Single-parameter utility

In mechanism design a single-parameter mechanism is a mechanism in which the preferences of each agent are represented by a single number. For example, an auction for a single item is single-parameter, since the preferences of each agent are represented by his evaluation of the item. In contrast, a combinatorial auction for two or more related items is not single-parameter, since the preferences of each agent are represented by his evaluation 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.

For every agent ${\displaystyle i}$ , there is a publicly-known 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]: 228–230}

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

This is a very special case of the the general case (handled e.g. by VCG mechanisms) in which each agent can assign a different value to every outcome in ${\displaystyle X}$ .

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 ${\displaystyle Outcome(v)}$  or ${\displaystyle Outcome(v_{i},v_{-i})}$ .

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.