Most of us try to persuade our children to act 'appropriately' by using religious precepts such as "God wants us to tell the truth". Most children (and adults) have an instinct to 'blend' such parental instruction with their previous attitudes, gradually modifying their behavior under adult guidance until it conforms, more or less, to adult expectations. Children with Asperger Syndrome (AS) tend to take a 'black-or-white' approach to life, either rejecting such a precept entirely, or accepting it entirely and becoming very upset about 'white lies'. AS children may take to religion very easily, but it's absolutist tendencies may drive an AS child to extreme positions that do not help him deal with real life.
At least for older AS children, modern game theory provides a possible alternative. The central result of game theory, as it applies to 'moral' behavior, is that successful people often sacrifice short-term advantage to build a reputation that is valuable to them over a long period. It may seem strange to bring such novel and complex ideas to what 'ought' to be simple, but if traditional methods don't work, it seems reasonable to try new ones.
An early struggle many parents have is how their child should be advised to respond to the various selfish acts of other children. "Never hit back. If the other child keeps hitting you, report it to authority." is the usual approach, but we all know this doesn't work. I think most parents hope their child will blend this exhortation with his natural instinct to lash back, to produce some reasonable approximation of a successful strategy. So what is this successful strategy we are hoping for?
Modern game theory demonstrates clearly that a 'tit for tat' strategy would work best if there was perfect understanding between the children. My own mathematical work tries to explore the effects of partial understanding between them. One interesting piece of research indicates that the person being hit estimates the blow as three times harder than the estimate of the hitter, on average. Thus our 'tit for tat' strategy should really be "hit him back about a third as hard as he hit you". Even this is too much: if both parties adopt this strategy, an accidental bump will result in an endless string of equal blows. If both parties reduce their retaliation to a quarter of the provocation, then each blow will in fact be softer than the previous one, until they are so obviously futile that the pair separate out of boredom with the exchange.
This is indeed a complex strategy to try to explain to a child, but it does have the great advantage of working in the real world. If your child is hit by another child, he has basically three choices:
Not retaliate at all - turn the other cheek. The obvious danger of this is that he acquires a reputation as someone who can be hit without danger to the hitter, and thus a safe place to vent frustrations. The first requirement for a successful strategy is that it should effectively discourage the worst behavior of other people
Hit back harder. This is good for the reputation, but has obvious problems when the other child has the same strategy. The fight will escalate until one child either dies or changes strategy. The second requirement for a successful strategy is that it produce an acceptable result when it is applied by both sides.
Hit back softer. Of course, if you are faced with a total maniac, he may escalate faster than you de-escalate, but then nothing would have worked in this case. The real reason this strategy works is that those 'against' whom it does not work have usually been eliminated one way or another without you ever meeting them.
The principles behind this example can be seen to underlie almost all the significant relationships we have in life.