Constrained equilibrium in a bidding problem

There are a number of situations where firms vie for some constrained total quantity by bidding individual quantities but where the allocation is determined on a ‘pro-rated’ basis. One such example is a licenses-on-demand method of import quota allocation. Bidders are constrained to bid at most the total quantity, and if the sum of all bids exceeds this total, the awards are granted proportional to the fractional amounts that the bidders request. In this note, we identify the (Nash) equilibrium of the model, present some examples, and do a sensitivity analysis of the bids as the total amount available changes. We also look into changes of the bids when a new participant enters the bidding process. In an economic context, such allocation methods are shown to be biased in favor of the least efficient firms.