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I.e, if agent <math>i</math> wins by declaring a certain value, then he can also win by declaring a higher value (when the declarations of the other agents are the same).
The monotonicity property can be generalized to randomized mechanisms, which return a probability-distribution over the space <math>X</math>.<ref name=agt07/>{{rp|334}} The WMON property implies that for every agent <math>i</math> and every <math>v_i,v_i',v_{-i}</math>, the function:
:<math>Prob[Outcome(v_i, v_{-i}) \in W_i]</math>
is a weakly-increasing function of <math>v_i</math>.
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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, 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>.
== Deterministic implementation ==
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