We reject the hypothesis of two distinct male alliance strategies in Shark Bay 
. Instead, we found a continuum of second-order alliance sizes from 6–14 males. Both large and small second-order alliances may be stable for years [3
We did not find a simple relationship between second-order alliance size and first-order alliance stability. The two groups that began consorting females during the study (BL and RR) had the lowest alliance stability values. Low alliance stability in younger groups is expected in systems with competition for social partners.
We found consistent low-level associations between particular male groups. Any association between males in different groups is surprising as the two levels of alliance formation we documented previously are clearly based on access to females. Our permutation analysis asked the question, ‘If a male is with at least one of his allies, do they exhibit preferences when associating with other alliances?’ We found that in some cases they do. These associations are not related to food resources as we restricted analyses to non-foraging groups. The fights involving greater than two groups suggest that these associations, like the second-order alliances, are employed in conflicts over females.
These fights were chaotic as they involved 13–23 males, so the only evidence of one group supporting another is based on who remained together when normal behaviour resumed after the fight.
Four of the five trios that did not have second-order alliance partners were older males that were adults consorting females in the 1980s as part of second-order alliances. The second-order alliance history of the other trio (PGH) is unknown. The need to have allies for female defence may explain the third-order associations that older male trios form with second-order alliances (e.g. CB with FCB) or each other (BB with SK) after their former allies have disappeared.
The dolphins' fission–fusion grouping pattern as well as variation in second-order alliance size may favour males having third-order alliance relationships. Second-order alliance partners may not always be present when rivals appear and even some third-order allies may not be enough (e.g. example 3).
Only humans and Shark Bay bottlenose dolphins are known to have multiple-level male alliances within a social network. It is unlikely a coincidence that humans and dolphins also have in common the largest brains, relative to body size, among mammals. Our evidence for a third level of alliance formation in the dolphins should refocus attention on the potential cognitive burdens for individuals embedded in such a system, where decisions at one level may have impacts at other levels (reviewed in [3