This article looks at a more technical way to determine how to assign scores in score voting by voting tactically. This approach can be extrapolated to approval voting as well.
Analysis suggests that best score voting strategy is to give a max score to all candidates you like better than the expected value of your favored frontrunner, and min-score the others. With approval voting, that just means to vote for the candidates you like better than the expected value of the favored frontrunner.
Say your sincere normalized preferences for the candidates in the 2008 Democratic presidential primary were:
Then you would assign probabilities to the winners (based on then-current polling and primary results) like:
The rest effectively 0%
If you wanted to be really unbiased, you could even look at the trading prices on web sites where people play the “candidate stock market” (i.e. gamble). But going with this rough estimate, you calculate my expected value from the winner to be:
(0.45 × 8) + (0.43 × 3) + (0.12 × 1) = 5.01
So if you were maximally strategic, you would want to vote for every candidate you like more than 5.01, as follows:
Dodd, Gravel, Obama, Richardson
With score voting, you would give these candidates a maximum score, and all the others a minimum score.