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For example, in a [[second-price auction]], the critical value for agent <math>i</math> is the highest bid among the other agents.
In single-parameter environments, deterministic truthful mechanisms have a very specific format.<ref name=agt07/>{{rp|334}} Any deterministic truthful mechanism is fully specified by the set of functions c. Agent <math>i</math> wins if and only if his bid is at least <math>c_i(v_{-i})</math>, and in that case, he pays exactly <math>c_i(v_{-i})</math>.
In randomized truthful mechanisms, the payment per player must be of the form:
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== Deterministic implementation ==
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