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== Combinatorial preferences ==
One complication in multi-issue voting is that there may be dependencies between agents' preferences on different issues. For example, suppose the issues to be decided on are different kinds of food that may be given in a meal. Suppose the bread can be either [[Black bread|black]] or [[White bread|white]], and the main dish can be either [[hummus]] or [[tahini]]. An agent may want either black bread with hummus or white bread with tahini, but not the other way around. This problem is called '''non-separability'''. There are several approaches for eliciting
# If there are only few issues, it is possible to ask each voter to rank all possible combinations of candidates. However, the number of combinations increases exponentially in the number of issues, so it is not practical when there are many issues. There is some research on languages for concise representation of preferences.<ref>{{Cite journal |last=Lang |first=Jérôme |date=2007-01-06 |title=Vote and aggregation in combinatorial domains with structured preferences |url=https://dl.acm.org/doi/abs/10.5555/1625275.1625496 |journal=Proceedings of the 20th international joint conference on Artifical intelligence |series=IJCAI'07 |location=San Francisco, CA, USA |publisher=Morgan Kaufmann Publishers Inc. |pages=1366–1371 |doi=}}</ref>
# It is possible to ask for each voters' favorite alternative in each issue separately. This option is simpler, but might lead to multiple-election paradoxes, where the collective decision is worst for all agents. For example, suppose there are three issues, and for each issue there are two candidates - 1 and 0. Suppose Alice's top choice is (1, 1, 0), Bob's top choice is (1, 0, 1), and Chana's top choice is (0, 1, 1), and all agents' last choice is (1, 1, 1). A majority voting in each issue separately would lead to the outcome (1,1,1), which is worst for all voters.<ref>{{Cite journal |last=Lacy |first=Dean |last2=Niou |first2=Emerson M.S. |date=2000-01-01 |title=A Problem with Referendums |url=http://journals.sagepub.com/doi/10.1177/0951692800012001001 |journal=Journal of Theoretical Politics |language=en |volume=12 |issue=1 |pages=5–31 |doi=10.1177/0951692800012001001 |issn=0951-6298}}</ref>
# In '''sequential voting''',<ref>{{Cite journal |last1=Lang |first1=Jérôme |last2=Xia |first2=Lirong |date=2009-05-01 |title=Sequential composition of voting rules in multi-issue domains |url=https://www.sciencedirect.com/science/article/pii/S0165489608001261 |journal=Mathematical Social Sciences |series=Special Issue: Voting Theory and Preference Modeling |language=en |volume=57 |issue=3 |pages=304–324 |doi=10.1016/j.mathsocsci.2008.12.010 |issn=0165-4896 |s2cid=35194669}}</ref><ref>{{Cite journal |last=Xia |first=Lirong |last2=Conitzer |first2=Vincent |last3=Lang |first3=Jérôme |date=2011-06-05 |title=Strategic sequential voting in multi-issue domains and multiple-election paradoxes |url=https://doi.org/10.1145/1993574.1993602 |journal=Proceedings of the 12th ACM conference on Electronic commerce |series=EC '11 |location=New York, NY, USA |publisher=Association for Computing Machinery |pages=179–188 |doi=10.1145/1993574.1993602 |isbn=978-1-4503-0261-6}}</ref> the issues are decided in order, so that each agent can vote on an issue based on the outcomes in previously-decided issues. This method is useful when there is a natural order of dependence on the issues. However, if some issues depend on decisions in future issues, the voters will have a hard time deciding what to vote.<ref>{{Cite journal |last=Conitzer |first=Vincent |last2=Lang |first2=Jérôme |last3=Xia |first3=Lirong |date=2009-07-11 |title=How hard is it to control sequential elections via the agenda? |url=https://dl.acm.org/doi/abs/10.5555/1661445.1661463 |journal=Proceedings of the 21st International Joint Conference on Artificial Intelligence |series=IJCAI'09 |location=San Francisco, CA, USA |publisher=Morgan Kaufmann Publishers Inc. |pages=103–108 |doi=}}</ref>
# In '''iterative voting''',<ref>{{Cite journal |last=Meir |first=Reshef |last2=Polukarov |first2=Maria |last3=Rosenschein |first3=Jeffrey |last4=Jennings |first4=Nicholas |date=2010-07-04 |title=Convergence to Equilibria in Plurality Voting |url=https://ojs.aaai.org/index.php/AAAI/article/view/7624 |journal=Proceedings of the AAAI Conference on Artificial Intelligence |language=en |volume=24 |issue=1 |pages=823–828 |doi=10.1609/aaai.v24i1.7624 |issn=2374-3468}}</ref><ref>{{Cite journal |last=Kavner |first=Joshua |last2=Meir |first2=Reshef |last3=Rossi |first3=Francesca |last4=Xia |first4=Lirong |date=2023-01-20 |title=Convergence of Multi-Issue Iterative Voting under Uncertainty |url=http://arxiv.org/abs/2301.08873 |journal=arXiv:2301.08873 [cs]}}</ref> we ask for each voters' favorite alternative in each issue separately, but allow them to revise their vote based on other people's votes. Voters are allowed to update only one issue at a time. The problem is that the iterative dynamics might not converge. However, in certain special cases, a [[Nash equilibrium]] exists.<ref name=":4" /> Iterative voting can improve the social welfare and prevent some of the multiple-election paradoxes; this was shown both by computer simulations<ref>{{Cite journal |last=Bowman |first=Clark |last2=Hodge |first2=Jonathan K. |last3=Yu |first3=Ada |date=2014-06-01 |title=The potential of iterative voting to solve the separability problem in referendum elections |url=https://doi.org/10.1007/s11238-013-9383-2 |journal=Theory and Decision |language=en |volume=77 |issue=1 |pages=111–124 |doi=10.1007/s11238-013-9383-2 |issn=1573-7187}}</ref> and by laboratory experiments.<ref>{{Cite journal |last=Grandi |first=Umberto |last2=Lang |first2=Jérôme |last3=Ozkes |first3=Ali I. |last4=Airiau |first4=Stéphane |date=2022-12-10 |title=Voting behavior in one-shot and iterative multiple referenda |url=https://doi.org/10.1007/s00355-022-01436-0 |journal=Social Choice and Welfare |language=en |doi=10.1007/s00355-022-01436-0 |issn=1432-217X}}</ref>
A survey on voting in combinatorial domains is given by Lang and Xia.<ref>{{Cite journal |last=Lang |first=Jérôme |last2=Xia |first2=Lirong |date=2016 |title=Voting in Combinatorial Domains |url=https://hal.science/hal-01493535 |language=en |pages=197 |doi=10.1017/CBO9781107446984.010}}</ref>
== Generalizations ==
Lackner, Maly and Rey extend the concept of perpetual voting to [[participatory budgeting]].<ref>{{Cite journal |last1=Lackner |first1=Martin |last2=Maly |first2=Jan |last3=Rey |first3=Simon |date=2021-05-03 |title=Fairness in Long-Term Participatory Budgeting |url=https://dl.acm.org/doi/abs/10.5555/3463952.3464161 |journal=Proceedings of the 20th International Conference on Autonomous Agents and MultiAgent Systems |series=AAMAS '21 |location=Richland, SC |publisher=International Foundation for Autonomous Agents and Multiagent Systems |pages=1566–1568 |isbn=978-1-4503-8307-3}}</ref>
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