Skip to main content
Erschienen in: Social Choice and Welfare 3/2018

29.03.2018 | Original Paper

A concept of sincerity for combinatorial voting

verfasst von: Francesco De Sinopoli, Claudia Meroni

Erschienen in: Social Choice and Welfare | Ausgabe 3/2018

Einloggen

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

A basic problem in voting theory is that all the strategy profiles in which nobody is pivotal are Nash equilibria. We study elections where voters decide simultaneously on several binary issues. We extend the concept of conditional sincerity introduced by Alesina and Rosenthal (Econometrica 64(6):1311–1341, 1996) and propose an intuitive and simple criterion to refine equilibria in which players are not pivotal. This is shown to have a foundation in a refinement of perfection that takes into account the material voting procedure. We prove that in large elections the proposed solution is characterized through a weaker definition of Condorcet winner and always survives sophisticated voting.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Anhänge
Nur mit Berechtigung zugänglich
Fußnoten
1
An example is given by the November 2016 California ballot, in which citizens have been asked to vote on seventeen issues, including marijuana legalization, gun control, drug prices, and condoms in porn.
 
2
In the example above, for instance, a voter may want marijuana legalization and gun control to pass together, but prefer neither issue to be implemented rather than each issue passing alone for fear of armed stoned people.
 
3
Such a concept has been employed also in Ingberman and Rosenthal (1997).
 
4
Oddness of the number of players is assumed just to guarantee that a pure strategy profile induces a pure outcome. In this case preference orders over outcomes are sufficient to study the main concepts that we introduce. We could replace this assumption with any deterministic tie-breaking rule.
 
5
A player’s best response depends only on the aggregate behavior of the opponents, i.e., on the sum of the positive votes that each issue gets from the others. Henceforth, we will often use a vector to summarize this information, and we will refer to it as “pivotal event” if the player is decisive for some issue.
 
6
Lacy and Niou (2000) and Ahn and Oliveros (2012) observe that if preferences are separable in every issue then voting accordingly to the most preferred outcome is a dominant strategy for a player. This is an immediate corollary of the characterization in Proposition 1.
 
7
Separable preferences do not imply the existence of a Condorcet winner. Take for example three players with the following separable preferences:
$$\begin{aligned} A&\succ _1\varnothing \succ _1 AB \succ _1 B,\\ B&\succ _2\varnothing \succ _2 AB \succ _2 A,\\ AB&\succ _3 A\succ _3 B\succ _3\varnothing . \end{aligned}$$
The dominant strategy profile \(s=((1,0),(0,1),(1,1))\) induces the local Condorcet winner AB, which however is not a Condorcet winner as the majority of the players prefer outcome \(\varnothing \) to it.
 
8
The issue sincere strategy profile with respect to outcome B is \(s=((0,0),(0,0),(1,1))\), so B is not an issue sincere outcome.
 
9
If every player plays according to \(s'\), strategies (0, 0) and (0, 1) are equivalent for the first two players and they are strictly better than strategies (1, 0) and (1, 1), which would induce outcome AB. Hence, close-by \(s'\), the first two strategies are the only best replies for them. Clearly, these strategies induce a different outcome only when the first two players are pivotal for issue B, that is, B takes one vote from the opponents. In this case, if issue A takes two votes then they strictly prefer strategy (0, 1), otherwise they strictly prefer strategy (0, 0). For \(s'\) to be perfect, then, the probability of the pivotal event (2, 1) has to be sufficiently greater than the probability of the events (0, 1) and (1, 1).
 
10
We use the term b-strategy as a reminiscence of behavioral strategy. In fact, b-strategies would precisely be behavioral strategies if we described a combinatorial voting game as an extensive form one.
 
11
In particular, to the b-strategy (xy) with \(x,y\in (0,1)\) corresponds the completely-mixed strategy \(xy(1,1)+x(1-y)(1,0)+y(1-x)(0,1)+(1-x)(1-y)(0,0)\).
 
12
This result is in general not new, as Wichardt (2008) shows that an extensive-form game without perfect recall may not have any Nash equilibrium in behavioral strategies. However, his example cannot be framed in the context of combinatorial voting.
 
13
The only strategies that survive iterated dominance are strategy (1, 1) for players 1 and 2 and strategies (1, 1) and (1, 0) for player 3.
 
14
In particular, outcome \(\varnothing \) is a local Condorcet winner of the game, while the outcome that survives the process, AB, is the Condorcet winner. See the example in Table 5 in Lacy and Niou (2000) for a game in which iterated dominance eliminates the Condorcet winner and selects a local Condorcet winner.
 
15
The same relation would hold if uncertainty were introduced in the vote counting. As a matter of fact, under our assumption of strict preferences an equivalent definition of b-perfection could be stated based on a vanishing probability of misrecording votes, in a way similar to Laslier and Van der Straeten (2016).
 
16
Note that to the b-strategy (xx) corresponds the mixed strategy \(x^2(1,1)+x(1-x)(1,0)+x(1-x)(0,1)+(1-x)^2(0,0)\), which gives positive weight also to the pure strategies that are not best replies.
 
17
If player 3 plays (0, 0), player 2 weakly prefers strategy (0, 0) to both strategies (1, 0) and (0, 1). The condition on player 1’s strategy assures that player 2 prefers strategy (0, 0) also to strategy (1, 1) (i.e., \(9\ge 3x+9z+10(1-x-y-z)\)).
 
18
Among the equivalent definitions of perfect equilibrium proposed in the literature, we will use the extension of Definition 6 to mixed strategies (obtained just substituting in that definition s with \(\sigma \)).
 
19
In particular, the utility of the first three strategies is \(\frac{112\varepsilon -992\varepsilon ^2+2352\varepsilon ^3-1664\varepsilon ^4}{3(1 - 8 \varepsilon + 11\varepsilon ^2)}\), while the utility of strategy (1, 1) is \(\frac{22\varepsilon -92\varepsilon ^2-78\varepsilon ^3+316\varepsilon ^4}{3(1 - 8 \varepsilon + 11\varepsilon ^2)}\).
 
Literatur
Zurück zum Zitat Ahn DS, Oliveros S (2012) Combinatorial voting. Econometrica 80(1):89–141CrossRef Ahn DS, Oliveros S (2012) Combinatorial voting. Econometrica 80(1):89–141CrossRef
Zurück zum Zitat Alesina A, Rosenthal H (1996) A theory of divided government. Econometrica 64(6):1311–1341CrossRef Alesina A, Rosenthal H (1996) A theory of divided government. Econometrica 64(6):1311–1341CrossRef
Zurück zum Zitat Brams SJ, Kilgour DM, Zwicker WS (1997) Voting on referenda: the separability problem and possible solutions. Electoral Stud 16(3):359–377CrossRef Brams SJ, Kilgour DM, Zwicker WS (1997) Voting on referenda: the separability problem and possible solutions. Electoral Stud 16(3):359–377CrossRef
Zurück zum Zitat Brams SJ, Kilgour DM, Zwicker WS (1998) The paradox of multiple elections. Soc Choice Welf 15(2):211–236CrossRef Brams SJ, Kilgour DM, Zwicker WS (1998) The paradox of multiple elections. Soc Choice Welf 15(2):211–236CrossRef
Zurück zum Zitat De Sinopoli F, Ferraris L, Iannantuoni G (2013) Electing a parliament. Soc Choice Welf 40(3):715–737CrossRef De Sinopoli F, Ferraris L, Iannantuoni G (2013) Electing a parliament. Soc Choice Welf 40(3):715–737CrossRef
Zurück zum Zitat Debreu G, Scarf H (1963) A limit theorem on the core of an economy. Int Econ Rev 4(3):235–246CrossRef Debreu G, Scarf H (1963) A limit theorem on the core of an economy. Int Econ Rev 4(3):235–246CrossRef
Zurück zum Zitat Edgeworth FY (1881) Mathematical psychics. Kegan Paul, London Edgeworth FY (1881) Mathematical psychics. Kegan Paul, London
Zurück zum Zitat Farquharson R (1969) Theory of voting. Yale University Press, New Haven Farquharson R (1969) Theory of voting. Yale University Press, New Haven
Zurück zum Zitat Ingberman DE, Rosenthal H (1997) Median voter theorems for divisible governments. Mimeo, New York Ingberman DE, Rosenthal H (1997) Median voter theorems for divisible governments. Mimeo, New York
Zurück zum Zitat Kohlberg E, Mertens JF (1986) On the strategic stability of equilibria. Econometrica 54(5):1003–1037CrossRef Kohlberg E, Mertens JF (1986) On the strategic stability of equilibria. Econometrica 54(5):1003–1037CrossRef
Zurück zum Zitat Lacy D, Niou EM (2000) A problem with referendums. J Theor Polit 12(1):5–31CrossRef Lacy D, Niou EM (2000) A problem with referendums. J Theor Polit 12(1):5–31CrossRef
Zurück zum Zitat Laslier JF, Van der Straeten K (2016) Strategic voting in multi-winner elections with approval balloting: a theory for large electorates. Soc Choice Welf 47(3):559–587CrossRef Laslier JF, Van der Straeten K (2016) Strategic voting in multi-winner elections with approval balloting: a theory for large electorates. Soc Choice Welf 47(3):559–587CrossRef
Zurück zum Zitat Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4(1):24–55CrossRef Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4(1):24–55CrossRef
Zurück zum Zitat Wichardt PC (2008) Existence of nash equilibria in finite extensive form games with imperfect recall: a counterexample. Game Econ Behav 63(1):366–369CrossRef Wichardt PC (2008) Existence of nash equilibria in finite extensive form games with imperfect recall: a counterexample. Game Econ Behav 63(1):366–369CrossRef
Metadaten
Titel
A concept of sincerity for combinatorial voting
verfasst von
Francesco De Sinopoli
Claudia Meroni
Publikationsdatum
29.03.2018
Verlag
Springer Berlin Heidelberg
Erschienen in
Social Choice and Welfare / Ausgabe 3/2018
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-018-1125-5

Weitere Artikel der Ausgabe 3/2018

Social Choice and Welfare 3/2018 Zur Ausgabe