**UFRJ**

# TEORIA DOS JOGOS-fernandobarrichelo

(Parte **1** de 6)

Game Theory

45-988 Competitive Strategy Project Carnegie Mellon University - GSIA

Prof. Jeffrey Williams Fall 2002

Game Theory for Managers Some review, applications and limitations of Game Theory

Luiz Fernando Barrichelo

Pittsburgh, PA 12/20/02 – Version 1

1 fernando@barrichelo.com.br

Game Theory

2.2. Example 2 – Sequential game – Strategic Entry Deterrence with 3 players ……………… | 8 |

2.5. Example 5 – Combining Sequential and Simultaneous Game and Dominant Strategy … | 13 |

2. Games – Structure and Examples ……………………………………………………………………. 7 2.1. Example 1 – Sequential game – Strategic Entry Deterrence ……………………………….. 7 2.3. Example 3 – Simultaneous Game – Prisoner’s Dilemma ………………………………….. 10 2.4. Example 4 – Simultaneous Game – Prisoner’s Dilemma in two business games …………. 1 2.6. Example 6 – Iterative Dominance and Nash Equilibrium …………………………………. 15 2.7. Example 7 – Uncertainty and Expected Payoff ……………………………………………. 17 2.8. Example 8 – Repeated Games and Mixed Strategies …………………………………...…. 18 2.9. Other games ……………………………………………………………………………...… 20

4.3.2. Understanding the composition of payoffs – second example ….………………… | 30 |

4.4.1. Example 1 – Kiwi Airlines ………………………………………………………… | 35 |

4.5. What the literature offers to managers …………………………………………………… | 37 |

Appendix

A – Chronology of Game Theory ……………………………………………………………….. 49 B – ComLabGames and Experimental Methods ………………………………………………… 56

Game Theory Introduction

This paper was written for the class 45-988 Competitive Strategy Project with Prof. Jeffrey Williams. Its goal is to present Game Theory to a manager with limited knowledge in this subject. Here I present a basic review of the theory, demonstrate some examples and discuss some difficulties that managers can have and how to address them.

Game Theory is a debatable theory in terms of real applicability, mainly because of its strong assumptions and lack of real published cases. However, it provides important insights for companies to compete more effectively and for managers to think more strategically. In an easy language, my goal with this paper is to show those advantages for managers1.

Section 1 presents a summary of definitions and assumptions of Game Theory.

Section 2 shows how a game is structured and provides examples of different kinds of games and solutions. The objective of this section is not to cover all game-theoretic analysis, but to create a common language and understanding to the following sections. Most of the examples are simplified just to teach the reasoning of Game Theory.

Section 3 presents some benefits of Game Theory.

Section 4 discusses important issues about applications and limitations that a manager and a student must know. This is the main part of this paper, but also does not solve all problems of Game Theory. The objective of this section is to motivate the discussion and to provide some advices for a basic reader about how to face some limitations.

Section 5 concludes with some check lists for practical use.

Appendix A presents a Chronology of Game Theory. Basically, it is a replication of my source and it is here just to provide the reader with more sources, besides my reference section.

Appendix B shows the site ComLabGames, a useful tool for professors and students, even managers, to design and test game-theoretic situations.

1 As Game Theory has many sources, I will continue with my research to complement this work and to write other versions. 3

Game Theory 1. Game Theory2 – definitions and assumptions

In the business world, companies make decisions and actions to maximize their payoffs, often represented by profits, revenues, market share, market value, etc. In most competitive situations, companies’ payoffs depend not only on their own actins, but also on the actions of other competitors that are pursuing their own objectives.

“Game theory is the study of interactions among players whose payoffs depend on one another’s choice and who take that interdependence into account when trying to maximize their respective payoffs” (Ghemawat, 1999)

It seems obvious to analyze the consequences of our own actions and competitors’ reactions. An intuitive way to do this analysis is to use some information about competitors, anticipate their likely actions (or reactions) and choose the best moves.

However, it is not an easy task – managers need data for payoff calculations, experience about competitors’ behavior and some methodology. It is in the methodology part where Game Theory helps managers structure their reasoning. This theory serves as a framework for competitive analysis that takes into account the interaction of strategies.

“Game Theory offers a scientific approach to strategic decision-making. In place of the anecdotes, cases, stories, and examples that are commonly offered as advice to negotiators, Game Theory gives a systematically structure view” (McMillan, 1992).

It imposes an assumption typically made in economics: each player acts to maximize his payoff. Moreover, as companies’ payoffs depend on the actions of the other companies, Game Theory assumes the each player reasons to figure out all competitors’ options and to anticipate what they are going to do.

To do so, you need to put in your rival’s shoes and reason what he would do to maximize his payoffs. So, there is another assumption: you should know your competitors’ motivation, payoffs and capabilities. In fact, it assumes that your competitors knows yours either.

2 Appendix A shows a chronology of Game Theory related to academic papers 4

Game Theory

In summary, Game Theory assumes that: • Each player has clear preferences and acts to maximize his utility (utility maximization and rationality)

• Each player knows the other’s utilities and preferences (common knowledge)

In short, all strategic players are assumed to be rational and self -interested. This means that players want to get as much utilities for themselves as possible, and are quite clever in figuring out how best to accomplish this objective.

Game Theory is a good tool for understanding how decisions affect each other. Dixit and Skeath (1999) stated the three uses of Game Theory:

• “Explanation – when the situation involves interaction of decision makers with different aims, Game Theory supplies the key to understanding the situation and explains why it happened.

• Prediction – when looking ahead to situations where multiple decisions makers will interact strategically, people can use Game Theory to foresee what actions they will take and what outcomes will result.

• Advice or Prescription – Game Theory can help one participant in the future interaction, and tell him which strategies are likely to yield good results and which ones are liable to lead to disaster.”

Next section presents some examples of games and solutions following Game Theory’s assumptions.

Game Theory 2. Games –Structure and Examples

Many books and articles start explaining Game Theory with analogies with sports, games, cards, movies, etc. I acknowledge that these analogies are helpful to create a general understanding, to provide some insights and to think strategically in multiple environments in life. However, I decided to limit all examples in this paper to company’s situations because I want the reader to focus on his business set, generating insights for his practical use.

First, it is important to establish a common language and general understanding of a game. Some names the reader should have in mind (LeClair, 1995):

• Player A rational and self interested decision-making entity.

• Strategy A rule that tells the player which action to choose at each instant of the game.

• Outcome The result of every possible sequence of actions by the players.

• Payoff The personal satisfaction obtained from a particular outcome.

• Equilibrium The "best" combination of choices.

As you will see, a formal game theory model consists of the following (Rasmusen, 1989 in LeClair, 1995): • a set of players, a set of actions and strategies for each player,

• the information available to any player at any point in the game,

• the outcome that results from every possible sequence of actions by players,

• a ranking of each outcome to every player (the player's payoffs), and

• a solution (or equilibrium) concept

All those definitions will be better understood with the next examples. The objective of this section is to explain the mechanics and reasoning of types of interactions. All examples are very simplified, not necessary based on past real cases, and the payoffs are given. Also, there are not complex math in the solutions. Since the goal of this paper is to discuss some issues about benefits, applications and limitations in order to give some advices to managers, I didn’t include all elements and examples as in a formal game theory course.

If you do not know much about Game Theory, this section provides you with a good overview, so that you can deep your knowledge in specific books. People who know the theory can skip this section.

Game Theory

2.1. Example 1 – Sequential game - Strategic Entry Deterrence3 Game theory distinguishes two major categories of games: simultaneous games, where all participants make their decisions simultaneously, or sequential games, where participants react to each other's actions in turn.

The Sears Tower in Chicago is currently the tallest building in the United States. This status endows the building with a special form of prestige, enabling its owners to command higher rents than in otherwise similar office buildings. Suppose that an Entrant is considering whether to build an even taller building. Suppose also it knows that any firm that has permanent ownership of the tallest building will earn a large economic profit. Its concern, naturally, it that Sears (or some other firm) may build a still taller building, which would substantially diminish the Entrant’s payoff.

It is a sequential game because Entrant must choose first, and Sears will know the Entrant’s choice to make its decision. The game can be modeled in a game-tree shown in picture 1, called extensive form4, that shows all possible options and outcomes of each option. There is a label with numbers in each element, called node.

Picture 1

You can see that the Entrant (node 1) must decide between Enter and Don’t Enter in this market, i.e., to build a taller tower or not. If it chooses Don’t Enter, the game ends in the node 2. If its chooses Enter, then Sears (node 3) has two options, Don’t build it (node 4) or Build a higher building (node 5).

3 Example from Frank, 2000, with some modifications to fit some explanations

4 All schematic examples were drawn in the software from the site ComLabGames http://www.cmu.edu/comlabgames/ an open tool created by Miller and Prasnikar from Carnegie Mellon University, Graduate School of Industrial Administration. The software is for instructors and students to design, run, and analyze the outcomes of games played over the Internet. See more details in Appendix X

Game Theory

Payoffs are necessary for players to make their decisions. Since game theory involves formal reasoning, we must have a device for thinking of utility maximization in mathematical terms.

If Entrant does not enter, nothing changes in the current situation – Sears gets a payoff of 100, and Entrant gets zero (node 2). If the Entrant enters and Sears don’t compete by building an even higher tower, then Entrant has advantage and captures a payoff of 60, while Sears gets 40 (node 4). If Sears build a higher one, then Entrant loses money with a payoff of -50 and Sears get 30 (node 5).

Sears naturally wants Entrant not to enter because it prefers the payoff of 100 (node 2), but this decision depends only on the Entrant. How does the Entrant must decide? It must use the concept of Backward Induction, that is looking ahead and reason back.

For example, looking to the Sears’ choices, and assuming that Sears wants to maximize its payoff, Sears will prefer to not build a higher building because the payoff of 40 (don’t build) is greater then 30 (build). Entrant knows that Sears will think in this way, so the Entrant’s payoff will be 60 (node 4). Then, Entrant knows that, if it chooses Don’t Enter, it will get zero, if chooses Enter, it will get 60. As a result, it will prefer Enter and build a tower. The equilibrium of this game is in node 4.

Note that there are some simplifications, since there are many alternatives in real life. For example, the Entrant could build a small building, or Sears could build another tower even if Entrant does not enter, or it could build a small one if Entrant enters. However, those options do not capture this competitive situation and they are irrelevant in this analysis.

2.2. Example 2 – Sequential game - Strategic Entry Deterrence with 3 players5 In this example, there are three players, an Entrant and two retailers, respectively called Big Monopolist and Small Monopolist, which currently hold regional monopolies in the localities they serve. The schematic representation in a game-tree is shown in picture 2.

5 Example from Miller and Prasnikar, 2004, with some modifications to fit some explanations 8

Game Theory

Picture 2

The Entrant can decide to enter or not in the market. If it chooses not to enter, the absence of competition provides the Big and Small Monopolist with payoffs of $20 and $10 million, respectively, in present value. In this case, payoff of Entrant is zero (node 4).

Now the reader knows how to interpret all options and respective payoffs by looking the game tree, so we will simplify and do not explain all business situations anymore. The objective here is to explain the mechanics of games and then next chapter discuss some important issues.

To solve the problem, if you are the Entrant, the reasoning is the same – the backward induction. Check game tree and payoffs to verify that:

• If Entrant chooses Enter First Market, Big Monopolist will choose Collude to get payoff of 15

• If Entrant chooses Enter Second Market, Small Monopolist will choose Collude to get payoff of 4 So, you predict Sears’ choices and you know that you have 3 possible rational outcomes:

• Zero if Stay Out (node 4)

• 5 if Enter First Market (node 5)

• 6 if Enter Second Market (node 8)

Therefore, the strategy that maximizes the Entrant’s payoffs is to enter in the second market because the Entrant knows that the Small Monopolist will choose to collude.

Notice that, in this case, Big Monopolist does not play the game. It shows that not always all players act in the decision, but we need to consider them in the game tree to predict the best choices.

Game Theory

The reasoning to solve more complex sequential game trees is the same. You start from the bottom and go up doing simplification in the tree since you know the likely choices of all players using the assumptions of utility maximization, rationality and common knowledge.

Example 3 – Simultaneous Game – Prisoner’s Dilemma One of the most frequently discussed examples of pursuing self-interest is the so-called prisoner’s dilemma. The mathematician A.W.Tucker is credit with having discovered this simple game, whose name derives from the anecdote originally used to illustrate it (Frank, 2000). Two prisoners are held in separate cells for a serious crime that they did, in fact, commit. The prosecutor, however, has only enough hard evidence to convict them of a minor offence, for which the penalty is a year in jail. Each prisoner is told that if one confesses while the other remains silent, the confessor will go free while the other will spend 20 years in prison. If both confess, they will get an intermediate sentence of 5 years.

(Parte **1** de 6)