Published on July 14, 2025 2:08 PM GMT
Dear all, this little note (that will be included as a section in the next version of “The ideal political workflow”) is a summary of the relation between the Storable Votes literature and Social Choice. I want to test it before my next attempt to publish the article, so please, feel free to comment on it as harshly as you please, or to suggest improvements or extensions.
Solving multi-alternative multi-period independent voting
Social choice theory extends the theory of individual rational choice to collective decision-making. Its foundational result is the Arrow Impossibility Theorem (Arrow, 1951), that shows that for a social choice among more than two alternatives, there is not any algorithm mapping the many ordinal preferences of the citizens into a single social ordering of the alternatives that fulfil a series of common-sense requirements.
The typical interpretation of the Arrow theorem is that it reflects a fundamental limitation for democratic procedures. This led to an extensive literature (economic, mathematical and philosophical) including a long research program on vote trading (Casella and Macé, 2021). A dissenting tradition (Castellanos, 2005; Vasiljev, 2008) argues that the theorem simply proves that ordinal preferences are not expressive enough and that proper social aggregation depends on allowing individuals to express preference intensity by cardinal voting.
In fact, under truthful revelation of preferences (the case considered by Arrow), normalized cardinal utility makes social choice trivial. Suppose every citizen truthfully provides a normalized cardinal valuation for each alternative: that is, valuations are positive, and for each citizen the sum of the valuations for all alternatives is one. Then, the sum of the valuations by alternative is a consistent measure of the social value for every alternative (Dhillon, Bouveret and Lemaître, 1999).
While the social choice problem does not exist under normalized cardinal voting, preference falsification remains a problem in multi-alternative elections with a single issue: under majoritarian voting rules, the rational voter tries to avoid her vote to be “wasted” in an alternative with low chances of being chosen by engaging in “strategic voting”, which leads to a coordination game and implies the misrepresentation of preferences (Myatt, 2007). This problem can be addressed when there are multiple independent elections.
The typical framework for the analysis of multiperiod voting is the “independent identically distributed” (i.i.d) setting, where there is a sequence of social decisions and in each period and agents have a random and independently drawn utility on every possible social alternative. Voters maximize the (discounted) present value of future utility across periods.
Alessandra Casella (2005) proposed the Storable Votes (SV) voting mechanism, where participants in a sequence of elections are given additional votes in each period, so they can signal the intensity of their preferences:[1] shortly after the introduction of SV, Jackson & Sonnenschein (2007) proved that connecting decisions over time can resolve the issue of incentive compatibility in a broad range of social choice problems. The result was a meta-mechanism not suitable for direct implementation.
While SV was an obvious improvement over pure “period-by-period” majoritarian systems, two limitations were evident. First, in the original SV the number of votes can grow without limit. This implies that an “steady state” is hard to compute for that system. Additionally, in SV (Casella, 2017) the majority can try to block the victories of the minority, leading to a complex strategic scenario: “a hide-and-seek game between majority and minority voters that corresponds to a decentralized version of the Colonel Blotto game".
This author addressed both problems with Storable votes “pay-as-you-win” (SV-PAYW) voting mechanism (Macías, 2024). Compared to SV, the following modifications are proposed in SV-PAYW: i) there is a total fixed number of votes that can be stored (in the first period all participants receive the same amount of votes), ii) in each election players cast a number of votes on the social alternatives, iii) the alternative receiving more votes wins the election, and only votes casted on the winning alternative are withdrawn from the votes accounts of each player, and v) finally the withdrawn votes are pro-rata redistributed among all players (this equal redistribution is what makes this a “one man, one vote system”).
SV-PAYW can be interpreted as an auction of social alternatives using as currency some tokens (named “votes”) that are equally redistributed to the electors after its use. The problem of "wasted votes" does not arise in auctions because the bidder must not to pay when the auctioned good is not obtained. SV-PAYW imports this feature of auctions to the field of social choice.
Simultaneously, the “Karma system” was proposed in the field of resource sharing. In the Karma system a resource sharing community issues a specific token (named “karma”) and in each period the resource is allocated by means of an auction carried out in karmas. The properties of Karma systems have been studied in a series of recent papers (Elokda, Bolognani, Censi et al.,2024; Elokda, Nax, Bolognani et al., 2024; Elokda, Cenedese, Zhang et al.,2025) and the results show that the system substantially increases welfare over other allocation systems when valuations of the resource in each period are private. If “winning” a specific election is considered as the scarce resource and the circulating votes are interpreted as “karmas”, SV-PAYW is a particular case of Karma system (that can be named “Karma voting”[2]) and the results about Karma systems are likely to be adaptable to voting theory.
Finally, the latest development in this literature (Ghosh and Pivato, 2025) is Quadratically Normalized Utilitarian Voting (QNU), a multiple election multi alternative mechanism that “maximizes a weighted utilitarian social welfare function”, while at the cost of randomization among alternatives (the mechanism does nor choose simply the “best” alternative in each election).
All of this suggests that the voting problem is already solved for the case of multiple independent elections. The rest of this article [“The Ideal Political Workflow”] is about the remaining problems in distributed governance.
References
Arrow, K.J.(1951), Social Choice and Individual Values. New York: Wiley,
Casella, A. (2005). Storable Votes, Games and Economic Behavior. https://doi.org/10.1016/j.geb.2004.09.009
Casella, A., Macè, A. (2021). Does Vote Trading Improve Welfare?, Annual Review of Economics.https://doi.org/10.1146/annurev-economics-081720-114422
Castellanos, D. (2005). Arrow's impossibility theorem is not so impossible and Condorcet's paradox is not so paradoxical: the adequate definition of a social choice problem. https://ideas.repec.org/p/col/000089/002025.html
Dhillon, A., Bouveret, S.,Lemaître, M. (1999). Relative Utilitarianism. Econometrica. https://doi.org/10.1111/1468-0262.00033
Elokda, E., Bolognani, S., Censi, A., Dörfler, F., & Frazzoli, E. (2024). A self-contained karma economy for the dynamic allocation of common resources. Dynamic Games and Applications, 14(3), 578-610.
Elokda, E., Nax, H., Bolognani, S., & Dörfler, F. (2024). Dynamic Resource Allocation with Karma: An Experimental Study. arXiv preprint arXiv:2404.02687.
https://doi.org/10.48550/arXiv.2404.02687
Elokda, E., Cenedese, C., Zhang, K., Censi, A., Lygeros, J., Frazzoli, E., & Dörfler, F. (2025). CARMA: Fair and efficient bottleneck congestion management via nontradable karma credits. Transportation Science, 59(2), 340-359.
Ghosh, R. and Pivato, M., (2025) Quadratically Normalized Utilitarian Voting http://dx.doi.org/10.2139/ssrn.5173219
Jackson M.O.,Sonnenschein H.F. (2007). Overcoming incentive constraints by linking decisions, Econometrica. https://doi.org/10.1111/j.1468-0262.2007.00737.x
Myatt, D.P. (2007). On the Theory of Strategic Voting, The Review of Economic Studies.https://doi.org/10.1111/j.1467-937X.2007.00421.x
[1] SV was designed to overcome the disenfranchisement of minorities in simple majority voting, but SV works as partial cardinalization, that is the natural framework for multi-alternative voting (as Arrow implicitly proved).
[2] Karma is a far more fortunate name than SV-PAYW, and it underscores the relation between the voting and resource sharing literatures, so I heartly embrace this change of name.
Discuss
