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:^[1]^: 228

$v_{i}$ 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 $X$ .^[1]^: 334 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.^[1]^: 334 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:^[1]^: 229

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:

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

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:

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:^[1]^: 232

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.

References

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

[agt07-1] Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.

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