Cooperation in heterogeneous groups, where individuals differ in resources, productivity, and behavioural responsiveness, underpins collective action across social and biological systems. Introspection dynamics is a learning rule suited to such asymmetric settings: at each time step a randomly selected player compares their current payoff to the payoff an alternative action would have given, and switches with a probability increasing in the difference. We extend introspection dynamics to include mutation, where a selected player adopts an action with payoff-independent, player-specific probabilities, together with player-specific selection intensities. From the extended dynamics we rederive and generalise the results of Couto and Pal for additive games, those in which the payoff difference a player evaluates when considering a switch is independent of the other players’ actions: the stationary distribution remains a product measure, and each player’s long-run cooperation probability has an explicit form in terms of their own payoffs, selection intensity, and mutation probabilities alone. We consider the heterogeneous public goods game, where $$N$$ N players may differ in their contributions, public goods multipliers $$r_i$$ r i , and selection intensities $$\beta _i$$ β i ; the long-run cooperation probability admits a closed form with $$N$$ N terms rather than $$2^N$$ 2 N . Several structural consequences follow: a player-specific cooperation threshold at $$r_i = N$$ r i = N under symmetric mutation, a neutral-drift regime in which cooperation is governed entirely by mutation bias, and a mutation-selection balance in which aggregate cooperation is affine in the mutation rate, interpolating between selection and neutrality. Mutation also regularises the strong-selection limit $$\beta _i\rightarrow \infty $$ β i → ∞ , where the mutation-free dynamics degenerate.

Theory, software, and research — game theory for the real world.

Many faculty say this is not the job they signed up for. They need leaders who see this and who are looking out for them.

In January 2025, the PSF submitted a proposal to the US government National Science Foundation under the Safety, Security, and Privacy of Open Source Ecosystems program to address structural vulnerabilities in Python and PyPI. It was the PSF’s first time applying for government funding, and navigating the intensive process was a steep learning curve for our small team to climb. Seth Larson, PSF Security Developer in Residence, serving as Principal Investigator (PI) with Loren Crary, PSF Deputy Executive Director, as co-PI, led the multi-round proposal writing process as well as the months-long vetting process. We invested our time and effort because we felt the PSF’s work is a strong fit for the program and that the benefit to the community if our proposal were accepted was considerable. We were honored when, after many months of work, our proposal was recommended for funding, particularly as only 36% of new NSF grant applicants are successful on their first attempt. We became concerned, however, when we were presented with the terms and conditions we would be required to agree to if we accepted the grant. These terms included affirming the statement that we “do not, and will not during the term of this financial assistance award, operate any programs that advance or promote DEI, or discriminatory equity ideology in violation of Federal anti-discrimination laws.” This restriction would apply not only to the security work directly funded by the grant, **but to any and all activity of the PSF as a whole**. Further, violation of this term gave the NSF the right to “claw back” previously approved and transferred funds. This would create a situation where money we’d already spent could be taken back, which would be an enormous, open-ended financial risk. Diversity, equity, and inclusion are core to the PSF’s values, as committed to in our mission statement: > _The mission of the Python Software Foundation is to promote, protect, and advance the Python programming language, and to support and facilitate the growth of**a diverse and international community** of Python programmers._ Given the value of the grant to the community and the PSF, we did our utmost to get clarity on the terms and to find a way to move forward in concert with our values. We consulted our NSF contacts and reviewed decisions made by other organizations in similar circumstances, particularly The Carpentries. In the end, however, the PSF simply can’t agree to a statement that we won’t operate any programs that “advance or promote” diversity, equity, and inclusion, as it would be a betrayal of our mission and our community. We’re disappointed to have been put in the position where we had to make this decision, because we believe our proposed project would offer invaluable advances to the Python and greater open source community, protecting millions of PyPI users from attempted supply-chain attacks. The proposed project would create new tools for automated proactive review of all packages uploaded to PyPI, rather than the current process of reactive-only review. These novel tools would rely on capability analysis, designed based on a dataset of known malware. Beyond just protecting PyPI users, the outputs of this work could be transferable for all open source software package registries, such as NPM and Crates.io, improving security across multiple open source ecosystems. In addition to the security benefits, the grant funds would have made a big difference to the PSF’s budget. The PSF is a relatively small organization, operating with an annual budget of around $5 million per year, with a staff of just 14. $1.5 million over two years would have been quite a lot of money for us, and easily the largest grant we’d ever received. Ultimately, however, the value of the work and the size of the grant were not more important than practicing our values and retaining the freedom to support every part of our community. The PSF Board voted unanimously to withdraw our application. Giving up the NSF grant opportunity—along with inflation, lower sponsorship, economic pressure in the tech sector, and global/local uncertainty and conflict—means the PSF needs financial support now more than ever. We are incredibly grateful for any help you can offer. If you're already a PSF member or regular donor, you have our deep appreciation, and we urge you to share your story about why you support the PSF. Your stories make all the difference in spreading awareness about the mission and work of the PSF. How to support the PSF: * Become a Member: When you sign up as a Supporting Member of the PSF, you become a part of the PSF. You’re eligible to vote in PSF elections, using your voice to guide our future direction, and you help us sustain what we do with your annual support. * Donate: Your donation makes it possible to continue our work supporting Python and its community, year after year. * Sponsor: If your company uses Python and isn’t yet a sponsor, send them our sponsorship page or reach out to sponsors@python.org today. The PSF is ever grateful for our sponsors, past and current, and we do everything we can to make their sponsorships beneficial and rewarding.