu of responses to all possible signals the informed player could emit. If it is ex ante efficient for the informed player to emit a certain signal according to his type, this is the signaling equilibrium. Nevertheless, it is a necessary condition for an equilibrium that abiding by this signal (e.g. a high debt level or a high dividend) is costly to the informed player, in this case the manager, and therefore not ex post efficient when the information asymmetry is resolved. This implies that in such signaling models the “Nash equilibrium is an equilibrium for the overall game, but it is not an equilibrium over every properly defined subgame within this overall game”^[49], i.e. it is not a subgame-perfect equilibrium.^[50]

3.2 Enhancing Financial Theory with Game Theoretic Modeling

Rigorous game theoretic modeling provides analytic tools to overcome those difficulties implicit in the first models of asymmetric information. Such implicit inaccuracies are averted by explicitly specifying the studied game in extensive form and reversing the order of moves, as Thakor demonstrated with the Ross incentive signaling model.^[51]

Reversing the order of moves means that the informed player is assumed to move first by signaling based on his private information. The uninformed players then interpret this signal and revise their prior beliefs using Bayes’ rule.^[52] This structure explicitly avoids the problems of forcing the informed player to abide by ex post inefficient strategies and provides subgame perfect equilibria. Explicitly game-theoretic signaling models also have the advantage that everything is made precise: the sequence of moves, possible off-equilibrium paths and reactions etc. This eliminates any ad hoc solutions, providing an extensive analysis that does not encounter the problems inherent in earlier signaling models.^[53]

However, those models have provided important insights into the implications of asymmetric information for finance. Therefore, it is important to describe them, as well as recent, more rigorous game-theoretic treatments of financial issues.

4 Selected Applications of Game Theory in Finance

4.1 Dividends as Signals of Future Cash Flows

One of the most extensively researched issues in finance has been what Black termed the dividend puzzle.^[54] Although MM had demonstrated that dividend policy does not matter in perfect financial markets, firms continued to pay out, on average, half of their earnings as dividends, subjecting their shareholders to substantial amounts of taxes.^[55]

To this day substantial progress has been made in researching the tax effects and the information effects of dividends. The empirical evidence on tax effects is mixed and inconclusive^[56], but the signaling literature focusing on the information contained in the announcement of dividends has produced important insights. MM already suggested in their original paper that dividends might convey important information about a firm's prospects, but it was not until 20 years later that initial attempts were undertaken to understand this issue.^[57]

Bhattacharya was the first to use Spence’s job-market signaling equilibrium to explain why firms pay dividends.^[58] He introduced asymmetric information to financial markets by assuming that the inside managers of firms possess private information about the profitability of an investment. By committing to a high level of dividends, they can signal this to the market, which rewards the firm with a higher valuation. This constitutes a Nash equilibrium of the game: if the investment does not turn out to be profitable, the firm has to resort to outside financing to pay the dividend, thereby incurring deadweight transaction costs. Those costs keep the managers of bad firms from mimicking good firms^[59], thereby ensuring incentive compatibility, i.e. providing no incentives to misrepresent.^[60]

[...]

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^[1] The MM capital structure irrelevancy propositions were published in Modigliani/Miller (1958) and Modigliani/Miller (1963), the dividend irrelevancy propositions in Modigliani/Miller (1961). For a discussion of the MM propositions ‘40 years later’ see Miller (1998).

^[2] Ross/Wersterfield/Jaffe (2002), p. 397.

^[3] See Allen/Morris (2001), p. 23; MM assume perfect and complete markets.

^[4] See Modigliani/Miller (1958) pp. 272-276 and p. 296.

^[5] See e.g. Allen/Michaely (1995), pp. 802-832 for a good survey of the literature on dividend policy.

^[6] See Allen/Morris (2001), p. 23; Dixit/Skeath (1999), p. XIX; Kreps (1990) pp. 5-7.

^[7] See Allen/Morris (2001); Thakor (1991); Thakor (1989) for surveys.

^[8] Kreps (1990), p. 6.

^[9] See Kreps (1990), p. 6-7.

^[10] For a thorough introduction to game theory see Dixit/Skeath (1999); Kreps (1990); Fudenberg/Ti­role (1991). The first substantial text of game theory is Neumann/Morgenstern (1947).

^[11] Dixit/Skeath (1999), p. 3.

^[12] See Dixit/Skeath (1999), pp. 2-12.

^[13] See Dixit/Skeath (1999), pp. 1-2; Kreps (1990), pp. 5-7.

^[14] Brickley/Smith/Zimmermann (2000), p. 85.

^[15] See Brickley/Smith/Zimmermann (2000), p. 85; Dixit/Skeath (1999), pp. 23-24; Kreps (1990), p. 9; see Thakor (1991), pp. 90-91 for ideas on the application of cooperative game theory in finance.

^[16] This may involve a time difference of moves. But as long as players are not informed about each others actions, it is essentially a game with simultaneous moves. Actual chronology is only important if it influences the information distribution. See Kreps (1990), p. 18.

^[17] See Dixit/Skeath (1999), pp. 18-19.

^[18] See Dixit/Skeath (1999), p. 80; Kreps (1990), pp. 10-13. See exhibit 1 in the appendix for an example of a simultaneous move game in normal or strategic form.

^[19] See Dixit/Skeath (1999), p. 46; Kreps (1990), pp. 13-21. See exhibit 2 in the appendix for an example of a sequential move game in extensive form. The tree structure is also called arborescence.

^[20] The strategic form can be transformed into extensive form and vice versa; see Kreps (1990), pp. 21-25. For combinations of both game forms see Dixit/Skeath (1999), pp. 178-206.

^[21] Dixit/Skeath (1999), p. 25.

^[22] See Dixit/Skeath (1999), p. 25-26 and p. 48-49; Kreps (1990), pp. 21-22.

^[23] Ordinal rankings are the most common; expected profits and other forms are also used.

^[24] Dixit/Skeath (1999), pp. 26-27; expected utility approaches from decision theory are useful to include risk-aversion, see Kreps (1990), p. 23; Dixit/Skeath (1999), pp. 173-176 and pp. 219-223.

^[25] Dixit/Skeath (1999), p. 27.

^[26] Evolutionary game theory works without the rationality assumption. But as it yields similar results, it can be regarded as a backdoor justification for this assumption. See Dixit/Skeath (1999), pp. 347.

^[27] Dixit/Skeath (1999), p. 30.

^[28] Dixit/Skeath (1999), p. 82.

^[29] As opposed to a mixed strategy which specifies that actual moves will be made randomly from a set of pure strategies with specific probabilities. See Dixit/Skeath (1999), pp. 124-160.

^[30] See Dixit/Skeath (1999), pp. 82-83; Kreps (1990), pp. 28-36; for different ways to find equilibria see Dixit/Skeath (1999), pp. 89-118; Brickley/Smith/Zimmermann (2000), pp. 87-88.

^[31] Dixit/Skeath (1999), p. 83.

^[32] Or to play the “best response” to another player’s dominant strategy; see Dixit/Skeath (1999), pp. 83-93; Kreps (1990), pp. 26-28; see exhibit 3 in the appendix for a game with dominant strategies.

^[33] Dominance arguments by definition always lead to a Nash equilibrium, as each player is choosing the best action without incentives to change. See Dixit/Skeath (1999), p. 84.

^[34] See Dixit/Skeath (1999), pp. 49-53; Kreps (1990), pp. 53-56. Rollback again leads to a Nash equilibrium, as each player is choosing the best action given the other player’s supposed actions.

^[35] See Feess (2000), pp. 579-582; Dixit/Skeath (1999), pp. 397-427. The seminal paper introducing asymmetric information to economic analysis was Akerlof (1970).

^[36] See Feess (2000), p. 580. This is also true for any simultaneous move game. Although there is complete information, the players do not know which action their opponents are choosing.

^[37] The name comes from insurance, where the insurer is not able to observe the actions of the insured person completely. See Feess (2000), pp. 580-581; Dixit/Skeath (1999), p. 399; Thakor (1989), p. 39.

^[38] Again the name stems from insurance, where the insured person is better informed about the insured risk than the insurer. See Feess (2000), p. 580; Dixit/Skeath (1999), p. 399; Thakor (1989), p. 39.

^[39] Dixit/Skeath (1999), p. 404.

^[40] See Dixit/Skeath (1999), pp. 404-405 and pp. 412-416.

^[41] See Feess (2000), pp. 581-582.

^[42] See Jensen/Smith (1985); Fama (1980); Jensen/Meckling (1976); Ross (1973).

^[43] Allen/Morris (2001), p. 23.

^[44] Myers (1984), p. 576.

^[45] See Miller/Rock (1985); John/Williams (1985); Bhattacharya (1979); Ross (1977).

^[46] Thakor (1991), p. 75.

^[47] See Thakor (1991), pp. 74-75; Kreps (1990), pp. 82-83.

^[48] Thakor (1991), p. 74.

^[49] Thakor (1991), p. 74.

^[50] See Allen/Morris (2001), p. 24; Dixit/Skeath (1999), pp. 195-199; Thakor (1991), p. 74.

^[51] See Thakor (1991), pp. 73-86; Ross (1977), pp. 23-40.

^[52] See Dixit/Skeath (1999), pp. 170-173 and p. 419 for an explanation of Bayes’ rule.

^[53] See Thakor (1991), pp. 73-76; Thakor (1989), p. 49.

^[54] See Black (1976), pp. 5-8.

^[55] See Ross/Wersterfield/Jaffe (2000), p. 494-522; Allen/Michaely (1995), pp. 793-802; Jensen/ Smith (1984), pp. 18-20; Modigliani/Miller (1961), pp. 411-433.

^[56] See Ross/Wersterfield/Jaffe (2000), pp. 510-511; Fama/French (1998), pp. 819-821; Mann (1989), pp. 4-12; Litzenberger/Ramaswamy (1982), pp. 429-443; Miller (1977), pp. 264-266.

^[57] See Allen/Michaely (1995), pp. 818, Modigliani/Miller (1961), pp. 411-433.

^[58] See Allen/Morris (2001), pp. 23-24; Bhattacharya (1979); Spence (1974).

^[59] The attributes „good“ (high returns) and „bad“ (low returns) are common to describe the quality of firms (i.e. its type) in signaling models. See e.g. Thakor (1991), pp. 76; Thakor (1989), pp. 40-41.

^[60] See Allen/Morris (2001), p. 24; Allen/Michaely (1995), pp. 818-819; Thakor (1989), pp. 40-41; Bhattacharya (1979), pp. 259-260.