A voting protocol is said to be strongly participatory if for any player i and any strategy profile either the outcome is i‘s preferred one or has a strategy which would ensure her a better outcome, and VNMstrategy proof if at any preference profile the set of sincere strategies of each player is a VNM-stable set. It is shown that the proportional lottery (PL) modular voting protocol is both strongly participatory and VNMstrategy proof.

Stefano Vannucci

English

Research Papers in Economics

    
Università degli Studi di Siena 
DIPARTIMENTO DI ECONOMIA POLITICA 
  
STEFANO VANNUCCI 
 
  
  
  
 
  
  
The Proportional Lottery Protocol is Strongly 
Participatory and VNM-Strategy-Proof 
 
 
  
  
 
 
  
 
 
 
n. 490 -  Novembre  2006 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Abstract - A voting protocol is said to be strongly participatory if for any player  i and any strategy profile 
either the outcome is i‘s preferred one or i has a strategy which would ensure her a better outcome, and VNM-
strategy proof  if at any preference profile the set of sincere strategies of each player is a VNM-stable set. It is 
shown that the proportional lottery (PL) modular voting protocol is both strongly participatory and VNM-
strategy proof. 
 
Jel Classification: D70,D71 
 
 
 
 
 
 
Stefano Vannucci, Dipartimento di Economia Politica, Università degli Studi di Siena 1I n t r o d u c t i o n
It is widely agreed that, ideally, a nice voting protocol should provide prospec-
tive voters with incentives to participate and to cast votes consistently with
their ‘true’ preferences. But then, does there exist a voting protocol invari-
ably providing each voter with a positive incentive to participate and cast a
forthright vote, independently of the size of the relevant outcome set and of the
population of voters?
In the extant literature the foregoing ‘forthrightness’ and ‘participation’ is-
sues have been typically addressed separately, focussing in turn on ‘participa-
tory’ and ‘strategy-proof’ protocols. Thus, a voting protocol is usually said to
be participatory if casting a sincere vote can never induce a worse outcome than
abstention, and strategy-proof if a sincere vote can never induce a worse out-
come than an insincere, manipulative one. Notice that, if voters are allowed to
express indiﬀerence between distinct outcomes, then under that formulation the
‘participation’ requirement may also be regarded as a consequence of ‘strategy-
proofness’ by taking abstention to be equivalent to a special case of manipulative
voting behaviour. Hence, the existence problem stated above might arguably
reduce eventually to addressing the ‘strategy-proofness’ issue. Now, it is well-
known that, by the Gibbard-Satterthwaite theorem, dictatorial mechanisms are
the only available nonmanipulable or ‘strategy-proof’ protocols when every con-
ceivable proﬁle of preference rankings is allowed and the range of possible voting
outcomes includes three items at least. Observe that dictatorial protocols are
clearly participatory according to the deﬁnition introduced above, even under
ballot formats that preclude the expression of nontrivial indiﬀerence. It fol-
lows that, if the existence issue concerning forthright and participatory protocols
is thus formulated via the ‘participation’ and ‘strategy-proofness’ requirements
then the answer is after all already known, thanks to the Gibbard-Satterthwaite
theorem. Yes, forthright participatory voting protocols do exist, but they are
unfortunately neither nice nor interesting.
However, I would like to suggest that the foregoing formulation, far from be-
ing compelling, is indeed rather inadequate. Concerning incentives to vote, the
‘participation’ requirement as deﬁned above is clearly exceedingly weak: it only
requires that voting is never worse than abstaining, allowing for the possibility
that those two options be invariably indiﬀerent. That is why dictatorships turn
out to be ‘participatory’ in that weak sense. But, as a matter of fact, dictatorial
mechanisms are hardly endowed with positive incentive to participate for all
voters since the dictator’s ballot is by deﬁnition the only one to count. Clearly,
a stronger property is needed if the notion of positive incentives to participate is
to be properly expressed in our model. Here, I opt for the strongest possible ver-
sion of such a notion one may possibly conceive of, namely the requirement that
there be for each player and under any circumstance the possibility to improve
any improvable outcome by participating. This strong requirement I call strong
participation, and it is immediately checked that dictatorships are as expected
not strongly participatory. Hence, any poss i b l ep o s i t i v ea n s w e rt oo u rs e a r c hf o r
voting protocols enjoying both (strong) participation and forthrightness clearly
 
 
 
 
   1also requires some relaxation of the standard ‘strategy-proofness’ requirement.
In that vein, I introduce a mild weakening of the usual strategy-proofness re-
quirement based upon the notion of Von Neumann-Morgenstern stability and
therefore denoted as VNM-strategy-proofness. VNM-strategy-proofness refers
to voting protocols that require submission of information on preferences plus
possibly some extra-input: as a result at each preference proﬁle an entire set of
‘sincere’ strategies may be identiﬁed for each voter, and that set is typically not
a singleton. VNM-strategy-proofness simply requires the set of sincere strate-
gies of each player to be a stable standard of behaviour,n a m e l yaVNM-stable set
with respect to the dominance relation on the player’s strategy set. It should be
emphasized that strategy-proofness itself, asking the (unique) sincere strategy
of any player to be a dominant strategy i.e. the essentially unique maximum
of the dominance relation on the player’s strategy set, may be regarded as a
specialization of VNM-strategy-proofness.1
Thus, the present paper combines a signiﬁcant strenghtening of the partic-
ipatory requirements with a relatively mild weakening of ‘strategy-proofness’.
Its main result consists in showing that the proportional lottery protocol (a
modular-arithmetical deterministic version of random dictatorship) is indeed
both strongly participatory and ‘VNM-strategy-proof’. Thus, under the new
suggested formulation (strong) participation, forthrightness and a reasonably
uniform allocation of decision power are after all consistent.
The paper is organized as follows. Section 2 introduces the basic formal
framework, and deﬁnes strong participation and VNM-strategy-proofness. Sec-
tion 3 is devoted to the main result of the paper on strong participation and
VNM-strategy-proofness of the proportional lottery protocol. Section 4 includes
some comments on the sparse related literature with a few concluding remarks.
2 Strong participation and VNM-strategy-proofness
Let N denote the countable population of all possible voters, Pf(N) the set of
all ﬁnite subsets of N, X the ﬁnite set of all possible outcomes and LX the set
of all linear orders - i.e. total transitive and antisymmetric binary relations - on
X.F o re a c hv o t e ri ∈ N, Ri = LX denotes the set of her possible preferences
on X: as usual, the actual preference of a voter is regarded as a nonveriﬁable
p r i v a t ec h a r a c t e r i s t i co ft h el a t t e r .
A signalling function of type Σ for LX is an injective function σ : Σ →
P(LX), while a set U is said to be σ-direct if there exists another set V such
that U = V × Σ,a n dsuperdirect on LX if it is σ-direct with respect to some
signalling function σ for LX. For any signalling function σ : Σ → P(LX) and
preference <i∈ LX, a σ-signal x ∈ Σ is said to be sincere at <i if <i∈ σ(x).
1One should consider the possible existence of equivalent strategies. Therefore, strictly
speaking, strategy-proofness is a special case of VNM-strategy-proofness as deﬁned in this
paper on the class of protocols consisting of reduced game forms (where equivalent strategies
are identiﬁed, and the sincere strategy is identiﬁed with its equivalence class). Alternatively,
one might slightly modify the deﬁnition of VNM-strategy proofness, requiring that the set
consisting of sincere strategies and their equivalent strategies be a VNM-stable set.
   2A (uniform, strategic) voting protocol for Pf(N) on X is a family
G =
©
GN =( N,X,(Si)i∈N,h N)
ª
N∈Pf(N) of strategic game forms (where
N,X denote the player and outcome sets, respectively, (Si)i∈N is the proﬁle of
strategy sets, and hN ∈ XΠi∈NSi is the (surjective) outcome function) such that
for all N ∈ Pf(N) and i ∈ N there exists s∗
i ∈ Si satisfying hN((sj)j∈N\{i},s ∗
i)=
hN\{i}((sj)j∈N\{i}) for all (sj)j∈N\{i} ∈
Y
j∈N\{i}
Sj .2
Av o t i n gp r o t o c o lG =
©
GN =( N,X,(Si)i∈N,h N)
ª
N∈Pf(N) is said to be
order-enriched if both N and X are endowed with linear orders 6Nand 6X,
respectively, superdirect on L :=
[
N∈Pf(N)
Y
i∈N
Ri if for any N ∈ Pf(N), and
any i ∈ N the strategy set Si is superdirect on Ri, symmetric if Si = Sj for
any N ∈ Pf(N), and any i,j ∈ N. Moreover, a voting protocol G which is
superdirect on L (w.r.t. signalling function σ) is Pareto eﬃcient (w.r.t. σ) if
for any y ∈ X,N ∈ Pf(N), preference proﬁle (<i)i∈N ∈ LN
X, proﬁle (xi)i∈N of
sincere σ-signals at (<i)i∈N, and preference proﬁle (<0
i)i∈N such that <0
i∈ σi(x)
for any i ∈ N :
if there exists z ∈ X\{y} such that z <0
i y for each i ∈ N, then y/ ∈
hN[Πi∈N(V ×{ xi})].
We shall be henceforth interested in order-enriched, superdirect, and sym-
metric voting protocols.
Let us now turn to the deﬁnitions of the two basic requirements which con-
stitute the focus of the present paper.
Av o t i n gp r o t o c o lG for for Pf(N) is (strongly) participatory w.r.t. L if
for all N ∈ Pf(N) with |N| > 2, for all i ∈ N, (<i)i∈N ∈
Y
i∈N
Ri and s0 ∈
Y
j∈N\{i}
Sj there exists si ∈ Si such that hN((si,s 0)) <i hN\{i}(s0) (respectively,
such that either hN((si,s 0)) Âi hN\{i}(s0) or
©
hN((si,s 0))
ª
∪
©
hN\{i}(s0)
ª
⊆
top(<i) where Âidenotes the asymmetric component of <iand top(<i) the set
of all <i-maximal outcomes).
Let G =
©
GN =( N,X,(Si)i∈N,h N)
ª
N∈Pf(N) b eav o t i n gp r o t o c o lf o rPf(N)
that is superdirect w.r.t. a signalling function σ : Σ → P(LX) :a s t r a t e g y
si =( t,x) ∈ Si = T × Σ of player i ∈ N ∈ Pf(N) is sincere at <i∈ Ri if x
is sincere at <ii.e. if <i∈ σi(x):we denote S∗
i (<i) ⊆ Si denote the set of all
sincere strategies at <iof player i in GN.F o re a c hi ∈ N ∈ Pf(N), and <i∈ Ri an
individual dominance relation 4<i(GN) ⊆ Si ×Si is deﬁned as follows: for any
si,t i ∈ Si, si4<i(GN)ti iﬀ hN(si,s −i) <i hN(ti,s −i) for each s−i ∈
Y
j∈N\{i}
Sj
and there exists s−i ∈
Y
j∈N\{i}
Sj such that hN(si,s −i) Âi hN(ti,s −i).Then,
2Thus, it is required that for any set of participants all players have the option of an entirely
‘neutral’ i.e. inconsequential abstention.
   3G is said to be VNM-strategy-proof on
Y
i∈N
Ri if for any N ∈ Pf(N),i∈ N,
<i∈ Ri,S ∗
i (<i) is a VNM-stable set of (Si,4<i(GN)), i.e. S∗
i (<i) satisﬁes the
following properties:
(i) Internal stability : for any si,t i ∈ S∗
i (<i),n o ts i4<i(GN)ti;
(ii) External stability: for any ti ∈ Si\S∗
i (<i) there exists si ∈ S∗
i (<i) such
that si4<i(GN)ti.
3 The proportional lottery protocol: main re-
sult
We are interested in ﬁnding out whether strong participation and VNM-strategy-
proofness are consistent properties of a voting protocol. In view of some well-
known results on strategy-proof protocols on restricted domains, the most nat-
ural candidates for a positive solution are those protocols which arise from suit-
able ‘combinations’ of dictatorial mechanisms (see Danilov and Sotskov (2002)).
In that vein, we focus on the proportional lottery protocol, a special determin-
istic version of ‘random dictatorship’ which relies on modular arithmetic.
The proportional lottery (PL) protocol
GPL =
©
GN
PL =( N,X,(Si)i∈N,h N
PL)
ª
N∈Pf(N) for Pf(N) on the domain
L =
[
N∈Pf(N)
Y
i∈N
LX
i of all proﬁles of linear orders on X is deﬁned as follows:
for any N = {1,..,n} ∈ Pf(N),i∈ N,a n d(<i)i∈N ∈
Y
i∈N
LX
i , Si = X × Z+,
and for any s =( ( x1,z 1),..,(xn,z n)) ∈
Y
i∈N
Si, hN
PL(s)=xi∗ where i∗ =
P
i∈N zi(modn).
Remark 1 Notice that PL is a Pareto-eﬃcient voting protocol. To see this,
observe that PL as deﬁned above is by deﬁnition superdirect on L w.r.t. the
signalling function σ mapping each outcome x ∈ X into the set of linear orders
<∈ LX such that top(<)={x}. Therefore, for any y ∈ X, N ∈ Pf(N),
preference proﬁle (<i)i∈N ∈
Y
i∈N
Ri, proﬁle (xi)i∈N of sincere σ-signals at (<i
)i∈N, if preference proﬁle (<0
i)i∈N is such that <0
i∈ σ(xi) for any i ∈ N then
by construction top(<0
i)=top(<i) for all i ∈ N . Therefore, if there exists
z ∈ X\{y} such that z <0
i y for each i ∈ N, then y/ ∈ top(<0
i)={xi} i.e. y 6= xi
for any i ∈ N whence by deﬁnition y/ ∈ hN
PL[Πi∈N(V ×{ xi})] as required.
Remark 2 Notice that PL is clearly ‘coalitionally anonymous’ namely under
PL coalitions of the same size can enforce the same outcome-subsets. To check
this, notice that each player can subvert any outcome through a judicious choice
of her strategy: therefore, while the grand coalition N can obviously enforce any
single outcome, any other coalition is unable to enforce any proper subsets of
   4outcome. That circumstance is by no means coincidental. It can be shown that
any strongly participatory protocol must induce such an allocation of decision
power among coalitions. The details however will not be spelled here.
Proposition 3 The proportional lottery protocol
GPL =
©
GN
PL =( N,X,(Si)i∈N,h N
PL)
ª
N∈Pf(N) for Pf(N) is strongly partic-
ipatory and VNM-strategy-proof on L.
Proof. Let N ∈ Pf(N), (<i)i∈N ∈
Y
i∈N
Ri ,
s0 =( ( x1,z 1),..,(xj−1,z j−1),(xj+1,z j+1),..,(xn,z n)) ∈
Y
i∈N\{j}
Si ,j∈ N,
and x = top(<j) / ∈ h
N\{j}
PL (σ).Then, consider
X
i∈N\{j}
zi(modn): of course, there
exists z ∈ Z+ such that (z +
X
i∈N\{j}
zi)(modn)=j. Then, posit sj =( x,z).
Clearly, hN
PL((sj,s 0)) = x. Thus, GPL is indeed strongly participatory.
Next, take a preference proﬁle (<i)i∈N ∈
Y
i∈N
Ri with N ∈ Pf(N),a n d
consider the set
Y
i∈N
S∗
i (<i) of sincere strategy proﬁles of GN
PL at (<i)i∈N ∈
D.L e t s =( ( x1,z 1),..,(xn,z n)),t =( ( x1,z0
1),..,(xn,z0
n)) ∈
Y
i∈S
S∗
i (<i),a n d
i ∈ N.I f zi(modn)=z0
i(modn) then hN(s)=hN(t)=x ∈
Y
i∈N
S∗
i (<i
) for any sN\{i} ∈
Y
j∈N\{i}
Sj ,t h e r e f o r e- b yd e ﬁnition- not si4<i(GN
PL)ti.
If zi(modn) 6= z0
i(modn) then -by deﬁnition of the PL protocol- there exist
s0
N\{i},s 00
N\{i} ∈
Y
j∈N\{i}
Sj such that hN(si,s 0
N\{i})=xi 6= hN(ti,s 0
N\{i}) and
hN(ti,s 00
N\{i})=xi 6= hN(si,s 00
N\{i}),w h e n c enot si4<i(GN
PL)ti again. More-
over, let t ∈
Y
i∈N
Si as t r a t e g yp r o ﬁle that is not sincere at (<i)i∈N.T h e n
by deﬁnition there exists i ∈ N such that ti =( yi,z i) / ∈ S∗
i (<i),n a m e l y
yi 6= xi = top(<i). Thus, consider strategy si =( xi,z i): clearly, for any
sN\{i} =( ( x1,z 1),..,(xi−1,z i−1),(xi+1,z i+1),..,(xn,z n)) ∈
Y
j∈N\{i}
Sj either
hN(si,s N\{i})=xi and hN(ti,s N\{i})=yi hence hN(si,s N\{i}) Âi hN(ti,s N\{i})
or hN(si,s N\{i})=hN(ti,s N\{i}) 6= xi. It follows that si4<i(GN
PL)ti as re-
quired.
Remark 4 It should also be emphasized here that ‘Strong Participation’ and
‘VNM-Strategy-Proofness’ are indeed two mutually independent properties for
voting protocols. To check this, consider ﬁrst any standard direct dictatorial
mechanism GDIC(j)with Si = X as strategy set for each player i. Clearly,
   5GDIC(j) is reduced and strategy-proof hence in particular VNM-strategy-proof,
but it is obviously not strongly participatory. Next, take the ‘permuted’ vari-
ant of PL induced by the outcome functions hN
PPL(.) as deﬁned by the follow-
ing rule: for any strategy proﬁle s, hN
PPL(s)=xπ(xi∗)+1(modn) where i∗ = P
i∈N zi(modn) and π : X → {1,..,|X|} is a function that respects 6Xi.e.
π(x) 6 π(y) if and only if x 6X y. It can be immediately checked that such a
‘permuted’ PL is strongly participatory but not VNM-strategy-proof.
4 Related literature and concluding remarks
Under the heading of ‘strategy-proofness’, problems pertaining to incentives for
forthrightness of voters’ behaviour have spawned a huge literature. A discussion
of some selected topics most germane to our present concerns as well as a pre-
sentation of the proportional lottery mechanism (under the simpler denotation
‘lottery mechanism’) is provided e.g. in Danilov and Sotskov (2002).
The paradoxical fact that participation may be occasionally self-damaging
- the ‘No Show Paradox’- under some standard voting protocols (e.g. plurality
voting with runoﬀ)w a sﬁrst noticed by Brams and Fishburn (1983), Fishburn
and Brams (1984)). Subsequently Moulin (1988) addressed the issue in a fairly
general setting: he introduced a Participation axiom for voting mechanisms
requiring that by participating and submitting a sincere preference ranking a
voter can never induce a worse outcome than by abstaining. Then, he pro-
ceeded to show that when there are at least four outcomes no voting mechanism
selecting Condorcet winners when the latter exist do satisfy Participation (a
‘No Show Paradox’ again), while (essentially) simple scoring mechanisms (with
no runoﬀ) always do. Pérez (2001) extends those results to Condorcet voting
correspondences, while Saari (1994) relates No Show Paradox-type results to
manipulability and (non)monotonicity problems. Holzman(1988/1989) studies
Participation under mechanisms with special majority quotas and proves that if
the cardinality of the outcome set is four or larger, then the only winning quota
consistent with Participation is quasi-unanimity i.e. (n − 1)/n.
As mentioned in the Introduction above, however, Participation amounts
to a rather weak participation incentive. Indeed, dictatorial mechanisms are
perfectly consistent with the Participation requirement. That is why this paper
focusses on the more demanding Strong Participation property, requiring that
under any circumstance each player be able to positively improve the outcome
by participating and choosing some ‘forthright’ voting strategy. It would cer-
tainly be of some interest a characterization of all voting protocols enjoying
both VNM-Strategy-Proofness and Strong Participation (or some other weaker
property of ’intermediate’ strenght i.e. stronger than Participation) over the
universal domain as well as over other interesting large domains. I leave it as a
topic for further research.
   6References
[1] Brams, S., and P.C.Fishburn: Paradoxes of Preferential Voting, Mathematics
Magazine 56 (1983), 207-214.
[2] Danilov, V.I., and A.I. Sotskov: Social Choice Mechanisms. Springer, Berlin
2002.
[3] Fishburn, P.C., and S. Brams: Manipulability of Voting by Sincere Trunca-
tion of Preferences, Public Choice 44 (1984), 397-410.
[4] Holzman, R.: To Vote or Not to Vote: What is the Quota?, Discrete Applied
Mathematics 22 (1988/1989), 133-141.
[5] Moulin, H.: Condorcet’s Principle Implies the No Show Paradox, Journal of
Economic Theory 45 (1988), 53-64.
[6] Pérez, J.: The Strong No Show Paradoxes are a Common Flaw in Condorcet
Voting Correspondences, Social Choice and Welfare 18 (2001), 601-616.
[7] Saari, D.G.: Geometry of Voting. Springer, Berlin 1994.
   7

The Proportional Lottery Protocol is Strongly Participatory and VNM-Strategy-Proof

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