Excerpt

## Table of Contents

List of Illustrations

List of Tables

1 Introduction

2 Examples of Game Theories

2.1 Zero-Sum Games

2.2 Nonzero-Sum Games

2.3 Extensive Form Games

2.4 Cooperative Games

2.5 Bargaining Games

3 Conclusion

4 Literature

## 1 Introduction

Negotiation has been since ever omnipresent. It can confront us in our daily life, for example the question who is doing the dishes today or in much more complex issues, such as in economics or politics.^{1} „Simply defined, negotiation is the process of attempting to get what one wants, through agreement with one or more other parties.“^{2}

There are different tools with which negotiations can be modeled. One of them is game theory. „GAME THEORY aims to help us understand situations in which decision-makers interact. A game in the everyday sense—“a competitive activity… in which players contend with each other according to a set of rules.”^{3} According to the definition, game theory is an appropriate tool to apply for negotiations and get the best outcome of it. It can make it easier for us to analyze our own needs, the least acceptable agreement, and desires, the most desired outcome, but also the ones of our opponent. Consequently, game theory allows us to estimate the strategy the other party is going to use.

Moreover game theory reaches back far in history. There had been recognized some game theoretic situations even in the bible.^{4} But the major development started in the 1920s. First, game theory was considered to be a mathematic discipline. The two mathematicians, John von Neumann and Emile Borel worked on game theory and subsequently the book *Theory of games and economic behavior* was published by von Neumann and Oskar Morgenstern in 1944.^{5} Later on, in the 1950s, game theory was not only considered to be a mathematician discipline, but it has been also applied in „economic theory and political science, and psychologists began studying how human subjects behave in experimental games“.^{6}

## 2. Examples of Game Theories

The aim of the following is to give some insights and a general overview of game theory in a simple way.

### 1.1 Zero-Sum Games

Zero-Sum Games occur in „situations in which one person’s gain is equivalent to another’s loss, so the net change in wealth or benefit is zero. A zero-sum game may have as few as two players, or millions of participants.“^{7}

An example of a Zero-Sum Game would be the game ** Matching Pennies **.

^{8}

There are two players and both have a coin. Player 1 wins, if the pennies match and player 2 wins if the pennies don’t match. To solve this game, the players have to choose randomly, because if they would know what the opponent is going to choose, they would never agree, as the win-lose chance is 50/50. This kind of strategy is called mixed strategy.^{9}

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Table 1: Matching Pennies (see: Peters (2008))

Furthermore this game could be transferred to an economic issue, like launching a new product „by an established producer and a new firm“.^{10} The established producer would prefer that the product of the new firm isn’t similar to his one, so that his customer won’t buy the competitor’s one. On the other side, the newcomer would like the product to be similar to the one of the established producer, so he can be on the safe side and won’t make any losses.^{11}

Nonzero-Sum Games

A Nonzero-Sum Game is a „situation where one decision maker's gain (or loss) does not necessarily result in the other decision makers' loss (or gain).“^{12}

A famous example of a Nonzero-Sum Game is the ** Prisoner’s Dilemma **.

^{13}Two criminals are suspected having committed a crime and held in separate cells, to be interrogated alone. If one confesses and the other one keeps quiet, the confessing one will be released and the other one must go to jail for 12months. If both keep quiet, both will be sent for 1 month each to prison and if both confess they will be sentenced to 8 months imprisonment.

At first sight, everyone would think that both would be best off, if they keep quiet. But, as they are interrogated separately, everyone has to take care of his personal interest. Considering player 1’s point of view, he would be twice better off if he confesses, no matter if player 2 keeps quite or confesses. So, in this case, confessing is a strictly dominant strategy, as everyone focuses on his individual best outcome.

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Table 2: Prisoner’s Dilemma (see: Spaniel (2014))

This game is also applicable in an economic issue. A situation with a similar structure could be a Duopoly in its simplest form. “Two ﬁrms produce the same good, for which each ﬁrm charges either a low price or a high price.”^{14} Both firms would be best off if they would choose a high price on the first sight, but nevertheless the strictly dominant choice in this case is, if they decide to offer the good for a low price, as each firm cares of his own profit and doesn’t know which strategy the opponent is going to choose.

Another Nonzero-Sum Game is ** The Battle of Sexes **.

^{15}In this game, the players prefer cooperating than not cooperating, but they don’t concur which is the best outcome.

A woman and a man want to spend the night together, but they don’t have any means to communicate. The woman prefers going to a ballet and the man to a soccer game. Now they have to make a decision independently, but both will be equally unhappy if they don’t see each other. The two optimal solutions are if both would decide either to go to the ballet or to the soccer game, otherwise they will be unhappy.

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Table 3: The Battle of Sexes (see Peters (2008))

Maybe they have talked the day before for hours about soccer, and each of them remembers that, so they will decide both for the soccer game and thus for one of the optimal solutions. Otherwise it is difficult to predict this game.

„The Battle of the Sexes game is metaphoric for problems of coordination.”^{16}

### 1.2 Extensive Form Games

Until now we have discussed “one-shot games”, where the players decided simultaneously and on their own. “In parlor games as well as in games derived from real-life economic or political situations, this is often not what happens. Players may move sequentially, and observe or partially observe each others’ moves. Such situations are better modelled by^{17} extensive form games.”

An example for an extensive form game is ** The Sequential Battle of Sexes **.

^{18}The situation is almost the same as in the previous example, except that the man is doing his choice first and the woman knows what he is choosing and chooses afterwards. This game can be modelled by a decision tree.

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Figure 1: The decision tree of sequential Battle of the Sexes (see: Peters (2008))

In this decision tree player 1 is the man and player 2 the woman. The first number is the payoff the man and the second one of the woman. Working backwards, is the best way to analyze this game. If player 1 decides for S (B), player 2 will decide also for S (B), because this is the optimal choice. Knowing the choice behavior of player 2, player 1 should decide for S, his best outcome. “The solution described above is an example of a so-called backward induction (or subgame perfect).“^{19}

Another economic example for an extensive game form is *The Entry Deterrence.* ^{ 20 } “An old question in industrial organization is whether an incumbent monopolist can maintain his position by threatening to start a price war against any new firm that enters the market.“^{21}

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Figure 2: The game of entry deterrence (see: Peters (2008))

There are two players, the entrant and the incumbent. The entrant can either entrance the market (E), or stay out of it (O). If he stays out of it, the market profit, 100, remains in the hand of the incumbent. However, he can also enter and if he enters, the incumbent has to decide whether he is colluding (C) or fighting (F). Assuming that the incumbent is colluding, the market profit will be divided almost equally. Deciding for the other choice, namely fighting, the entrant loses 10 and the incumbent’s outcome will be zero. Working backward, it is obvious, that the entrant will enter the market and the incumbent will collude, as he still wants to make profit.

### 1.3 Cooperative Games

In general “in a cooperative game the focus is on payoffs and coalitions, rather than on strategies.“^{22}

An example for a cooperative game is the ** Dentist Game: A Permutation Game **.

^{23}The following situation is given:

„Mr. Adams, Mrs. Benson, and Mr. Cooper have appointments with the dentist on Monday, Tuesday, and Wednesday, respectively. This schedule not necessarily matches their preferences, due to different urgencies and other factors. These preferences (expressed in numbers) are given in Table 1.3“^{24} (see: Peters (2008))

By forming a coalition and changing the appointments, the patients can gain by helping each other.

If Mr. Adams and Mr. Cooper change their appointments, they would obtain a better outcome, namely 18 instead of 6. In this situation every patient can achieve a better outcome by cooperating. The possible results and outcomes are given in Table 1.4.

In economic context an imaginable situation could be, if a firm should produce a component on its own or look for a supplier who can deliver that component and thus cooperate with him.

**[...]**

^{1} See Godin (2009) pp. 1-2 (holds for the paragraph)

^{2} Godin (2009) p. 1

^{3} Osborne (2000) p.1

^{4} See Peters (2008) p. 1-2

^{5} See Osborne (2009) pp. 2- 3 (holds fort he paragraph)

^{6} Osborne (2000) p. 3

^{7} Investopedia, Definition : Zero-Sum Game

^{8} See Peters (2008) p.16-17

^{9} See Spaniel (2014), Video: Matching Pennies (holds for the paragraph)

^{10} Osborne (2000) p.17

^{11} See Osborne (2000) p.17 (holds for the paragraph)

^{12} Business Dictionary, Definition Nonzero-Sum Game

^{13} See Spaniel (2014), Video: The Prisoner’s Dilemma and Strict Dominance (holds for the paragraph)

^{14} Osborne (2000) p. 14

^{15} See Peters (2008) p. 6-7 (holds for the paragraph)

^{16} Peters (2008) p.7

^{17} Peters (2008) p.8

^{18} Peters (2008) p. 8-9 (holds for the paragraph)

^{19} Peters (2008) p. 9

^{20} See Peters (2008) p.9-10 (holds for the paragraph)

^{21} Peters (2008) p.9

^{22} Peters (2008) p.12

^{23} See Peters (2008) p. 14 (holds for the paragraph)

^{24} Peters (2008) p. 14

- Quote paper
- Elena Ristova (Author), 2014, Negotiations and Game Theory. Understanding situations in which decision-makers interact, Munich, GRIN Verlag, https://www.grin.com/document/703072

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