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Tilburg University

Game theory and applications in finance

van Gulick, G.

Publication date:

2010

Document Version

Publisher's PDF, also known as Version of record

Link to publication in Tilburg University Research Portal

Citation for published version (APA):

van Gulick, G. (2010). Game theory and applications in finance. CentER, Center for Economic Research.

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Game Theory and Applications

in Finance

Proefschrift

ter verkrijging van de graad van doctor aan de Univer-siteit van Tilburg, op gezag van de rector magnificus, prof.dr. Ph. Eijlander, in het openbaar te verdedigen ten overstaan van een door het college voor promoties aangewezen commissie in de aula van de Universiteit op vrijdag 25 juni 2010 om 14.15 uur door

Gerwald van Gulick

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Preface

The duke had a mind that ticked like a clock and, like a clock, it regularly went cuckoo.

Terry Pratchett, Wyrd Sisters

Four years of work as a Ph.D. student have gone into this thesis. In this Preface I would like to look back on those years, and express my gratitude to some people who were important to me in that period.

The first person I like to thank is Ruud Hendrickx. In the spring of 2005 I had almost finished my studies Econometrics and Operations Research at Tilburg University. All that remained was to write a Masters Thesis. Since I had fulfilled the criteria to graduate in any of the three tracks the studies had to offer, the sheer amount of choices for a subject for a Masters Thesis overwhelmed me. When talking to Ruud after a night of playing chess at TSV Rochade, he suggested having a meeting with Peter Borm and himself, convinced that game theory could be an appropriate subject for me.

I had followed the course Game Theory by Peter Borm a year earlier. I thought it was a great course, but because I was organising the “Landelijke Econometristen Dag” at that time, and because there was only one course on game theory in the entire studies, I did not give writing a thesis on it as much thought as I should have. The meeting with Ruud and Peter was a short one, but I was walking out with a paper on “Deposit Games”. Chapter 3 in this thesis is called “Deposit Games”; need I say more?

Ruud and Peter were not my only supervisors of my Master’s Thesis. Because deposit games do not deal with game theory exclusively but also with finance, I had a third supervisor: Anja De Waegenaere. She would be one of my two supervisors for the next four years; the other being Henk Norde.

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When my thesis was finished I listen to the advice of my father, and I made the decision to become a Ph.D. student. However, the university had other plans: I was to obtain an M.Phil. degree first. In recent times such a requirement is unheard of, simply because the M.Phil. degree is not recognised by the dutch law. However, that did not bother the university at all, and I spent one year obtaining said degree.1 I then applied to become a Ph.D. student, but it did not help that

I was focusing on both game theory and finance. While game theory is part of Econometrics and Operations Research, Finance has its own department. So all positions were filled with Ph.D. students on one of those areas, and I was told to apply at the other department. I was terrified that this deadlock was going to ruin my plans, but fortunately some people in the Econometrics and Operations Research department made a lot of effort to get me accepted as a Ph.D. student. Many thanks!

As I already mentioned, my supervisors during the M.Phil. and Ph.D. were Anja and Henk. I would like to express my gratitude to them for all the help I have received and their faith in me. They both have their own specialisations which coincide with the two major subjects of this thesis; while Anja is more involved in finance, Henk has a background that is more in game theory. Because of this I always managed to write down something that at most one of them would understand, which has helped a great deal in improving my explanatory skills and writing the chapters in a clear and structured way.

Thanks go to Marco Slikker, Hans Reijnierse and Erwin Charlier as well, for their valuable input in this thesis. I would also like to thank the other members of my committee for reading my thesis and providing me with valuable comments. Besides my supervisors, the committee consists of Josep-Maria Izquierdo Aznar, P´eter Cs´oka, Hans Reijnierse, Marco Slikker and Jeroen Suijs.

I would like to thank David, Elleke, Gert, Henk, John, Marieke and Peter for having the pleasure to cooperate with them in various courses, such as Statistiek voor HBO, Ori¨entatie Econometrie and Analyse. The course ‘Independent Mod-elling’ has provided me with the opportunity to get some students to see the light, as well as having a laugh with the other supervisor. The same can be said for the oral exams of Game Theory.

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PREFACE ix

was room for a lot of people despite my tendency to take up a lot of space at the table. Usually games, puzzles, political discussions or something more creative followed in the Trie-angel. For this I thank Annemiek, Bart, Edwin D., Edwin, Elleke, Feico, Gijs, Hans, Hein, Henk, Herbert, Iris, Jacob, Josine, John, Kim, Lisanne, Maaike, Marieke, Mark, Marloes, Mirjam, Mohammed, Peter, Rene, Roy, Ruud B., Ruud, Salima, Soesja and Willem.

Besides work related activities, I also got to know my colleagues outside office hours. A team consisting of Cristian, Edwin, Henk, Jacob, John, Marco, Mo-hammed and myself won the TEV football tournament in 2009. Earlier that year, I had won the TEV indoor sports afternoon together with Cristian, Edwin, John and Josine. Less successful, but not less fun, were earlier attempts at sport event that also included Chris, Elleke, Herbert and Ralph.

There have also been less competitive activities, such as games and Sinterk-laas; another testament to that are several huge jigsaw puzzles with my signature on them. For these subdepartment activities I thank Cristian, Edwin, Elleke, Gijs, Iris, Josine, John, Kim, Lisanne, Maaike, Marieke, Mark V., Marloes, Mikel, Mirjam, Oriol, Ralph, Romeo, Ruud, Salima and Soesja.

Some of my coworkers I got to know even better during conference trips. In 2007 I joined John and Ruud for my first conference visit; Hans, Henk and Marco also attended this conference. At this conference, Henk and Marco proposed to start work on the monoclus, on what has become Chapter 5. When we had some energy left after an exhausting day at the conference, we enjoyed the cool sea breeze in Madrid, as well as deer from the mountains of Toledo. A flag that badly needed ironing was the kickoff for many years of well-documented jokes. Sadly, this conference was tainted by one emotionally scarring event: at the very first day in Madrid, my travel company refused to warn me sufficiently about the broken cover of the camera, and they still take great pleasure in blaming me for the lack of pictures of Madrid.

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was inspirational, because no one other than John Nash took an interest in my work, and we had a little talk about it.

After the conference I took a plane to Boston, where I visited Jeroen, whom I have seen way too little the last four years. I stayed at his place and together we saw the sights of this almost European city and got our share of American history. I thank him for his hospitality.

In 2009 I went with Edwin, John and Ruud to a conference in Saint Petersburg. The conference was never as special as the city to be honest. It was a pity Ruud and I had to wait outside in line for a couple of hours at the Hermitage while it was very cold and windy; at the same time Edwin and John were nice and warm inside a police station. But still it was well worth it. And there was vodka too.

I would like to thank everyone who visited those conferences with me for mak-ing work feel like holiday.

In the four years I have been at the department of Econometrics and Operations Research I have had an impressive amount of office mates. There have been eleven: Gijs, John, Tim, Barı¸s, Gaia, Juan-Juan, Mich´el`e, Guillaume, Marco, Soesja and Roy. I thank all of them for their company, even if I did not see some of them that much; I have met Tim only once. I can only hope I have had such a large list of office mates because of reorganisations within the department.

One other office mate deserves a special mention, because he has been my office mate for three of those four years: Barı¸s C¸ ift¸ci. Whenever he was around, there was the initial state of amazement about the (research related) drawings on the whiteboard. We discussed Dutch and Turkish news, as well as IND applications, which never seemed to make sense. He has been in a permanent state of learning Dutch, and I now know two Turkish words, none of which are appropriate to write down here. Barı¸s had three phones to handle, so small wonder he was being called all the time. It always seemed to be Esen though.

The list of secretaries is almost as long as the list of office mates. I would like to thank Annemiek, Heidi, Jolande, Karin, Korine, Loes, Marloes V. en Nicole for being the backbone of our department.

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PREFACE xi

and Victor for the trips we had to tournaments in Riga, London and the EUDC in Istanbul.

I have been playing chess at TSV Rochade for quite some time now, and the club has been mentioned in the beginning of this Preface as well. I would like to thank everyone at the club. In particular I would like to thank Ben, Clemens, Fabian, Mark, Remco and Rick for always reminding me that every event in life has a chance of 50%.

I thank Frank O., Freek, Jan, Paul S., Lindy, Mich`ele and Willianne for the games we played and the weekends we spent. I hope many more will follow.

I would like to thank Bart, Chantal, Dennis, Gerda and Roel for organising a great conference with me.

I thank Dennis and Edwin for visiting London with me on several occasions to watch some NFL. Additional thanks go to Edwin for bringing his passport on all these occasions, so I never had to travel alone.

Thanks go to everyone at BC Theseus where I play bridge, but to Thomas in particular. Also I want to show my gratitude to everyone in The Silver Order for making me enjoy playing online games.

Finally, I would like to thank everyone in my family, including Mark and Jiri. I would like thank my living grandparents, Paulette, Truus and Wim, for helping me out where they can, for instance when I bought my apartment, and for always expressing an interest in everything I do. I have inherited a great deal from them, but can also find inspiration in what they do. Paulette is very independent and always stands up to injustice. Truus always has faith that things turn out alright, no matter how many irrational fears she has. And Wim for what he has accomplished, but more so for his resilience, refusal to be down and ability to always keep learning (again), no matter the conditions.

I would like to express my gratitude to Paul and Kitty, my parents. I could elaborate on their support, my childhood, coping with my inability to explain what my thesis is about, and what they mean to me. Unfortunately, it will always do injustice; I just lack the words.

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Contents

1 Introduction 1 1.1 On game theory . . . 1 1.2 Overview. . . 5 2 Preliminaries 13 2.1 Basic notation . . . 13

2.2 Games with transferable utility . . . 14

3 Deposit games with reinvestment 19 3.1 Introduction . . . 19

3.2 Deposit problems and deposit games with reinvestment . . . 21

3.3 Deposit games with and without reinvestment . . . 28

3.4 Term dependent deposit games . . . 32

3.5 Capital dependent deposit games . . . 36

3.6 Reinvestment and debt . . . 43

3.7 Reinvestment without debt revisited . . . 47

3.8 Conclusions . . . 51

4 Fuzzy cores and fuzzy balancedness 53 4.1 Introduction . . . 53

4.2 Notation . . . 55

4.3 Continuous fuzzy games . . . 56

4.4 Main result . . . 62

4.5 Fuzzy deposit games . . . 70

4.6 Conclusions . . . 74

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5 On the monoclus and airport games 75

5.1 Introduction . . . 75

5.2 On the monoclus in general . . . 78

5.3 Vectors of subbalanced weights . . . 83

5.4 Special types of vsws . . . 94

5.4.1 Extending vsws. . . 94

5.4.2 vsws with a special geometric representation. . . 107

5.5 The monoclus of an airport game . . . 112

5.6 The monoclus of an airport game with up to 4 players. . . 129

5.7 The monoclus of an airport game with 4 players . . . 131

5.8 A painting story. . . 144

5.9 Conclusions . . . 156

6 Excess Based Allocation of Risk Capital 157 6.1 Introduction . . . 157

6.2 Model . . . 161

6.2.1 Risk Capital Allocation Problems . . . 161

6.2.2 The Aumann-Shapley value . . . 165

6.2.3 Excess Based Allocation . . . 170

6.3 Properties . . . 180

6.4 Continuity; a proof . . . 187

6.5 Monotonicity . . . 192

6.6 Linear programming problems . . . 197

6.7 Conclusions . . . 208

7 On handling ruin as a business sector 211 7.1 Introduction . . . 211

7.2 Ruin in a business sector . . . 213

7.3 Bankruptcy problems and transfers . . . 217

7.4 Fairness . . . 222

7.5 Conclusions . . . 229

Bibliography 231

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CONTENTS xv

Index 241

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Chapter 1

Introduction

Everything starts somewhere, though many physicists disagree. But people have always been dimly aware of the problem with the start of things. They wonder how the snowplough driver gets to work, or how the makers of dictionaries look up the spelling of words.

Terry Pratchett, Hogfather

1.1

On game theory

Game theory has a wide field of application. Ever since the book Theory of Games and Economic Behavior by Von Neumann and Morgenstern (1944), the field of game theory has been used to capture and analyse the main features of situations that involve interaction between individuals (called players).

Strategic behaviour is as old as human life. A particularly nice example is given below.

Example 1.1.1 Hern´an Cort´es (1485-1547) has made his mark on history by conquering Mexico. He did so while being vastly outnumbered. Two things con-tributed most to his victory: his ability to make alliances with native peoples, and the fact that he understood how fear is a major weapon in any battle, both in the enemy and in his own men.

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Fight Flee Fight (12,12) (1, 0)

Flee (0, 1) (34,34)

Table 1.1: Chances of dying in Example 1.1.1. Fight Flee Fight (1 2, 1 2) (1, 1) Flee (1, 1) (1, 1)

Table 1.2: Chances of dying when the ships are burnt in Example 1.1.1. When Cort´es landed in America, he was faced with an overwhelming number of hostile natives. He and his men had to make a choice between fighting and fleeing. Each individual had to make this choice, however, the outcome depends on the choices others make. If everybody fights, there is a chance of dying in battle, but also the battle can be won and you can survive. If you decide to flee while others fight, they hold off the enemy for long enough for you to escape with your life. However, if everybody flees, then the enemy can pursue and kill everyone without any trouble at all; there is a chance you make it to the ships however. If we consider two soldiers, their options are given in Table 1.1. The numbers corresponding to those options are the chances of survival for the soldiers, so (0, 1) means that soldier 1 lives and soldier 2 dies for sure. The numbers have been chosen arbitrarily, although in accordance with the story above.

Whatever the other soldier decides to do, fleeing is the best strategy here if you are minimising your chance to die. This is a modification of the well-known prisoner’s dilemma. Your chances of surviving decrease when you both decide to run, but still it is the rational thing to do.

Cort´es did not wait around for his men to figure this out on their own. Rather, he decided to modify the situation a bit. He decided to burn his ships. Now, anyone fleeing would have nowhere to go and be killed for sure. The options for two soldiers are given in Table 1.2.

Clearly, every soldier’s best chances of survival are to stand and fight.

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1.1. On game theory 3

model, but is nice to tell. The natives watched Cort´es burn his ships, and thought he must have been very confident of his chances of victory. They decided that no matter what his reasons where to be this confident, they must have been very compelling.

Without fighting at all, they fled. ⊳

Example 1.1.1 illustrates one of the two main approaches in game theory: non-cooperative game theory. A player has several options to choose from, and how he does so is called a strategy. His value does not depend on his strategy alone, but also on the strategies of other players. Even though communication can be allowed, there are no binding agreements. So, every player maximises his own utility. Later, we see that the exclusion of binding agreements is in contrast to another popular model in game theory.

When considering applications in finance, many situations might come to mind that can be modelled using non-cooperative game theory. In many cases, binding agreements are explicitly forbidden. For instance, think of cartel forming and price agreements. However, non-cooperative games only play a minor role when we consider applications in finance in this thesis. Although we do not use a formal non-cooperative game in the remainder of this thesis, the line of thought that originates from non-cooperative game theory, as outlined in Example1.1.1, has had its influence on this thesis. For example, in Chapter7we develop a framework for cooperation for a business sector, such that the strategies of individual companies are clear, when they are faced with the choice whether to participate or not in a scheme we propose: participation is a dominant strategy.

The approach we focus on in the remainder of this thesis is cooperative game theory, where players are able to sign binding agreements. This opens the way for coalitions of players to cooperate and gain benefits. These benefits then have to be divided among the cooperating players. The following example illustrates the link between cooperation, allocation, and finance.

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combination 1 2 3 1, 2 1, 3 2, 3 1, 2, 3

interest 20 40 100 120 140 180 330

Table 1.3: Interest for each combination of friends in Example1.1.2. the offer looks very appealing to them. However, when they look at the fine print they find that it is required to deposit at least 5000 euro to qualify for the 6% interest rate. If they deposit only half that amount, i.e. 2500 euro, then they still get a 4% interest rate.

It seems that cooperation between the friends is needed in order to qualify for the 6% interest rate. Quickly, they discard their motto “Keep your friends and money separated” and consider their options, which are given in Table 1.3.

Note that friend 3 has enough money to qualify for the 4% interest rate by himself, and every combination of two friends can also obtain that rate. Only if they all cooperate, are they able to get the 6% they so desire. They decide that pooling all their money is clearly the best option, and make the deposit of 5500 euro.

A year passes, and even though (or maybe because) the bank has been bailed out in the meantime, they get their 330 euro interest. Because the deposit was made in name of friend 3, he has to transfer money to the other friends’ accounts. Friend 1 jokingly proposes to split the interest equally: (110, 110, 110). Everybody sees the humour of this, and then friend 2 suggests, quite seriously, that everyone should get 6% on the amount of money they deposited: (60, 120, 150). The debate looks settled, but after a little thinking player 3 does not agree. He was able to get 4% on his own, while the others got 2%, and besides, the deposit was made in his name, and since he has the online access codes, he has control over the interest. He proposes that the other friends get 5%, and he takes the rest: (50, 100, 180). That is still more for anyone than any combination of friends could have made by themselves, he reasons. Friend 1 is furious about this, and reinstates his claim of 110 euro. This time he is serious, he explains: this claim also yields more for anyone than any combination of friends could have made. Before long, they are involved in an argument. After they settled this hours later, they never watch their favourite television show together again.

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1.2. Overview 5

have helped if they had made (binding) arrangements before the money was de-posited. But even in the latter case, how do you divide the interest? Cooperative game theory tries to answer this question. We come back to this in Chapter 3, where we extend the problem in this example to more persons, multiple periods

and more deposit options. ⊳

In Example 1.1.2, all three proposed allocations of the interest among the three friends were stable, i.e. no combination of friends could do better on their own. This is an important concept in cooperative game theory. The main question in cooperative game theory is, whether it is possible to find stable allocations for a situation. Because we know that stable allocations need not exist in general, this question continues to be relevant.

However, in the example we saw that there could be multiple stable allocations, so we need to choose one. There are two ways to do this. One way this can be done is to look at the game itself, so that any game yielding the numbers in Example 1.1.2 gives rise to the same allocation. For these allocations it does not matter what the underlying problem was that resulted in this game. This results in allocations that possess certain desirable properties for all games. The nucleolus is one such solution concept. The alternative way is that a solution concept is tailored to a specific class of problems. These solution concepts potentially take more information into account. Because in Example1.1.2the values were obtained from interest rates, one can propose to use this information and allocate on this basis.

Cooperative game theory itself consists broadly of two main models. The first one is applicable when the proceeds of cooperation do not have the same value for all individuals. This is the model of games with non-transferable utility (NTU games). The other model is that of games with transferable utility (TU games), like Example 1.1.2. In these games, the players agree on the value of what has to be divided. In this case, one can think of money. Because models in finance generally deal with money exclusively, we only deal with TU games in this thesis.

1.2

Overview

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game theory exclusively, while others deal with the application of game theoretic concepts to financial applications. This thesis is structured as follows. The first chapter after the introduction and preliminaries deals with a financial problem where a classical game theoretic model can be applied in the form of deposit games. In the next two chapters, game theory itself is the main focus, and applications in finance are scarce. These chapters deal with an extension of standard cooperative games called fuzzy games, and with an allocation scheme called the monoclus. Then we revisit applications in finance, where we focus on risk capital allocation and on bankruptcy. These chapters are not straightforward applications of existing game theoretic models, but rather the rationale from game theory is followed to develop frameworks to solve these applications in finance.

Deposit problems combine the two major elements of this thesis: game theory and finance. A deposit problem is a decision problem where an individual has a certain amount of capital at his disposal to deposit at a bank, without any risk. This amount can fluctuate over time. The aim of the individual is to maximise the return on his investment. A group of individuals is interested in investing their capital jointly. By pooling their capital, new investment opportunities can open up. These new investment opportunities in turn can lead to higher revenues. It is clear that this problem has its origins in finance. The proceeds of their investments as a group then have to be divided among the individuals. This allocation problem is an example of classical cooperative game theory, where it is clear how the situation gives rise to a value for each coalition of players. It is a generalisation of the investment situation and allocation problem in Example 1.1.2.

In Chapter 3, we extend the approach of Borm, De Waegenaere, Rafels, Suijs, Tijs, and Timmer (2001), who also study deposit problems. By allowing reinvest-ment of intermediate returns we increase the realism of their work; the amount of capital available depends on previous investment decisions. De Waegenaere, Suijs, and Tijs (2005)also develop a model that allows for reinvestment of intermediate returns, but their approach is more general because they allow for a broader range of investment products than deposits.

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1.2. Overview 7

The other case, called capital dependent deposit games, is the natural counterpart of term dependent deposit games; the term of each deposit is exactly one period, and it is the interest rate that varies depending on the amount deposited. If we assume the interest rate is higher the more capital is deposited, then a population monotonic allocation scheme (pmas) exists. An allocation scheme assigns a num-ber to every player in every coalition, and we go into this concept in more depth later on. The key in the construction of this pmas is to allocate each individual his share of the (intermediate) revenue at each period of time, rather than allocating the aggregate revenue after the final period. This insight can prove valuable to other forms of investment too.

In a further extension, we also allow for debt: individuals can borrow money at a cost. We investigate the relation between superadditive games and deposit problems, with and without debt.

The next two chapters of this thesis focus on game theory, without providing any direct applications in finance that warrants the highlighted approaches. That is not to say that these applications do not exist, but rather indicates that the ideas proposed in these chapters are born from a much more general setting.

In Chapter 4 we extend the concept of balancedness to fuzzy games. In a standard cooperative game, players that choose to combine efforts form a coalition. If a player is in a coalition, he is assumed to put in his maximal effort. However, in certain settings it makes sense to consider partial cooperation of players. For instance, if the participation of a player is money or time, it makes sense to consider situations where the different levels of participation of a player are considered. Fuzzy games are used to study the situations where the participation level of each player can range from no cooperation to full cooperation.

The fuzzy core is the natural extension of the core for fuzzy games. We extend the result that the core of a game is non-empty if and only if the game is balanced. To this end we first extend the notion of balancedness to fuzzy games. We use limited fuzzy games to study the behaviour of continuous fuzzy games, and prove that the fuzzy core of continuous fuzzy games is non-empty if and only if the fuzzy game is balanced. We show that this result can be extended to all fuzzy games with a non-empty fuzzy core.

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on single-valued solution concepts. However, we also consider allocation schemes to be interesting. Finding a value for each player in each coalition is important because of coalition forming, which in turn is important for the stability of an allocation. When forming the grand coalition, a player considers his alternatives, and these include the allocation to this player in any smaller coalition; if the allocation in a smaller coalition is higher than the allocation in the grand coalition, this player is interested in forming this coalition rather than the grand coalition.

Slikker and Norde (2009) introduce an allocation scheme rule: the monoclus. The basic idea underlying the monoclus is similar to the nucleolus; to lexicograph-ically maximise a set of monotonicities respectively excesses. The monoclus of a game is a pmas if the game allows for a pmas. In that case, even if a player exhibits the behaviour described above, the monoclus is stable since each player is allocated less costs if we add more players to the coalition we are evaluating. This allocation scheme is the focus of Chapter 5.

Besides deriving some properties of the monoclus, we apply the monoclus to a specific class of games that allows for a pmas: airport games. In an airport game, several airlines operate from the same airport but with different types of airplanes. Because larger airplanes require a larger runway when taking off or landing, each airline requires its own length of runway. One runway with maximal length suffices for all airlines since smaller planes can always land on a runway that is longer than the minimal length they require, so all airlines can jointly use this runway. Hence, cooperation between the airlines can be beneficial. The question we deal with is how to divide the costs of constructing this runway over all airlines. Airport games and the underlying airport problems have been widely studied; for an overview we refer to Thomson (2007).

To answer this question, we require insight into vectors of subbalanced weights (vsws). Norde and Reijnierse (2002) show that vsws provide important informa-tion about the class of games with a pmas. We look into the specific structure of vsws when considering airport games, and show which vsws are of particular interest for the monoclus of an airport game.

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1.2. Overview 9

The remainder of this thesis then revisits applications in finance. The inter-est in particular is on risk. Wherever risk is taken, there is a possibility of an unfavourable turn of events. In the case of financial risk, there is always the pos-sibility of default. Proper risk management is becoming increasingly important in decision making within society and within a firm. We are becoming more aware of the consequences of - what is often considered - a stroke of bad luck, and how it is possible to incorporate those in decision making theory and lessening their impact. The problem is particularly relevant for firms that operate in the financial sector, such as banks and insurance companies. For instance, in the United States of America as well as in Europe, it has become necessary to bail out big banks and insurers in an attempt to prevent a collapse of the current financial system (or even the real economy). This was the case with ING, for instance, while DSB Bank has gone bankrupt altogether. The impact on the lives of countless indi-viduals that trusted these firms with their money is large, and regulation is being discussed to prevent such a thing from happening again.

Regulators therefore require that financial institutions hold a level of capital, referred to as risk capital, to cope with this risk. The risk capital needs to be invested safely in order to act as a buffer. Similar toArtzner, Delbaen, Eber, and Heath (1999), we consider settings where the amount of risk capital is determined by means of a coherent risk measure. Although retaining risk capital reduces the chances of bankruptcy, there are several open problems.

Two related problems to preventing ruin of firms are the focal points in the final two chapters of this thesis. Ideas often used in game theory are applied to solve these problems.

Withholding risk capital is costly, since it implies that the capital cannot be used for potentially profitable investments. Because it is costly, it affects the performance of an investment. Therefore, if one wants to evaluate the performance of different divisions of a firm, it is important that the total risk capital is allocated to the individual lines of business in a ‘fair’ way. How this should be done is not immediately clear when determining the total risk capital; the underlying intuition is that because the risks of different divisions are typically not perfectly correlated, some hedge potential may arise from combining different divisions.

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company has to deal with in the coming year are dependent on the weather, and can be divided into three states: either there is a drought, there is heavy rain and thunderstorms, or the weather is calm. If the weather is calm, we do not ex-pect fire insurance nor car insurance to have to deal with heavy losses. However, if there is a drought, then fires are more likely to occur, so the division fire insurance runs a risk of incurring more losses because of the drought. If there is heavy rain, fires are less likely, but due to slippery roads and obscured vision, car accidents might occur more frequently. This could result in more losses for the car insurance division. Each of the divisions, if they were to operate on their own, would have to reserve capital based on the worst state. But now that these two divisions are within one insurance company, the company only has to reserve the capital of the worst case for one of them. But what costs does each of the divisions bear?

Our goal in Chapter6is to propose an allocation mechanism in settings where the risk capital is determined by a coherent risk measure. We evaluate alloca-tions considering the excess of every combination of portfolios with respect to the allocation, i.e. the expectation of the losses in excess of the risk capital that is allocated to the combination of portfolios. The allocation mechanism we propose is the one that minimises the excesses in a lexicographical manner. We show that this Excess Based Allocation (eba) has some advantages over other allocation mechanisms proposed in the literature, such as an allocation on the basis of the marginal risk contribution, which, in game-theoretic terms, is equivalent to the Aumann-Shapley value (see Aumann and Shapley (1974)). For instance, we show that eba is defined for a broader class of risk capital allocation problems than the Aumann-Shapley value. Also, we study the behaviour of the mechanisms by evaluating various continuity and monotonicity properties.

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1.2. Overview 11

If all companies in a certain branch retain risk capital, it is highly unlikely that all of them will go bankrupt at the same time. It is much more likely that while some companies might not be able to avert bankruptcy, other companies might, or are even able to make a profit. In Chapter 7, we provide a framework such that the business sector supports the customers of bankrupt firms. There is a natural divide between two types of outcomes, one where this is enough capital in the entire business sector to prevent default for all firms, and the other where this is not the case. We model the need for capital in both types as a bankruptcy problem (cf. Aumann and Maschler (1985)). We use a bankruptcy rule, which in the latter case allocates the available capital over the claimants, such that they get at least zero, and at most their claim. In the former case, a bankruptcy rule can also be used to determine the amount needed from each firm to reimburse customers.

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Chapter 2

Preliminaries

The trouble was that he was talking in philosophy, but they were listening in gibberish.

Terry Pratchett, Small Gods

2.1

Basic notation

We denote the set of all natural numbers by N. The set of all real numbers is denoted R, the set of non-negative real numbers by R+ and the set of positive real

numbers by R++.

For a finite set N we denote its power set, i.e. the collection of all its subsets, by 2N and the number of elements by |N|. An element of RN is denoted by a

vector x = (xi)i∈N.

For all x ∈ RN and for all S ⊆ N we denote the aggregate P

i∈Sxi of x with

respect to S by xS. Similarly, for all random variables X = (Xi)i∈N we denote

XS =Pi∈SXi.

We denote (y)+ = max{y, 0} for all y ∈R. We define x+ = ((x

i)+)i∈N for all

x ∈RN.

Let K be a finite set. For all x ∈ RK, the ordering that arises from x by

arranging the coordinates of x in a non-increasing fashion is θ(x) = y, where y ∈ RK. Formally, σ is a permutation defined by a bijection σ : {1, 2, . . . , |K|} → K.

The collection of all permutations σ of K is denoted Π(K). Next, θ :RN R|K|

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is defined by θi(x) = xσ(i) for all i ∈ K, where the permutation σ is such that

xσ(1) ≥ xσ(2) ≥ . . . ≥ xσ(|K|). The function θ(·) is continuous.

For any two vectors x, y ∈ RK we denote y > x if y

i > xi for all i ∈ K, we

denote y ≥ x if yi ≥ xi for all i ∈ K and we denote y x if yi ≥ xi for all i ∈ K,

but there exists at least one j ∈ K such that yj > xj.

Let K = {1, 2, . . . , k}. For any two vectors x, y ∈ RK, x is said to be

lexico-graphically strictly smaller than y if there exists an i ∈ K such that xi < yi, and

for all j < i it holds that xj = yj. We denote this by x <lex y. We say that x

is lexicographically smaller than y, denoted by x ≤lex y, if x = y or x <lex y. If

θ(x) <lex θ(y) for x, y ∈RK, then there is an i such that the biggest i − 1 elements

of x are equal to the corresponding i − 1 biggest elements of y, and x is strictly smaller than y for the i-th biggest element. The remaining elements are the |K|−i smallest elements of x and y, and no specific relation between x and y is required for these elements.

We use Conv{A} to denote the convex hull of a finite collection of vectors A ⊂ RN. We denote the convex cone, generated by a finite collection of vectors

A, by CC{A}.

For a finite probability space Ω, probability distribution π ∈ (0, 1]Ω where we

require P

ω∈Ωπ(ω) = 1, and X ∈RΩ, the expected value of X is defined by

E[X] =X

ω∈Ω

X(ω) · π(ω).

2.2

Games with transferable utility

A cooperative game with transferable utility, or TU-game, is a pair (N, v) where N = {1, 2, . . . , n} is the set of players, and v : 2N R is the characteristic

function that assigns a value to each coalition S ⊆ N, representing the payoff to this coalition. By definition, v(∅) = 0. We refer to a cooperative game with transferable utility as ‘a game’, and denote the game by v instead of (N, v) when no confusion arises. For all S ⊆ N, S 6= ∅ the subgame with respect to S is defined as (S, v|S) with v|S(T ) = v(T ) for all T ⊆ S.

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2.2. Games with transferable utility 15

A game (N, v) is called additive if

v(S) =X

i∈S

v({i}),

for all S ⊆ N. The additive game (N, v) corresponding to a vector y ∈ RN is

defined

v(S) =X

i∈S

yi,

for all S ⊆ N. A game (N, v) is convex if for all i ∈ N and all S ⊂ T ⊆ N\{i} it holds

v(S ∪ {i}) − v(S) ≤ v(T ∪ {i}) − v(T ). For a game (N, v), we denote the preimputation set by

P I(N, v) = ( x ∈RN X i∈N xi = v(N) ) .

Any x ∈ P I(N, v) is called efficient. If furthermore x is individually rational, i.e. xi ≥ v({i}) for all i ∈ N, then x is in the imputation set, defined by

I(N, v) = ( x ∈RN X i∈N

xi = v(N), xi ≥ v({i}) for all i ∈ N

) . The core of a cooperative game (N, v) is defined by

C(v) = ( x ∈RN X i∈N xi = v(N), ∀S ⊆ N : X i∈S xi ≥ v(S) ) , and a core allocation is a vector x ∈ C(v).

A cooperative game (N, v) is called balanced if P

S⊆N λ(S)v(S) ≤ v(N) for all

functions λ : 2N R

+ satisfying

P

S⊆N :i∈Sλ(S) = 1 for all i ∈ N. A game is

called totally balanced if every subgame (S, v|S) is balanced. Bondareva (1963)

and Shapley (1967)derived the following result about the core and balancedness.

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The marginal vector mσ(v) of a game (N, v) corresponding to the permutation

σ is defined by

σ(i)(v) = v({σ(1), σ(2), . . . , σ(i)}) − v({σ(1), σ(2), . . . , σ(i − 1)}), for all i ∈ N.

The Shapley value as introduced by Shapley (1953)is defined by Φ(N, v) = 1

|N|! X

σ∈Π(N )

mσ(v), i.e. the average over all marginal vectors.

The excess of a coalition S ⊆ N with respect to the vector x is defined by e(S, x) = v(S) − xS.

This value represents the additional payoff to coalition S compared to x, if S is formed. The excess measures the dissatisfaction of S with the proposed vector x, where lower is better. The aim of the (pre)nucleolus is to minimise the highest excesses. If we consider only efficient allocations, we obtain the prenucleolus of the game (N, v), defined by

ψ(N, v) = { x ∈ P I(N, v)| θ(x) ≤lex θ(y) for every y ∈ P I(N, v)} .

The prenucleolus is always a singleton set. The nucleolus of the game (N, v) is defined by

n(N, v) = { x ∈ I(N, v)| θ(x) ≤lexθ(y) for every y ∈ I(N, v)}

if I(N, v) 6= ∅. The nucleolus is a single point whenever it is defined. Furthermore, the nucleolus is always in the core of the game, if the core is non-empty.

According toSprumont (1990)an allocation scheme (xS)

S⊆N,S6=∅ with xS ∈RS

for all S ⊆ N, S 6= ∅ is called a population monotonic allocation scheme (pmas) for a cooperative game (N, w) if it satisfies efficiency and monotonicity. Here (xS)

S⊆N,S6=∅ satisfies efficiency if

X

i∈S

xSi = w(S),

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2.2. Games with transferable utility 17

If (xS)

S⊆N,S6=∅ is a pmas, then for all S ⊆ N, S 6= ∅, xS belongs to the core of the

subgame (N, v|S) restricted to S and hence (N, v) is totally balanced.

An extension of a game with transferable utility is a fuzzy game with trans-ferable utility (N, r), where N is the player set and r : [0, 1]N R is the

char-acteristic function. Players are assumed to be infinitely divisible, hence coalitions consist of fractional players, i.e. s ∈ [0, 1]N. By definition r(0) = 0. We denote

as =Pi∈Nsi · ai.

Similar to the core of a game, the fuzzy core of a fuzzy game r is defined by F Core(r) = {x ∈RN|x

s≥ r(s) for all s ∈ [0, 1]N,

X

i∈N

xi = r(1, 1, . . . , 1)}.

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Chapter 3

Deposit games with reinvestment

The Yen Buddhists are the richest religious sect in the universe. They hold that the accumulation of money is a great evil and a burden to the soul. They therefore, regardless of personal hazard, see it as their unpleasant duty to acquire as much as possible in order to reduce the risk to innocent people.

Terry Pratchett Witches Abroad

3.1

Introduction

A deposit problem is a decision problem where an individual has a certain amount of capital at his disposal to deposit at a bank. This individual aims to maximise the return on his investment. When we allow for cooperation between individuals by means of joint investment, they can increase their joint return. However, this gives rise to the additional question of how to allocate the proceeds among the individuals. In this chapter, we analyse cooperation in deposit situations and in particular, we explore whether an allocation exists such that all players will want to cooperate and form the grand coalition.

Deposit games have previously been studied byBorm, De Waegenaere, Rafels, Suijs, Tijs, and Timmer (2001), Tan (2000) and De Waegenaere, Suijs, and Tijs (2005). Borm et al. (2001) define a deposit as a fixed amount of capital at a bank for a certain amount of time, and use a revenue function that describes the

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revenue of a deposit in a fixed end period, after which no more deposits can be made. In particular, they do not allow for reinvestment of intermediate revenue. Tan (2000)extends this approach and adds the possibility of borrowing.

The approach of De Waegenaere et al. (2005) is more general. They allow reinvestment and a broader range of investment products than deposits. Among other things, the money invested in a single investment product is allowed to change every period, rather than being a fixed amount of capital. The drawback of this approach is that because of its generality, it is harder to draw firm conclusions. Lemaire (1983)andIzquierdo and Rafels (1996)analyse related issues. Lemaire (1983) gives an overview of the use of game theory in financial issues. Izquierdo and Rafels (1996) analyse games that are related to capital dependent deposit games.

In this chapter, which is based onVan Gulick, Borm, De Waegenaere, and Hen-drickx (2010), we take the approach that reinvestment should be possible, while still allowing for a particular form of the revenue function. We start by modelling the decisions of individuals during a discrete and finite number of periods of time. Any deposit will lead to a non-negative revenue. This non-negativity condition is justified because, by the nature of deposits, it is always possible to privately save any amount of money at no cost. There will be no default risk involved in any deposit.

We allow coalitions of individuals to deposit money jointly by cooperating. Clearly, the revenue of a coalition will never be lower than the sum of all individual revenues. We are interested in finding a core element, i.e. we want an allocation for the grand coalition such that no coalition has any incentive to split off.

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3.2. Deposit problems and deposit games with reinvestment 21

decision is on the amount of capital to deposit; depositing over different lengths of time does not add any beneficial options to the investor. We show that, if the rate of return in capital dependent deposit games is increasing in the amount of capital deposited, a core element exists. In fact, we can explicitly construct a population monotonic allocation scheme a la Sprumont (1990).

In a further extension of the model, we allow individuals to also have debt, and we show that all non-negative superadditive games are deposit games of this type.

The remainder of this chapter is structured as follows. In section 3.2 we in-troduce deposit problems and deposit situations with reinvestment and define corresponding deposit games. In section3.3 we compare the current model to the one analysed by Borm et al. (2001) and show that it indeed is an extension. In the next two sections we analyse two specific subclasses of deposit games. Term dependent deposit games are analysed in section3.4 and in section 3.5 capital de-pendent deposit games are considered. We show how superadditive games can be rewritten as deposit games with reinvestment and debt in section 3.6, and elabo-rate on this issue in section3.7, where we revisit deposit games with reinvestment, but without debt.

3.2

Deposit problems and deposit games with

reinvestment

Similar to Borm et al. (2001), a deposit is defined as a positive, fixed amount of capital c that is at a bank during a prearranged and consecutive number of periods t1, t1 + 1, . . . , t2, where 1 ≤ t1 ≤ t2 ≤ τ . Here τ is the final period in which a

deposit can be made. The scope of a deposit problem is thus a discrete and finite time span {1, . . . , τ }. The time interval over which money is deposited is called the term of the deposit

T = {t1, t1+ 1, . . . , t2}.

The set of all possible terms of a deposit is given by

T =

{

T ⊆ {1, . . . , τ } |∃t1, t2 ∈ {1, . . . , τ } : T = {t1, t1+ 1, . . . , t2}

}

. (3.1)

A deposit with capital c and term T = {t1, t1+ 1, . . . , t2} can be represented as a

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c is deposited and at the start of period t2 + 1 the amount c is returned. Note

that since capital is returned at t2+ 1, we extend our model to include the period

τ + 1.

To illustrate this, we look at a deposit where τ = 3. A possible deposit can be written as (0, 3, 0, −3), which means that three units of capital (c = 3) are deposited during periods t1 = 2 and t2 = 3, and this is returned at the beginning

of period 4, which is t2+ 1.

The set of all possible deposits is denoted

∆ =δ ∈ Rτ +1|∃c > 0, T ∈ T : δ = c · h(T ) ,

(3.2) where we define the function h : T →Rτ +1 as a deposit of 1 unit of capital over

term T = {t1, t1+ 1, . . . , t2}, so for all t ∈ {1, 2, . . . , τ + 1} and all T ∈ T we have

ht(T ) =      1, if t = t1, −1, if t = t2+ 1, 0, otherwise. (3.3)

We assume there is a revenue function P : ∆ →Rτ +1+ that assigns to each deposit

δ ∈ ∆ a non-negative revenue in each period of time in {1, . . . , τ + 1}. No explicit formula for the revenue function is assumed at this time. Borm et al. (2001) allow revenue to be obtained only in period τ + 1. This corresponds to Pt(δ) = 0

for all t < τ + 1 and all deposits δ ∈ ∆. If we consider again τ = 3 and the deposit δ = (0, 3, 0, −3) and assume that we get a 4% interest on it, the revenue is P (δ) = (0, 0, 0.12, 0.12). Note that the revenue from t = 3 is not included in the revenue from t = 4. In fact, this revenue at t = 3 is now available to deposit and will return as such to the individual.

To avoid complications, we assume that money has to be deposited first, and only in a later period a revenue can be attained.

Assumption 3.2.1 For all δ = c · h({t1, t1 + 1, . . . , t2}) ∈ ∆ and all t ≤ t1 we

have Pt(δ) = 0.

We also assume that arbitrage is excluded; it is not possible to obtain an infinite revenue using a finite amount of capital.

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3.2. Deposit problems and deposit games with reinvestment 23

the consumption of the individual. The endowments are exogenous and express the net income an individual has available to use for deposits. The second source is payback of capital that was previously deposited. The third source is the revenue from deposits.

The endowments of an individual are given by m ∈Rτ. Although the

endow-ment can be negative at some point in time, we assume that for an individual at each time period the cumulative endowment is non-negative. This boils down to the following assumption.

Assumption 3.2.2 For all t ∈ {1, . . . , τ } we have Pt

s=1ms≥ 0.

A deposit problem with final period τ , set of deposits ∆, revenue function P and endowments m is denoted by (τ, ∆, P, m).

Because an individual has limited endowments, there are only certain combi-nations of deposits that he can invest in. A portfolio of deposits is denoted by a function f : ∆ → N ∪ {0}, which expresses how many units of each deposit are used. A portfolio of deposits an individual is able to finance with his endow-ments is called feasible. Since the endowment of an individual in a given period is the amount of capital that is available in that period after consumption, and because there is no upper bound on the amount of money that can be deposited, and moreover, the return on any deposit is non-negative, we assume an individual will, in any time period, invest all available capital in deposits. Note that we take into account the possibility of reinvestment of previously deposited capital and returns. Also note it is possible to get a feasible collection of deposits using the same deposit more than once. It is for instance possible to open two accounts at the bank and deposit the same amount in each account.

The set of all feasible portfolios for an endowment vector m ∈Rτ is thus

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(3.3). Consequently this part of the expression can be negative if less capital is deposited in new deposits than is paid back from previous deposits.

The right hand side of the equality consists of two parts. The first part are the endowments, which can be negative. The second part is the sum of the revenue over all deposits at time t, which is non-negative. Thus the equality states that the net change in investment in deposits is equal to the endowments plus revenue. Before we proceed, two remarks are in order. First, even though ∆ is non-denumerable, either side of the equality in the feasibility condition is bounded. At t = 1 there is no revenue by Assumption 3.2.1, so the right hand side of the equality in the feasibility condition is finite. This implies that a feasible portfolio f (δ) is non-zero only for a denumerable number of deposits δ ∈ ∆ with period 1 included in their term. Because of the exclusion of arbitrage, it is impossible to get an unbounded revenue with a finite amount of capital. Therefore, also in the next period, t = 2, the revenue is bounded, and again only a denumerable amount of deposits including period 2 can be non-zero. This argument holds for all periods, so f can only be non-zero for a denumerable amount of deposits. Thus, the sum is well defined.

Second, in the definition of F (m) it may be more natural to consider the restriction as an inequality,P

δ∈∆f (δ)δt ≤ mt+Pδ∈∆f (δ)Pt(δ) for all t. However,

without loss of generality we can assume the expression to be an equality, because the non-negativity of the revenue function makes it possible to carry over any amount of capital to the next period without losses.

We now define the optimisation problem for an individual in the same way as Borm et al. (2001). The natural objective is to maximise the total revenue. However, the fact that at any period there might be some revenue presents us with some difficulties. Because of the non-negativity of the revenue function it is optimal to carry over all available capital (including intermediate revenue) to period τ + 1. Hence the natural objective is to maximise the total revenue at time τ + 1.

The total capital of an individual at time τ + 1 consists of three parts: the revenue of deposits at time τ + 1, which is P

δ∈∆f (δ)Pτ +1(δ), the endowments of

the individual, and any intermediate revenue. The latter two are both obtained before period τ + 1, and are thus equal to the payback of deposits at τ + 1. This payback of deposits at τ + 1 equals −P

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3.2. Deposit problems and deposit games with reinvestment 25

construction of deposits and equation (3.3) in particular. Thus the total capital at time τ + 1 as a function of a feasible portfolio f ∈ F (m) equals

Π(f ) =X

δ∈∆

f (δ)[P (δ) − δ]τ +1.

Observe that we have f + g ∈ F (m1+ m2) for any two portfolios f ∈ F (m1) and

g ∈ F (m2), and

Π(f ) + Π(g) =X

δ∈∆

(f (δ) + g(δ))[P (δ) − δ]τ +1 = Π(f + g). (3.4)

With m ∈Rτ define the maximal revenue in period τ + 1 by means of a feasible

portfolio as π(m) = sup ( Π(f ) − τ X t=1 mt f ∈ F (m) ) .

We assume that for all m ∈Rτ +1 this supremum exists. The only concern here is

the value of the supremum being infinity. This can by definition only happen if there is the possibility of arbitrage, which is assumed to be excluded. Note that π(m) ≥ 0.

Within a cooperative framework, where a group of agents N = {1, . . . , n} com-bines efforts, it is possible for individuals to form coalitions and deposit money jointly. If for instance interest rates, defined as the revenue divided by the amount of capital deposited, are higher when a larger sum of money is invested, or money is deposited over a longer period, it might be attractive for individuals to co-operate. In this way, coalitions can possibly make more money than the indi-viduals can when they act alone. In this way we consider a deposit situation (N, τ, ∆, P, (m(i))i∈N), where agent i ∈ N has endowment vector m(i) ∈Rτ

avail-able to deposit. The availavail-able capital m(S) to a coalition S ⊆ N, S 6= ∅ is simply the sum of all capital available to each member of the coalition. So for a coalition S ⊆ N, S 6= ∅

m(S) =X

i∈S

m(i).

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We define the deposit game with reinvestment corresponding to a deposit situation (N, τ, ∆, P, (m(i))i∈N) by the transferable utility game (N, v) with characteristic

function v as given by equation (3.5) for all S ⊆ N, S 6= ∅. By assumption v(∅) = 0. We add the qualification ‘with reinvestment’ to make the distinction with deposit games as defined by Borm et al. (2001). Throughout the remainder, where no confusion arises this qualification is dropped.

The next theorem states that all deposit games are superadditive. The intu-ition behind this is that all players in a coalintu-ition are at least able to invest in the same deposits as they would optimally do as smaller coalitions.

Theorem 3.2.3 Let (N, v) be a deposit game corresponding to deposit situation (N, τ, ∆, P, (m(i))i∈N). Then (N, v) is superadditive.

Proof: Let S, T ⊆ N be such that S ∩ T = ∅. Then v(S) + v(T ) = sup ( Π(fS) − τ X t=1 mt(S) fS ∈ F (m(S)) ) + sup ( Π(fT) − τ X t=1 mt(T ) fT ∈ F (m(T )) ) = sup ( Π(fS) + Π(fT) − τ X t=1 mt(S) − τ X t=1 mt(T ) fS ∈ F (m(S)), fT ∈ F (m(T )) ) = sup ( Π(fS+ fT) − τ X t=1 mt(S ∪ T ) fS ∈ F (m(S)), fT ∈ F (m(T )) ) ≤ sup ( Π(f ) − τ X t=1 mt(S ∪ T ) f ∈ F (m(S ∪ T )) ) = v(S ∪ T ),

where the inequality follows from the fact that

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3.2. Deposit problems and deposit games with reinvestment 27

where we use that S ∩ T = ∅. 

The natural question that arises is how to divide the maximal joint revenue over the individual players. To tackle this question we analyse core allocations of the corresponding game. The core of a cooperative game (N, v) is defined in Chapter 2. A core allocation is stable against coalitional deviations because every coalition is allocated at least what it can obtain by depositing on its own.

The following example shows that within the general setting of deposit games as developed above, where we have not put any restrictions on the revenue function P other than the exclusion of arbitrage opportunities and Assumption 3.2.1, core elements need not exist.

Example 3.2.1 Consider a three player deposit situation (N, τ, ∆, P, (m(i))i∈N)

with two periods, so τ = 2, N = {1, 2, 3} and the set of deposits ∆ is given by equation (3.2). Furthermore, assume that these three players are identical, in the sense that they each have the same endowments. Assume that the endowments of player i ∈ N are given by m(i) = (300, −50). Note that m(i) satisfies Assumption 3.2.2.

The setting we consider consists of only two deposits with non-zero revenue. The first one is a two-year bond of 500 yielding an interest rate of 6% per period. The second one is a two-year bond of 250 yielding 1% interest. So, in our model the revenue function is given by

P (δ) =      (0, 2.5, 2.5), if δ = (250, 0, −250), (0, 30, 30), if δ = (500, 0, −500), (0, 0, 0), otherwise.

This revenue function satisfies Assumption 3.2.1.

We now calculate the value of each coalition. Obviously, for every two-player coalition {i, j} it is optimal to buy one two-year bond of 500. The remaining 100 in the first period can not be used to get any revenue, but can be carried over to the next period (without interest). In the second period, the coalition still has 500 deposited, has 100 carried over, receives endowments of −100 and receives a revenue of 30. This intermediate revenue is carried over. The optimal feasible portfolio f for any two-player coalition is thus given by

f (δ) = (

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This gives all two-player coalitions a value of

v({i, j}) = (30 − (−500)) + (0 − (−30)) − (600 − 100) = 60.

To clarify further, had the endowments been (300, −60), the deposit (500, 0, −500) would still have been feasible. In period t = 2 the endowments of the coalition is only 480, however with the intermediate return of 30 from the mentioned deposit itself, 510 would be available. With endowments (300, −70) however, the deposit would not have been feasible since at t = 2 only 490 can be invested in deposits.

If we return to the situation where the endowments are (300, −50), observe that any single player i ∈ N has insufficient capital to deposit 500. Clearly depositing in the two-year bond of 250 is optimal, together with a deposit that carries over the intermediate revenue from t = 2 to t = 3. This results in a value

v({i}) = 5.

The grand coalition can buy one two-year bond of 500 and one of 250 in t = 1 and carries over the remaining capital of 150. The intermediate revenue at period t = 2 is also carried over to period t = 3. This leads to

v(N) = 65.

If we want to construct a core element x ∈ R3for this deposit game, it needs to

satisfy several inequalities. For the two-player coalitions, these are: x1+ x2 ≥ 60,

x1+ x3 ≥ 60 and x2+ x3 ≥ 60. If we add these three inequalities together, we get

2(x1+ x2 + x3) ≥ 180, or x1+ x2 + x3 ≥ 90. However, efficiency tells us that a

core allocation x should satisfy x1+ x2+ x3 = 65 < 90. So the core is empty for

this particular deposit game. ⊳

3.3

Deposit games with and without

reinvest-ment

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3.3. Deposit games with and without reinvestment 29

The scope of a deposit problem without reinvestment is a finite and discrete time span {1, . . . , ρ}. The set of all possible terms of a deposit T is denoted by equation (3.1) with τ = ρ. Borm et al. (2001) denote deposits by d = c · e(T ), where

et(T ) =

(

1, if t1 ≤ t ≤ t2,

0, otherwise, (3.6)

for all T = {t1, t1+ 1, . . . , t2} ∈ T , t ∈ {1, . . . , τ }. The set of all deposits is given

by

D = {d ∈Rρ+|∃c ≥ 0, T ∈ T : d = c · e(T )} .

The revenue function is denoted by R : D → R+ with R(0) = 0. R(d) is the

revenue of a deposit d, which is received in period ρ + 1. Let N denote the set of all players. The endowments of player i ∈ N are denoted by a vector ω(i) ∈ Rρ.

ωt(i) denotes the total amount of capital available to player i ∈ N to deposit in

period t. For S ⊆ N, S 6= ∅ we define ω(S) = P

i∈Sω(i). A deposit situation

without reinvestment is denoted by (N, ρ, D, R, (ω(i))i∈N).

Finally the value of a coalition S ⊆ N in the corresponding deposit game without reinvestment is defined by

w(S) = sup        ℓ X k=1 R(dk) ∃ℓ ∈N, ∃d1, . . . , d∈ D : ∀t ∈ {1, 2, . . . , τ } : ℓ X k=1 dkt ≤ ωt(S)        .

Theorem 3.3.1 Every deposit game without reinvestment is a deposit game with reinvestment.

Proof: Consider a deposit game without reinvestment (N, w) corresponding to a deposit situation (N, ρ, D, R, (ω(i))i∈N). We proceed to construct a deposit

situation with reinvestment (N, τ, ∆, P, (m(i))i∈N) for which the corresponding

deposit game (N, v) is equal to (N, w).

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to δ = c · h(T ) ∈ ∆. In particular, from equations (3.3) and (3.6) we find for all T ∈ T ht(T ) =      e1(T ), if t = 1, et(T ) − et−1(T ), if t ∈ {2, . . . , τ }, −eτ(T ), if t = τ + 1.

Moreover, define the revenue function P : ∆ →Rτ +1+ by Pt(c · h(T )) =

(

R(c · e(T )), if t = τ + 1,

0, otherwise,

for all c > 0 and all T ∈ T . For all i ∈ N, the endowments m(i) ∈ Rτ are

recursively defined by mt(i) =

(

ω1(i), if t = 1,

ωt(i) − ωt−1(i), if t ∈ {2, . . . , τ }.

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3.3. Deposit games with and without reinvestment 31

where at the fourth equality we use that because the feasible portfolio is non-zero for a denumerable number of deposits only, the supremum over the revenue of these denumerable many deposits is equal to the supremum of the sum of the revenue of any finite number of deposits. If we use the definition of a deposit, the previous expression is equal to

sup        ℓ X k=1 Pτ +1(ck· h(Tk)) ∃ℓ ∈N ∪ {0}, ∃c1, . . . , c> 0, ∃T1, . . . , T∈ T : ∀t ∈ {1, . . . , τ } : ℓ X k=1 ck· ht(Tk) = mt(S)        = sup              ℓ X k=1 Pτ +1(ck· h(Tk)) ∃ℓ ∈N ∪ {0}, ∃c1, . . . , cℓ > 0, ∃T1, . . . , T∈ T : ∀t ∈ {1, . . . , τ } : t X r=1 ℓ X k=1 ck· hr(Tk) = t X r=1 mr(S)              = sup              ℓ X k=1 Pτ +1(ck· h(Tk)) ∃ℓ ∈N ∪ {0}, ∃c1, . . . , cℓ > 0, ∃T1, . . . , T∈ T : ∀t ∈ {1, . . . , τ } : ℓ X k=1 ck t X r=1 hr(Tk) = t X r=1 mr(S)              = sup        ℓ X k=1 Pτ +1(ck· h(Tk)) ∃ℓ ∈N ∪ {0}, ∃c1, . . . , cℓ > 0, ∃T1, . . . , Tℓ ∈ T : ∀t ∈ {1, . . . , τ } : ℓ X k=1 ck· e t(Tk) = ωt(S)        .

Because R(0) = 0, it is always possible to make at least one deposit, i.e. any deposit with c = 0, hence we can rewrite the last expression as

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= w(S),

so the deposit games (N, v) and (N, w) are equal. 

3.4

Term dependent deposit games

From Example 3.2.1 on page 27 it is clear that in general it can be difficult in a deposit situation to find a suitable allocation of the total revenue among the indi-viduals. In particular, we did not impose any structure on the revenue function, making it difficult to derive conclusions in general about the revenues, and thus values, of all coalitions. In this section and the next we consider two different simplifying restrictions.

The return on deposits typically depends on the term of the investment and/or on the amount of capital invested. Consider for example saving accounts at a bank; these deposits give a fixed interest rate over the capital that is deposited. Each period, this interest rate is paid over the amount of capital deposited. However, if the individual is willing to make a longer commitment, that is, he agrees to deposit his money over multiple periods, the bank typically offers a higher interest rate. On the other hand, the return of an investment can also be affected by the amount of capital invested. The latter effect may be particularly relevant in, for example, investment in real estate, where a higher return may become feasible only if more capital is available.

In this section, we focus at the effect of term dependence by considering a setting where the effect of the amount of capital that is deposited is kept constant for each possible term. In section 3.5, we will investigate the effect of capital dependence, by considering a setting where the effect of the term of the investment is kept constant.

In reality of course, a combination of both types of deposits exist, but we focus on these two types separately, better to highlight how the two aspects work through in the allocation process.

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3.4. Term dependent deposit games 33

Definition 3.4.1 A revenue function P : ∆ →Rτ +1

+ is called term dependent if

it holds for all t ∈ {1, . . . , τ + 1}, for all deposits δ ∈ ∆ and for all α > 0, that Pt(αδ) = αPt(δ). If the underlying revenue function is term dependent, also the

corresponding deposit situation and deposit game are called term dependent. Note that in a term dependent deposit situation, for all terms T ∈ T , all ℓ ∈ N and all c1, . . . , c> 0 we have

ℓ X k=1 Pck· h(T )= Ph(T ) ℓ X k=1 ck.

So without loss of generality we can assume that all deposits with the same term t ∈ T can be combined into an aggregate portfolio deposit. Hence, in a term dependent deposit situation, decisions are made on the term of the deposits and not on the amount, since the rate of return is fixed for each possible term.

The following lemma shows that if we have a feasible portfolio for some endow-ments, then if we scale these endowments the portfolio that consists of the scaled deposits is feasible for the new problem, and furthermore its revenue is also scaled in the same manner.

Lemma 3.4.2 Let m ∈ Rτ, f ∈ F (m) and λ > 0 and define g(λ · δ) = f (δ) for

all δ ∈ ∆. Then g ∈ F (λ · m) and Π(g) = λ · Π(f ).

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Moreover, Π(g) =X δ∈∆ g(δ)[P (δ) − δ]τ +1 = X λδ∈∆ g(λδ)[P (λδ) − λδ]τ +1 = λ X λδ∈∆ f (δ)[P (δ) − δ]τ +1 = λ X δ∈∆ f (δ)[P (δ) − δ]τ +1 = λΠ(f ),

which proves the second statement. 

We now show that for each term dependent deposit game there is a core allocation. For this we use the notion of balancedness, and its relation to the core as in Theorem 2.2.1.

Theorem 3.4.3 Every term dependent deposit game with reinvestment is totally balanced.

Proof: Let (N, v) be a deposit game corresponding to a term dependent deposit situation (N, τ, ∆, P, (m(i))i∈N). First we show that (N, v) is balanced. Take

λ : 2N R

+ such that PS⊆N :i∈Sλ(S) = 1 for all i ∈ N. Then for every

t ∈ {1, . . . , τ } X S⊆N λ(S)mt(S) = X S⊆N λ(S)X i∈S mt(i) = X i∈N X S⊆N :i∈S λ(S)mt(i) =X i∈N mt(i) X S⊆N :i∈S λ(S) (3.7) =X i∈N mt(i) = mt(N).

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