A Game Theoretic Analysis of Liquidity Events in Convertible Instruments

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.