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In [[mechanism design]],
== Notation ==
There is a set <math>X</math> of possible outcomes.
There are <math>n</math> agents which have different valuations for each outcome.
For every agent <math>i</math>, there is a publicly-known subset <math>W_i \subset X</math> which are the "winning outcomes" for agent <math>i</math> (e.g, in a single-item auction, <math>W_i</math> contains the outcome in which agent <math>i</math> wins the item).▼
▲
For every agent, there is a number <math>v_i</math> which represents the "winning-value" of <math>i</math>. The agent's valuation of the outcomes in <math>X</math> can take one of two values:<ref name=agt07>{{Cite Algorithmic Game Theory 2007}}</ref>{{rp|228}}
* <math>v_i</math> for each outcome in <math>W_i</math>;
* 0 for each outcome in <math>X\setminus W_i</math>.
▲This is a very special case of the the general case (handled e.g. by [[VCG mechanism]]s) in which each agent can assign a different value to every outcome in <math>X</math>.
The vector of the winning-values of all agents is denoted by <math>v</math>.
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For every agent <math>i</math>, the vector of all winning-values of the ''other'' agents is denoted by <math>v_{-i}</math>. So <math>v \equiv (v_i,v_{-i})</math>.
A [[social choice]] function is a function that takes as input the value-vector <math>v</math> and returns an outcome <math>x\in X</math>. It is denoted by <math>\text{Outcome}(v)</math> or <math>\text{Outcome}(v_i,v_{-i})</math>.
== Monotonicity ==
<div id="monotonicity" ></div>
The [[Monotonicity (mechanism design)#wmon|'''weak monotonicity''' property]] has a special form in single-parameter domains. A social choice function is weakly-monotonic if for every agent <math>i</math> and every <math>v_i,v_i',v_{-i}</math>, if:
: <math>\text{Outcome}(v_i, v_{-i}) \in W_i</math> and
: <math>v'_i \geq v_i > 0</math> then:
: <math>\text{Outcome}(v'_i, v_{-i}) \in W_i</math>
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:
== Critical value ==▼
:<math>\Pr[\text{Outcome}(v_i, v_{-i}) \in W_i]</math>
For every social-choice function, for every agent <math>i</math> and for every vector <math>v_{-i}</math>, there is a '''critical value''' <math>c_i(v_{-i})</math>, such that agent <math>i</math> wins if-and-only-if his bid is at least <math>c_i(v_{-i})</math>.▼
is a weakly-increasing function of <math>v_i</math>.
▲=== Critical value ===
▲For every weakly-monotone social-choice function, for every agent <math>i</math> and for every vector <math>v_{-i}</math>, there is a '''critical value''' <math>c_i(v_{-i})</math>, such that agent <math>i</math> wins if-and-only-if his bid is at least <math>c_i(v_{-i})</math>.
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>.
== Deterministic implementation ==
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The mechanism should work in the following way:<ref name=agt07/>{{rp|229}}
* Ask the agents to reveal their valuations, <math>v</math>.
* Select the outcome based on the social-choice function: <math>x = \text{Outcome}[v]</math>.
* Every winning agent (every agent <math>i</math> such that <math>x \in W_i</math>) pays a price equal to the critical value: <math>\text{Price}_i(x, v_{-i}) = -c_i(v_{-i})</math>.
* Every losing agent (every agent <math>i</math> such that <math>x \notin W_i</math>) pays nothing: <math>\text{Price}_i(x, v_{-i}) = 0</math>.
This mechanism is truthful, because the net utility of each agent is:
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In a randomized mechanism, every agent <math>i</math> has a probability of winning, defined as:
: <math>w_i(v_i,v_{-i}) := \Pr[\text{Outcome}(v_i,v_{-i})\in W_i]</math>
and an expected payment, defined as:
: <math>\mathbb{E}[
In a single-parameter ___domain, a randomized mechanism is truthful-in-expectation if-and-only if:<ref name=agt07/>{{rp|232}}
* The probability of winning, <math>w_i(v_i,v_{-i})</math>, is a weakly-increasing function of <math>v_i</math>;
* The expected payment of an agent is:
: <math>\mathbb{E}[
Note that in a deterministic mechanism, <math>w_i(v_i,v_{-i})</math> is either 0 or 1, the first condition reduces to weak-monotonicity of the Outcome function and the second condition reduces to charging each agent his critical value.
== Single-parameter vs. multi-parameter domains ==
When the utilities are not single-parametric (e.g. in [[combinatorial auction]]s), the mechanism design problem is much more complicated. The [[VCG mechanism]] is one of the only mechanisms that works for such general valuations.
== See also ==
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== References ==
{{reflist}}
[[Category:Mechanism design]]
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