Revenue equivalence theorem

Revenue equivalence is a concept in auction theory that states that given certain conditions, any mechanism that results in the same outcomes (i.e. allocates items to the same bidders) also has the same expected revenue.

There is a set ${\displaystyle X}$ of possible outcomes.

There are ${\displaystyle n}$ agents which have different valuations for each outcome. The valuation of agent ${\displaystyle i}$ (also called its "type") is represented as a function:

which expresses the value it has for each alternative, in monetary terms.

The agents have Quasilinear utility functions; this means that, if the outcome is ${\displaystyle x}$ and in addition the agent receives a payment ${\displaystyle p_{i}}$ (positive or negative), then the total utility of agent ${\displaystyle i}$ is:

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