On the existence of personal equilibria

Abstract

We consider an investor who, while maximizing his/her expected utility, also compares the outcome to a reference entity. We recall the notion of personal equilibrium and show that, in a multistep, generically incomplete financial market model such an equilibrium indeed exists, under appropriate technical assumptions.

Summary

This paper addresses the question of whether "personal equilibria" exist in financial markets, a concept where an investor's optimal strategy is determined in relation to a reference point that is an independent copy of their own outcome. The authors focus on multi-step, generically incomplete financial market models, which are more realistic than the single-step complete market models previously studied. They aim to provide a theoretical guarantee that the notion of personal equilibrium is not empty even in complex financial settings. The authors employ a dynamic programming approach, building on previous work on optimal investment problems. They recursively solve a one-step optimization problem and then apply these solutions iteratively to construct an optimal strategy for the multi-step market. The core of their argument relies on proving the existence of fixed points, which requires involved mathematical arguments. They demonstrate the continuous dependence of strategies on past events, allowing them to apply Schauder's fixed-point theorem in a Banach space of continuous functions. The key finding is that personal equilibria indeed exist in multi-step, generically incomplete financial market models under specific technical assumptions. Moreover, the authors prove the existence of a *preferred* personal equilibrium, which represents the best-performing equilibrium strategy. This result matters to the field of behavioral economics and financial modeling because it provides a theoretical foundation for using personal equilibrium as a viable concept in more complex and realistic market scenarios. It moves beyond the limitations of single-step models and opens the door for further research on the properties and implications of these equilibria.

Key Insights

• •The paper extends the concept of personal equilibrium from single-step, complete market models to more realistic multi-step, incomplete market models.
• •They establish the existence of both personal equilibria and *preferred* personal equilibria, addressing the question of whether these equilibria are non-empty in complex settings.
• •The proof involves a dynamic programming approach coupled with Schauder's fixed-point theorem, requiring the demonstration of continuous dependence of optimal strategies on past events and preferences.
• •A key technical contribution is Proposition 3.3, which demonstrates that if a value function V satisfies certain assumptions, the resulting optimized value function v also satisfies similar assumptions, albeit with a reduced H\"older exponent. This proposition is applied recursively in the dynamic programming approach.
• •The assumptions on the utility function U and the gain-loss function ν are crucial. Specifically, ν is assumed to be concave, strictly increasing, and linear on the negative axis, but bounded from above on the whole real line, which deviates from previous work.
• •The paper leverages the existence of an auxiliary random variable to create an independent copy of the market's driving factors, facilitating the definition of the reference point for the investor.
• •The authors acknowledge that their work does not address the uniqueness of personal equilibria or preferred personal equilibria, indicating a potential area for future research.

Practical Implications

• •This research provides a theoretical justification for using personal equilibrium as a behavioral model in financial markets, particularly in scenarios with multiple time steps and incomplete information.
• •Financial modelers and economists can use this framework to develop more realistic models of investor behavior that incorporate reference-dependent preferences.
• •The results could be used to design investment strategies or financial products that are better aligned with investors' psychological biases and reference points.
• •Future research could explore the properties of preferred personal equilibria, including their uniqueness and sensitivity to changes in model parameters. Further investigation could be done to understand the impact of different specifications of the gain-loss function, incorporating functions like ν0 (x) = α1x, x < 0, ν0 (x) = α2x, x ≥ 0 with 0 < α2 < α1.
• •The mathematical techniques developed in the paper, such as the use of Schauder's fixed-point theorem and the recursive application of Proposition 3.3, could be applied to other optimization problems in finance and economics.

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