Optimal equilibria of the best shot game

TL;DR

This paper studies the best shot game on networks, where players choose between two actions, and proposes a mechanism that converges to an optimal equilibrium minimizing the number of nodes choosing action 1, regardless of network complexity.

Contribution

It introduces an implementable mechanism that guarantees convergence to an optimal equilibrium in the best shot game on any network, even without knowledge of the network structure.

Findings

The mechanism converges to the optimal equilibrium in the limit of infinite time.

It works across arbitrary network structures.

It does not require prior knowledge of the network or equilibrium.

Abstract

We consider any network environment in which the "best shot game" is played. This is the case where the possible actions are only two for every node (0 and 1), and the best response for a node is 1 if and only if all her neighbors play 0. A natural application of the model is one in which the action 1 is the purchase of a good, which is locally a public good, in the sense that it will be available also to neighbors. This game typically exhibits a great multiplicity of equilibria. Imagine a social planner whose scope is to find an optimal equilibrium, i.e. one in which the number of nodes playing 1 is minimal. To find such an equilibrium is a very hard task for any non-trivial network architecture. We propose an implementable mechanism that, in the limit of infinite time, reaches an optimal equilibrium, even if this equilibrium and even the network structure is unknown to the social planner.