Single parameter utility

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 subset W i ⊂ 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:

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 ≡ ( v i , v − i ) .

A social choice function is a function that takes as input the value-vector v and returns an outcome x ∈ X . It is denoted by O u t c o m e ( v ) or O u t c o m e ( 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:

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:

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:

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.

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:

and an expected payment, defined as:

In a single-parameter domain, a randomized mechanism is truthful-in-expectation if-and-only if:

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.