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on Utility Models and Prospect Theory |
| By: | Lucas Bordeu; Javier Castro |
| Abstract: | This study presents a rigorous mathematical approach to the optimization of round and betting policies in Blackjack, using Markov Decision Processes (MDP) and Expected Utility Theory. The analysis considers a direct confrontation between a player and the dealer, simplifying the dynamics of the game. The objective is to develop optimal strategies that maximize expected utility for risk profiles defined by constant (CRRA) and absolute (CARA) aversion utility functions. Dynamic programming algorithms are implemented to estimate optimal gambling and betting policies with different levels of complexity. The evaluation is performed through simulations, analyzing histograms of final returns. The results indicate that the advantage of applying optimized round policies over the "basic strategy" is slight, highlighting the efficiency of the last one. In addition, betting strategies based on the exact composition of the deck slightly outperform the Hi-Lo counting system, showing its effectiveness. The optimized strategies include versions suitable for mental use in physical environments and more complex ones requiring computational processing. Although the computed strategies approximate the theoretical optimal performance, this study is limited to a specific configuration of rules. As a future challenge, it is proposed to explore strategies under other game configurations, considering additional players or deeper penetration of the deck, which could pose new technical challenges. |
| Date: | 2025–04 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2505.00724 |
| By: | Dongmei Zhu; Ashley Davey; Harry Zheng |
| Abstract: | We study S-shaped utility maximisation with VaR constraint and unobservable drift coefficient. Using the Bayesian filter, the concavification principle, and the change of measure, we give a semi-closed integral representation for the dual value function and find a critical wealth level that determines if the constrained problem admits a unique optimal solution and Lagrange multiplier or is infeasible. We also propose three algorithms (Lagrange, simulation, deep neural network) to solve the problem and compare their performances with numerical examples. |
| Date: | 2025–06 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2506.10103 |
| By: | Francesco Menoncin; Elena Vigna |
| Abstract: | We solve a dynamic portfolio optimization problem in the accumulation phase of a defined contribution (DC) pension plan with a stochastic wage driven by a non-hedgeable random source. The incomplete financial market consists of a riskfree and a risky assets, and the stochastic wage has non-zero correlation with the risky asset. The optimization problem defined with a constant absolute risk aversion (CARA) utility function is solved via dynamic programming in closed form with constant riskfree interest rate and constant correlation between wage and risky asset. We also show that the application of the martingale approach provides an approximated solution based on the least square method, and we highlight the difference between the optimal and the approximated solutions. A numerical application investigates (i) the impact on the optimal investment strategy of the correlation between wage and risky asset, (ii) the comparison with the complete market case, and (iii) the relationship between the optimal and the approximated solutions. The main conclusion drawn is that failing to model the imperfect correlation between wage and risky asset in a DC pension scheme leads to investment policies that are far away the optimal ones, and to distorted outcomes in terms of final wealth. |
| Keywords: | DC pension scheme; incomplete market; stochastic wage; dynamic programming; optimal portfolio |
| Date: | 2025 |
| URL: | https://d.repec.org/n?u=RePEc:cca:wpaper:740 |
| By: | Giovanni Mastrobuoni; Emily Owens |
| Abstract: | We propose a new framework to investigate whether criminals exhibit strategic behavior in response to the criminal law and their enforcement. Unique data on commercial robberies in Milan allow us to examine the decisions robbers make regarding weapon choice, number of accomplices, and the type of business targeted. Our analysis explores the relationship between these decisions, the expected return from the robbery, and the probability of arrest, considering the constraints imposed by Italian law, which prescribes differential punishments based on certain criminal choices. We find some evidence that robbers act in accordance with expected utility maximization, particularly when operating in groups. |
| Keywords: | Police, Crime, Robberies, Strategic Behavior. |
| Date: | 2025 |
| URL: | https://d.repec.org/n?u=RePEc:cca:wpaper:741 |
| By: | Paweł Doligalski; Piotr Dworczak; Mohammad Akbarpour; Scott Duke Kominers* |
| Abstract: | Policymakers often distort goods markets to effect redistribution—for example, via price controls, differential taxation, or in-kind transfers. We investigate the optimality of such policies alongside the (optimally-designed) income tax. In our framework, agents differ in both their ability to generate income and their consumption preferences, and a planner maximizes a social welfare function subject to incentive and resource constraints. We uncover a generalization of the Atkinson-Stiglitz theorem by showing that goods markets should be undistorted if the heterogeneous consumption tastes (i) do not affect the marginal utility of disposable income, (ii) do not enter into the social welfare weights and (iii) are statistically independent of ability. We also show, however, that market interventions play a role in the optimal resolution of the equity-efficiency trade-off if any of the three assumptions is relaxed. In a special case of our model with linear utilities, binary ability, and continuous willingness to pay for a single good, we characterize the globally optimal mechanism and show that it may feature meanstested consumption subsidies, in-kind transfers, and differential commodity taxation |
| Date: | 2025–04–02 |
| URL: | https://d.repec.org/n?u=RePEc:bri:uobdis:25/787 |
| By: | Echenique, Federico; Núñez, Matías |
| Abstract: | We describe a sequential mechanism that fully implements the set of efficient outcomes in environments with quasi-linear utilities. The mechanism asks agents to take turns in defining prices for each outcome, with a final player choosing an outcome for all: Price and Choose. The choice triggers a sequence of payments from each agent to the preceding agent. We present several extensions. First, payoff inequalities may be reduced by endogenizing the order of play. Second, our results extend to a model without quasi-linear utility, to a setting with an outside option, robustness to max-min behavior, and caps on prices. (JEL C72, D11, D44, D71, D82) |
| Keywords: | Economics, Applied Economics, Economic Theory, Banking, finance and investment, Applied economics, Economic theory |
| Date: | 2025–05–01 |
| URL: | https://d.repec.org/n?u=RePEc:cdl:econwp:qt5dw4g7k5 |
| By: | Roberto Gomez Cram; Howard Kung; Hanno Lustig; David Zeke |
| Abstract: | Unfunded fiscal shocks are a significant source of risk premia in Treasury markets when central banks and governments decide to insulate taxpayers and expose bondholders' wealth to government funding needs. We illustrate this bond risk premium mechanism analytically in a two-agent model featuring monetary-fiscal interactions and a fraction of constrained agents. Surprise government transfer spending devalues real Treasury payoffs through fiscal inflation, while fiscal redistribution makes these high marginal utility states for bond investors, leading to risky government debt. We show that this fiscal redistribution mechanism can quantitatively explain the nominal term premium in a TANK framework. |
| JEL: | E62 G12 |
| Date: | 2025–05 |
| URL: | https://d.repec.org/n?u=RePEc:nbr:nberwo:33769 |
| By: | Bushra Shehnam Ashraf; Thomas S. Salisbury |
| Abstract: | We study optimal consumption and retirement using a Cobb-Douglas utility and a simple model in which an interesting bifurcation arises. With high wealth, individuals plan to retire. With low wealth they plan to never retire. At a critical level of initial wealth they may choose to defer this decision, leading to a continuum of wealth trajectories with identical utilities. |
| Date: | 2025–06 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2506.02155 |
| By: | Arnon Archankul; Jacco J. J. Thijssen |
| Abstract: | We consider singular control in inventory management under Knightian uncertainty, where decision makers have a smooth ambiguity preference over Gaussian-generated priors. We demonstrate that continuous-time smooth ambiguity is the infinitesimal limit of Kalman-Bucy filtering with recursive robust utility. Additionally, we prove that the cost function can be determined by solving forward-backward stochastic differential equations with quadratic growth. With a sufficient condition and utilising variational inequalities in a viscosity sense, we derive the value function and optimal control policy. By the change-of-coordinate technique, we transform the problem into two-dimensional singular control, offering insights into model learning and aligning with classical singular control free boundary problems. We numerically implement our theory using a Markov chain approximation, where inventory is modeled as cash management following an arithmetic Brownian motion. Our numerical results indicate that the continuation region can be divided into three key areas: (i) the target region; (ii) the region where it is optimal to learn and do nothing; and (iii) the region where control becomes predominant and learning should inactive. We demonstrate that ambiguity drives the decision maker to act earlier, leading to a smaller continuation region. This effect becomes more pronounced at the target region as the decision maker gains confidence from a longer learning period. However, these dynamics do not extend to the third region, where learning is excluded. |
| Date: | 2025–05 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2505.07761 |
| By: | Hans Buehler; Blanka Horvath; Yannick Limmer; Thorsten Schmidt |
| Abstract: | This paper addresses the challenge of model uncertainty in quantitative finance, where decisions in portfolio allocation, derivative pricing, and risk management rely on estimating stochastic models from limited data. In practice, the unavailability of the true probability measure forces reliance on an empirical approximation, and even small misestimations can lead to significant deviations in decision quality. Building on the framework of Klibanoff et al. (2005), we enhance the conventional objective - whether this is expected utility in an investing context or a hedging metric - by superimposing an outer "uncertainty measure", motivated by traditional monetary risk measures, on the space of models. In scenarios where a natural model distribution is lacking or Bayesian methods are impractical, we propose an ad hoc subsampling strategy, analogous to bootstrapping in statistical finance and related to mini-batch sampling in deep learning, to approximate model uncertainty. To address the quadratic memory demands of naive implementations, we also present an adapted stochastic gradient descent algorithm that enables efficient parallelization. Through analytical, simulated, and empirical studies - including multi-period, real data and high-dimensional examples - we demonstrate that uncertainty measures outperform traditional mixture of measures strategies and our model-agnostic subsampling-based approach not only enhances robustness against model risk but also achieves performance comparable to more elaborate Bayesian methods. |
| Date: | 2025–06 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2506.07299 |