Games for cautious players: the equilibrium in secure strategies

A non-cooperative solution, the Equilibrium in Secure Strategies (EinSS), is defined that extends the Nash equilibrium in pure strategies when it does not exist and is meant to solve games where players are "cautious", i.e. looking for secure positions and avoiding threats. This concept abstracts and unifies various ad hoc solutions already formulated in various applied economic games that have been discussed extensively in the literature. It complements usefully mixed strategy Nash equilibria that are usually not explicit and difficult to interpret in these games. Like the Nash equilibrium, the EinSS is a static concept, and the basic requirement of excluding at equilibrium some deviations remains. But it also appeals to dynamic intuitions, tolerating at equilibrium the possibility of some deviations, which would be blocked by counter-deviations punishing the deviator. This is in line with the "objection-counter- objection" rationale first introduced in cooperative games. A general existence theorem is provided and then applied to the price-setting game in Hotelling location model, to Tullock's rent-seeking contests and to Bertrand-Edgeworth duopoly. Finally competition in the insurance market game is re-examined and the Rothchild-Stiglitz- Wilson contract shown to be an EinSS even when the Nash equilibrium breaks down.