  Following the recent literature on make take fees policies, we consider an
exchange wishing to set a suitable contract with several market makers in order
to improve trading quality on its platform. To do so, we use a principal-agent
approach, where the agents (the market makers) optimise their quotes in a Nash
equilibrium fashion, providing best response to the contract proposed by the
principal (the exchange). This contract aims at attracting liquidity on the
platform. This is because the wealth of the exchange depends on the arrival of
market orders, which is driven by the spread of market makers. We compute the
optimal contract in quasi explicit form and also derive the optimal spread
policies for the market makers. Several new phenomena appears in this multi
market maker setting. In particular we show that it is not necessarily optimal
to have a large number of market makers in the presence of a contracting
scheme.
