Applying Game Theory in Finance

Table of  Contents  

  List of Abbreviations  II

1  Introduction  1

1.1  Problem and Objective of the Paper  1

1.2  Organization of the Paper  1

2  Game Theoretic Foundations  2

2.1  Basic Definitions  2

2.2  Nash Equilibrium Dominance and Rollback  4

2.3  Asymmetric Information  6

3  Game Theory in the Context of Finance  7

3.1  First Game Theoretic Concepts in Finance  7

3.2  Enhancing Financial Theory with Game Theoretic Modeling  8

4  Selected Applications of Game Theory in Finance  9

4.1  Dividends as Signals of Future Cash Flows  9

4.2  Signaling and Agency Models of Capital Structure Decisions  12

4.3  Other Areas of Game Theory Application in Finance  14

5  Critique Concluding Remarks  15

Appendix   17

  References  20

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  List of Abbreviations  

  eds. editors  

  ed. edition  

  IPO Initial Public Offering  

  MM MODIGLIANI & MILLER  

  NPV Net Present Value  

  no. number  

  vol. volume  

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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 cons idered the beginning point of modern managerial finance.” 2 The ir 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 informa- tion or complete contracts, while trying to retain an exp lanatory value in the spirit of the original MM papers. 5 This incorporation of more realistic elements, such as strategic interaction and asym- metric information, brought several problems to financial economists’ models: they re- quired 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 econo m- ics 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 con- cepts. 6 Nowadays, there are few issues in finance research which have not been mod- eled 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 demon- strate 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 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.

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of strategy. Special consideration is given to the incorporation of asymmetric informa- tion, 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 dem- onstrates 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 sho w-

ing the real application value of game theory with important historic and current exa m- ples. It provides a discussion of published papers using game theoretic concepts to en- hance the understanding of two unresolved issues in finance: dividend and capital struc- ture 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 re- marks 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 de- ductively.” 8 These models provide three advantages: (1) a clear and precise la nguage to transfer insights from one context to another, (2) the possibility to cross-check those in- sights 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 situa- 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/TI- ROLE (1991). The first substantial text of game theory is NEUMANN/MORGENSTERN (1947). 11 DIXIT /SKEATH (1999), p. 3.

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tions are called games of strategy, i.e. any interactive games which involve not only pure chance or skill, but also some amount of ratio nal 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 agree- ments are not enforceable; co-operation may arise only because such cooperative beha v- ior 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 ana lyzing 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 simultane-

ous 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 strate- gic 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 conse- quences, 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) displa ying 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

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 in- fluences 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 exa m- ple of a simultaneous move game in normal or strategic form.

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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 ac- tion”. 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 un- der 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 calcula- tions of what actions will best serve those interests”. 25 The rationality assumption is es- sential, 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 de- pending on the specifics of the game. The two most important solution techniques for games with simultaneous moves are Nash equilibrium analysis and dominance.

19 See DIXIT /SKEATH (1999), p. 46; KREPS (1990), pp. 13-21. See exhibit 2 in the appendix for an exa m- ple 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.