Utility functions of countries
Caveat: my familiarity with the international agreement literature is limited to that of environmental and resource economics problems.

For those cases instead of a utility function the usual approach is to formulate the problem in terms of costs and benefits for countries' economies as a whole.

When it comes to environmental problems this usually boils down to individual benefits of pollution versus collective damages, or its inverse individual abatement costs versus collective benefits. For resource problems it is individual profits (or consumption) from harvest versus the collective diminishing returns in terms of stock growth or increasing harvest costs.

Active authors in this field are a.o. Scott Barrett, Carlo Carraro, Michael Finus in terms of environment and Marco Lindroos and Pedro Pintassilgo for fisheries.