Calculated Boldness: Optimizing Financial Decisions with Illiquid Assets

TL;DR

This paper develops a dynamic programming approach to optimize betting strategies in games of chance with external capital, revealing how risk preferences vary with capital allocation and challenging conventional views on super-Kelly betting.

Contribution

It introduces a novel Kelly optimization framework incorporating external capital, with a close approximation method and insights into risk behavior in multi-round betting scenarios.

Findings

Optimal strategies depend on the ratio of in-game to external capital.

Risk aversion varies with capital ratio, influencing betting aggressiveness.

Super-Kelly betting may be rational when external resources are considered.

Abstract

We consider games of chance played by someone with external capital that cannot be applied to the game and determine how this affects risk-adjusted optimal betting. Specifically, we focus on Kelly optimization as a metric, optimizing the expected logarithm of total capital including both capital in play and the external capital. For games with multiple rounds, we determine the optimal strategy through dynamic programming and construct a close approximation through the WKB method. The strategy can be described in terms of short-term utility functions, with risk aversion depending on the ratio of the amount in the game to the external money. Thus, a rational player's behavior varies between conservative play that approaches Kelly strategy as they are able to invest a larger fraction of total wealth and extremely aggressive play that maximizes linear expectation when a larger portion of their capital is locked away. Because you always have expected future productivity to account for as external resources, this goes counter to the conventional wisdom that super-Kelly betting is a ruinous proposition.