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 ${\displaystyle X}$ of possible outcomes.

There are ${\displaystyle n}$ agents which have different valuations for each outcome.

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

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

For every agent, there is a number ${\displaystyle v_{i}}$ which represents the "winning-value" of ${\displaystyle i}$. The agent's valuation of the outcomes in ${\displaystyle X}$ can take one of two values:[1]: 228 

• ${\displaystyle v_{i}}$ for each outcome in ${\displaystyle W_{i}}$;
• 0 for each outcome in ${\displaystyle X\setminus W_{i}}$.

The vector of the winning-values of all agents is denoted by ${\displaystyle v}$.

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

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

Monotonicity

The weak monotonicity property has a special form in single-parameter domains. A social choice function is weakly-monotonic if for every agent ${\displaystyle i}$ and every ${\displaystyle v_{i},v_{i}',v_{-i}}$, if:

${\displaystyle {\text{Outcome}}(v_{i},v_{-i})\in W_{i}}$ and

${\displaystyle v'_{i}\geq v_{i}>0}$ then:

${\displaystyle {\text{Outcome}}(v'_{i},v_{-i})\in W_{i}}$

I.e, if agent ${\displaystyle i}$ 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 ${\displaystyle X}$.[1]: 334  The WMON property implies that for every agent ${\displaystyle i}$ and every ${\displaystyle v_{i},v_{i}',v_{-i}}$, the function:

${\displaystyle \Pr[{\text{Outcome}}(v_{i},v_{-i})\in W_{i}]}$

is a weakly-increasing function of ${\displaystyle v_{i}}$.

Critical value

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

For example, in a second-price auction, the critical value for agent ${\displaystyle i}$ is the highest bid among the other agents.

In single-parameter environments, deterministic truthful mechanisms have a very specific format.[1]: 334  Any deterministic truthful mechanism is fully specified by the set of functions c. Agent ${\displaystyle i}$ wins if and only if his bid is at least ${\displaystyle c_{i}(v_{-i})}$, and in that case, he pays exactly ${\displaystyle c_{i}(v_{-i})}$.

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:[1]: 229 

• Ask the agents to reveal their valuations, ${\displaystyle v}$.
• Select the outcome based on the social-choice function: ${\displaystyle x={\text{Outcome}}[v]}$.
• Every winning agent (every agent ${\displaystyle i}$ such that ${\displaystyle x\in W_{i}}$) pays a price equal to the critical value: ${\displaystyle {\text{Price}}_{i}(x,v_{-i})=-c_{i}(v_{-i})}$.
• Every losing agent (every agent ${\displaystyle i}$ such that ${\displaystyle x\notin W_{i}}$) pays nothing: ${\displaystyle {\text{Price}}_{i}(x,v_{-i})=0}$.

This mechanism is truthful, because the net utility of each agent is:

• ${\displaystyle v_{i}-c_{i}(v_{-i})}$ if he wins;
• 0 if he loses.

Hence, the agent prefers to win if ${\displaystyle v_{i}>c_{-i}}$ and to lose if ${\displaystyle v_{i}<c_{-i}}$, 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 ${\displaystyle i}$ has a probability of winning, defined as:

${\displaystyle w_{i}(v_{i},v_{-i}):=\Pr[{\text{Outcome}}(v_{i},v_{-i})\in W_{i}]}$

and an expected payment, defined as:

${\displaystyle \mathbb {E} [{\text{Payment}}_{i}(v_{i},v_{-i})]}$

In a single-parameter domain, a randomized mechanism is truthful-in-expectation if-and-only if:[1]: 232 

• The probability of winning, ${\displaystyle w_{i}(v_{i},v_{-i})}$, is a weakly-increasing function of ${\displaystyle v_{i}}$;
• The expected payment of an agent is:

${\displaystyle \mathbb {E} [{\text{Payment}}_{i}(v_{i},v_{-i})]=v_{i}\cdot w_{i}(v_{i},v_{-i})-\int _{0}^{v_{i}}w_{i}(t,v_{-i})dt}$

Note that in a deterministic mechanism, ${\displaystyle w_{i}(v_{i},v_{-i})}$ 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.

See also

• Single peaked preferences

References

1. Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.