"It appears that only five of the ten votes were read and counted; this may have been done partly as a further precaution against corruption (doubling the odds against a bribe doing any good to its giver)."
To "double the odds for" something means to take an action that makes it twice as likely to happen. From this, it follows that doubling the odds against something makes it only half as likely to happen.
In this case, I think the logic is that, if one of the votes is corrupt and all ten votes are counted, then the bribe will definitely have been effective. In other words, the odds of its being effective are 1:1 or 100%. If only the half votes are counted, there is a only 50% chance that the corrupt vote will be among them; the odds of the bribe being effective have dropped to 2:1. The odds against effectiveness have doubled.