An $\alpha-$MaxMin Axiomatisation of Temporally-Biased Multiple Discounts

This article completes an axiomatic approach of utilities streams. The approach is more precisely based upon the robust pre-orders that open the scope for $\alpha$-MaxMin representations. A general $T$-steps Temporal Bias axiom is first introduced, that encapsulates stationarity and $1$-step present bias, aka quasi-hyperbolic discounting, as special cases. A detailed characterisation of the sets of probabilities that represent the weights of the future values of the utilities stream is then completed. This is first achieved for the close future pre-order where a generalised picture of present biases in brought into evidence. This is complemented for the distant future pre-order where it proved that, under the same system of axioms, the weights of the tail of the utility stream now correspond to Banach limits, who, in the evaluation of distant future, can be considered as the counterpart of the geometric discount rates in the evaluation of close future. The whole result is eventually given in an explicit $\alpha$-Maxmin representation.