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 $$X$$ of possible outcomes.

There are $$n$$ agents which have different valuations for each outcome.

In general, each agent can assign a different and unrelated value to every outcome in $$X$$.

In the special case of single-parameter utility, each agent $$i$$ has a publicly known outcome proper subset $$W_i \subset X$$ which are the "winning outcomes" for agent $$i$$ (e.g., in a single-item auction, $$W_i$$ contains the outcome in which agent $$i$$ wins the item).

For every agent, there is a number $$v_i$$ which represents the "winning-value" of $$i$$. The agent's valuation of the outcomes in $$X$$ can take one of two values:
 * $$v_i$$ for each outcome in $$W_i$$;
 * 0 for each outcome in $$X\setminus W_i$$.

The vector of the winning-values of all agents is denoted by $$v$$.

For every agent $$i$$, the vector of all winning-values of the other agents is denoted by $$v_{-i}$$. So $$v \equiv (v_i,v_{-i})$$.

A social choice function is a function that takes as input the value-vector $$v$$ and returns an outcome $$x\in X$$. It is denoted by $$\text{Outcome}(v)$$ or $$\text{Outcome}(v_i,v_{-i})$$.

Monotonicity
The weak monotonicity property has a special form in single-parameter domains. A social choice function is weakly-monotonic if for every agent $$i$$ and every $$v_i,v_i',v_{-i}$$, if:
 * $$\text{Outcome}(v_i, v_{-i}) \in W_i$$ and
 * $$v'_i \geq v_i > 0$$ then:
 * $$\text{Outcome}(v'_i, v_{-i}) \in W_i$$

I.e, if agent $$i$$ 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 $$X$$. The WMON property implies that for every agent $$i$$ and every $$v_i,v_i',v_{-i}$$, the function:
 * $$\Pr[\text{Outcome}(v_i, v_{-i}) \in W_i]$$

is a weakly-increasing function of $$v_i$$.

Critical value
For every weakly-monotone social-choice function, for every agent $$i$$ and for every vector $$v_{-i}$$, there is a critical value $$c_i(v_{-i})$$, such that agent $$i$$ wins if-and-only-if his bid is at least $$c_i(v_{-i})$$.

For example, in a second-price auction, the critical value for agent $$i$$ 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 $$i$$ wins if and only if his bid is at least $$c_i(v_{-i})$$, and in that case, he pays exactly $$c_i(v_{-i})$$.

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, $$v$$.
 * Select the outcome based on the social-choice function: $$x = \text{Outcome}[v]$$.
 * Every winning agent (every agent $$i$$ such that $$x \in W_i$$) pays a price equal to the critical value: $$\text{Price}_i(x, v_{-i}) = -c_i(v_{-i})$$.
 * Every losing agent (every agent $$i$$ such that $$x \notin W_i$$) pays nothing: $$\text{Price}_i(x, v_{-i}) = 0$$.

This mechanism is truthful, because the net utility of each agent is: Hence, the agent prefers to win if $$v_i>c_{-i}$$ and to lose if $$v_i<c_{-i}$$, which is exactly what happens when he tells the truth.
 * $$v_i - c_i(v_{-i})$$ if he wins;
 * 0 if he loses.

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 $$i$$ has a probability of winning, defined as:
 * $$w_i(v_i,v_{-i}) := \Pr[\text{Outcome}(v_i,v_{-i})\in W_i]$$

and an expected payment, defined as:
 * $$\mathbb{E}[\text{Payment}_i(v_i,v_{-i})]$$

In a single-parameter domain, a randomized mechanism is truthful-in-expectation if-and-only if:
 * The probability of winning, $$w_i(v_i,v_{-i})$$, is a weakly-increasing function of $$v_i$$;
 * The expected payment of an agent is:
 * $$\mathbb{E}[\text{Payment}_i(v_i,v_{-i})] = v_i\cdot w_i(v_i,v_{-i}) - \int_{0}^{v_i} w_i(t,v_{-i}) dt$$

Note that in a deterministic mechanism, $$w_i(v_i,v_{-i})$$ 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 auctions), the mechanism design problem is much more complicated. The VCG mechanism is one of the only mechanisms that works for such general valuations.