All-purpose minimal sufficient networks in the threshold game

This paper considers a multi-player stag hunt where players are either available for action or not, and where players additionally differ in their degree of conservatism, i.e. in the threshold of players that need to act along with them before they see benefits in collective action. Minimal sufficient networks, which depending on their thresholds allow players to achieve just enough interactive knowledge about each other’s availability to act, take the form of hierarchies of cliques (Chwe, RES, 2000).
We show that any typical threshold game has a plethora of such networks, so that players seem to face a large degree of strategic uncertainty over which network to use. The plethora of networks includes cases where the structure of the network infects players into acting more conservatively than is reflected in their thresholds.
An extreme case of this is the core-periphery network, where each player acts as
conservatively as the most conservative player in the population. Because of this
feature, the core-periphery network is minimal sufficient for all possible populations.
Players can thus solve the strategic uncertainty arising from the multiplicity of
minimal sufficient networks by using the all-purpose core-periphery network.