Nash equilibrium of identical agents facing the Unilateralist’s Curse

This paper is an addendum to the ‘Unilateralist’s Curse’ paper of Nick Bostrom, Thomas Douglas and Anders Sandberg [BDS12]. It demonstrates that if there are identical agents facing a situation where any one of them can implement a policy unilaterally, then the best strategies they can implement are also Nash equilibriums. It also notes that if this Nash equilibrium involves probabilistic reactions to observations, then it is a weak Nash equilibrium and a single agent is free to change all their nontrivial probabilistic decisions, without changing the expected utility of the outcome. The Unilateralist’s Curse paper analyses how to make decisions when there is a certain policy under consideration, and many different agents who could each unilaterally implement that policy. If each agent simply followed their own estimate’s of the value of that policy, we would be in a situation similar to the winner’s curse in auctions: the policy would get implemented if the most optimistic agent thought it was a good idea. Thus in these situations, agents must take care to construct a decision process that counteracts this effect and makes the agents less likely to go ahead on personal, marginally optimistic, information. The problem is isomorphic, in reverse, to policies that require unanimity: there the policy’s implementation is dictated by the opinion of the most pessimistic agent. This paper looks at a specific simplified version of this problem. It assumes that all the agents have identical preferences (they judge each outcome as equally good or equally bad), that they are equally likely to see any given piece of evidence about the value of the policy, and that they can’t communicate. They will attempt to construct the best (probabilistic) strategy they can, given these constraints. Because they are identical, they will all construct the same probabilistic strategy. This paper demonstrates that if this is indeed the best strategy (or even a local maxima), then it is a Nash equilibrium: it cannot be improved by unilateral changes by a single agent. If the strategy is probabilistic (given certain observations, the agent is neither entirely certain to implement the policy, nor entirely certain to refrain), then it is a weak Nash equilibrium – a single agent can change their strategy without making the situation worse. Indeed, a single agent can change all the non-trivial