In dit verslag wordt het gedrag van een dief die huizen besteelt wiskundig bekeken. Het dievenspel is de aanleiding van dit verslag. In dit spel bepaalt een dief zijn optimale strategie voor het bestelen van één huis uit verschillende huizen met verschillende waarden. Daarnaast is er een agent die op zoek is naar zijn optimale strategie om de huizen te bewaken. Dit verslag duikt in de speltheorie om het dievenspel te ontleden. Het dievenspel wordt opgelost en vervolgens uitgebreid naar het geval dat de dief meerdere huizen kan bestelen en er meerdere agenten zijn. Er worden twee gevallen onderscheiden, waar huizen dezelfde waarde hebben en waar huizen verschillende waarde hebben. Voor de situatie met huizen van dezelfde waarde wordt er gebruik gemaakt van de theorie over stopproblemen om een optimale strategie voor de dief te bepalen. Eerder wordt de theorie van stopproblemen uitgelicht. Hier worden interessante voorbeelden en optimale stopregels besproken. Voor de situatie met huizen van verschillende waarde zal de strategie van de dief en de agent numeriek worden geoptimaliseerd. Er zal blijken dat het voordeliger is voor de dief om meer huizen van lagere waarde te bestelen en voor de agenten om meer huizen van hogere waarden te bewaken.

In this thesis report we consider a search and rescue problem in which one or multiple targets/objects are hidden in some playing field, and must be rescued/found by a searcher. The targets are for example: earthquake survivors, lost hikers or prisoners held by an adversary, and are hidden in some section of the play field. Searching any of the sections of the play field has a certain probability of failing; the searcher might get lost, trapped or captured herself. The goal is to find the search that maximises the probability of finding all targets, and to find the hiding spot that minimises it. We define and solve the search and rescue game on a play field for which movement between any two section of the play field is always possible. We also define and solve the game played on a tree for which movement is limited by the tree structure. Due to the complexity of this game, we restrict ourselves to one target. Finally, we extend the game by replacing the play field with specific types of graphs that are not trees. For some of these graphs, we have found (partial) solutions.

With the growing wealth and economy of a country, there are an increasing amount of small and big businesses. Every company has its own marketing strategy that it uses in order to lure customers away from their competition and increase their sales. Choosing the perfect time to advertise or discount several products is of essence for a company to gain more money than their competition. These type of marketing games are all slight variations of duels. The purpose of this report is to research how this duel is played most optimal when there are two or more participants. Several types of two-player duels shall be analysed first in order to understand and analyse a three-player duel.

In this report, we will look at the connection between the Fibonacci and Euler numbers. By using a combinatorial argument including the Fibonacci and Euler numbers, we will prove our main theorem:  Fn·En ≥ n! From the main theorem and the asymptotics of these numbers, we will conclude that π ≤ 2ϕ. We follow the proof in the article of Alejandro H. Morales, Igor Pak & Greta Panova, but we will give a more detailed proof and some extra facts about the Golden Ratio, ϕ, and the Fibonacci and Euler numbers. Finally, the article discusses the number of linear extensions of certain partially ordered sets, or posets. We see that there exist a two-dimensional poset Un and complement poset Ûn, both with n elements, such that the number of linear extensions are respectively En and Fn. We conclude that the Fibonacci and Euler numbers are related to each other.    

This thesis provides a fresh perspective on the (vertex) integrity of graphs, serving as a measure of network robustness. The study begins by introducing fundamental concepts and methods for evaluating the integrity of different graph families. An Integer Linear Programming (ILP) model, specifically designed for assessing integrity, is then presented. By applying this model, integrity values are calculated for various graph families, and patterns within the results are identified. These patterns contribute to establishing boundaries or determining exact values of integrity for the analyzed graph families. The analyzed graph families encompass Glued Paths, generalized Theta graphs, (Double) Cone graphs, Fan graph, Lollipop graphs, generalized Barbell graphs, (Dutch) Windmill graphs, Paley graphs, and Kneser graphs. Additionally, two conjectures are formulated: one concerning a lower bound for the integrity of all Paley graphs, and another addressing the exact integrity values of specific Kneser graphs. The ILP model proves to be a valuable tool for further exploration of graph family integrity, offering opportunities for additional research and expanding our understanding of network robustness.

Interest rate products form a large segment of over-the-counter derivatives. When the interest rate became negative, for the first time, in July 2009, interest rate models needed to adjust. Where first a log-normal model, as the Brace Gatarek Musiela (BGM) model, might have seemed logical for interest-rate products, as they were bounded by zero, now a normally distributed model, as the Hull-White model, could be considered more practical. To our knowledge, no comparison of the Hull-White model and the Brace Gatarek Musiela model has been made on the Expected Positive Exposure (EPE) (and thus Credit Valuation Adjustment (CVA)) of a swap portfolio. Therefore, this thesis compares the Hull-White model with the BGM model on the EPE of a swap portfolio. First, we show how both models can be simulated with Monte Carlo simulation and calibrated to caplets, after which we validate the used simulation. Finally, both models are compared on the convergence, computation time and EPE. It was found that the Hull-White model had a faster convergence and computation time than the BGM model for our implementation. Moreover, it was shown that the Hull-White model and BGM model have significantly different swap EPEs, except for far in-the-money (ITM) swaps and single payment swaps. Therefore, the models used for the EPE of a swap portfolio have a model risk.

Almost everyone is familiar with games such as poker and checkers, but games can also be found in non-entertaining settings, such as competing companies in a market or conflict resolutions between countries. However, what happens when players want to avoid working together? This thesis provides two game-theoretical models of boycotts based on existing literature and is complemented with new insights. We discuss the dynamic process of a consumer boycott, where the effectiveness can be determined by the predefined consumers’ and firm’s thresholds or by the maximum boycott duration. Besides, we consider a static model using the Shapley value to analyze the impacts of boycotts, which are balanced for 2-player boycotts but not when three players are involved. We extended the existing models by finding supermodularity as a sufficient requirement for the boycott to never be profitable for the involved players, by introducing a direct formula for the coalition value in a boycott game (the boycott-contraction), and by analyzing real-world data, where we observe the effect of boycotts on trading activities with uninvolved players.

Cooperative game theory studies multi-agent environments where agents are able to make binding agreements. A lot has been written about dividing goods or other positive gains among the agents. This study investigates ways to distribute tasks with a negative utility in a strategyproof way. The intended application is a group of people or companies who can distribute such tasks between them to benefit from each other. The agents value tasks being done, but would rather not do it themselves. They can however, distribute the tasks to mutually benefit. Agents value tasks differently and also have different costs for them. This study investigates the theoretical properties of this problem. Particularly, we look at the Core, which is the set of solutions where agents have no incentive to form coalitions between them and ignore the result of the mechanism. Then, two algorithms are proposed to solve the problem. Finally, experiments are done to predict what results would occur in practice.

The Competitive Investor Game from Bell & Cover (1980) and the 푘-Player Ranking Game from Alpern & Howard (2017) are analysed in thesis. Optimal strategies have been derived and the related proofs have been given a new look. The Symmetric Multiplayer Ranking Game is considered as the general interpretation of financial competition among hedge funds. This study focused on the hedge funds that manage a Long/Short U.S. Equity strategy. Some minor evidence has been found to support the hypothesis that the studied hedge funds manage a strategy that has the objective to beat the competition in terms of annual performance in order to achieve the highest ranking.

Temporal networks, like physical contact networks, are networks whose topology changes over time. However, this representation does not account for group interactions, when people gather in groups of more than two people, that can be represented as higher-order events of a temporal network. The prediction of these higher-order events is usually overlooked in traditional temporal network prediction methods, where each higher-order event of a group is regarded as a set of pairwise interactions between every pair of individuals within the group. However, pairwise interactions only allow to partially capture interactions among constituents of a system. Therefore, we want to be able to predict the occurrence of such higher-order interactions one step ahead, based on the higher-order topology observed in the past, and to understand which types of interactions are the most influential for the prediction. We find that the similarity in network topology is relatively high at two time steps with a small time lag between them and that this similarity decreases when the time lag increases. This motivates us to propose a memory-based model that can predict a higherorder temporal network at the next time step based on the network observed in the past. In particular, the occurrence of a group event will be predicted based on the past activity of this target group and of other groups that form a subset or a superset of the target group. Our model is network-based, so it has a relatively low computational cost and allows for a good interpretation of its underlying mechanisms. We propose as a baseline the memory-based method for the traditional pairwise network prediction problem. In this baseline model, the predicted higher-order events at a prediction time step are then deduced from the predicted pairwise network at the same prediction time step. We evaluate the prediction quality of all models in eight real-world physical contact networks and find that our model outperforms the baseline model. We also analyze the contribution of group events of different orders to the prediction quality. We find that the past activity of the target group is the most important factor for the prediction. Moreover, the past activity of groups of a larger size has, in general, a lower impact on the prediction of events of an arbitrary size than groups of a smaller size.

In this report, I investigate strategic decision making in the Formula E racing series for Porsche. Formula E is an electric car circuit racing series, where the main tasks of race strategy are allocating energy consumption across the race and timing mandatory "attack mode" activations (similar to small-scale pitstops). I work on a flawed existing tool that simulates Formula E races with the goal of identifying optimal strategies. By learning from Porsche's experience and comparing the tool against related literature, I investigate how to model a Formula E race. To do this, I outline a method to measure the realism of our simulated races. By defining a race as a set of stochastic processes approximated with empirical data, I find a dissimilarity metric for the underlying probability distributions of two collections of races. My dissimilarity metric is based on Kantorovich's formulation of the optimal transport problem as applied to probability measures. The cost function used within this dissimilarity metric is based on how differently two drivers' individual races evolve from start to end, using gaps between times of arrival. I introduce various alterations to the race model, based on experience and related literature. Through applying my metric to different versions of my race model, I evaluate these alterations and find improvements. Doing so shows measurable gains in the race model were achieved; this is also checked by measuring the prediction error of simulations versus the true observed race. My approach yields a scalar metric that allows fast and straightforward tuning of any race model. It uses a high degree of aggregation to successfully compare otherwise not trivially comparable data. While already delivering a significantly improved model, my dissimilarity metric in conjunction with the prediction error-based metrics can help to find further improvements in the future. This is also independent of the used simulation method or format, since my metric only requires realisations of true and simulated races as input.

Melvin Dresher beschouwde in zijn boek uit 1961 over speltheorie een getallenraadspel over N getallen. Hij liet zien hoe de optimale strategieën van beide spelers kon worden gevonden met behulp van lineair programmeren. Later toonde Selmer Johnson oplossingen voor N<=11 en merkte op dat de berekeningen steeds complexer werden bij toenemende N. Deze thesis beschrijft technieken om het spel op te lossen voor N<=980. Het algoritme van Dresher vormt nog steeds de basis, maar er zijn enkele aanpassingen. Niet alle voorwaarden worden bijvoorbeeld in het begin gegenereerd. Voorwaarden worden alleen op het moment dat ze nodig zijn gevonden met een subroutine. Dit is een vorm van delayed column generation. Er wordt ook gebruik gemaakt van heuristieken, zoals het schatten van de optimale kansverdeling van één van de spelers. Dit werk verschaft meer inzicht in het patroon van de oplossingen van dit spel en of enkele eerder geformuleerde vermoedens van Selmer Johnson en Edgar Gilbert over dit spel kloppen.

The Game of Cycles, invented by Francis Su (2020, p.51) is an impartial game played on a graph, where players take turns marking an edge according to a set of rules. Together with the game, there also came a conjecture that gives a condition for whether a specific position is winning or losing. Proving or disproving this conjecture is the main focus of this research, which we end up succeeding in by giving a counter-example, thus disproving the conjecture. We do this by first showcasing some relevant background knowledge from game theory in chapter 1. In chapter 2 we then introduce the Game of Cycles and its rules, as well as some of the previous results others have found. We continue in chapter 3 by creating a python script to brute-force the game for us and it is here that we find a counter-example to the main conjecture, of which we prove that it is indeed a counter-example. We close off with chapter 4 by looking at a simplification of the game, where it is played on trees instead of any graph. Here we prove that the main conjecture does hold for a special family of trees and state a conjecture for the solution of any tree.

This thesis presents an insight in the Riemann zeta function and the prime number theorem at an undergraduate mathematical level. The main goal is to construct an explicit formula for the prime counting function and to prove the prime number theorem using the zeta function and a Tauberian theorem. The Riemann zeta function can be continued analytically to the whole complex plane except at s = 1. Two proofs of this continuation were given by Bernhard Riemann in his famous article ``Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse'' from 1859. Those proofs are studied in detail in this thesis after introducing all the required foreknowledge on the gamma function. The prime counting function π(x) counts the number of primes less than or equal to x. An explicit formula for π(x) in terms of the nontrivial zeros of the zeta function will be constructed in a similar way as Riemann did in his article. Finally, the prime number theorem will be proved. This theorem describes the asymptotic distribution of the primes among the natural numbers. Using the analytic continuation of the zeta function and a Tauberian theorem, the prime number theorem can be proved quite easily with only basic theory from complex analysis.

Intention aware routing system is a route-planning algorithm for electric vehicles that minimizes overall travel time by taking into consideration congestion at charging stations. This thesis extends this algorithm to allow choices to be made based on prices at charging stations. The goal of this thesis is to find a way to minimize maximum congestion while maximizing overall profit across the stations. To achieve this an optimal price has to be calculated. To this end, a formula is devised and applied to several graphs.

This thesis captures the calibration of a FX hybrid model: The FX Black-Scholes Hull-White model. The main focus is on the calibration of the parameters in the Hull-White process: The mean reversion and the volatility parameter. The latter is commonly calibrated as a time-dependent parameter, whilst the mean reversion parameter is not. This thesis covers the calibration of the mean reversion as a time-dependent parameter. A known method from the Literature is researched, where we calibrate the mean reversion independently from the volatility parameter to the ratio of two swaptions with the same expiry but different tenor. In our research this method is extended to the negative interest rate environment by assuming that the swap rate follows a shifted lognormal distribution instead of a lognormal distribution. We show that a specific set of swaptions can be chosen, so that the calibration problem is simplified. This choice leads to sequential calibration of convex optimization problems. Numerical results of calibration to artificial and market data are presented, where we compare our method to picking the mean reversion parameter arbitrarily. The findings suggest that the choice of mean reversion parameter affects the calibration procedure. Therefore, we argue that a calibration method for the mean reversion would be appropriate. Besides the focus on the Hull-White process, the calibration of the volatility parameter in the FX Black-Scholes process of the hybrid model is investigated. For the calibration of the FX volatility parameter ATM options on FX rates are used. Numerical results of calibration to artificial and market data are shown. A typical problem of calibration in the industry is discussed: Precision for late maturities. The results in this thesis suggest that this problem could be solved. For research on homogeneity constraints on the time-dependent parameters, the performance of delta-hedging is investigated. The delta-hedging is performed in a simple Black-Scholes world with time-dependent volatility. We show that given a fixed amount hedges beforehand and interest rate zero, there exists an optimal distribution of time points that will lead to equal variance on the profit and loss for every volatility function that has equal implied volatility from t0 to maturity T. Using this strategy, the homogeneity of the volatility parameter has no impact on the performance of delta-hedging. Therefore, in this thesis no homogeneity constraints are used for the calibration of the time-dependent parameters.

Automated Market Makers (AMMs) are a novel type of market makers that eliminate the need for a coun-terparty in a trade. This thesis analyses the properties of several types of AMMs, and in particular the con-centrated liquidity market maker. An axiomatic definition of AMMs is provided, and two types of constant function market makers (CFMMs), called the constant sum market maker (CSMM) and constant product market maker (CPMM), are explored. The concentrated liquidity market maker, which improves liquidity provision compared to the CPMM, is thoroughly analyzed, and it is shown that it’s trading function can be formulated as a composition of functions. This thesis also conjectures that the concentrated liquidity trading function can be approximated by taking an infinite number of compositions. Additionally, a simulation study is conducted using transaction mocking. The simulation study supports the conjecture, and brings several other noteworthy properties of the concentrated liquidity market maker to light.

In deze Bachelor thesis hebben we het gedrag bestudeerd van 3 typen kankercellen die een rol spelen bij prostaatkanker. Dit doen we met behulp van speltheorie met als spelers de dokter tegenover de kankercellen. Met de replicator dynamica hebben we eerst gekeken naar de verdeling van cellen in een tumor zonder dat er medicijn was toegevoegd. We hebben de stabiliteit van de evenwichtspunten hiervan bepaald met behulp van de eigenwaardes, de trace en determinant van het gelineariseerde systeem en de Routh-Hurwitz methode. Vervolgens hebben we de replicator dynamica omgeschreven tot Lotka-Volterra vergelijkingen en medicijn en resistentie toegevoegd aan de differentiaalvergelijkingen. Hiermee hebben we de dynamica van de cel populatie na therapie bekeken. Tot slot hebben we geprobeerd een goede balans te vinden in de dosis medicijn om de tumorcellen zoveel mogelijk te doden, maar de levenskwaliteit van de patiënt zo goed mogelijk te houden. We hebben hierbij drie strategieën vergeleken; maximale getolereerde dosis medicijn (MTD), Nash strategie, Stackelberg strategie. De Stackelberg strategie bleek hierin het meest succesvol. De conclusie is dat er bij de Stackelberg strategie het minste medicijn nodig is om te kunnen leven met de tumor.