We present a controlled laboratory environment in which we use an ultimatum game to generate two endogenous fairness indices. We use these as alternatives to the more conventional exogenous measure, the offer index, in a model of offer-acceptance which includes measures of social value orientations and risk attitudes as variables for explaining the acceptance or rejections of offers in an ultimatum game. In particular we are interested in providing an explanatory model which can support situations in which the likelihood to accept unfair offers (as measured by the offer index) will exceed the likelihood of rejecting a fair offer (again, as measured by the offer index). The offer index in the ultimatum game setting is the amount offered by a sender divided by the total endowment of the sender. Our endogenous fairness indices meet our condition of the likelihood of acceptance of an unfair offer exceeding the likelihood of rejecting a fair offer even though the explanatory power of the offer-acceptance models with the endogenous fairness indices is not significantly different from that with the exogenous fairness index.