complete functionality for fair multiparty computation In this paper I found the statement "We reduce the fairness problem in MPC to the problem of fair decryption [...] Fair decryption was shown to be a complete functionality for fair MPC in [GIM+10]". 

How is decryption considered to be complete functionality for fair multiparty computation? What do you mean by complete functionality?
The functionality that this paper refers to when speaking about "fair decryption" is probably the "fair reconstruction" functionality from [GIM+10].

In secure computation we say some cryptographic primitive is complete if we can use it to implement any ideal functionality (in some setting). One can think of complete primitives as the sufficient requirements for secure computation. For example, we know that one can securely compute any desired functionality if one has access to oblivious transfer as a building block.

Roughly speaking, [GIM+10] shows that fair reconstruction is complete as follows:
All parties first execute any unfair secure computation protocol that outputs (non-malleable) secret shares of the output.
The parties then run the fair reconstruction function, which returns the reconstructed secret.