In example 1, Becky must appoint an independent valuer. In example 2, the "mechanism failed" because Becky has not.
So isn't Becky liable for failing to appoint a valuer?
Why "May & Butcher ought to apply and the contract would be void"?
Poole, Shaw-Mellors. Contract Law Concentrate (4 ed 2019). p 32.
Practical example 
Alex agrees to sell his bicycle to Becky, price to be determined by independent valuers, one to be appointed by each of the parties. The terms are agreed as ‘those usually operating in the sale of bicycles’.
The agreement may leave an essential matter (such as price) either undecided or impossible to calculate so that the courts will not enforce that agreement.
When the price is missing
If there is no mechanism in the contract for fixing the price (i.e. nothing is said about the price), then statute may provide for a ‘reasonable price’.
Section 8(1) of the Sale of Goods Act (SGA) 1979 provides that, in a contract of sale, the contract may fix the price for goods or provide a mechanism for fixing the price. Where, however, there is no such mechanism, the buyer must pay a reasonable price: s. 8(2). Similarly, with contracts for services, s. 15 of the Supply of Goods and Services Act (SGSA) 1982 states that where there is no mechanism for fixing the charge for a service, a reasonable charge is payable (B2B contract) and s. 51 of the Consumer Rights Act (CRA) 2015 contains a similar provision requiring a reasonable price to be paid for a service in B2C contracts. Section 8 SGA also extends to cover contracts of sale in the B2C context—there being (unlike with contracts for services) no corresponding provision in the CRA.
However, if there is a mechanism for fixing the price but it has not been implemented, s. 8 SGA, s. 15 SGSA, and s. 51 CRA cannot apply to allow the implication of a reasonable price.
May & Butcher Ltd v R (1934) (HL)
FACTS: The contract provided for the price to be agreed upon by the parties from time to time. There was no party agreement.
HELD: The contract contained a mechanism for fixing the price but this had failed. Therefore, the argument that the agreement should be construed as an agreement to sell at a fair or reasonable price was rejected.
Practical example 
The agreement between Alex and Becky provides a mechanism for fixing the price. If that has worked and there is an agreed price, then the terms are certain. But what if Alex has appointed a valuer but Becky has not and is instead claiming that a ‘reasonable price’ should apply? Technically the mechanism has failed, May & Butcher ought to apply and the contract would be void.