I had the same intuition as RhysSouthan that most people who acquire the second vote in a Demeny voting structure would use the two votes for the same party/candidate/policy . I think an important facet here is that the salience of the vote being for the 'future generation' may nudge people on the margin to use both votes for the policy/party that best benefits the future generation, whereas without receiving the second vote they may not have voted this way. The Kochi University of Technology Research Institute of Future Design have some papers that show making future generations salient increases pro-social behaviour towards future generations. Thus, I think this the above hypothesis is plausible. Also, as Tyler showed above, even marginal changes of the second vote (from the first) that benefits future generations would be a good thing. I do think motivated reasoning is a big concern here though ( people manipulating their beliefs to think that their vote is good for future generations, when in fact it is not), and it would be interesting to see if there is any evidence of this.
The same institute actually just published a paper using a lab experiment to investigate the psychological effects of Demeny Voting. (I need to re-read the paper, as it has changed from their working paper version, which I read recently). However, the paper investigates the voting behaviour of people who do not receive the second vote (single-ballot voters). They hypothesise that these single-ballot voters may be more likely to use their vote for the benefit of future generations in a Demony Voting structure due to either:
a) The cost of behaving ethically decreases as their vote has less impact on the overall outcome. However, they do not find any evidence of this in their data (NPV Treatment: Demony voting, but participants are not told why a second vote was allocated to some people). The authors argue that single ballot voters may perceive the voting structure to be unfair, inducing them to vote more egotistically in this treatment.
b) A bandwagon effect (conformity): These voters expect that more votes will be used on the policy that benefits future generations, as a result of others getting a second vote (and told to use it for the interest of future generations) "promoting the ethical voting of single ballot voters."
They find evidence supporting the bandwagon effect. When people are told why a second vote is given in a Demony Voting structure (PV treatment), single ballot voters are more likely to vote for the policy that benefits the future compared to the NPV treatment. (Note: I am not sure a bandwagon effect is the best model to explain this result).
The authors contend that "these results suggest that the success of the new voting scheme suggested by Demeny should depend on whether the rationale behind giving some voters a second ballot is explained and understood."
It is important to highlight that this is a single lab experiment, and external validity concerns are legitimate (to the authors credit they make sure to talk about this at the end of the paper). Field experiments, the use of observational data, and further lab experiments are all necessary to get a better understanding of how Demony Voting affects voter's behaviour, but it's exciting to see academics interested in what I believe is an important question.