  The dilemma in cooperation is one of the major concerns in game theory. In a
public-goods game, each individual pays a cost for cooperation, or to prevent
defection, and receives a reward from the collected cost in a group. Thus,
defection is beneficial for each individual, while cooperation is beneficial
for the group. Now, groups (say, countries) consisting of individual players
also play games. To study such a multi-level game, we introduce a hierarchical
public-goods (HPG) game in which two groups compete for finite resources by
utilizing costs collected from individuals in each group. Analyzing this HPG
game, we found a hierarchical prisoner's dilemma, in which groups choose the
defection policy (say, armaments) as a Nash strategy to optimize each group's
benefit, while cooperation optimizes the total benefit. On the other hand, for
each individual within a group, refusing to pay the cost (say, tax) is a Nash
strategy, which turns to be a cooperation policy for the group, thus leading to
a hierarchical dilemma. Here, the reward received by one group increases with
the population, as does the collected cost. In spite of this, we find that
there exists an optimal group size that maximizes its payoff. Furthermore, when
the population asymmetry between two groups is large, a smaller group will
choose a cooperation policy (say, disarmament) to avoid excessive response from
the larger group, which leads to the resolution of the prisoner's dilemma
between the groups. The relevance of the HPG game to policy selection in
society and the optimal size in human or animal groups are discussed
accordingly.
