Thesis (M.S. in Operations Research)--Naval Postgraduate School, Dec. 2004. / Thesis Advisor(s): James N. Eagle, W. Matthew Carlyle. Includes bibliographical references (p. 35). Also available online.

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Abstract Game contingent claims (GCCs), as introduced by Kifer (2000), are a generalization of American contingent claims where the writer has the opportunity to terminate the contract, and must then pay the intrinsic option value plus a penalty. In complete markets, GCCs are priced using no-arbitrage arguments as the value of a zero-sum stochastic game of the type described in Dynkin (1969). In incomplete markets, the neutral pricing approach of Kallsen and Kühn (2004) can be used. In Part I of this thesis, we introduce GCCs and their pricing, and also cover some basics of mathematical finance. In Part II, we present a new algorithm for valuing game contingent claims. This algorithm generalises the least-squares Monte-Carlo method for pricing American options of Longstaff and Schwartz (2001). Convergence proofs are obtained, and the algorithm is tested against certain GCCs. A more efficient algorithm is derived from the first one using the computational complexity analysis technique of Chen and Shen (2003). The algorithms were found to give good results with reasonable time requirements. Reference implementations of both algorithms are available for download from the author’s Github page https://github.com/del/ Game-option-valuation-library

Tong Kwong-Bun. / Thesis submitted in: December 2002. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 77-[80]). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- An Overview --- p.1 / Chapter 1.2 --- Tree Search --- p.2 / Chapter 1.2.1 --- Minimax Algorithm --- p.2 / Chapter 1.2.2 --- The Alpha-Beta Algorithm --- p.4 / Chapter 1.2.3 --- Alpha-Beta Enhancements --- p.5 / Chapter 1.2.4 --- Selective Search --- p.9 / Chapter 1.3 --- Construction of Evaluation Function --- p.16 / Chapter 1.4 --- Contribution of the Thesis --- p.17 / Chapter 1.5 --- Structure of the Thesis --- p.19 / Chapter 2 --- The Probabilistic Forward Pruning Framework --- p.20 / Chapter 2.1 --- Introduction --- p.20 / Chapter 2.2 --- The Generalized Probabilistic Forward Cuts Heuristic --- p.21 / Chapter 2.3 --- The GPC Framework --- p.24 / Chapter 2.3.1 --- The Alpha-Beta Algorithm --- p.24 / Chapter 2.3.2 --- The NegaScout Algorithm --- p.25 / Chapter 2.3.3 --- The Memory-enhanced Test Algorithm --- p.27 / Chapter 2.4 --- Summary --- p.27 / Chapter 3 --- The Fast Probabilistic Forward Pruning Framework --- p.30 / Chapter 3.1 --- Introduction --- p.30 / Chapter 3.2 --- The Fast GPC Heuristic --- p.30 / Chapter 3.2.1 --- The Alpha-Beta algorithm --- p.32 / Chapter 3.2.2 --- The NegaScout algorithm --- p.32 / Chapter 3.2.3 --- The Memory-enhanced Test algorithm --- p.35 / Chapter 3.3 --- Performance Evaluation --- p.35 / Chapter 3.3.1 --- Determination of the Parameters --- p.35 / Chapter 3.3.2 --- Result of Experiments --- p.38 / Chapter 3.4 --- Summary --- p.42 / Chapter 4 --- The Node-Cutting Heuristic --- p.43 / Chapter 4.1 --- Introduction --- p.43 / Chapter 4.2 --- Move Ordering --- p.43 / Chapter 4.2.1 --- Quality of Move Ordering --- p.44 / Chapter 4.3 --- Node-Cutting Heuristic --- p.46 / Chapter 4.4 --- Performance Evaluation --- p.48 / Chapter 4.4.1 --- Determination of the Parameters --- p.48 / Chapter 4.4.2 --- Result of Experiments --- p.50 / Chapter 4.5 --- Summary --- p.55 / Chapter 5 --- The Integrated Strategy --- p.56 / Chapter 5.1 --- Introduction --- p.56 / Chapter 5.2 --- "Combination of GPC, FGPC and Node-Cutting Heuristic" --- p.56 / Chapter 5.3 --- Performance Evaluation --- p.58 / Chapter 5.4 --- Summary --- p.63 / Chapter 6 --- Conclusions and Future Works --- p.64 / Chapter 6.1 --- Conclusions --- p.64 / Chapter 6.2 --- Future Works --- p.65 / Chapter A --- Examples --- p.67 / Chapter B --- The Rules of Chinese Checkers --- p.73 / Chapter C --- Application to Chinese Checkers --- p.75 / Bibliography --- p.77

Thesis (Ph. D.)--University of California, San Diego, 2006. / Vita. Includes bibliographical references.

Approved for public release; distribution in unlimited. / In this thesis, a new fictitious play (FP) procedure is presented to solve two-person zero-sum (TPZS) Blotto games. The FP solution procedure solves TPZS games by assuming that the two players take turns selecting optimal responses to the opponent's strategy observed so far. It is known that FP converges to an optimal solution, and it may be the only realistic approach to solve large games. The algorithm uses dynamic programming (DP) to solve FP subproblems. Efficiency is obtained by limiting the growth of the DP state space. Blotto games are frequently used to solve simple missile defense problems. While it may be unlikely that the models presented in this paper can be used directly to solve realistic offense and defense problems, it is hoped that they will provide insight into the basic structure of optimal and near-optimal solutions to these important, large games, and provide a foundation for solution of more realistic, and more complex, problems. / Captain, Republic of Korea Air Force

Advances in digital signal processing (DSP) and computing technology have resulted in the emergence of increasingly adaptive radar systems. It is clear that the Electronic Attack (EA), or jamming, of such radar systems is expected to become a more difficult task. The reason for this research was to address the issue of jamming adaptive radar systems. This required consideration of adaptive jamming systems and the development of a methodology for outlining the features of such a system is proposed as the key contribution of this thesis. For the first time, game-based optimization methods have been applied to a maritime counter-surveillance/counter-targeting scenario involving conventional, as well as so-called ‘smart’ noise jamming.Conventional noise jamming methods feature prominently in the origins of radar electronic warfare, and are still widely implemented. They have been well studied, and are important for comparisons with coherent jamming techniques.Moreover, noise jamming is more readily applied with limited information support and is therefore germane to the problem of jamming adaptive radars; during theearly stages when the jammer tries to learn about the radar’s parameters and its own optimal actions.A radar and a jammer were considered as informed opponents ‘playing’ in a non-cooperative two-player, zero-sum game. The effects of jamming on the target detection performance of a radar using Constant False Alarm Rate (CFAR)processing were analyzed using a game theoretic approach for three cases: (1) Ungated Range Noise (URN), (2) Range-Gated Noise (RGN) and (3) False-Target (FT) jamming.Assuming a Swerling type II target in the presence of Rayleigh-distributed clutter, utility functions were described for Cell-Averaging (CA) and Order Statistic (OS) CFAR processors and the three cases of jamming. The analyses included optimizations of these utility functions, subject to certain constraints, with respectto control variables (strategies) in the jammer, such as jammer power and spatial extent of jamming, and control variables in the radar, such as threshold parameter and reference window size. The utility functions were evaluated over the players’ strategy sets and the resulting matrix-form games were solved for the optimal or ‘best response’ strategies of both the jammer and the radar.

Advances in digital signal processing (DSP) and computing technology have resulted in the emergence of increasingly adaptive radar systems. It is clear that the Electronic Attack (EA), or jamming, of such radar systems is expected to become a more difficult task. The reason for this research was to address the issue of jamming adaptive radar systems. This required consideration of adaptive jamming systems and the development of a methodology for outlining the features of such a system is proposed as the key contribution of this thesis. For the first time, game-based optimization methods have been applied to a maritime counter-surveillance/counter-targeting scenario involving conventional, as well as so-called ‘smart’ noise jamming.Conventional noise jamming methods feature prominently in the origins of radar electronic warfare, and are still widely implemented. They have been well studied, and are important for comparisons with coherent jamming techniques.Moreover, noise jamming is more readily applied with limited information support and is therefore germane to the problem of jamming adaptive radars; during theearly stages when the jammer tries to learn about the radar’s parameters and its own optimal actions.A radar and a jammer were considered as informed opponents ‘playing’ in a non-cooperative two-player, zero-sum game. The effects of jamming on the target detection performance of a radar using Constant False Alarm Rate (CFAR)processing were analyzed using a game theoretic approach for three cases: (1) Ungated Range Noise (URN), (2) Range-Gated Noise (RGN) and (3) False-Target (FT) jamming.Assuming a Swerling type II target in the presence of Rayleigh-distributed clutter, utility functions were described for Cell-Averaging (CA) and Order Statistic (OS) CFAR processors and the three cases of jamming. The analyses included optimizations of these utility functions, subject to certain constraints, with respectto control variables (strategies) in the jammer, such as jammer power and spatial extent of jamming, and control variables in the radar, such as threshold parameter and reference window size. The utility functions were evaluated over the players’ strategy sets and the resulting matrix-form games were solved for the optimal or ‘best response’ strategies of both the jammer and the radar.

Cette thèse est une contribution à la théorie économique sur trois aspects: la dynamique de prix dans les marchés financiers avec asymétrie d’information, la mise à jour des croyances et les raffinements d'équilibre dans les jeux de signaux, et l'introduction de l'ambiguïté dans la théorie du prix limite. Dans le chapitre 2, nous formalisons un jeu d'échange à somme nulle entre un secteur mieux informé et un autre qui l'est moins, pour déterminer de façon endogène, la dynamique du prix sous-jacent. Dans ce modèle, joueur 1 est informé de la conjoncture (L) mais est incertain de la croyance de joueur 2, car ce dernier est seulement informé à travers un message (M) qui est lié à cette conjoncture. Si L et M sont indépendants, alors le processus de prix sera une Martingale Continue à Variation Maximale (CMMV) et joueur 1 peut disposer de cet avantage informationnel. Par contre, si L et M ne sont pas indépendants, joueur 1 ne révèlera pas son information pendant le processus, et il ne bénéficiera donc pas de son avantage en matière d'information. Dans le chapitre 3, je propose une définition de l'équilibre de Test d'hypothèse (HTE) pour des jeux de signaux généraux, avec des joueurs non-Bayésiens qui sont soumis à une règle de mise à jour selon le modèle de vérification d'hypothèse caractérisé par Ortoleva (2012). Un HTE peut être différent d'un équilibre séquentiel de Nash en raison d'une incohérence dynamique. Par contre, dans le cas où joueur 2 traite seulement un message à probabilité nulle comme nouvelle inespérée, un HTE est un raffinement d'équilibre séquentiel de Nash et survit au critère intuitif dans les jeux de signaux généraux mais pas inversement. Nous fournissons un théorème d'existence qui couvre une vaste classe de jeux de signaux qui sont souvent étudiés en économie. Dans le chapitre 4, j'introduis l’ambiguïté dans un modèle d'organisation industrielle classique, dans lequel l'entreprise déjà établie est soit informée de la vraie nature de la demande agrégée, soit soumise à une incertitude mesurable classique sur la conjoncture, tandis qu'un éventuel nouvel arrivant fait face à une incertitude a la Knight (ambiguïté) concernant cette conjoncture. Je caractérise les conditions sou lesquelles le prix limite émerge en équilibre, et par conséquent l'ambigüité diminue la probabilité d'entrée. L'analyse du bien-être montre que le prix limite est plus nocif dans un marché où la demande escomptée est plus élevée que dans un autre où celle-ci est moindre. / This thesis contributes to the economic theory literature in three aspects: price dynamics in financial markets with asymmetric information belief updating and equilibrium refinements in signaling games, and introducing ambiguity in limit pricing theory. In chapter 2, we formulate a zero-sum trading game between a better informed sector and a less 1nformed sector to endogenously determine the underlying price dynamics. In this model, player 1 is informed of the state (L) but is uncertain about player 2's belief about the state, because player 2 is informed through some message (M) related to the state. If L and M are independent, then the price proces s will be a Continuous Martingale of Maximal Variation (CMMV), and player 1 can benefit from his informational advantage. However, if L and M are not independent, player 1 will not reveal his information during the trading process, therefore, he does not benefit from his informational advantage. In chapter 3, I propose a definition of Hypothesis Testing Equilibrium (HTE) for general signaling games with non-Bayesian players nested, by an updating rule according to the Hypothesis Testing model characterized by Ortoleva (2012). An HTE may differ from a sequential Nash equilibrium because of dynamic inconsistency. However, in the case in which player 2 only treats a zero-probability message as an unexpected news, an HTE is a refinement of sequential Nash equilibrium and survives the intuitive Critenon in general signaling games but not vice versa. We provide an existence theorem covering a broad class of signaling games often studied in economics. In chapter 4, I introduce ambiguity in a standard industry organization model, in which the established firm is either informed of the true state of aggregate demand or is under classical measurable uncertainty about the state, while the potential entrant is under Knightian uncertainty (ambiguity) about the state. I characterize the conditions under which limit pricing emerges in equilibria, and thus ambiguity decreases the probability of entry. Welfare analysis shows that limit pricing is more harmful in a market with higher expected demand than in a market with lower expected demand.