A Game Theoretic Analysis of Liquidity Events in Convertible Instruments

TL;DR

This paper models liquidity events in convertible instruments as a game, analyzing equilibrium existence and providing algorithms for optimal strategies, especially for SAFE contracts used in startup financing.

Contribution

It introduces a general game theoretic model for liquidity events in convertible instruments, including SAFE contracts, and develops polynomial algorithms for equilibrium computation.

Findings

Pure strategy Nash equilibria may not always exist.

Optimal pure strategy Nash equilibria exist for uniform SAFE contracts.

Polynomial time algorithms are provided for equilibrium computation.

Abstract

Convertible instruments are contracts, used in venture financing, which give investors the right to receive shares in the venture in certain circumstances. In liquidity events, investors may have the option to either receive back their principal investment, or to receive a proportional payment after conversion of the contract to a shareholding. In each case, the value of the payment may depend on the choices made by other investors who hold such convertible contracts. A liquidity event therefore sets up a game theoretic optimization problem. The paper defines a general model for such games, which is shown to cover all instances of the Y Combinator Simple Agreement for Future Equity (SAFE) contracts, a type of convertible instrument that is commonly used to finance startup ventures. The paper shows that, in general, pure strategy Nash equilibria do not necessarily exist in this model, and there may not exist an optimum pure strategy Nash equilibrium in cases where pure strategy Nash equilibria do exist. However, it is shown when all contracts are uniformly one of the SAFE contract types, an optimum pure strategy Nash equilibrium exists. Polynomial time algorithms for computing (optimum) pure strategy Nash equilibria in these cases are developed.