By Ollie Stone-Lee BBC News Online politics staff

Efforts at Lords reform probably ran into gridlock because MPs wrongly second guessed how their colleagues would vote, according to an expert in game theory.

A majority might have been secured for a mostly elected house had a crucial handful of MPs not thought their first choice would be preferred to no change at all, says Andrew Lilico, a game theorist at Europe Economics.

In Tuesday's votes, the House of Lords backed an all-appointed second chamber but MPs failed to give a majority for any of the seven options on offer.

Game theory is a mathematical method of decision-making, the key solution concept of which was the brainchild of John Forbes Nash, the man made famous in Oscar-winning film A Beautiful Mind.

Using those methods, Mr Lilico examined how the Commons votes failed to produce a majority for any of the seven options.

Split decision

MPs were allowed to vote for as many options as they liked.

The votes started with the extremes of an all-appointed and all-elected second chamber - and ended with the choice of a 50-50 split of elected and appointed peers.

Seven options

100% elected 80% appointed, 20% elected 20% appointed, 80% elected 60% appointed, 40% elected 40% appointed, 60% elected 50% appointed, 50% elected 100% appointed

Click here to see how they voted

The idea of having 80% of peers elected and 20% appointed came within just three votes of getting a majority.

But the other mostly elected option, having 60% of peers elected, was defeated by 253 votes in favour, 316 against.

Mr Lilico told BBC News Online: "The most likely cause of the problem here was that there were fairly even numbers of supporters of both the majority-elected-but-not-100% options.

"Each of them thought their option would be preferred to no change.

"In fact it turned out that the 60% vote came second and probably only the 80% elected option was really preferred to no change - a recipe for gridlock."

Voting order crucial?

Mr Lilico pointed to the example of an MP who wanted 60% of peers elected, but whose second choice would be 80% elected.

If such an MP decided the most likely choices were their first and second preferences, that could cause problems.

That MP could decide not to back the 80% elected option when that vote arrived (which was before the 60% elected vote), thinking that if the 80% elected option were defeated, their favourite option (60% elected) would be accepted later.

Mr Lilico explained: "If there were even two MPs in that position, then the order of votes was crucial to the outcome.

"If the 80% elected option had been voted for after the 60% option, such people might have voted the other way," he said.