  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.
