Single-parameter utility
Single-parameter utility
In mechanism-design, an agent is said to have single-parameter utility if his valuation of the possible outcomes can be represented by a single number. For example, in an auction for a single item, the utilities of all agents are single-parametric, since they can be represented by their monetary evaluation of the item. In contrast, in a combinatorial-auction for two or more related items, the utilities are usually not single-parametric, since they are usually represented by their evaluations to all possible bundles of items.
Notation There is a set <math>X</math> of possible outcomes.
There are <math>n</math> agents which have different valuations for each outcome.
In general, each agent can assign a different and unrelated value to every outcome in <math>X</math>.
In the special case of single-parameter utility, each agent <math>i</math> has a publicly known outcome proper 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: <math>v_i</math> for each outcome in <math>W_i</math>; 0 for each outcome in <math>X\setminus W_i</math>.
The vector of the winning-values of all agents is denoted by <math>v</math>.
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 **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>. The WMON property implies that for every agent <math>i</math> and every <math>v_i,v_i',v_{-i}</math>, the function: :<math>\Pr[\text{Outcome}(v_i, v_{-i}) \in W_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. 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 It is known that, in any domain, weak monotonicity is a necessary condition for implementability. I.e, a social-choice function can be implemented by a truthful mechanism, only if it is weakly-monotone.
In a single-parameter domain, weak monotonicity is also a sufficient condition for implementability. I.e, for every weakly-monotonic social-choice function, there is a deterministic truthful mechanism that implements it. This means that it is possible to implement various non-linear social-choice functions, e.g. maximizing the sum-of-squares of values or the min-max value.
The mechanism should work in the following way: 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: <math>v_i - c_i(v_{-i})</math> if he wins; 0 if he loses. Hence, the agent prefers to win if <math>v_i>c_{-i}</math> and to lose if <math>v_i<c_{-i}</math>, which is exactly what happens when he tells the truth.
Randomized implementation A randomized mechanism is a probability-distribution on deterministic mechanisms. A randomized mechanism is called truthful-in-expectation if truth-telling gives the agent a largest expected value.
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}[\text{Payment}_i(v_i,v_{-i})]</math>
In a single-parameter domain, a randomized mechanism is truthful-in-expectation if-and-only if: 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}[\text{Payment}_i(v_i,v_{-i})] = v_i\cdot w_i(v_i,v_{-i}) - \int_{0}^{v_i} w_i(t,v_{-i}) dt</math>
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.