How would I derive new payoffs for "Rawlsian" Nash players by assuming that they act as though there is a probability that they could be either a member of group A or group B. That is, for each strategy combination, there is a probability of getting the payoff for group A, and payoff for getting the payoff for group B. This isn't an issue of "equal bargaining strength", this is an assumption about how payoffs are evaluated. Using the values below:

FIGURE 4. Payoffs for the Sovereign-Constituency Transgression Game 8 Moves first Induced subgame between A and B (payoffs: S,A,B) Acquiesce Challenge Transgress against Acquiesce both A Challenge Transgress against A Acquiesce A Challenge 8 Transgress against 3 Acquiesce A Challenge B Transgress Acquiesce Challenge against neither Acquiesce A Challenge