Single-parameter utility

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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

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There is a set   of possible outcomes.

There are   agents which have different valuations for each outcome.

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

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

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

  •   for each outcome in  ;
  • 0 for each outcome in  .

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

For every agent  , the vector of all winning-values of the other agents is denoted by  . So  .

A social choice function is a function that takes as input the value-vector   and returns an outcome  . It is denoted by   or  .

Monotonicity

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The weak monotonicity property has a special form in single-parameter domains. A social choice function is weakly-monotonic if for every agent   and every  , if:

  and
  then:
 

I.e, if agent   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  .[1]: 334  The WMON property implies that for every agent   and every  , the function:

 

is a weakly-increasing function of  .

Critical value

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For every weakly-monotone social-choice function, for every agent   and for every vector  , there is a critical value  , such that agent   wins if-and-only-if his bid is at least  .

For example, in a second-price auction, the critical value for agent   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   wins if and only if his bid is at least  , and in that case, he pays exactly  .

Deterministic implementation

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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,  .
  • Select the outcome based on the social-choice function:  .
  • Every winning agent (every agent   such that  ) pays a price equal to the critical value:  .
  • Every losing agent (every agent   such that  ) pays nothing:  .

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

  •   if he wins;
  • 0 if he loses.

Hence, the agent prefers to win if   and to lose if  , which is exactly what happens when he tells the truth.

Randomized implementation

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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   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:[1]: 232 

  • The probability of winning,  , is a weakly-increasing function of  ;
  • The expected payment of an agent is:
 

Note that in a deterministic mechanism,   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

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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

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References

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  1. ^ 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.