Markovian equilibria in dynamic spatial legislative
bargaining: Existence with three players ∗
Jan Zápal
CERGE-EI & IAE-CSIC and Barcelona GSE
[email protected]
July 11, 2016
Abstract
The paper proves, by construction, the existence of Markovian equilibria in
a dynamic spatial legislative bargaining model. Three players bargain over onedimensional policies in an infinite horizon. In each period, a sequential protocol of
proposal-making and voting, with random proposer recognitions and a simple majority, produces a policy that becomes the next period’s status-quo. An equilibrium
exists for any profile of proposer recognition probabilities, any profile of players’ ideal
policies, and any discount factor. In equilibrium, policies converge to the median’s
ideal policy, players moderate and propose policies close to the median’s ideal in an
attempt to constraint future proposers, but the tendency to moderate is a strategic
substitute as the opponent of a moderating player does not moderate.
JEL Classification: C73, C78, D74, D78
Keywords: dynamic decision-making; endogenous status-quo; spatial bargaining;
legislative bargaining
∗
Some of the results presented in this paper originally appeared in my Ph.D. dissertation (Zapal, 2012,
chapter 2) and were previously circulated as a working paper entitled ‘Simple equilibria in dynamic bargaining games over policies’. I owe special thanks to my advisors Ronny Razin and Gilat Levy. Further,
I would like to thank Marco Battaglini and two anonymous referees, as well as Avidit Acharya, Vincent Anesi, Enriqueta Aragones, David Baron, Daniel Cardona, John Duggan, Jean Guillaume Forand,
Tasos Kalandrakis, Antoine Loeper, Fabio Michelucci, Francesco Nava, Salvatore Nunnari, Clara Ponsati,
Ronny Razin, Francesco Squintani and seminar and conference participants at IAE-CSIC, University of
Waterloo, University of the Balearic Islands, the 2013 Barcelona GSE Summer Forum Workshop on
Dynamic Decisions and the 2015 EEA-ESEM Annual Meeting in Toulouse for helpful comments and
discussions. Some of the presented ideas took shape while visiting W. Allen Wallis Institute of Political
Economy at the University of Rochester and their hospitality is appreciated. Financial support from
the Post-Doc Research Fund of Charles University in Prague is gratefully acknowledged. All remaining
errors are my own.
1
Introduction
Legislative policy-making often concerns policies that are continuing in nature, evolve and change over time. Any change to a continuing policy is
negotiated under the shadow of the extant legislation, leads to a revision
of the extant legislation, and results in a shift of the status-quo. Dynamic
legislative bargaining models reflect these features. The models embed a sequential protocol of proposal-making and voting from static non-cooperative
legislative bargaining models as a stage game in an infinite horizon dynamic
strategic situation. Two consecutive rounds of negotiations are linked; the
decision from the former becomes the status-quo for the latter.
Starting with Baron (1996), the dynamic legislative bargaining literature
has been steadily growing. For a spatial setting, bargaining over policies,
Baron (1996) develops partial equilibrium characterization and provides intuition for the strategic forces at play. For a distributive setting, bargaining
over the allocation of benefits, Kalandrakis (2004) is the first to characterize a Markov equilibrium. In the absence of applicable existence theorems
for Markovian equilibria, his characterization constitutes an existence proof.
Due to the lack of similar characterization for the spatial model, and in the
continuing absence of applicable existence theorems, the existence and properties of Markov equilibria in the dynamic spatial model remain unknown.
In this paper we prove, using constructive arguments, the existence of
Stationary Markov Perfect equilibrium (SMPE) in a dynamic spatial legislative bargaining model. Three legislators repeatedly set policy in a onedimensional policy space. Their preferences are quadratic around policy
ideals, around bliss points. In each of an infinite number of periods, one legislator is randomly recognized to propose a policy. The legislature decides
either to adopt the policy proposal or to maintain the status-quo policy under a simple majority vote. The winning policy determines the legislators’
utility for the period and becomes the status-quo for the next period.
The equilibrium construction relies on two classes of (pure) stationary
Markov proposal strategies. A proposal strategy maps the status-quo into
a policy proposal. A proposal strategy in the first class depends on a single
parameter, the policy a player proposes when the status-quo gives her ample
bargaining power. In the static setting this parameter would be the player’s
bliss point. In the dynamic setting we call this parameter the strategic bliss
1
point, the policy maximizing, in equilibrium, the dynamic utility of a player.
Because a proposal strategy of a player is fully determined by her strategic
bliss point, a profile of strategic bliss points fully determines a profile of proposal strategies, generates the dynamic utility of each player and induces a
different profile of dynamic utility maximizers, a different profile of strategic
bliss points. The SMPE construction using the proposal strategies in the
first class can be seen as finding a fixed-point of this operation.1
An SMPE cannot be always constructed using the proposal strategies in
the first class because those can be optimal only if the dynamic utilities are
single-peaked. The proposal strategies in the second class can be optimal
even for non-single-peaked dynamic utilities and allow us to prove the general SMPE existence result. That it is possible to construct an SMPE using
the proposal strategies from the two classes is the main insight of the paper.
Moderation and its strategic substitute nature are at the core of our
equilibrium construction. A player moderates when she proposes her strategic bliss point, a more moderate policy—closer to the median—than her
(static) bliss point. Moderation is driven by strategic considerations. The
proposing player anticipates her proposal’s impact on the future policies. A
moderate proposal constrains future proposers, most importantly the current proposer’s opponents, to propose moderate policies as well. Moderation
is a strategic substitute; when a player’s opponents do moderate, they are
constraining themselves and the player has no incentive to moderate; when
a player’s opponents do not moderate, the player has an incentive to do so.
The equilibrium extent of moderation is a result of two opposing forces.
The first force is standard and pushes proposed policies towards the players’
ideal policies. The second force is strategic and pushes proposed policies towards the median’s ideal policy. The second force gains prominence and the
equilibrium extent of moderation increases with the patience of the players
and with the probability of recognition of their opponents.
Our work is the most closely related to the dynamic spatial legislative
bargaining literature. Baron (1996) is the first to study policy determination with an endogenous status-quo. We study a three-player version of
1
The proposal strategies in the first class closely resemble those of Romer and Rosenthal (1979). A player proposes her (strategic) bliss point for any status-quo for which it
is accepted, and otherwise proposes the policy closest to it, from among those that are
acceptable.
2
Baron’s (1996) model.2 The partial equilibrium characterization developed
in Baron (1996) uncovers moderation as an equilibrium feature. By providing complete equilibrium construction, we uncover not only moderation,
but also its strategic substitute nature.3 Kalandrakis (2015) considers a
model identical to ours but assumes equidistant players’ bliss points and
equal recognition probabilities.4 Forand (2014) explores an electoral competition model between two parties and a median voter. His key assumption,
incumbent policy commitment, creates a link between periods and makes
his model closely related to the model in Baron (1996), with an additional
restriction that only two players possess agenda setting power and alternate
in the proposer role.5 In a companion paper (Zapal, 2015), we study extension of the model considered here to a multi-player environment. The
equilibrium existence result we prove there is limited to a certain class of
games and utilizes only the first class of proposal strategies discussed above.
The insight that the proposal strategies in the second class can be used to
complete general existence result is specific to this paper.
General characterization and existence results for Stationary Markov
Perfect equilibria in dynamic legislative bargaining games are scarce. Kalandrakis is the first to characterize an SMPE in a dynamic distributive legislative bargaining model with three (Kalandrakis, 2004) or more than five
(Kalandrakis, 2010) players. Duggan and Kalandrakis (2012) prove general
SMPE existence result for a large class of dynamic bargaining games, relying on ‘smoothing’ due to shocks in players’ preferences and between-period
status-quo transitions. The shocks significantly complicate equilibrium characterization and are absent in our model. Diermeier and Fong (2011) provide
2
We do not restrict the policy space to R+ , which is of little consequence, and we use
quadratic stage utilities, which is necessary and cannot be dispensed with. See discussion
following Theorem 1. Baron (1996) does not assume quadratic utilities, but assumes that
the dynamic median voter theorem applies. Hence, assuming quadratic utilities does not
decrease the generality of the model as compared with Baron (1996).
3
Baron (1996) includes informal discussion of an example of full equilibrium characterization for a five player game (his Table 1) with symmetric extent of moderation by all
non-median players. For three players, Kalandrakis (2015, Proposition 1) shows that symmetric equilibria of the type characterized by Baron (1996) cannot exist. Our Proposition
2 has the same implication and Proposition A4 in Appendix shows that all non-median
players moderating is incompatible with a large class of equilibria.
4
Footnote 13 explores the relationship between his mixed strategy and our pure strategy equilibrium constructions.
5
A working paper version of Forand (2014) (Forand, 2010) draws the analogy between
his electoral model and the legislative bargaining model of Baron (1996). Footnote 14
explores the relationship between his and our equilibrium constructions.
3
an algorithm that delivers an SMPE in a model with persistent agenda setter, discrete policy space and vanishing impatience (see Anesi and Duggan,
2015a, for more general results). Anesi and Duggan (2015b), utilizing the
‘simple solutions’ developed by Anesi and Seidmann (2015), construct a continuum of SMPE in a general dynamic spatial legislative bargaining model
with patient players. Their construction requires a policy space with enough
dimensions and fails in the one-dimensional model considered here.
We proceed as follows. The next section introduces our model, notation and solution concept. Section 3 provides the main result, explains the
intuition behind its constructive proof and provides further results concerning the equilibria constructed. Section 4 presents concluding remarks. All
proofs are relegated to Appendix.
2
Model
A game G = hx, r, δ, Xi is fully specified by x, r, δ, and X all satisfying the
assumptions we introduce next, and which are maintained throughout. A
stage utility of i ∈ N = {1, 2, 3} from policy p is ui (p) = −(p − xi )2 where xi
is the bliss point of i. x = {x1 , x2 , x3 } denotes the profile of bliss points of
the players and we assume all the bliss points are distinct. It is thus without
loss of generality to assume x2 = 0, x3 < 0 and x1 ≥ |x3 |.6 Hence player 1
is more extreme than player 3 and player 2 is the median.
In each discrete period of infinite horizon, i ∈ N is recognized to propose
policy p ∈ X, where X ⊆ R is a closed convex interval. If X ( R then
we require X to be symmetric about x2 and include x1 . r = {r1 , r2 , r3 }
with ri ≥ 0 ∀i ∈ N is the profile of probabilities of recognition such that
P
i∈N ri = 1. Given the status-quo x ∈ X and policy proposal p ∈ X
by recognized i ∈ N , a majoritarian vote between x and p follows. The
winning alternative determines the utility of the players and becomes the
new status-quo. The utility of player i ∈ N from an infinite path of policies
p = {p0 , p1 , . . .} is
Ui (p) =
∞
X
δ t ui (pt )
(1)
t=0
6
The game is shift invariant and setting x2 = 0 amounts to setting the origin. Any
game with x1 < |x3 | can be accommodated by exchanging the non-median players.
4
where δ ∈ [0, 1) is the common discount factor.7
A pure stationary Markov strategy of each i ∈ N is a pair (p̂i , v̂i ). The
first element, p̂i : X → X, is the proposal strategy specifying a policy
proposal given status-quo x, p̂i (x). The second element, v̂i : X 2 → {yes, no},
is the voting strategy specifying a vote given status-quo x and proposal p,
v̂i (x, p). A profile of pure stationary Markov strategies is σ̂ = ((p̂i , v̂i ))i∈N .
Any σ̂ induces a continuation value function of player i ∈ N , Vi : X → R.
Vi (x|σ̂) denotes the expected utility of i from an infinite future of play
according to σ̂, starting with status-quo x, before the identity of the proposer
in the next period has been determined. Denote the dynamic (expected)
utility of i from accepted x, Ui : X → R, by Ui (x|σ̂) = ui (x) + δVi (x|σ̂).
The social acceptance set for given x ∈ X, A(x|σ̂), is the set
A(x|σ̂) = {p ∈ X|2 ≤ |{i ∈ N |v̂i (x, p) = yes}|}.
(2)
Definition 1 (Stationary Markov Perfect Equilibrium). A stationary Markov
perfect equilibrium (SMPE) is a profile of stationary Markov strategies σ̂ ∗ =
((p̂∗i , v̂i∗ ))i∈N such that, ∀i ∈ N , ∀x ∈ X and ∀p ∈ X,
p̂∗i (x) ∈ arg max ui (s) + δVi (s|σ̂ ∗ )
s∈A(x|σ̂ ∗ )
and v̂i∗ (x, p) = yes if and only if
Ui (p|σ̂ ∗ ) ≥ Ui (x|σ̂ ∗ ).
Definition 1 requires strategies to be best-responses and adds several
restrictions on the proposal and voting strategies. Concerning the latter,
we assume that all players use the stage-undominated voting strategies of
Baron and Kalai (1993) when voting between the proposed p ∈ X and the
status-quo x ∈ X and vote deferentially, that is, vote for p when indifferent
between p and x.8 Concerning the former, we assume that proposals with
zero probability of acceptance are never made.9 This no-delay property
7
Specification of G does not require an initial status-quo, which we leave unspecified.
Stage-undominated voting is a standard assumption in voting literature and rules
out implausible equilibria that can support arbitrary outcomes that are accepted because
no voter is pivotal. Assuming deferential voting avoids any open set complications.
9
Given status-quo x, the proposing player whose utility maximizing proposal is x can
obtain this utility either by proposing x or by making a proposal she knows would be
rejected. We assume she does the former. This assumption does not change the set of
8
5
implies that Vi (x|σ̂) is the unique solution to the recursion10
Vi (x|σ̂) =
X
rj [ui (p̂j (x)) + δVi (p̂j (x)|σ̂)] .
(3)
j∈N
Our equilibrium concept is a pure stationary Markov perfect equilibrium
in stage-undominated deferential voting and no-delay proposal strategies,
which we refer to more succinctly as to an SMPE.
3
SMPE existence
Theorem 1. For any G, an SMPE exists.
The proof of Theorem 1 is in Appendix A1. For any G, it constructs a
profile of strategies σ̂ that constitutes an SMPE. In the remainder of this
section we demonstrate key features of the construction.
The first step of the construction is an observation that for any profile
σ̂ such that v̂i (x, p) = yes if and only if Ui (p|σ̂) ≥ Ui (x|σ̂) ∀i ∈ N and
∀(x, p) ∈ X 2 , a policy proposal p ∈ X is accepted under status-quo x ∈ X if
and only if U2 (p|σ̂) ≥ U2 (x|σ̂). That is, if and only if it is preferred by the
median player. This ‘dynamic median voter theorem’ significantly simplifies
the SMPE construction. It implies that the social acceptance set A(x|σ̂)
under status-quo x ∈ X is {p ∈ X|U2 (p|σ̂) ≥ U2 (x|σ̂)}; it is the set of
policies player 2 prefers to x.
The dynamic median voter theorem critically depends on the quadratic
stage utilities. The median is the player with x2 since player 2 is decisive
in the vote over any (deterministic) policies x ∈ X and p ∈ X. However, in
G voting between x and p means voting over lotteries over future outcomes.
That the decisiveness of the median extends from deterministic to stochastic
alternatives, under quadratic preferences, is a well known result (Banks and
Duggan, 2006; Cho and Duggan, 2003). Equally well known is the fact that
this result does not extend beyond quadratic utilities (see example following
proof of Lemma 2.1 in Banks and Duggan, 2006).
equilibria that are observationally (outcome) equivalent and is standard in the dynamic
bargaining literature.
10
For a profile of proposal strategies (p̂i )i∈N , the right hand side of (3) provides continuation value function calculated assuming that proposals generated by (p̂i )i∈N are always
accepted. When we say that a continuation value function, and dynamic utility, is induced
by a profile of proposal strategies, via (right hand side of) (3), we refer to this procedure.
6
The SMPE construction proceeds by defining a class of proposal strategies with a particular threshold property, which we call c-strategies. For
status-quo x ∈ X, the c-strategy of i ∈ {1, 3} states that i proposes either
her strategic bliss point x̂i , when |x̂i | ≤ |x|, or the policy in [−|x|, |x|] with
minimal distance to x̂i , when |x̂i | > |x|. The c-strategy of player 2 states
that she proposes x̂2 for any status-quo x ∈ X. Formally, the c-strategy pi
of i ∈ N is11
pi (x|x̂i ) =



min {x̂1 , |x|}


if i = 1
x̂2



max {x̂ , −|x|}
3
if i = 2 .
(4)
if i = 3
A strategic bliss point x̂i fully determines the proposal strategy of i. Any
profile of strategic bliss points x̂ = {x̂1 , x̂2 , x̂3 } thus induces a profile of
proposal strategies. A useful property of any x̂ = {x̂1 , x̂2 , x̂3 } with x̂1 ≥
x̂2 = 0 ≥ x̂3 is that it induces V2 via (3) such that the social acceptance
set given status-quo x ∈ X is [−|x|, |x|]. Because, ∀i ∈ N and ∀x ∈ X,
p(x|x̂i ) ∈ [−|x|, |x|] when x̂1 ≥ x̂2 = 0 ≥ x̂3 , the SMPE construction reduces
to finding x̂, if possible, such that the proposal strategies induced by x̂
satisfy Definition 1.
Showing that an SMPE can be constructed using the c-strategies, and under what conditions, is the crux of the SMPE construction. The c-strategies
do not deliver SMPE for any G, but when the c-strategies do not, an alternative class of d-strategies, discussed below, does. That it is possible to
construct SMPE for any G using either the c-strategies or the d-strategies is
the main insight of the paper leading to Theorem 1.
To understand the SMPE construction using the c-strategies, consider
G0
with x = {1, 0, −1}, r = { 31 , 13 , 31 }, δ =
9
10
and X = R. From the proof of
Theorem 1, one of the profiles of strategic bliss points that constitutes an
SMPE is x̂ = {x1 , 0, x3 (1 − 2δr1 )} = {1, 0, − 25 }. Figure 1 shows the profile
of proposal strategies induced by x̂.
We call x̂i a strategic bliss point since it is the policy i proposes for any
status-quo x such that x̂i ∈ [−|x|, |x|], that is, when i is not constrained by
the median’s acceptance. Not being constrained, i can propose the policy
maximizing her dynamic utility Ui , her strategic bliss point. In Figure 1,
11
The c-strategies are in the spirit of the strategies used by Baron (1996). See his third
equilibrium property on page 321 and the discussion leading to his Proposition 5.
7
Figure 1: G 0 : c-strategies for x = {1, 0, −1} and x̂ = {1, 0, − 25 }
pi (x|x̂i )
x1 = x̂1 = 1
p1 (x|x̂1 )
1
x2 = x̂2 = 0
p2 (x|x̂2 )
x̂3 = − 52
p3 (x|x̂3 )
x3 = −1
-1
-1
0
1
x
this happens when x ∈
/ (−1, 1) for i = 1 and x ∈
/ (− 25 , 25 ) for i = 3.
Because x̂i maximizes dynamic utility Ui = ui + δVi while xi maximizes
static utility ui , x̂i and xi in general differ. Consider player 3 and status-quo
x = −1. We claim p3 (−1) = − 52 while the policy maximizing u3 is x3 = −1.
With x = −1, A = [−1, 1] so that x3 = −1 would be accepted, if proposed.
The reason x̂3 = − 25 6= x3 = −1 is that in the dynamic setting player 3
anticipates the impact of her proposal on the distribution of future policies.
Two such distributions, induced by proposing x3 = −1 and p3 (−1) = − 52 ,
are indicated by the (red) circles left of x = 0 in Figure 1. By proposing
p3 (−1) = − 52 in lieu of proposing x3 = −1, player 3 fails to maximize
her static utility but brings the future policy of player 1 from p1 (−1) = 1
to p1 (− 25 ) = 25 . Player 3 moderates and it is optimal for her to do so; the
moderate policy she proposes increases her future utility by constraining the
future policy of player 1 (at the cost of current static utility). The incentive
to moderate is purely strategic; absent the intertemporal link created by
persistent policies, player 3 would propose x3 = −1.
Furthermore, we claim that it is optimal for player 1 not to moderate
and her strategic bliss point x̂1 = 1 = x1 . Certainly, the strategic force
to moderate is present for player 1 as well. Consider status-quo x = 1.
We claim player 1 proposes p1 (1) = 1 in lieu of moderating and proposing,
using the same extent of moderation as player 3, p0 = 52 . Given x = 1 both
p1 (1) = 1 and p0 =
2
5
would be accepted and induce the distribution over
8
future policies indicated by the (blue) circles to the right of x = 0 in Figure
1. Player 1 does not moderate because proposing p1 (1) = 1 or p0 =
identical policy by player 3, p3 (1) =
p3 ( 25 )
=
− 25 .
2
5
induces
In order to constrain the
future policy of player 3, player 1 would have to moderate to some policy
in [0, 25 ), which is too costly for her in terms of current utility. In other
words, moderation is a strategic substitute; when player 3 moderates, the
best response for player 1 is not to moderate, and when player 3 does not
moderate, player 1 best responds by moderating.
The following proposition shows that moderation is a strategic substitute
in any SMPE in the c-strategies; in any SMPE exactly one player does and
exactly one player does not moderate. The equilibrium extent of moderation
of the player who does is given by the desired extent of moderation, which
equals x̃1 = max{0, x1 (1 − 2δr3 )} and x̃3 = min{0, x3 (1 − 2δr1 )}. To state
the proposition, denote by σ(x̂) the profile of proposal strategies induced
by x̂ along with a profile of stage-undominated deferential voting strategies
constructed from dynamic utilities induced via (3).12
Proposition 1. Suppose δ ∈ (0, 1) and ri > 0 for i ∈ {1, 3}. Suppose σ(x̂)
with x̂ = {x̂1 , x̂2 , x̂3 } constitutes an SMPE. Then either x̂ = {x̃1 , 0, x3 } or
x̂ = {x1 , 0, x̃3 }.
The equilibrium extent of moderation of the moderating i increases with
her patience, with δ, and with the probability of recognition of her opponent,
with r−i ; δ and r−i determine the strength of the strategic force to moderate.
When 2δr−i ≥ 1, the strategic force is so strong that i moderates to the
largest extent possible, she proposes pi (x|0) = 0 = x2 for any status-quo
x ∈ X. Proposition 1 is silent about the possibility of co- or non-existence of
SMPE in the c-strategies the following proposition clarifies,
using parameter
√
shorthand T1 =
√ 1
2 1−δr3 −1
when δr3 ≥
1
2
and T1 =
δr3
√
1− δr3
when δr3 < 12 .
Proposition 2. Suppose δ ∈ (0, 1) and ri > 0 for i ∈ {1, 3}.
1. σ(x̂) with x̂ = {x1 , 0, x̃3 } constitutes an SMPE if and only if x̃1 ≥ |x̃3 |.
2. σ(x̂) with x̂ = {x̃1 , 0, x3 } constitutes an SMPE if and only if x̃1 ≤ |x̃3 |
and
x1
|x3 |
≤ T1 .
12
Proposition 1 assumes δ ∈ (0, 1) and ri > 0 for i ∈ {1, 3} in order to avoid special
cases that complicate the exposition and add little in terms of additional insights.
9
Proposition 2 implies that the moderating player in any SMPE in the
c-strategies is the one with the smaller |x̃i | and if |x̃1 | = |x̃3 |, two SMPE
might exist. In G 0 above, x̃1 = |x̃3 | =
2
5
and
x1
|x3 |
= 1 ≤ T1 ≈ 1.49. Hence,
in addition to an SMPE based on x̂ = {1, 0, − 25 } from Figure 1, there
is a ‘mirror’ SMPE based on x̂ = { 25 , 0, −1}.13 Moreover, Proposition 2
implies that there does not exist an SMPE in the c-strategies based on
x̂ = {x̂1 , x̂2 , x̂3 } with x̂1 = |x̂3 |; the equilibrium extent of moderation is
never symmetric.
The intuition why σ(x̂) with x̂ = {x1 , 0, x̃3 } does not constitute an
SMPE when x̃1 < |x̃3 | is the following. x̃i is the best-response extent of
moderation of i to a non-moderating −i, leaving open the question of what is
the best-response extent of moderation of i to a moderating −i. When x̃1 ≥
|x̃3 |, player 3 moderates more than what player 1 would like to constrain
player 3 to if player 3 did not moderate; player 1 has no incentive to constrain
player 3 further. When x̃1 < |x̃3 |, player 3 moderates less than what player
1 would like; player 1 still has an incentive to moderate to constrain player
3 further. That is, when x̃1 < |x̃3 |, the only remaining possibility for an
SMPE in the c-strategies is the one based on x̂ = {x̃1 , 0, x3 }.
However, Proposition 2 shows that even x̂ = {x̃1 , 0, x3 } fails to induce an
SMPE when
x1
|x3 |
> T1 . An example of such failure is G 00 with x = {2, 0, −1},
1 1 9
, 2 , 20 }, δ =
r = { 20
9
10
and X = R. In G 00 , player 1 has the more extreme
x1 > |x3 |, but r3 is large enough to bring x̃1 =
38
100
below |x̃3 | =
any SMPE in the c-strategies has to be based on x̂ =
x̂ =
38
{ 100
, 0, −1}
fails to induce an SMPE since
x1
|x3 |
91
100 .
Hence,
38
, 0, −1},
{ 100
but
= 2 > T1 ≈ 1.84.
The intuition behind the failure needs to consider player 1.
Given
status-quo x ≥ 2, x1 = 2 would be accepted and moderating, proposing
p1 (x|x̃1 ) = x̃1 =
38
100 ,
involves large static-utility cost, making deviation to
x1 profitable. It is only when the status-quo x is close to x̃1 that player 1
optimally moderates to x̃1 ; x1 would no longer be accepted and the staticutility cost of x̃1 , relative to the policies acceptable under x, is modest.
G 0 with general δ ∈ [0, 1) is the game studied in Kalandrakis (2015). Since T1 ≥ 1
∀δr3 ∈ [0, 1), Proposition 2 implies that it admits two pure SMPE in c-strategies with
x̂ = {1, 0, −(1 − 23 δ)} and x̂ = {1 − 23 δ, 0, −1}. Those have been independently derived by
Kalandrakis (2015) (see his Proposition 2). Moreover, the values 1 − 32 δ and −(1 − 23 δ)
are the boundaries of his ‘mixing regions’. Therefore, G 0 with δ ∈ [0, 1) admits at least
three SMPE; two in pure c-strategies considered here and one in mixed strategies derived
in Kalandrakis (2015). Interestingly, the mixed strategy SMPE is Pareto inefficient while
the c-strategy SMPE is Pareto efficient (see Kalandrakis, 2015, for details).
13
10
This suggests a way to construct an SMPE in games akin to G 00 . The
key idea starts with x̂ = {x̃1 , 0, x3 } and finds a point of adjustment xa such
that, for player 1, the cost of moderation is modest for any status-quo in
[−|xa |, |xa |] and it is large for any status-quo not in [−|xa |, |xa |]. We then allow player 1 to use a d-strategy; to propose p1 (x|x̂1 ) for any x ∈ [−|xa |, |xa |],
to moderate, and to propose p1 (x|x1 ) for any x ∈
/ [−|xa |, |xa |], not to moderate. Let p1 (x|x̂1 , xa ) be the policy proposed by player 1 using her dstrategy with (x̂1 , xa ) given x ∈ X. For i ∈ {2, 3} the d- and c-strategies
coincide. Thus, any profile of strategic bliss points with an adjustment
x̂a = {(x̂1 , xa ), x̂2 , x̂3 } induces a profile of proposal strategies. Figure 2
shows a d-strategy profile that induces an SMPE. The value of xa in the
figure equals x̃a , where x̃a is a complex expression of the model parameters
derived in the proof of Theorem 1 (see Definition A3) evaluated at G 00 .14
In analogy to Propositions 1 and 2, we present two propositions for
the d-strategies. The first one restricts the d-strategies to satisfy x̂1 6= x1
and |xa | ∈ (min{x̂1 , x1 }, sup X). This condition separates the two strategy
classes; the c-strategies are a strict subset of the d-strategies and, if and only
if x̂a satisfies the condition, x̂ such that the c-strategies with x̂ replicate the
d-strategies with x̂a does not exist (see Lemma A5 in the Appendix).
Proposition 3. Suppose δ ∈ (0, 1) and ri > 0 for i ∈ {1, 3}. Suppose
σ(x̂a ) with x̂a = {(x̂1 , xa ), x̂2 , x̂3 }, x̂1 6= x1 and |xa | ∈ (min{x̂1 , x1 }, sup X)
constitutes an SMPE. Then x̂a = {(x̃1 , xa ), 0, x3 } either with unique |xa | ∈
14
Figure 2 lets us explain the relation between Forand (2014)’s equilibria and ours.
The pertinent features of his electoral model are parties L and R with bliss points 0
and 1, policy space [0, 1], the parties (endogenously) alternating in proposing (opposition)
and the proposer having an option to pass (stay out of an election). The bounds on
extremism in Forand’s Proposition 3, l∗ and r∗ , are closely related to x̃3 and x̃1 ; when
r1 = r3 = 21 , l∗ with x̃3 and r∗ with x̃1 possess identical δ → 1 limit (adapting the bliss
my
my
l∗
r∗
points and using quadratic stage utilities). Hence, Forand’s (σL
, σR
) and (σL
, σR
) and
our x̂ = {x1 , 0, x̃3 } and x̂ = {x̃1 , 0, x3 } profiles are closely related and, also, constitute an
equilibrium based on l∗ or r∗ being closer to the median’s bliss point M . Importantly,
and differently from our model, one of Forand’s profiles always constitutes an equilibrium.
The reason is a combination of his parties passing, party L when x ≤ M and party R
when x ≥ M , with policy space bounded by the parties’ bliss points. Heuristically, in
terms of Figure 2, this implies policy space [−1, 2], party 1 proposing only for x < 0
and party 3 proposing only for x > 0. That is, Forand need not consider d-strategies for
equilibrium existence. Moreover, the consistent equilibria supporting convergence paths
towards M underlying Forand’s Proposition 4, unlike our d-strategies, always start with a
moderating step to l∗ or r∗ and are built on one party’s moderating step (discontinuity)
being reciprocated in future by its opponent. In terms of Figure 2, the entire set of
discontinuities in the proposal strategies of these equilibria is restricted to [−x̃1 , x̃1 ].
11
Figure 2:
G 00 :
d-strategies for x = {2, 0, −1},
x̂a
=
38
{( 100
, 2 − 59
q
13
119 ), 0, −1}
pi (x|x̂i )
p1 (x|x̂1 , xa )
x1 = 2
1
x̂1 =
38
100
p2 (x|x̂2 )
x2 = x̂2 = 0
x3 = x̂3 = −1
-1
-2
−xa -1
0
1
xa
2
p3 (x|x̂3 )
x
(|x3 |, x1 ) or with an arbitrary |xa | ∈ [x1 , sup X).
Proposition 3 shows that any SMPE in the d-strategies, unless it is
an SMPE in the c-strategies, essentially embeds the proposal strategies
induced by x̂ = {x̃1 , 0, x3 }, but allows player 1 to propose p1 (x|x1 ) for
x∈
/ [−|xa |, |xa |], for any extreme status-quo when moderation to x̃1 is too
costly.
Proposition 4. Suppose δ ∈ (0, 1) and ri > 0 for i ∈ {1, 3}.
1. σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 }, where |xa | = x̃a ∈ (|x3 |, x1 ), constitutes an SMPE if and only if x̃1 ≤ |x̃3 | and
x1
|x3 |
> T1 .
2. σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 }, where |xa | ∈ [x1 , sup X), constitutes
an SMPE if and only if x̃1 ≤ |x̃3 | and
x1
|x3 |
= T1 .
The sufficient conditions for SMPE existence from Proposition 2 and 4
exhaust the entire set of possibilities, which is, in fact, why Theorem 1 can
be proven using only the c- and d-strategies.
4
Discussion
The existence result in Theorem 1 relies on several assumptions concerning the primitives of the model — quadratic stage utilities, homogeneous
12
discounting, simple majority voting and three players. The quadratic preferences are both convenient, by facilitating the algebraic manipulations
throughout, and necessary, by ensuring that the player with the median
bliss point is decisive. The latter property can be alternatively achieved by
giving, for example, the player with the median bliss point veto power. In
this alternative model, with general stage utilities, a construction analogous
to the one used above would be to construct a profile of proposal strategies
using the desired extent of moderation of the player for whom the incentive
to moderate is the strongest.15 The same applies to homogeneous discounting. With heterogeneous discount factors, the player with the median bliss
point need not be decisive, but can be made so by assumption.
The player with the median bliss point is the only decisive player due to
the simple majority. The only more demanding quota voting rule with three
players is unanimity. With unanimity, straightforward argument shows that
in any SMPE, starting with any status-quo located between the bliss points
of the players, the equilibrium policy has to remain constant. That is, any
status-quo between the players’ bliss point is absorbing in a similar way
as the median’s bliss point is absorbing with a simple majority. However,
a preliminary investigation of the model with unanimity suggests that the
way policies approach the absorbing set over time can be complex, unlike in
the model with a simple majority.
With more than three players the c-strategies used in this paper can be
used to construct an SMPE in a certain class of games (Zapal, 2015). The
construction relies on an algorithm that constructs strategic bliss points.
For three players, the strategic bliss points the algorithm produces coincide
with those in Proposition 2. For more than three players, the algorithm and
the entire analysis is significantly more complex and, yet, may fail to deliver
an SMPE. With three players, this paper uses the d-strategies to complete
the existence proof. Whether the same is feasible for games with more
than three players remains an open question, the key difficulty being the
(potentially many) discontinuities in the d-strategies. Similarly, it remains
an open question whether the mixed strategy SMPE construction in Kalandrakis (2015) generalizes beyond his three-player game with equidistant
15
The desired extent of moderation is found by taking a profile of strategies with
infinite strategic bliss points of the two non-median players and finding the maximizer of
the induced dynamic utilities.
13
bliss points and equal recognition probabilities.
While Theorem 1 establishes an existence result, Propositions 1 through
4 provide an answer to the uniqueness question, if only for the class of strategies considered. G 0 underlying Figure 1 admits two pure SMPE, however,
the propositions show that this multiplicity is specific to games with identical desired extent of moderation, x̃1 = |x̃3 |, or to games with
x1
|x3 |
= T1 . As
a result, any multiplicity of SMPE in the c- or d-strategies can be perturbed
away, by perturbing the bliss points or the recognition probabilities, and is
non-generic.
Furthermore, Propositions 1 and 3 jointly imply that any SMPE in the
c- or d-strategies has at least one non-median player whose equilibrium proposal strategy is a c-strategy with a strategic bliss point equal to her static
bliss point. At least one non-median player not moderating is at the core of
the strategic substitute nature of moderation and extends beyond the c- and
d-strategy SMPE. In Appendix A6 we show that, for games with 3 (Proposition A4) or more (Proposition A5) players, there does not exist a consistent
SMPE, defined as in Forand (2014), with all non-median players moderating. Whether this property extends to all SMPE; whether moderation is
always a strategic substitute, remains an open question.16
On a more substantive level, common themes emerging from our analysis
are convergence to the policy preferred by the median player, convergence
path alternation of policies around this policy and asymmetric tendency
for moderation towards this policy, even in symmetric games. With the
exception of the first theme, these predictions concern dynamics of policies
and cannot be addressed in static legislative bargaining models. Moderation,
strategic manipulation of policy in an attempt to influence future policies, is
a standard observation in dynamic political economy literature. Its strategic
substitute nature is, to our knowledge, novel.
References
Anesi, V. and J. Duggan (2015a). Dynamic bargaining and external stability
with veto players. mimeo.
16
Absent any restriction on the strategies considered, the strategies can induce dynamic
utilities exhibiting complex patterns, rendering any attempt to build a contradiction between all players moderating and SMPE futile.
14
Anesi, V. and J. Duggan (2015b). Existence and indeterminacy of Markovian equilibria in dynamic bargaining games. CEDEX Discussion Paper
Series No. 2015-01.
Anesi, V. and D. J. Seidmann (2015). Bargaining in standing committees
with an endogenous default. Review of Economic Studies 82 (3), 825–867.
Banks, J. S. and J. Duggan (2006). A social choice lemma on voting over
lotteries with applications to a class of dynamic games. Social Choice and
Welfare 26 (2), 285–304.
Baron, D. P. (1996). A dynamic theory of collective goods programs. American Political Science Review 90 (2), 316–330.
Baron, D. P. and E. Kalai (1993). The simplest equilibrium of a majorityrule division game. Journal of Economic Theory 61 (2), 290–301.
Cho, S.-J. and J. Duggan (2003).
Uniqueness of stationary equilibria
in a one-dimensional model of bargaining. Journal of Economic Theory 113 (1), 118–130.
Diermeier, D. and P. Fong (2011). Legislative bargaining with reconsideration. Quarterly Journal of Economics 126 (2), 947–985.
Duggan, J. and A. Kalandrakis (2012). Dynamic legislative policy making.
Journal of Economic Theory 147 (5), 1653–1688.
Forand, J. G. (2010). Two-party competition with persistent policies. University of Waterloo Department of Economics Working Paper Series No.
10-011.
Forand, J. G. (2014). Two-party competition with persistent policies. Journal of Economic Theory 152, 64–91.
Kalandrakis, A. (2004). A three-player dynamic majoritarian bargaining
game. Journal of Economic Theory 116 (2), 294–322.
Kalandrakis, A. (2010). Minimum winning coalitions and endogenous status
quo. International Journal of Game Theory 39 (4), 617–643.
Kalandrakis, A. (2015). Pareto efficiency in the dynamic one-dimensional
bargaining model. forthcoming in Journal of Theoretical Politics.
15
Romer, T. and H. Rosenthal (1979). Bureaucrats versus voters: On the
political economy of resource allocation by direct democracy. Quarterly
Journal of Economics 93 (4), 563–587.
Zapal, J. (2012). Dynamic Group Decision Making. Ph. D. thesis, London
School of Economics and Political Science.
Zapal, J. (2015). Simple Markovian equilibria in dynamic spatial legislative
bargaining. mimeo.
A1
Proof of Theorem 1
We prove Theorem 1 using the following procedure. First, we state and prove
a dynamic median voter theorem (Proposition A1). Second, we formally
define two classes of proposal strategies, the c-strategies and the d-strategies
(Definitions A1 and A2). Third, we prove several useful technical properties
of the value functions and of the dynamic utilities for any profile of the cstrategies and of the d-strategies (Lemma A1). Fourth, we state condition
E on the parameters of G. The condition guarantees that an SMPE in the
c-strategies exists (Proposition A2). Finally, we show that when E fails, an
SMPE can be constructed using the d-strategies (Proposition A3).
Proposition A1. For any profile of pure stationary Markov strategies σ̂
such that, ∀i ∈ N and ∀(x, p) ∈ X 2 , v̂i (x, p) = yes if and only if Ui (p|σ̂) ≥
Ui (x|σ̂), a policy proposal p ∈ X is accepted under status-quo x ∈ X if and
only if U2 (p|σ̂) ≥ U2 (x|σ̂).
Proof. The proposition is an implication of Banks and Duggan (2006) (Cho
and Duggan, 2003, prove a similar result). We present full proof in order to
demonstrate the dependence of the result on the quadratic utilities used.
Fix a profile of pure stationary Markov strategies σ̂. Consider two policies p0 ∈ X and p00 ∈ X generating stochastic sequence, via σ̂, of policies
p = {p0 , p1 , . . .} and p0 = {p00 , p01 , . . .} respectively. The utility of i ∈ N
from voting either for p0 or p00 is
Ui (p0 |σ̂) = E
Ui (p00 |σ̂)
=E
hX∞
hXt=0
∞
t=0
16
−δ t (pt − xi )2
i
i
−δ t (p0t − xi )2 .
(A1)
Differentiating Ui (p0 |σ̂) − Ui (p00 |σ̂) with respect to xi gives
i
h X∞
∂[Ui (p0 |σ̂) − Ui (p00 |σ̂)]
−δ t (p0t − pt )
=E 2
t=0
∂xi
(A2)
which is independent of xi and hence Ui (p0 |σ̂) − Ui (p00 |σ̂) is linear in xi .
Now assume U2 (p0 |σ̂) ≥ U2 (p00 |σ̂). Then Ui (p0 |σ̂) ≥ Ui (p00 |σ̂) for at least
one i ∈ {1, 3} and p0 is accepted. Conversely, if U2 (p0 |σ̂) < U2 (p00 |σ̂), then
Ui (p0 |σ̂) < Ui (p00 |σ̂) for at least one i ∈ {1, 3} and p0 is rejected.
Definition A1. The c-pure stationary Markov proposal strategy (c-strategy)
of i ∈ N is



min {x̂1 , |x|}


pi (x|x̂i ) = x̂2



max {x̂ , −|x|}
3
if i = 1
if i = 2
if i = 3
where x̂i is the strategic bliss point of i. A profile of strategic bliss points is
x̂ = {x̂1 , x̂2 , x̂3 }.
Definition A2. The d-pure stationary Markov proposal strategy (d-strategy)
of i ∈ N is

p (x|x̂ ) if x ∈ [−|x |, |x |]
1
1
a
a
p1 (x|x̂1 , xa ) =
p (x|x ) if x ∈
/ [−|xa |, |xa |]
1
1

x̂
if i = 2
2
pi (x|x̂i ) =
max {x̂ , −|x|} if i = 3
3
where x̂i is the strategic bliss point of i and xa is the point of adjustment. A
profile of strategic bliss points with an adjustment is x̂a = {(x̂1 , xa ), x̂2 , x̂3 }.
An arbitrary x̂ or x̂a fully determines a profile of proposal strategies.
Denote by Vi (x|x̂) and Vi (x|x̂a ) the continuation value of player i ∈ N from
status-quo x ∈ X calculated using the proposal strategies induced by x̂
and x̂a , respectively, assuming that any proposal generated by the profile of
proposal strategies is accepted. That is
Vi (x|x̂) =
X
rj [ui (pj (x|x̂j )) + δVi (pj (x|x̂j )|x̂)]
(A3)
j∈N
and similarly for x̂a . Denote by Ui (x|x̂) and Ui (x|x̂a ) the corresponding
17
dynamic utilities. Straightforward modification of the argument used to
prove Proposition A1 shows that, for any x̂0 ∈ {x̂, x̂a }, ∀x ∈ X and ∀p ∈ X,
|{j ∈ N |Uj (p|x̂0 ) ≥ Uj (x|x̂0 )}| ≥ 2 if and only if U2 (p|x̂0 ) ≥ U2 (x|x̂0 ). The
set of policies player 2 weakly prefers to status-quo x ∈ X thus constitutes
the social acceptance set, which we denote, for x̂0 ∈ {x̂, x̂a }, by A(x|x̂0 ) =
{p ∈ X|U2 (p|x̂0 ) ≥ U2 (x|x̂0 )}.
For an arbitrary profile of strategic bliss points x̂ = {x̂1 , x̂2 , x̂3 } denote
by N D(x̂) = {−|x̂1 |, |x̂1 |, −|x̂3 |, |x̂3 |, 0} the points at which the proposal
strategies induced by x̂ are not differentiable and by D(x̂) = X \ N D(x̂) its
complement in X. With this notation,
∂pi (x|x̂i )
∂x
= p0i (x|x̂i ) exists ∀x ∈ D(x̂)
and ∀i ∈ N , and satisfies p0i (x|x̂i ) ∈ {−1, 0, 1}. For x ∈ D(x̂) denote
by C(x|x̂) = {i ∈ N |p0i (x|x̂i ) = 0} the set of players who at x are on
the constant part of their strategy. For i ∈ {1, 3} and x ∈ D(x̂) define
ri,nc (x|x̂) = ri if p0i (x|x̂i ) 6= 0 and ri,nc (x|x̂) = 0 if p0i (x|x̂i ) = 0. FiP
nally, define rnc (x|x̂) =
i∈{1,3} ri,nc (x|x̂). In words, rnc is the sum of
the recognition probabilities of the non-median players who are on a nonconstant part of their proposal strategies at x ∈ D(x̂). Define similar objects
for an arbitrary profile of strategic bliss points with an adjustment x̂a =
{(x̂1 , xa ), x̂2 , x̂3 }. The only definition that does not apply in an identical
way is that of N D(x̂) = {−|x̂1 |, |x̂1 |, −|x̂3 |, |x̂3 |, −|xa |, |xa |, 0, −|x1 |, |x1 |}.17
The lemma below establishes several properties of Vi (x|x̂0 ) induced by
x̂0 ∈ {x̂, x̂a }. To state the lemma we need x̂0 to satisfy Condition A1, the
adjustment point xa in x̂a to be defined as in Definition A3 and the strategic
bliss point x̂1 in x̂a to be defined as in Definition A4.
Condition A1. A profile of strategic bliss points x̂ = {x̂1 , x̂2 , x̂3 } satisfies
Condition A1 if x̂1 ≥ x̂2 = 0 ≥ x̂3 . A profile of strategic bliss points with
an adjustment x̂a = {(x̂1 , xa ), x̂2 , x̂3 } satisfies Condition A1 if x̂1 ∈ [0, |x3 |),
x̂2 = 0, x̂3 = x3 and |xa | ∈ (|x3 |, x1 ).
17
C, rnc , r1,nc and r3,nc are left undefined at the points in N D(x̂). Nevertheless, below
we often take one-sided limits at these points. For rnc , r1,nc and r3,nc this is standard
since these objects are functions. For C this is unusual since C(x|x̂) is a set of players. For
any x ∈ N D(x̂), define limz→x+ C(z|x̂) = C(z 0 |x̂) taking arbitrary z 0 ∈ U in an open U
such that inf U = x and U \ N D(x̂) is convex. Similarly, limz→x− C(z|x̂) = C(z 0 |x̂) taking
arbitrary z 0 ∈ U in an open U such that sup U = x and U \ N D(x̂) is convex. Because C
is constant on any interval induced by N D(x̂), both limits are well defined.
18
Definition A3. Suppose
x1
|x3 |
T1 =
> T1 and x1 (1 − 2δr3 ) < |x3 |, where


√ 1
2 1−δr3 −1
√
 √δr3
1− δr3
if δr3 <
1
2
if δr3 ≥
1
2
.
Then, a point of adjustment x̃a is defined by

q
x1 − 4δr3 x2 − δr3 (x1 − x3 )2
1
1−δr3
q
x̃a =
δr3
1
x −
2
2
1
1−δr3 x1 − 1−δr3 (x1 − x3 )
Defining x̃a only if
x1
|x3 |
if δr3 <
1
2
if δr3 ≥
1
2
.
> T1 ensures that x̃a ∈
/ C. The same condition
ensures that x̃a < x1 . Finally, x̃a > |x3 | rewrites as [x1 (1 − 2δr3 ) + x3 ]2 > 0
when δr3 < 21 , which holds, and as 2x1 (2δr3 − 1) − x3 > 0 when δr3 ≥ 12 ,
which also holds. Therefore, when x̃a is defined it satisfies x̃a ∈ (|x3 |, x1 )
required by Condition A1.
Definition A4. The desired extent of moderation of player i ∈ {1, 3} is
x̃1 = max{0, x1 (1 − 2δr3 )} and x̃3 = min{0, x3 (1 − 2δr1 )}.
The significance of x̃i in Definition A4 is that in any SMPE we construct
the strategic bliss point of one of the non-median players equals x̃i . Note
that x̃1 = x̃3 = 0 is not possible as it implies 1 − 2δr3 ≤ 0 and 1 − 2δr1 ≤ 0,
which implies δ(r1 + r3 ) ≥ 1 and is not possible.
Lemma A1. Let x̂0 ∈ {x̂, x̂a } satisfy Condition A1. If x̂0 = x̂a , let |xa | = x̃a
from Definition A3 and x̂1 = x̃1 from Definition A4. Then, ∀i ∈ N ,
1. Vi (|x| |x̂0 ) = Vi (−|x| |x̂0 ) ∀x ∈ X;
2. U1 (−|x| |x̂0 ) < U1 (|x| |x̂0 ), U2 (−|x| |x̂0 ) = U2 (|x| |x̂0 ) and U3 (−|x| |x̂0 ) >
U3 (|x| |x̂0 ) ∀x ∈ X \ {0};
3. if x̂0 = x̂, Ui is continuous on X;
4. if x̂0 = x̂a , U1 is continuous on X, Ui is continuous on X \{−|xa |, |xa |}
and Ui (|xa | |x̂a ) ≥ limx→|xa |+ Ui (x|x̂a ) for i ∈ {2, 3};
5. Ui0 (x|x̂0 ) exists ∀x ∈ D(x̂0 );
6. A(x|x̂0 ) = [−|x|, |x|] ∀x ∈ X.
19
Proof. To see part 1, any c-strategy with any strategic bliss point x̂i ∈ R
satisfies pi (−|x| |x̂i ) = pi (|x| |x̂i ) ∀i ∈ N and ∀x ∈ X. The same applies for
any d-strategy with any strategic bliss point and any point of adjustment.
The claim then follows from (A3). Part 2 follows easily from the symmetry,
∀i ∈ N , of Vi about 0, asymmetry, ∀i ∈ {1, 3}, of ui and symmetry of u2 .
To prove parts 3 and 4, as Ui (x|x̂0 ) = ui (x) + δVi (x|x̂0 ), it suffices to
prove the claimed properties for the continuation value functions. In order
to consider parts 3 and 4 jointly, we abuse the notation somewhat and write
p1 (x|x̂1 ) instead of p1 (x|x̂1 , xa ) when x̂0 = x̂a . By the symmetry of the
continuation value functions about 0 and by pi (x|x̂0i ) ∈ {−|x|, |x|} ∀x ∈
D(x̂0 ) and ∀i ∈ N \ C(x|x̂0 ), we can write Vi (x|x̂0 ) using (A3), ∀x ∈ D(x̂0 ),
as
P
j∈N
0
Vi (x|x̂ ) =
rj ui (pj (x|x̂0j )) + δ
0
0
j∈C(x|x̂0 ) rj Vi (pj (x|x̂j )|x̂ )
.
(x|x̂0 )
P
1 − δrnc
(A4)
Because pj (x|x̂0j ) is constant in x when j ∈ C(x|x̂0 ), because rnc (x|x̂0 ) and
C(x|x̂0 ) are constant in x on any interval induced by N D(x̂0 ), and because
pj (x|x̂0j ) is continuous in x ∀j ∈ N and ∀x ∈ D(x̂0 ), it follows that Vi (x|x̂0 )
is continuous in x on D(x̂0 ).
What remains are the claimed properties at x ∈ N D(x̂0 ). In the limit
arguments that follow, for f : X → R denote limx→z − f (x) = f (z − ) and
limx→z + f (x) = f (z + ). First, we claim that, ∀i ∈ N , Vi (0− |x̂0 ) = Vi (0|x̂0 ) =
Vi (0+ |x̂0 ).
Because Vi (0|x̂0 ) =
ui (0)
1−δ
∀i ∈ N and because Vi (0− |x̂0 ) =
Vi (0+ |x̂0 ) ∀i ∈ N by part 1, it suffices to prove that Vi (0+ |x̂0 ) =
ui (0)
1−δ
∀i ∈ N .
From (A4) we have
P
0
+
Vi (0 |x̂ ) =
j∈N
rj ui (pj (0+ |x̂0j )) + δ
+ 0
0
j∈C(0+ |x̂0 ) rj Vi (pj (0 |x̂j )|x̂ )
(0+ |x̂0 )
P
1 − δrnc
i (0)
ui (0) + δ(1 − rnc (0+ |x̂0 )) u1−δ
ui (0)
=
=
+
0
1 − δrnc (0 |x̂ )
1−δ
(A5)
1
1
where = follows from pj (0+ |x̂0j ) = 0 ∀j ∈ N and from the fact that if
j ∈ C(0+ |x̂0 ) then x̂0j = 0 and hence there has to exist an open neighbourhood
U 3 0 such that pj (x|x̂0j ) = 0 ∀x ∈ U .
Second, we claim that Vi (x− |x̂0 ) = Vi (x|x̂0 ) = Vi (x+ |x̂0 ) ∀i ∈ N for any
x ∈ {−|x̂1 |, |x̂1 |, −|x̂3 |, |x̂3 |, −|x1 |, |x1 |}. If x = 0 the claim follows by the
previous step, so consider x > 0. When x < 0 the argument is similar and
20
omitted. Using (A3) we can write, ∀i ∈ N ,
Vi (x|x̂0 ) =
X
rj ui (pj (x|x̂0j )) + δVi (pj (x|x̂0j )|x̂0 )
j∈N
1
=
X
X
rj ui (pj (x− |x̂0j )) + δ
rj Vi (pj (x− |x̂0j )|x̂0 )
(A6)
j∈C(x− |x̂0 )
j∈N
+ δrnc (x− |x̂0 )Vi (x|x̂0 )
= Vi (x− |x̂0 )(1 − δrnc (x− |x̂0 )) + δrnc (x− |x̂0 )Vi (x|x̂0 )
1
where = follows from i) pj (x|x̂0j ) = pj (x− |x̂0j ) ∀j ∈ N ; ii) Vi (pj (x|x̂0j )|x̂0 ) =
Vi (pj (x− |x̂0j )|x̂0 ) ∀j ∈ C(x− |x̂0 ), which holds since j ∈ C(x− |x̂0 ) implies that
there exists an open neighbourhood U 3 x on which pj is constant; and iii)
Vi (pj (x|x̂0j )|x̂0 ) = Vi (x|x̂0 ) ∀j ∈ N \ C(x− |x̂0 ). Similarly, ∀i ∈ N ,
Vi (x|x̂0 ) =
X
rj ui (pj (x|x̂0j )) + δVi (pj (x|x̂0j )|x̂0 )
j∈N
1
=
X
rj ui (pj (x+ |x̂0j )) + δ
rj Vi (pj (x+ |x̂0j )|x̂0 )
j∈C(x− |x̂0 )
j∈N
+δ
X
X
+
rj Vi (pj (x
|x̂0j )|x̂0 )
(A7)
+
0
0
+ δrnc (x |x̂ )Vi (x|x̂ )
j∈C(x+ |x̂0 )\C(x− |x̂0 )
= Vi (x+ |x̂0 )(1 − δrnc (x+ |x̂0 )) + δrnc (x+ |x̂0 )Vi (x|x̂0 )
1
where = follows from i) pj (x|x̂0j ) = pj (x+ |x̂0j ) ∀j ∈ N ; ii) Vi (pj (x|x̂0j )|x̂0 ) =
Vi (pj (x+ |x̂0j )|x̂0 ) ∀j ∈ C(x− |x̂0 ); iii) Vi (pj (x|x̂0j )|x̂0 ) = Vi (pj (x+ |x̂0j )|x̂0 ) ∀j ∈
C(x+ |x̂0 ) \ C(x− |x̂0 ), which holds since j ∈ C(x+ |x̂0 ) \ C(x− |x̂0 ) implies that
there exists an open neighbourhood U 3 x such that pj is constant on
U ∩ [x, ∞);18 and iv) Vi (pj (x|x̂0j )|x̂0 ) = Vi (x|x̂0 ) ∀j ∈ N \ C(x+ |x̂0 ).
The arguments above show, ∀i ∈ N , that Ui is continuous on X when
x̂0
= x̂, part 3, and that Ui is continuous on X \ {−|xa |, |xa |} when x̂0 = x̂a ,
part 4 except for properties at {−|xa |, |xa |}. What remains is to consider
x̂0 = x̂a = {(x̂1 , xa ), x̂2 , x̂3 } with x̂1 = x̃1 and |xa | = x̃a . By Condition A1
and Definitions A3 and A4 we have x̂1 = max{0, x1 (1 − 2δr3 )} ∈ [0, |x3 |),
x̂2 = 0, x̂3 = x3 , and |xa | = x̃a ∈ (|x3 |, x1 ). Suppose xa = x̃a . When
xa = −x̃a , the argument is similar and omitted.
18
As a matter of convention, for some a ∈ X, we use shorthand ‘∀x ∈ [a, ∞)’ or ‘on
[a, ∞)’ instead of the somewhat longer ‘∀x ∈ X such that x ≥ a’. If X = R, the statements
are equivalent. If X ( R, our use of the shorthand creates minimal chance of confusion.
21
From Definition A2 of the d-strategies and from x̂2 = 0 ≤ x̂1 < |x̂3 | <
xa , ∀i ∈ N , pi (x|x̂i ) is constant on [|x̂3 |, xa ]. Hence, ∀i ∈ N , Vi (x|x̂a ) is
a
a
constant on [|x̂3 |, xa ] and thus Vi (x−
a |x̂ ) = Vi (xa |x̂ ). By part 1, we also
a
a
have Vi (−x+
a |x̂ ) = Vi (−xa |x̂ ) ∀i ∈ N .
a
a
+ a
Next, we claim that V1 (xa |x̂a ) = V1 (x+
a |x̂ ) and Vi (xa |x̂ ) ≥ Vi (xa |x̂ )
P
∀i ∈ {2, 3}. Note that (A3) writes as Vi (x|x̂a ) = j∈N rj Ui (pj (x|x̂j )|x̂a )
∀x ∈ X and ∀i ∈ N . Since x̂2 < |x̂3 | < xa < x1 , there exists an open
neighbourhood U 3 xa such that pj (x|x̂j ) = x̂j ∀x ∈ U and ∀j ∈ {2, 3}.
a
a
+
a
Therefore Vi (xa |x̂a ) − Vi (x+
a |x̂ ) = r1 [Ui (p1 (xa |x̂1 )|x̂ ) − Ui (p1 (xa |x̂1 )|x̂ )].
We also have p1 (xa |x̂1 ) = x̂1 and p1 (x|x̂1 ) = x for any x ∈ (xa , x1 ) so that
a
a
+ a
Vi (xa |x̂a ) − Vi (x+
a |x̂ ) = r1 [Ui (x̂1 |x̂ ) − Ui (xa |x̂ )] ∀i ∈ N .
a
Next we calculate Ui (x̂1 |x̂a ) and Ui (x+
a |x̂ ). For the former, because
x̂2 ≤ x̂1 < |x̂3 |, we have
Ui (x̂1 |x̂a ) = ui (x̂1 ) + δ [r1 Ui (x̂1 |x̂a ) + r2 Ui (0|x̂a ) + r3 Ui (−x̂1 |x̂a )]
=
(A8)
ui (x̂1 )(1 − δr3 ) + δr2 Ui (0|x̂a ) + δr3 ui (−x̂1 )
1 − δ(r1 + r3 )
where we have used Ui (x|x̂a ) − Ui (−x|x̂a ) = ui (x) − ui (−x) ∀x ∈ X. For
the latter, ∀x ∈ (xa , x1 ), we have
Ui (x|x̂a ) = ui (x) + δ [r1 Ui (x|x̂a ) + r2 Ui (0|x̂a ) + r3 Ui (x̂3 |x̂a )]
=
ui (x) +
δr2
a
1−δr3 Ui (0|x̂ )
+
δr3
1−δr3
[ui (x3 ) + δr1 Ui (x̂1 |x̂a )]
(A9)
1 − δr1
where we have used Ui (x̂3 |x̂a ) =
ui (x̂3 )+δr1 Ui (x̂1 |x̂a )+δr2 Ui (0|x̂a )
.
1−δr3
Carrying out
the algebra gives, ∀i ∈ N ,
a
Ui (x̂1 |x̂a ) − Ui (x+
a |x̂ )
δr3
= ui (x̂1 ) −ui (xa ) − 1−δr
[ui (x3 ) − ui (−x̂1 )] . (A10)
3
(1 − δr1 )−1
When i = 1, x̂1 = x̃1 and xa = x̃a , straightforward algebra shows that
a
the right hand side of (A10) is zero so that U1 (x̂1 |x̂a ) = U1 (x+
a |x̂ ) and hence
a
a
+ a
V1 (xa |x̂a ) = V1 (x+
a |x̂ ). To prove that Ui (x̂1 |x̂ ) ≥ Ui (xa |x̂ ) ∀i ∈ {2, 3},
it suffices to show that the derivative of the right hand side of (A10) with
respect to xi is non-positive. The derivative reads
h
2 x̂1 − xa −
δr3
1−δr3 (x̂1
22
i
− |x3 |) .
(A11)
When
δr3
1−δr3
≤ 1, from x̂1 < |x3 | < xa , −xa < −|x3 | ⇔ x̂1 −xa < x̂1 −|x3 | < 0
δr3
and thus x̂1 − xa < (x̂1 − |x3 |) 1−δr
. When
3
δr3
1−δr3
> 1, δr3 > 12 . Substituting
x̂1 = x̃1 = 0 and xa = x̃a from Definition A3 into (A11), along with routine
a
algebra, shows that (A11) is non-positive. Thus, Ui (x̂1 |x̂a ) ≥ Ui (x+
a |x̂ )
a
∀i ∈ {2, 3} and hence Vi (xa |x̂a ) ≥ Vi (x+
a |x̂ ) ∀i ∈ {2, 3}.
To show part 5, that Ui0 (x|x̂0 ) exists ∀x ∈ D(x̂0 ) and ∀i ∈ N , from (A4)
we have, ∀x ∈ D(x̂0 ) and ∀i ∈ N ,
P
Vi0 (x|x̂0 )
=
j∈N
rj u0i (pj (x|x̂0j ))p0j (x|x̂0j )
1 − δrnc (x|x̂0 )
.
(A12)
We have p0j (x|x̂0j ) ∈ {−1, 1} ∀x ∈ D(x̂0 ) and ∀j ∈ N \ C(x|x̂0 ) ⊆ {1, 3},
p0j (x|x̂0j ) = 0 ∀j ∈ C(x|x̂0 ), p1 (x|x̂01 ) = |x| if 1 ∈
/ C(x|x̂0 ) and p3 (x|x̂03 ) = −|x|
if 3 ∈
/ C(x|x̂0 ). Using Ui0 (x|x̂0 ) = u0i (x) + δVi0 (x|x̂0 ) we thus have

2
0
−
1−δrnc (x|x̂0 ) [x − xi (1 − 2δr1,nc (x|x̂ ))]
0
0
Ui (x|x̂ ) =
2
−
[x − x (1 − 2δr (x|x̂0 ))]
1−δrnc (x|x̂0 )
i
3,nc
if x < 0
(A13)
if x > 0
so that Ui0 (x|x̂0 ) exists ∀x ∈ D(x̂0 ) and ∀i ∈ N .
2x
To show part 6, from the previous part we have U20 (x|x̂0 ) = − 1−δrnc
(x|x̂0 )
∀x ∈ D(x|x̂0 ). By the continuity of U2 at the points in N D(x̂0 )\{−|xa |, |xa |}
and the direction of discontinuities at {−|xa |, |xa |}, it follows that A(x|x̂0 ) =
{p ∈ X|U2 (p|x̂0 ) ≥ U2 (x|x̂0 )} = [−|x|, |x|] ∀x ∈ X.
Any profile of strategic bliss points (with an adjustment) x̂0 ∈ {x̂, x̂a }
induces, via (A3), Vi (x|x̂0 ) and Ui (x|x̂0 ) ∀x ∈ X and ∀i ∈ N . For each i ∈ N ,
Ui induced by x̂0 determines the voting strategy of i as v̂i (x, p|x̂0 ) = yes if and
only if Ui (p|x̂0 ) ≥ Ui (x|x̂0 ) ∀(x, p) ∈ X 2 . Denote the profile of proposal and
voting strategies induced by x̂0 by σ(x̂0 ) = ((pi (·|x̂0i ), v̂i (·, ·|x̂0 )))i∈N (where
p1 (·|x̂1 , xa ) replaces p1 (·|x̂01 ) if x̂0 = x̂a ). Under the supposition of Lemma
A1, x̂0 also determines the social acceptance set A(x|x̂0 ) = [−|x|, |x|] ∀x ∈ X
via the voting strategy of player 2.
Note that for any x̂ = {x̂1 , x̂2 , x̂3 } that satisfies Condition A1 we have,
∀x ∈ X and ∀i ∈ N , pi (x|x̂i ) ∈ [−|x|, |x|]. For any x̂a = {(x̂1 , xa ), x̂2 , x̂3 }
with x̂1 = x̃1 and xa = x̃a that satisfies Condition A1 we also have, ∀x ∈ X,
p1 (x|x̂1 , xa ) ∈ [−|x|, |x|] and pi (x|x̂i ) ∈ [−|x|, |x|] ∀i ∈ {2, 3}. Therefore,
any x̂0 ∈ {x̂, x̂a } that satisfies Condition A1 (and with x̂1 = x̃1 and xa = x̃a
if x̂0 = x̂a ) induces σ(x̂0 ) such that the policy proposed by any i ∈ N
23
for any x ∈ X under the proposal strategies in σ(x̂0 ) is accepted under
the voting strategies in σ(x̂0 ). This implies that, ∀x ∈ X and ∀i ∈ N ,
Vi (x|σ(x̂0 )) = Vi (x|x̂0 ), Ui (x|σ(x̂0 )) = Ui (x|x̂0 ), and A(x|σ(x̂0 )) = A(x|x̂0 ).
The following condition is a parametric condition on G. Depending on
whether it holds or fails, an SMPE in the c-strategies or the d-strategies
exists.
Definition A5. G satisfies condition E if
either x̃1 ≥ |x̃3 | or x̃1 < |x̃3 | and
x1
≤ T1 .
|x3 |
(E)
Proposition A2. Suppose G satisfies condition E. Then σ(x̂) constitutes
an SMPE, where x̂ = {x1 , 0, x̃3 } if x̃1 ≥ |x̃3 | and x̂ = {x̃1 , 0, x3 } if x̃1 < |x̃3 |.
Proof. Fix x̂ ∈ {{x1 , 0, x̃3 }, {x̃1 , 0, x3 }}. Lemma A1 applies to x̂ since it
satisfies Condition A1. The voting strategies in σ(x̂) thus satisfy Definition
1. What remains to be shown is that pi (x|x̂i ) ∈ arg maxp∈A(x|σ(x̂)) Ui (p|σ(x̂))
∀i ∈ N and ∀x ∈ X. Recall that A(x|σ(x̂)) = [−|x|, |x|] ∀x ∈ X.
For i = 2, by Lemma A1 parts 2 and 4 and from (A13), U20 (x|σ(x̂)) < 0
∀x ∈ [0, ∞) \ N D(x̂), U2 is continuous and symmetric about 0. Therefore,
∀x ∈ X, arg maxp∈[−|x|,|x|] U2 (p|σ(x̂)) = {0}. The proposal strategy induced
by x̂2 = 0 thus satisfies the required property.
For i ∈ {1, 3}, by Lemma A1 part 2, U1 (|x| |σ(x̂)) > U1 (−|x| |σ(x̂)) and
U3 (−|x| |σ(x̂)) > U3 (|x| |σ(x̂)) ∀x ∈ X \ {0} and hence i under status-quo
x ∈ X facing an acceptance set [−|x|, |x|] never finds it optimal to propose
p < 0 if i = 1 and never finds it optimal to propose p > 0 if i = 3.
Now consider x̂ = {x1 , 0, x̃3 }. Note that since x̃3 = min{0, x3 (1 − 2δr1 )},
|x̃3 | ≤ |x3 | ≤ x1 . By the supposition of the proposition x̃1 ≥ |x̃3 |, which
implies that x̃1 > 0 and hence x̃1 = x1 (1 − 2δr3 ).
We claim that p1 (x|x1 ) ∈ arg maxp∈A(x|σ(x̂)) U1 (p|σ(x̂)). To show this we
study U1 on [0, ∞). From Lemma A1, U1 is continuous on X. From (A13),
sgn[U10 (x|σ(x̂))] = sgn[−x+x1 (1−2δr3,nc (x|x̂))], where r3,nc (x|x̂) = r3 ∀x ∈
(0, |x̃3 |) and r3,nc (x|x̂) = 0 ∀x ∈ (|x̃3 |, x1 ) ∪ (x1 , ∞). Hence U10 (x|σ(x̂)) >
0 on (0, |x̃3 |) if x < x1 (1 − 2δr3 ) = x̃1 , which holds since |x̃3 | ≤ x̃1 ,
U10 (x|σ(x̂)) > 0 on (|x̃3 |, x1 ) if x < x1 , which holds, and U10 (x|σ(x̂)) < 0
on (x1 , ∞) if x > x1 , which holds. Therefore, arg maxp∈A(x|σ(x̂)) U1 (p|σ(x̂))
equals |x| ∀x ∈ [−x1 , x1 ] and equals x1 ∀x ∈ X \ [−x1 , x1 ], that is, it is
identical to the policy generated by the c-strategy induced by x̂1 = x1 .
24
We now claim that p3 (x|x̃3 ) ∈ arg maxp∈A(x|σ(x̂)) U3 (p|σ(x̂)). To show
this we study U3 on (−∞, 0]. From Lemma A1, U3 is continuous on X. From
(A13), sgn[U30 (x|σ(x̂))] = sgn[−x + x3 (1 − 2δr1,nc (x|x̂))], where r1,nc (x|x̂) =
r1 ∀x ∈ (−x1 , x̃3 ) ∪ (x̃3 , 0) and r1,nc (x|x̂) = 0 ∀x ∈ (−∞, −x1 ). Hence
U30 (x|σ(x̂)) > 0 on (−∞, −x1 ) if x < x3 , which holds since −x1 ≤ x3 ,
U30 (x|σ(x̂)) > 0 on (−x1 , x̃3 ) if x < x3 (1 − 2δr1 ), which holds since x̃3 ≤
x3 (1 − 2δr1 ), and U30 (x|σ(x̂)) < 0 on (x̃3 , 0) if x > x3 (1 − 2δr1 ), which holds
since x̃3 = x3 (1 − 2δr1 ) if x̃3 < 0. Therefore, arg maxp∈A(x|σ(x̂)) U3 (p|σ(x̂))
equals −|x| ∀x ∈ [x̃3 , |x̃3 |] and equals x̃3 ∀x ∈ X \ [x̃3 , |x̃3 |], that is, it is
identical to the policy generated by the c-strategy induced by x̂3 = x̃3 . As
a result σ(x̂) with x̂ = {x1 , 0, x̃3 } constitutes an SMPE.
Now consider x̂ = {x̃1 , 0, x3 }. By the supposition of the proposition
x̃1 < |x̃3 |, which implies that x̃1 < |x̃3 | ≤ |x3 | ≤ x1 and x̃3 = x3 (1 − 2δr1 ).
We first claim that p3 (x|x3 ) ∈ arg maxp∈A(x|σ(x̂)) U3 (p|σ(x̂)), which is the
easier case. We again study U3 on (−∞, 0]. From Lemma A1, U3 is continuous on X. From (A13), sgn[U30 (x|σ(x̂))] = sgn[−x + x3 (1 − 2δr1,nc (x|x̂))],
where r1,nc (x|x̂) = r1 ∀x ∈ (−x̃1 , 0) and r1,nc (x|x̂) = 0 ∀x ∈ (−∞, x3 ) ∪
(x3 , −x̃1 ). Hence U30 (x|σ(x̂)) > 0 on (−∞, x3 ) if x < x3 , which holds,
U30 (x|σ(x̂)) < 0 on (x3 , −x̃1 ) if x > x3 , which holds, and U30 (x|σ(x̂)) < 0
on (−x̃1 , 0) if x > x3 (1 − 2δr1 ) = x̃3 , which holds since x̃3 < −x̃1 . Therefore, arg maxp∈A(x|σ(x̂)) U3 (p|σ(x̂)) equals −|x| ∀x ∈ [x3 , |x3 |] and equals x3
∀x ∈ X \ [x3 , |x3 |], that is, it is identical to the policy generated by the
c-strategy induced by x̂3 = x3 .
We now claim that p1 (x|x̃1 ) ∈ arg maxp∈A(x|σ(x̂)) U1 (p|σ(x̂)). We again
study U1 on [0, ∞).
(A13) we know that
From Lemma A1, U1 is continuous on X.
sgn[U10 (x|σ(x̂))]
From
= sgn[−x + x1 (1 − 2δr3,nc (x|x̂))], where
r3,nc (x|x̂) = r3 ∀x ∈ (0, x̃1 ) ∪ (x̃1 , |x3 |) and r3,nc (x|x̂) = 0 ∀x ∈ (|x3 |, x1 ) ∪
(x1 , ∞).
Hence U10 (x|σ(x̂)) > 0 on (0, x̃1 ) if x < x1 (1 − 2δr3 ), which
holds since x̃1 = x1 (1 − 2δr3 ) if x̃1 > 0, U10 (x|σ(x̂)) < 0 on (x̃1 , |x3 |) if
x > x1 (1 − 2δr3 ), which holds since x̃1 ≥ x1 (1 − 2δr3 ), and, skipping momentarily (|x3 |, x1 ), U10 (x|σ(x̂)) < 0 on (x1 , ∞) if x > x1 , which holds. If
|x3 | = x1 then (|x3 |, x1 ) need not be considered and as above σ(x̂) with
x̂ = {x̃1 , 0, x3 } constitutes an SMPE.
To show that x̂ = {x̃1 , 0, x3 } induces an SMPE we need to show that
U1 (x̃1 |σ(x̂)) ≥ U1 (x|σ(x̂)) ∀x ∈ [|x3 |, x1 ] when |x3 | < x1 .
U10 (x|σ(x̂))
Note that
> 0 on (|x3 |, x1 ) if x < x1 , which holds. It thus suffices to show
25
that U1 (x̃1 |σ(x̂)) ≥ U1 (x1 |σ(x̂)). Since U1 is differentiable on (x̃1 , |x3 |) ∪
Ra
(|x3 |, x1 ) and continuous on X and since f (a) − f (b) = b f 0 (x)dx for any
differentiable f : X → X, we have
x̃+
1
Z
U1 (x̃1 |σ(x̂)) − U1 (x1 |σ(x̂)) =
|x3
Z
+
|−
U10 (x|σ(x̂))dx
|x3 |+
x−
1
(A14)
U10 (x|σ(x̂))dx.
From (A13) we have
Z
x̃+
1
|x3 |−
Z
|x3
|+
x−
1
2
U10 (x|σ(x̂))dx = − 1−δr
3
U10 (x|σ(x̂))dx = − 12
h
x2
2
h
x2
2
− x · x1 (1 − 2δr3 )
− x · x1
ix̃+
1
|x3 |−
i|x3 |+
(A15)
x−
1
and U1 (x̃1 |σ(x̂)) − U1 (x1 |σ(x̂)) can be rewritten, after some algebra, as
−
When δr3 ≥
equivalent to
1
1−δr3
(x̃1 − x1 )2 − δr3 (x1 − x3 )2 + 4δr3 x1 x̃1 .
1
x̃1
2 we have
√
δr
x1
√3
|x3 | ≤ 1− δr3 =
= 0 and U1 (x̃1 |σ(x̂)) − U1 (x1 |σ(x̂)) ≥ 0 is
T1 . When δr3 <
1
2
we have x̃1 = x1 (1 − 2δr3 )
and U1 (x̃1 |σ(x̂)) − U1 (x1 |σ(x̂)) ≥ 0 is equivalent to
T1 . In both cases
x1
|x3 |
(A16)
x1
|x3 |
≤
√ 1
2 1−δr3 −1
=
≤ T1 holds since G satisfies condition E. Therefore,
arg maxp∈A(x|σ(x̂)) U1 (p|σ(x̂)) equals |x| ∀x ∈ [−x̃1 , x̃1 ] and equals x̃1 ∀x ∈
X \ [−x̃1 , x̃1 ], that is, it is identical to the policy generated by the c-strategy
induced by x̂1 = x̃1 . As a result σ(x̂) with x̂ = {x̃1 , 0, x3 } constitutes an
SMPE.
Proposition A3. Suppose G does not satisfy condition E. Then σ(x̂a )
constitutes an SMPE, where x̂a = {(x̃1 , x̃a ), 0, x3 }.
Proof. Fix x̂a = {(x̃1 , x̃a ), 0, x3 }. Because G does not satisfy condition E,
we have x̃1 < |x̃3 | and
x1
|x3 |
> T1 . From x̃1 < |x̃3 |, x̃3 = x3 (1 − 2δr1 ). Since
x1 (1 − 2δr3 ) ≤ x̃1 < |x̃3 | ≤ |x3 |, x̃1 ∈ [0, |x3 |) and x1 (1 − 2δr3 ) < |x3 |.
The latter inequality and
x1
|x3 |
> T1 , from Definition A3 and the subse-
quent remark, imply that x̃a is well defined and x̃a ∈ (|x3 |, x1 ). Therefore,
Lemma A1 applies to x̂a since it satisfies Condition A1. The voting strategies in σ(x̂a ) thus satisfy Definition 1. What remains to be shown is that
26
pi (x|x̂i ) ∈ arg maxp∈A(x|σ(x̂)) Ui (p|σ(x̂a )) ∀i ∈ N and ∀x ∈ X. Recall that
A(x|σ(x̂a )) = [−|x|, |x|] ∀x ∈ X.
For i = 2, by Lemma A1 parts 2 and 4 and from (A13), U20 (x|σ(x̂a )) < 0
∀x ∈ [0, ∞) \ N D(x̂a ), U2 is continuous except at {−x̃a , x̃a }, U2 is syma
metric about 0, and U2 (x̃a |σ(x̂a )) ≥ U2 (x̃+
a |σ(x̂ )). Therefore, ∀x ∈ X,
arg maxp∈[−|x|,|x|] U2 (p|σ(x̂a )) = {0}.
The proposal strategy induced by
x̂2 = 0 thus satisfies the required property.
For i = 3, we study U3 on (−∞, 0]. From Lemma A1, U3 is continuous
a
except at {−x̃a , x̃a } and U3 (−x̃a |σ(x̂a )) ≥ U3 (−x̃−
a |σ(x̂ )). From (A13),
sgn[U30 (x|σ(x̂a ))] = sgn[−x + x3 (1 − 2δr1,nc (x|x̂a ))], where r1,nc (x|x̂a ) = r1
∀x ∈ (−x1 , −x̃a )∪(−x̃1 , 0) and r1,nc (x|x̂a ) = 0 ∀x ∈ (−∞, −x1 )∪(−x̃a , x3 )∪
(x3 , −x̃1 ). Hence U30 (x|σ(x̂a )) > 0 on (−∞, −x1 ) if x < x3 , which holds
since −x1 ≤ x3 , U30 (x|σ(x̂a )) > 0 on (−x1 , −x̃a ) if x < x3 (1 − 2δr1 ) = x̃3 ,
which holds since −x̃a < x3 ≤ x̃3 , U30 (x|σ(x̂a )) > 0 on (−x̃a , x3 ) if x < x3 ,
which holds, U30 (x|σ(x̂a )) < 0 on (x3 , −x̃1 ) if x > x3 , which holds, and
U30 (x|σ(x̂a )) < 0 on (−x̃1 , 0) if x > x3 (1 − 2δr1 ) = x̃3 , which holds since
x̃3 < −x̃1 . Therefore, arg maxp∈A(x|σ(x̂a )) U3 (p|σ(x̂a )) equals −|x| ∀x ∈
[x3 , |x3 |] and equals x3 ∀x ∈ X \ [x3 , |x3 |], that is, it is identical to the policy
generated by the d-strategy induced by x̂3 = x3 .
For i = 1, we study U1 on [0, ∞). From Lemma A1, U1 is continuous on
X. From (A13), sgn[U10 (x|σ(x̂a ))] = sgn[−x + x1 (1 − 2δr3,nc (x|x̂a ))], where
r3,nc (x|x̂a ) = r3 ∀x ∈ (0, x̃1 ) ∪ (x̃1 , |x3 |) and r3,nc (x|x̂a ) = 0 ∀x ∈ (|x3 |, x̃a ) ∪
(x̃a , x1 ) ∪ (x1 , ∞). Hence U10 (x|σ(x̂a )) > 0 on (0, x̃1 ) if x < x1 (1 − 2δr3 ),
which holds since x̃1 = x1 (1 − 2δr3 ) if x̃1 > 0, U10 (x|σ(x̂a )) < 0 on (x̃1 , |x3 |)
if x > x1 (1 − 2δr3 ), which holds since x̃1 ≥ x1 (1 − 2δr3 ), U10 (x|σ(x̂a )) > 0 on
(|x3 |, x̃a ) ∪ (x̃a , x1 ) if x < x1 , which holds, and U10 (x|σ(x̂a )) < 0 on (x1 , ∞)
if x > x1 , which holds. Recall U1 (x̃1 |σ(x̂a )) = U1 (x̃a |σ(x̂a )) we have shown
when proving Lemma A1 part 4. Therefore, arg maxp∈A(x|σ(x̂a )) U1 (p|σ(x̂a ))
equals |x| ∀x ∈ [−x̃1 , x̃1 ], equals x̃1 ∀x ∈ (−x̃a , −x̃1 ] ∪ [x̃1 , x̃a ), equals |x|
∀x ∈ [−x1 , −x̃a ) ∪ (x̃a , x1 ], and equals x1 ∀x ∈ X \ [−x1 , x1 ]. For the statusquo in {−x̃a , x̃a }, 1 is indifferent between two optimal proposals x̃1 and x̃a .
Hence, ∀x ∈ X, p1 (x|x̃1 , x̃a ) ∈ arg maxp∈A(x|σ(x̂a )) U1 (p|σ(x̂a )). As a result
σ(x̂a ) with x̂a = {(x̃1 , x̃a ), 0, x3 } constitutes an SMPE.
27
A2
Proof of Proposition 1
Lemma A2. Suppose σ̂ = ((p̂i , v̂i ))i∈N constitutes an SMPE. Then p̂2 (x) =
0 ∀x ∈ X and p̂i (0) = 0 for i ∈ {1, 3}.
Proof. Fix σ̂ = ((p̂i , v̂i ))i∈N that constitutes an SMPE. Proposal p̂i (0) by
any i ∈ N under status-quo x = 0 is accepted and by Proposition A1 satisfies
u2 (0) + δV2 (0|σ̂) ≤ u2 (p̂i (0)) + δV2 (p̂i (0)|σ̂).
Since V2 (0|σ̂) =
P
i∈N
(A17)
ri [u2 (p̂i (0))+δV2 (p̂i (0)|σ̂)], multiplying the inequality
by ri and summing over i ∈ N gives u2 (0) + δV2 (0|σ̂) ≤ V2 (0|σ̂), so that
u2 (0) ≤ (1 − δ)V2 (0|σ̂). Because Vi (x|σ̂) ≤ 0 ∀x ∈ X and ∀i ∈ N and
u2 (0) = 0, V2 (0|σ̂) = 0 and thus U2 (0|σ̂) = 0. Moreover, U2 (x|σ̂) < 0
∀x ∈ X \ {0} so that A(0|σ̂) = {0} and A(x|σ̂) 3 0 ∀x ∈ X and the lemma
follows.
Now, fix σ(x̂) that constitutes an SMPE. By Lemma A2, p2 (x|x̂2 ) = 0
∀x ∈ X, and hence x̂2 = 0. By the same lemma, pi (0|x̂i ) = 0 for i ∈ {1, 3}
and hence, from Definition A1, x̂1 ≥ 0 ≥ x̂3 . Therefore, by Lemma A1 and
the subsequent discussion, A(x|σ(x̂)) = [−|x|, |x|] ∀x ∈ X. Properties of
dynamic utilities in the following lemma are instrumental in what follows.
Lemma A3. Suppose σ(x̂) with x̂ = {x̂1 , x̂2 , x̂3 } and x̂1 ≥ x̂2 = 0 ≥ x̂3
satisfies, ∀i ∈ N and ∀x ∈ X, pi (x|x̂i ) ∈ A(x|σ(x̂)).
1. Ui (x|σ(x̂)) is continuous ∀x ∈ X and Ui0 (x|σ(x̂)) exists ∀x ∈ D(x̂),
∀i ∈ N .
2. For any x ∈ D(x̂) ∩ R+ , U10 (x|σ(x̂)) > 0 if x < c1 , U10 (x|σ(x̂)) = 0 if
x = c1 and U10 (x|σ(x̂)) < 0 if x > c1 , where c1 = x̃1 if x < |x̂3 | and
c1 = x1 if x > |x̂3 |.
3. For any x ∈ D(x̂) ∩ R− , U30 (x|σ(x̂)) > 0 if x < c3 , U30 (x|σ(x̂)) = 0 if
x = c3 and U30 (x|σ(x̂)) < 0 if x > c3 , where c3 = x̃3 if x > −x̂1 and
c3 = x3 if x < −x̂1 .
Proof. Fix σ(x̂) with the stated properties. Because pi (x|x̂i ) ∈ A(x|σ(x̂))
∀x ∈ X and ∀i ∈ N , Ui (x|σ(x̂)) = Ui (x|x̂) ∀x ∈ X and ∀i ∈ N . Part 1 then
follows, since x̂1 ≥ x̂2 = 0 ≥ x̂3 , by Lemma A1 parts 3 and 5.
28
To prove part 2, from (A13) we have
2
U10 (x|σ(x̂)) = − 1−δrnc
(x|x̂) [x − x1 (1 − 2δr3,nc (x|x̂))]
(A18)
when x > 0 and x ∈ D(x̂). (Note that 0 ∈
/ D(x̂) and thus x > 0 suffices.)
When x > |x̂3 |, r3,nc (x|x̂) = 0 and the sign of U10 (x|σ(x̂)) is determined by
the sign of x − x1 . When x < |x̂3 |, r3,nc (x|x̂) = r3 and the sign of U10 (x|σ(x̂))
is determined by the sign of x − x1 (1 − 2δr3 ). We have x1 (1 − 2δr3 ) = x̃1
when x1 (1 − 2δr3 ) ≥ 0. When x1 (1 − 2δr3 ) < 0, U10 (x|σ(x̂)) < 0 for any
x ∈ D(x̂)∩[0, |x̂3 |] and we can set c1 = x̃1 = 0. The proof of part 3 is similar
and omitted.
We now claim that x̂i ∈ {x̃i , xi } for i ∈ {1, 3}. We provide detailed
argument for i = 1. For i = 3 the argument is similar and omitted. Since
σ(x̂) constitutes an SMPE, p1 (x|x̂1 ) ∈ arg maxs∈[0,|x|] U1 (s|σ(x̂)) ∀x ∈ X,
where we have already used U1 (−s|σ(x̂)) < U1 (s|σ(x̂)) for any s > 0, which
follows by the symmetry of V1 (s|σ(x̂)) about 0. Notice that δ ∈ (0, 1) and
r3 > 0 imply x̃1 < x1 .
Suppose, towards a first contradiction, x̂1 < x̃1 . Since x̂1 ≥ 0, then,
by Lemma A3, there exists ¯ > 0 such that ∀ ∈ (0, ¯), U10 (x̂1 + |σ(x̂)) >
0 and hence p1 (x̂1 + |x̂1 ) = x̂1 ∈
/ arg maxs∈[0,x̂1 +] U1 (s|σ(x̂)). Suppose,
towards a second contradiction, x̂1 ∈ (x̃1 , x1 ). If |x̂3 | ≤ x̂1 , the argument
just made leads to a contradiction. If |x̂3 | > x̂1 , then, by Lemma A3,
there exists ¯ > 0 such that ∀ ∈ (0, ¯), U10 (x̃1 + |σ(x̂)) < 0 and hence
p1 (x̃1 + |x̂1 ) = x̃1 + ∈
/ arg maxs∈[0,x̃1 +] U1 (s|σ(x̂)). Suppose, towards a
third contradiction, x̂1 > x1 . Then, by Lemma A3, there exists ¯ > 0 such
that ∀ ∈ (0, ¯), U10 (x1 + |σ(x̂)) < 0 and hence p1 (x1 + |x̂1 ) = x1 + ∈
/
arg maxs∈[0,x1 +] U1 (s|σ(x̂)).
What remains to be shown is that σ(x̂) with x̂ = {x1 , 0, x3 } or x̂ =
{x̃1 , 0, x̃3 } cannot constitute an SMPE. Consider σ(x̂) with x̂ = {x1 , 0, x3 }.
Notice that δ ∈ (0, 1) and r1 > 0 imply x3 < x̃3 and hence −x1 ≤ x3 < x̃3 .
By Lemma A3, there exists ¯ > 0 such that ∀ ∈ (0, ¯), U30 (x̃3 − |σ(x̂)) > 0
and hence p3 (x̃3 − |x3 ) = x̃3 − ∈
/ arg maxs∈[x̃3 −,0] U3 (s|σ(x̂)). (Where we
have simplified the optimization problem for i = 3 in a similar fashion as
for i = 1 above.) Consider σ(x̂) with x̂ = {x̃1 , 0, x̃3 }. If |x̃3 | ≤ x̃1 , then,
by Lemma A3, there exists ¯ > 0 such that ∀ ∈ (0, ¯), U10 (x̃1 + |σ(x̂)) > 0
and hence p1 (x̃1 + |x̃1 ) = x̃1 ∈
/ arg maxs∈[0,x̃1 +] U1 (s|σ(x̂)). If, |x̃3 | > x̃1 ,
29
we have x3 < x̃3 < −x̃1 . Then, by Lemma A3, there exists ¯ > 0 such
that ∀ ∈ (0, ¯), U30 (x̃3 − |σ(x̂)) < 0 and hence p3 (x̃3 − |x̃3 ) = x̃3 ∈
/
arg maxs∈[x̃3 −,0] U3 (s|σ(x̂)). This concludes the proof of Proposition 1.
We remark here and use in the proof of Proposition 2 that i) if x̃1 <
|x̃3 |, then σ(x̂) with x̂ = {x1 , 0, x̃3 } cannot constitute an SMPE; and ii)
if x̃1 > |x̃3 |, then σ(x̂) with x̂ = {x̃1 , 0, x3 } cannot constitute an SMPE.
To see the first claim, suppose, towards a contradiction, that σ(x̂) with
x̂ = {x1 , 0, x̃3 } constitutes an SMPE and x̃1 < |x̃3 |. Then, by Lemma A3,
there exists ¯ > 0 such that ∀ ∈ (0, ¯), U10 (x̃1 + |σ(x̂)) < 0 and hence
p1 (x̃1 + |x1 ) = x̃1 + ∈
/ arg maxs∈[0,x̃1 +] U1 (s|σ(x̂)). To see the second
claim, suppose, towards a contradiction, that σ(x̂) with x̂ = {x̃1 , 0, x3 }
constitutes an SMPE and x̃1 > |x̃3 |. Then, by Lemma A3, there exists
¯ > 0 such that ∀ ∈ (0, ¯), U30 (x̃3 − |σ(x̂)) > 0 and hence p3 (x̃3 − |x3 ) =
x̃3 − ∈
/ arg maxs∈[x̃3 −,0] U3 (s|σ(x̂)).
A3
Proof of Proposition 2
Part 1, the if part, that σ(x̂) with x̂ = {x1 , 0, x̃3 } constitutes an SMPE when
x̃1 ≥ |x̃3 |, follows directly from Proposition A2. The only if part, that σ(x̂)
with x̂ = {x1 , 0, x̃3 } does not constitute an SMPE when x̃1 < |x̃3 |, follows
from the remark we made at the end of the proof of Proposition 1.
Part 2, the if part, that σ(x̂) with x̂ = {x̃1 , 0, x3 } constitutes an SMPE
when x̃1 ≤ |x̃3 | and
x1
|x3 |
≤ T1 , follows, when x̃1 < |x̃3 | and
from Proposition A2. When x̃1 = |x̃3 | and
x1
|x3 |
x1
|x3 |
≤ T1 ,
≤ T1 , an argument almost
identical to the one made in the proof of Proposition A2 (while considering
the x̂ = {x̃1 , 0, x3 } case) shows that σ(x̂) with x̂ = {x̃1 , 0, x3 } constitutes an
SMPE. The only if part, that σ(x̂) with x̂ = {x̃1 , 0, x3 } does not constitute
an SMPE when x̃1 > |x̃3 | or
x1
|x3 |
> T1 , follows, when x̃1 > |x̃3 |, from the
remark we made at the end of the proof of Proposition 1. To conclude,
we need to show that when
x1
|x3 |
> T1 , σ(x̂) with x̂ = {x̃1 , 0, x3 } cannot
constitute an SMPE. To see this, proof of Proposition A2, namely (A16)
and the subsequent discussion, shows that, for σ(x̂) with x̂ = {x̃1 , 0, x3 },
x1
|x3 |
> T1 is equivalent to U1 (x̃1 |σ(x̂)) < U1 (x1 |σ(x̂)). Thus σ(x̂) with
x̂ = {x̃1 , 0, x3 } cannot constitute an SMPE when
x̃1 ∈
/ arg maxs∈[0,x1 ] U1 (s|σ(x̂)).
30
x1
|x3 |
> T1 as p1 (x1 |x̃1 ) =
A4
Proof of Proposition 3
We first prove that the c-strategies are a subset of the d-strategies (Lemma
A4) and that the condition on the d-strategies used in Proposition 3 separates the two classes of strategies (Lemma A5).
Lemma A4. For any x̂c1 , p1 (x|x̂c1 ) = p1 (x|x̂d1 , xda ) ∀x ∈ X if x̂d1 = x̂c1 and
xda = sup X.
Proof. Definition A2 implies that p1 (x|x̂d1 , xda ) = p1 (x|x̂d1 ) ∀x ∈ X when
xda = sup X.
Lemma A5. There does not exist x̂c1 such that p1 (x|x̂c1 ) = p1 (x|x̂d1 , xda )
∀x ∈ X if and only if x̂d1 6= x1 and |xda | ∈ (min{x̂d1 , x1 }, sup X).
Proof. For the if part, by Definition A1, p1 (x|x̂c1 ) = min{x̂c1 , |x|} and thus,
∀x ∈ X, p1 (x|x̂c1 ) is continuous in x for any x̂c1 . Hence, if p1 (x|x̂d1 , xda ) is not
continuous in x, x̂c1 such that p1 (x|x̂c1 ) = p1 (x|x̂d1 , xda ) ∀x ∈ X does not exist.
Now, suppose x̂d1 6= x1 and |xda | ∈ (min{x̂d1 , x1 }, sup X). If |xda | = 0, then
x̂d1 < 0 and, by Definition A2, p1 (0|x̂d1 , xda ) = x̂d1 < 0. By the same definition
p1 (x|x̂d1 , xda ) = p1 (x|x1 ) ∀x ∈ X \ {0} and thus limx→0 p1 (x|x̂d1 , xda ) = 0. If
|xda | > 0, by Definition A2, p1 (|xda | |x̂d1 , xda ) = p1 (|xda | |x̂d1 ) = min{x̂d1 , |xda |}.
Because |xda | < sup X, by the same definition, limx→|xda |+ p1 (x|x̂d1 , xda ) =
limx→|xda |+ p1 (x|x1 ) = limx→|xda |+ min{x1 , |x|} = min{x1 , |xda |}. As x̂d1 6= x1 ,
min{x̂d1 , |xda |} = min{x1 , |xda |} only when |xda | ≤ x1 and |xda | ≤ x̂d1 , or, equivalently, |xda | ≤ min{x1 , x̂d1 }. Therefore, min{x̂d1 , |xda |} 6= min{x1 , |xda |} when
|xda | ∈ (min{x̂d1 , x1 }, sup X).
For the only if part, suppose, towards a contradiction, that x̂d1 = x1 or
|xda | ≤ min{x̂d1 , x1 } or |xda | ≥ sup X. If x̂d1 = x1 , p1 (x|x̂d1 , xa ) = p1 (x|x1 )
∀x ∈ X for any xa directly from Definition A2. If |xda | ≤ min{x̂d1 , x1 },
p1 (x|x̂d1 , xda ) = |x| = p1 (x|x1 ) ∀x ∈ [−|xda |, |xda |] and p1 (x|x̂d1 , xda ) = p1 (x|x1 )
∀x ∈ X \ [−|xda |, |xda |] from Definition A2. If |xda | ≥ sup X, p1 (x|x̂d1 , xda ) =
p1 (x|x̂d1 ) ∀x ∈ X from Definition A2. The lemma now follows by setting
x̂c1 = x̂d1 .
We now proceed via a series of steps and lemmas. Throughout, fix σ(x̂a )
with x̂a = {(x̂1 , xa ), x̂2 , x̂3 }, x̂1 6= x1 and |xa | ∈ (min{x̂1 , x1 }, sup X) that
constitutes an SMPE.
31
Step 1: x̂1 ≥ x̂2 = 0 ≥ x̂3 . By Lemma A2, p2 (x|x̂2 ) = 0 ∀x ∈ X, and
hence x̂2 = 0. By the same lemma, pi (0|x̂i ) = 0 for i ∈ {1, 3} and hence,
from Definition A2, x̂1 ≥ 0 ≥ x̂3 .
The subsequent steps make use of the following lemma. We explain its
usefulness after proving it.
Lemma A6. Consider two profiles σ̂ = ((p̂i , v̂i ))i∈N and σ̂ 0 = ((p̂0i , v̂i0 ))i∈N .
Suppose, ∀i ∈ N and ∀x ∈ X, p̂i (x) ∈ A(x|σ̂) and p̂0i (x) ∈ A(x|σ̂ 0 ). Consider
any U ⊆ X such that i) x ∈ U ⇒ [−|x|, |x|] ⊆ U ; and ii) ∀i ∈ N and ∀x ∈ U ,
p̂i (x) = p̂0i (x) ∈ [−|x|, |x|]. Then, ∀i ∈ N and ∀x ∈ U , Vi (x|σ̂) = Vi (x|σ̂ 0 ).
Proof. Because any proposal generated by σ̂ or σ̂ 0 is always accepted, it
suffices to show that, for an arbitrary sequence of proposers starting from
status-quo x ∈ U , the resulting sequences of policies pσ̂ = {pσ̂0 , pσ̂1 , . . .} and
0
0
0
pσ̂ = {pσ̂0 , pσ̂1 , . . .} generated by σ̂ and σ̂ 0 respectively are identical. Since
0
x ∈ U , for the first proposer i, p̂i (x) = p̂0i (x) = pσ̂0 = pσ̂0 ∈ [−|x|, |x|] ⊆ U . By
0
0
a similar argument pσ̂1 = pσ̂1 ∈ U and more generally pσ̂t = pσ̂t ∀t ∈ N+ .
Notice that the proposal strategies in σ(x̂a ) with x̂a = {(x̂1 , xa ), x̂2 , x̂3 }
coincide with the proposal strategies in σ(x̂) with x̂ = {x̂1 , x̂2 , x̂3 } on
[−|xa |, |xa |], and any proposal generated by either of the two profiles is
accepted; σ(x̂a ) since it constitutes an SMPE, σ(x̂) by Lemma A1 and the
subsequent discussion, since we already know x̂1 ≥ x̂2 = 0 ≥ x̂3 . Thus the
two profiles induce identical continuation value functions on [−|xa |, |xa |],
which in turn makes Lemma A3 applicable, on [−|xa |, |xa |], to the dynamic
utilities induced by σ(x̂a ). Moreover, by Lemma A1, A(x|σ(x̂)) = [−|x|, |x|]
∀x ∈ X. Since the dynamic utilities induced by σ(x̂a ) and σ(x̂) coincide on
[−|xa |, |xa |], A(x|σ(x̂a )) ⊇ [−|x|, |x|] ∀x ∈ [−|xa |, |xa |].
Step 2: x̂1 < x1 . Suppose, towards a contradiction, x̂1 > x1 . Since
x̂1 6= x1 , this suffices. Because |xa | > min{x̂1 , x1 }, we have |xa | > x1 .
Then, by Lemma A3, applicable on [−|xa |, |xa |] ) [−x1 , x1 ] by Lemma A6,
there exists ¯ > 0 such that ∀ ∈ (0, ¯), U10 (x1 + |σ(x̂a )) < 0 and hence
p1 (x1 + |x̂1 , xa ) = x1 + ∈
/ arg maxs∈A(x1 +|σ(x̂a )) U1 (s|σ(x̂a )).
The following lemma implies that A(x|σ(x̂a )) = [−|x|, |x|] ∀x ∈ X, since
x1 > x̂1 ≥ x̂2 = 0 ≥ x̂3 from the preceding steps implies that the proposal
strategies in σ(x̂a ) satisfy its requirements.
Lemma A7. Suppose i) σ̂ = ((p̂i , v̂i ))i∈N constitutes an SMPE; ii) ∀x ∈ X,
p̂2 (x) = 0 and p̂i (−|x|) = p̂i (|x|) for i ∈ {1, 3}; and iii) ∀x ∈ X and ∀y ∈ X
32
such that 0 ≤ x ≤ y, 0 ≤ p̂1 (x) ≤ p̂1 (y) and 0 ≥ p̂3 (x) ≥ p̂3 (y). Then,
∀x ∈ X, A(x|σ̂) = [−|x|, |x|].
Proof. Fix σ̂ = ((p̂i , v̂i ))i∈N that constitutes an SMPE. By the symmetry
of the proposal strategies about 0, U2 (−|x| |σ̂) = U2 (|x| |σ̂) ∀x ∈ X. By
Proposition A1 it thus suffices to prove U2 (x|σ̂) > U2 (y|σ̂) ∀x ∈ X and
∀y ∈ X such that 0 ≤ x < y. Fix x ∈ X and y ∈ X such that 0 ≤ x < y.
Since u2 (x) > u2 (y), it suffices to show that V2 (x|σ̂) ≥ V2 (y|σ̂). Consider
an arbitrary sequence of proposers and the resulting sequences of policies
px = {px0 , px1 , . . .} and py = {py0 , py1 , . . .} starting from status-quo x and y
respectively. Since p̂2 (x) = p̂2 (y) = 0, 0 ≤ p̂1 (x) ≤ p̂1 (y) and 0 ≥ p̂3 (x) ≥
p̂3 (y), |px0 | ≤ |py0 |. By a similar argument |px1 | ≤ |py1 | and more generally
P
P∞
t
z
t z 2
|pxt | ≤ |pyt | ∀t ∈ N+ . Since V2 (z|σ̂) = E[ ∞
t=0 δ u2 (pt )] = E[ t=0 −δ |pt | ]
for z ∈ {x, y}, where the expectation is over proposer sequences, it follows
that V2 (x|σ̂) ≥ V2 (y|σ̂).
Step 3: x̂1 = x̃1 . Steps 1 and 2 imply x̂1 ≥ 0 and x̂1 ≤ x1 respectively.
The remaining argument is identical to two contradictions, labelled first and
second, in the proof of Proposition 1. The former, x̂1 < x̃1 , uses open neighbourhood above x̂1 , while the latter, x̂1 ∈ (x̃1 , x1 ), uses open neighbourhood
above x̂1 or x̃1 , depending on whether |x̂3 | ≤ x̂1 or |x̂3 | > x̂1 . Each neighbourhood needs to be covered by Lemma A3, which is indeed the case due
to Lemma A6 and |xa | > min{x̂1 , x1 } = x̂1 .
Step 4: |x̂3 | > x̃1 . By Step 3, min{x̂1 , x1 } = x̃1 and thus |xa | > x̃1 .
Suppose, towards a contradiction, |x̂3 | ≤ x̃1 . Since δ ∈ (0, 1) and r3 > 0,
x̃1 < x1 . Then, by Lemma A3, applicable on [−|xa |, |xa |] ) [−x̃1 , x̃1 ] by
Lemma A6, there exists ¯ > 0 such that ∀ ∈ (0, ¯), U10 (x̃1 + |σ(x̂a )) > 0
and hence p1 (x̃1 + |x̃1 , xa ) = x̃1 ∈
/ arg maxs∈[0,x̃1 +] U1 (s|σ(x̂a )).
The following lemma is instrumental in proving Step 5 below. To understand the objects in the lemma, note that when |xa | ∈ (x̃1 , sup X), we have
p1 (|xa | |x̃1 , xa ) = x̃1 and limx→|xa |+ p1 (x|x̃1 , xa ) = min{|xa |, x1 }.
Lemma A8. Suppose σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x̂3 } such that |xa | ∈
(x̃1 , sup X) constitutes an SMPE. U1 (x̃1 |σ(x̂a )) = U1 (min{|xa |, x1 }|σ(x̂a )).
Proof. Fix σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x̂3 } where |xa | ∈ (x̃1 , sup X) that
constitutes an SMPE. Since x1 > x̃1 ≥ x̂2 = 0 ≥ x̂3 , A(x|σ(x̂a )) = [−|x|, |x|]
∀x ∈ X by Lemma A7.
33
When |xa | ∈ [x1 , sup X), we need to show U1 (x̃1 |σ(x̂a )) = U1 (x1 |σ(x̂a )).
From Definition A2 and since |xa | > x̃1 , p1 (|xa | |x̃1 , xa ) = p1 (|xa | |x̃1 ) =
min{x̃1 , |xa |} = x̃1 . Moreover, since A(|xa | |σ(x̂a )) = [−|xa |, |xa |] and |xa | ≥
x1 , x1 ∈ A(|xa | |σ(x̂a )). Hence U1 (x̃1 |σ(x̂a )) ≥ U1 (x1 |σ(x̂a )) since, under
status-quo |xa |, x1 would be accepted but x̃1 is proposed. At the same time,
since |xa | ∈ [x1 , sup X), there exists > 0 such that p1 (|xa | + |x̃1 , xa ) =
p1 (|xa | + |x1 ) = min{x1 , |xa | + } = x1 and x̃1 ∈ A(|xa | + |σ(x̂a )). Hence
U1 (x̃1 |σ(x̂a )) ≤ U1 (x1 |σ(x̂a )) since, under status-quo |xa | + , x̃1 would be
accepted but x1 is proposed.
What remains is, when |xa | ∈ (x̃1 , x1 ), U1 (x̃1 |σ(x̂a )) = U1 (|xa | |σ(x̂a )).
U1 (x̃1 |σ(x̂a )) ≥ U1 (|xa | |σ(x̂a )) follows since p1 (|xa | |x̃1 , xa ) = x̃1 and |xa | ∈
A(|xa | |σ(x̂a )). We now claim U1 (x̃1 |σ(x̂a )) ≤ U1 (|xa | |σ(x̂a )), which requires a more elaborate argument. First, notice that since σ(x̂a ) constitutes an SMPE, V1 (|xa | |σ(x̂a )) ≤ V1 (|xa |+ |σ(x̂a )). If V1 (|xa | |σ(x̂a )) >
V1 (|xa |+ |σ(x̂a )) then U1 (|xa | |σ(x̂a )) > U1 (|xa |+ |σ(x̂a )) and there exists
¯ > 0 such that ∀ ∈ (0, ¯), U1 (|xa | |σ(x̂a )) > U1 (|xa | + |σ(x̂a )). But then
p1 (|xa | + |x̃1 , xa ) = |xa | + ∈
/ arg maxs∈[0,|xa |+] U1 (s|σ(x̂a )). Second, we
claim V1 (|xa | |σ(x̂a )) − V1 (|xa |+ |σ(x̂a )) = c · [U1 (x̃1 |σ(x̂a )) − U1 (|xa | |σ(x̂a ))],
where c > 0. Suppose |x̂3 | > |xa |. Then, since i) p2 (x|0) = 0 ∀x ∈ X; ii)
p1 (|xa | |x̃1 , xa ) = x̃1 ; iii) p3 (|xa | |x̂3 ) = −|xa |; and iv) there exists ¯ > 0 such
that ∀ ∈ (0, ¯), p1 (|xa | + |x̃1 , xa ) = −p3 (|xa | + |x̂3 ) = |xa | + ; we have
V1 (|xa | |σ(x̂a )) − V1 (|xa |+ |σ(x̂a )) =
= r1 [U1 (x̃1 |σ(x̂a )) − U1 (|xa |+ |σ(x̂a ))]
(A19)
+ r3 [U1 (−|xa | |σ(x̂a )) − U1 (−|xa |− |σ(x̂a ))].
The term in the first square brackets rewrites as
U1 (x̃1 |σ(x̂a )) − U1 (|xa |+ |σ(x̂a )) =
= U1 (x̃1 |σ(x̂a )) − U1 (|xa | |σ(x̂a )) + U1 (|xa | |σ(x̂a )) − U1 (|xa |+ |σ(x̂a ))
= U1 (x̃1 |σ(x̂a )) − U1 (|xa | |σ(x̂a )) + δ[V1 (|xa | |σ(x̂a )) − V1 (|xa |+ |σ(x̂a ))]
since U1 (x|σ(x̂a )) = u1 (x) + δV1 (x|σ(x̂a )) ∀x ∈ X and u1 is a continuous
34
function. The term in the second square brackets rewrites as
U1 (−|xa | |σ(x̂a )) − U1 (−|xa |− |σ(x̂a )) =
= δ[V1 (−|xa | |σ(x̂a )) − V1 (−|xa |− |σ(x̂a ))]
= δ[V1 (|xa | |σ(x̂a )) − V1 (|xa |+ |σ(x̂a ))]
since u1 is a continuous function and V1 (x|σ(x̂a )) = V1 (−x|σ(x̂a )) ∀x ∈ X.
Thus (A19) rewrites as
V1 (|xa | |σ(x̂a )) − V1 (|xa |+ |σ(x̂a )) =
=
a
r1
1−δ(r1 +r3 ) [U1 (x̃1 |σ(x̂ ))
− U1 (|xa | |σ(x̂a ))].
(A20)
When |x̂3 | ≤ |xa |, the argument is similar, simpler (the term in the second
square brackets in (A19) equals zero) and omitted. Third, since we have
V1 (|xa | |σ(x̂a )) ≤ V1 (|xa |+ |σ(x̂a )) in an SMPE as well as V1 (|xa | |σ(x̂a )) −
V1 (|xa |+ |σ(x̂a )) = c · [U1 (x̃1 |σ(x̂a )) − U1 (|xa | |σ(x̂a ))] with c > 0, it must be
the case that U1 (x̃1 |σ(x̂a )) ≤ U1 (|xa | |σ(x̂a )), which was to be shown.
Step 5: |x̂3 | < |xa | and thus x̂3 = x3 . Suppose first, towards a contradiction, |xa | ≤ |x̂3 |. Since |xa | > x̃1 , by Lemma A3, applicable on [−|xa |, |xa |]
by Lemma A6, U1 (x|σ(x̂a )) is continuous ∀x ∈ [x̃1 , |xa |] and U10 (x|σ(x̂a )) < 0
∀x ∈ (x̃1 , |xa |) \ {x1 }. Thus, ∀x ∈ (x̃1 , |xa |], U1 (x̃1 |σ(x̂a )) > U1 (x|σ(x̂a )).
Therefore, ∀|xa | ∈ (x̃1 , |x̂3 |], U1 (x̃1 |σ(x̂a )) > U1 (min{|xa |, x1 }|σ(x̂a )), a contradiction to Lemma A8.
We now argue that x̂3 = x3 . We have x̃1 < |x̂3 | by Step 4 and |x̂3 | < |xa |
just shown. Suppose, towards a first contradiction, x3 < x̂3 . Then, by
Lemma A3, applicable on [−|xa |, |xa |] ) [x̂3 , −x̂3 ] by Lemma A6, there exists
¯ > 0 such that ∀ ∈ (0, ¯), U30 (x̂3 − |σ(x̂a )) < 0 and hence p3 (x̂3 − |x̂3 ) =
x̂3 ∈
/ arg maxs∈[x̂3 −,0] U3 (s|σ(x̂a )). Suppose, towards a second contradiction, x̂3 < x3 . Then, by Lemma A3, applicable on [−|xa |, |xa |] ) [x3 , −x3 ]
by Lemma A6, there exists ¯ > 0 such that ∀ ∈ (0, ¯), U30 (x3 − |σ(x̂a )) > 0
and hence p3 (x3 − |x̂3 ) = x3 − ∈
/ arg maxs∈[x3 −,0] U3 (s|σ(x̂a )).
We remark here and use in the proof of Proposition 4 that if x̃1 > |x̃3 |,
then σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 } such that |xa | ∈ (|x3 |, sup X) does
not constitute an SMPE. Suppose, towards a contradiction, that σ(x̂a ) with
x̂a = {(x̃1 , xa ), 0, x3 } such that |xa | ∈ (|x3 |, sup X) constitutes an SMPE
and x̃1 > |x̃3 |. Then, by Lemma A3, applicable on [−|xa |, |xa |] ) [x̃3 , −x̃3 ]
35
by Lemma A6, there exists ¯ > 0 such that ∀ ∈ (0, ¯), U30 (x̃3 − |σ(x̂a )) > 0
and hence p3 (x̃3 − |x3 ) = x̃3 − ∈
/ arg maxs∈[x̃3 −,0] U3 (s|σ(x̂a )).
Step 6: If
x1
|x3 |
< T1 , then σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 }, where |xa | ∈
(|x3 |, sup X), does not constitute an SMPE. Note that by Step 4 |x̂3 | > x̃1
and by Step 5 x̂3 = x3 . Therefore x̃1 < |x3 | and (x̃1 , |x3 |) used below is
well defined. By Lemma A6, ∀x ∈ [−|xa |, |xa |], U1 (x|σ(x̂a )) = U1 (x|σ(x̂)),
where x̂ = {x̃1 , 0, x3 }. From Lemma A3, U1 (x|σ(x̂)) is continuous ∀x ∈ X,
U10 (x|σ(x̂)) < 0 on (x̃1 , |x3 |), U10 (x|σ(x̂)) > 0 on (|x3 |, x1 ), and U10 (x|σ(x̂)) <
0 on (x1 , ∞). Moreover, proof of Proposition A2, namely (A16) and the
subsequent discussion, shows that
U1 (x1 |σ(x̂)). Hence, if
(x̃1 , ∞) and thus U1 (x̃1
x1
|x3 |
x1
|x3 |
< T1 is equivalent to U1 (x̃1 |σ(x̂)) >
< T1 , then U1 (x̃1 |σ(x̂)) > U1 (x|σ(x̂)) ∀x ∈
|σ(x̂a ))
> U1 (x|σ(x̂a )) ∀x ∈ (x̃1 , |xa |]. Therefore,
∀|xa | ∈ (|x3 |, sup X), U1 (x̃1 |σ(x̂a )) > U1 (min{|xa |, x1 }|σ(x̂a )), a contradiction to Lemma A8.
Step 7: If
x1
|x3 |
> T1 , then σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 }, where
|xa | ∈ (|x3 |, sup X), constitutes an SMPE only for unique |xa | ∈ (|x3 |, x1 ).
Straightforward algebra shows T1 > 1 ∀δr3 ∈ (0, 1). Hence
x1
|x3 |
> T1 implies
x1 > |x3 | making (|x3 |, x1 ) below well defined. By the analysis of U1 (x|σ(x̂))
with x̂ = {x̃1 , 0, x3 } in Step 6,
Moreover, we have
x1
|x3 |
U10 (x|σ(x̂))
> T1 implies U1 (x̃1 |σ(x̂)) < U1 (x1 |σ(x̂)).
< 0 on (x̃1 , |x3 |) and U10 (x|σ(x̂)) > 0 on
(|x3 |, x1 ). Thus, there exists unique x0 ∈ (|x3 |, x1 ) such that U1 (x̃1 |σ(x̂)) =
U1 (x0 |σ(x̂)), U1 (x̃1 |σ(x̂)) > U1 (x|σ(x̂)) ∀x ∈ (|x3 |, x0 ) and U1 (x̃1 |σ(x̂)) <
U1 (x|σ(x̂)) ∀x ∈ (x0 , x1 ]. Since U1 (x|σ(x̂a )) = U1 (x|σ(x̂)) ∀x ∈ [−|xa |, |xa |]
by Lemma A6, |xa | = x0 as any other |xa | ∈ (|x3 |, sup X) \ {x0 } implies
U1 (x̃1 |σ(x̂a )) 6= U1 (min{|xa |, x1 }|σ(x̂a )), a contradiction to Lemma A8.
Step 8: If
x1
|x3 |
= T1 , then σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 }, where |xa | ∈
(|x3 |, sup X), constitutes an SMPE only for arbitrary |xa | ∈ [x1 , sup X).
By Steps 6 and 7, for x̂ = {x̃1 , 0, x3 },
x1
|x3 |
= T1 implies U1 (x̃1 |σ(x̂)) =
U1 (x1 |σ(x̂)) as well as U1 (x̃1 |σ(x̂)) > U1 (x|σ(x̂)) ∀x ∈ (|x3 |, x1 ). Since
U1 (x|σ(x̂a )) = U1 (x|σ(x̂)) ∀x ∈ [−|xa |, |xa |] by Lemma A6, |xa | ∈ [x1 , sup X)
as any |xa | ∈ (|x3 |, x1 ) implies U1 (x̃1 |σ(x̂a )) > U1 (|xa | |σ(x̂a )), a contradiction to Lemma A8. That |xa | is arbitrary, conditional on |xa | ∈ [x1 , sup X),
follows from Proposition 4 part 2.
36
A5
Proof of Proposition 4
Part 1, the if part, that σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 } and |xa | = x̃a
constitutes an SMPE when x̃1 ≤ |x̃3 | and
and
x1
|x3 |
x1
|x3 |
> T1 , follows, when x̃1 < |x̃3 |
> T1 , from Proposition A3. When x̃1 = |x̃3 | and
x1
|x3 |
> T1 , an
argument almost identical to the one made in the proof of Proposition A3
shows that σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 } and |xa | = x̃a constitutes an
SMPE. The only if part, that σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 } and |xa | = x̃a
does not constitute an SMPE when x̃1 > |x̃3 | or
x1
|x3 |
≤ T1 , follows, when
x̃1 > |x̃3 |, from the remark we made after Step 5 of the proof of Proposition
3. When
x1
|x3 |
≤ T1 , Definition A3 leaves x̃a undefined. In fact, by Steps 6
and 8 of the proof of Proposition 3, σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 }, when
x1
|x3 |
< T1 , does not constitute an SMPE for any |xa | ∈ (|x3 |, sup X), which
is the interval |xa | must belong to by Proposition 3, and, when
x1
|x3 |
= T1 ,
constitutes an SMPE only for |xa | ∈ [x1 , sup X).
For Part 2, we first prove the easier only if part. That σ(x̂a ) with
x̂a = {(x̃1 , xa ), 0, x3 } and |xa | ∈ [x1 , sup X) does not constitute an SMPE
when x̃1 > |x̃3 | or
x1
|x3 |
6= T1 , follows, when x̃1 > |x̃3 |, from the remark we
made after Step 5 of the proof of Proposition 3. When
x1
|x3 |
< T1 , by Step
6 of the proof of Proposition 3, σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 } does not
constitute an SMPE for any |xa | ∈ (|x3 |, sup X), which is the interval |xa |
must belong to by Proposition 3. When
Proposition 3,
σ(x̂a )
with
x̂a
x1
|x3 |
> T1 , by Step 7 of the proof of
= {(x̃1 , xa ), 0, x3 } constitutes an SMPE only
for |xa | ∈ (|x3 |, x1 ).
What remains to be shown is that σ(x̂a ) with x̂a = {(x̃1 , xa ), 0, x3 },
where |xa | ∈ [x1 , sup X), constitutes an SMPE when x̃1 ≤ |x̃3 | and
Fix
x̂a
x1
|x3 |
= T1 .
= {(x̃1 , xa ), 0, x3 } with any |xa | ∈ [x1 , sup X).
First, by the symmetry of the proposal strategies, we have Vi (x|x̂a ) =
Vi (−x|x̂a ) ∀x ∈ X and ∀i ∈ N . Moreover, an argument analogous to the
proof of Lemma A7 shows that U2 (x|x̂a ) > U2 (y|x̂a ) ∀x ∈ X and ∀y ∈ X
such that 0 ≤ x < y, and thus A(x|x̂a ) = [−|x|, |x|] ∀x ∈ X. From Definition
A2, we thus have, ∀x ∈ X, p1 (x|x̃1 , xa ) ∈ A(x|x̂a ), p2 (x|0) ∈ A(x|x̂a ) and
p3 (x|x3 ) ∈ A(x|x̂a ). Thus, ∀x ∈ X and ∀i ∈ N , Vi (x|x̂a ) = Vi (x|σ(x̂a )),
Vi (x|x̂a ) = Vi (x|σ(x̂a )) and A(x|x̂a ) = A(x|σ(x̂a )).
Second, by Lemma A6, Vi (x|σ(x̂a )) = Vi (x|σ(x̂)) ∀x ∈ [−|xa |, |xa |]
and ∀i ∈ N , where x̂ = {x̃1 , 0, x3 }. Properties of the dynamic utilities
37
induced by σ(x̂) from Lemma A3 thus carry over to the dynamic utilities induced by σ(x̂a ), on [−|xa |, |xa |]. From the Proof of Proposition A2,
namely (A16) and the subsequent discussion, U1 (x̃1 |σ(x̂a )) = U1 (x1 |σ(x̂a )).
Since x̃1 < x1 and U2 (x̃1 |σ(x̂a )) > U2 (x1 |σ(x̂a )), an argument analogous to
the proof of Proposition A1 implies U3 (x̃1 |σ(x̂a )) > U3 (x1 |σ(x̂a )). Therefore, since Vi (|xa | |σ(x̂a )) = r1 Ui (x̃1 |σ(x̂a )) + r2 Ui (0|σ(x̂a )) + r3 Ui (x3 |σ(x̂a ))
and Vi (|xa |+ |σ(x̂a )) = r1 Ui (x1 |σ(x̂a )) + r2 Ui (0|σ(x̂a )) + r3 Ui (x3 |σ(x̂a )), we
have V1 (|xa | |σ(x̂a )) = V1 (|xa |+ |σ(x̂a ) and V3 (|xa | |σ(x̂a )) > V3 (|xa |+ |σ(x̂a ).
Moreover, Vi (x|σ(x̂a )) = Vi (|xa |+ |σ(x̂a )) ∀x ∈ (|xa |, ∞) and ∀i ∈ N , since
the proposal strategies are constant on (|xa |, ∞).
Third, from the preceding discussion U2 (x|σ(x̂a )) > U2 (y|σ(x̂a )) ∀x ∈ X
and ∀y ∈ X such that 0 ≤ x < y and hence we have, ∀x ∈ X, p2 (x|0) = 0 ∈
arg maxs∈A(x|σ(x̂a )) U2 (s|σ(x̂a )).
Fourth, from the preceding discussion U3 (x|σ(x̂a )) is continuous ∀x ∈
X \ {−|xa |, |xa |}, U3 (−|xa | |σ(x̂a ) > U3 (−|xa |− |σ(x̂a )), U30 (x|σ(x̂a )) > 0 on
(−∞, −|xa |)∪(−|xa |, x3 ) and U30 (x|σ(x̂a )) < 0 on (x3 , −x̃1 )∪(−x̃1 , 0). Thus,
∀x ∈ X, p3 (x|x3 ) = max{x3 , −|x|} ∈ arg maxs∈A(x|σ(x̂a )) U3 (s|σ(x̂a )).
Fifth, from the preceding discussion U1 (x|σ(x̂a )) is continuous ∀x ∈
X, U1 (x̃1 |σ(x̂a )) = U1 (x1 |σ(x̂a )), U10 (x|σ(x̂a )) > 0 on (0, x̃1 ) ∪ (x3 , x1 )
and U10 (x|σ(x̂a )) < 0 on (x̃1 , x3 ) ∪ (x1 , |xa |) ∪ (|xa |, ∞). Thus, for x ∈
[0, x̃1 ], x ∈ (x̃1 , x1 ) and x ∈ [x1 , ∞), arg maxs∈A(x|σ(x̂a )) U1 (s|σ(x̂a )) equals
{x}, {x̃1 } and {x̃1 , x1 } respectively. Therefore, ∀x ∈ X, p1 (x|x̃1 , xa ) ∈
arg maxs∈A(x|σ(x̂a )) U1 (s|σ(x̂a )), which concludes the proof.
A6
Consistent SMPE and all players moderating
This section defines consistent strategies and shows that a consistent SMPE
with all players proposing policies strictly more moderate than their static
bliss point for any status-quo does not exist. The definition of a consistent
proposal strategy is equivalent, modulo the notation, to the one used by
Forand (2014). See his work for an insightful motivation of consistency.
Definition A6. Given profile σ̂ = ((p̂i , v̂i ))i∈N , p̂i is consistent if, ∀x ∈ X
and ∀x0 ∈ X, p̂i (x) ∈ A(x0 |σ̂) and p̂i (x) 6= p̂i (x0 ) implies p̂i (x0 ) ∈
/ A(x|σ̂). A
consistent SMPE is an SMPE in consistent proposal strategies.
Proposition A4. There does not exist a consistent SMPE σ̂ = ((p̂i , v̂i ))i∈N
such that i) ∀x ∈ X, p̂i (−|x|) = p̂i (|x|) for i ∈ {1, 3}; ii) ∀x ∈ X and ∀y ∈ X
38
such that 0 ≤ x ≤ y, 0 ≤ p̂1 (x) ≤ p̂1 (y) and 0 ≥ p̂3 (x) ≥ p̂3 (y); and iii)
supx∈X p̂1 (x) < x1 and inf x∈X p̂3 (x) > x3 .
Proof. Suppose, towards a contradiction, that there exists a consistent SMPE
σ̂ = ((p̂i , v̂i ))i∈N such that the three requirements of the proposition hold.
By Lemma A2, p̂2 (x) = 0 ∀x ∈ X. Hence, by the first two requirements and
Lemma A7, we have A(x|σ̂) = [−|x|, |x|] ∀x ∈ X. Following lemma is the
main implication of consistency.
Lemma A9. Suppose i) σ̂ = ((p̂i , v̂i ))i∈N is a consistent SMPE; ii) ∀x ∈ X,
p̂i (−|x|) = p̂i (|x|) for i ∈ {1, 3}; and iii) ∀x ∈ X and ∀y ∈ X such that
0 ≤ x ≤ y, 0 ≤ p̂1 (x) ≤ p̂1 (y) and 0 ≥ p̂3 (x) ≥ p̂3 (y). If p̂i (x0 ) ∈ (−|x0 |, |x0 |)
for some i ∈ {1, 3} and some x0 ∈ X, then p̂i (x) = p̂i (x0 ) ∀x ∈ [|p̂i (x0 )|, |x0 |].
Proof. Fix σ̂ = ((p̂i , v̂i ))i∈N that constitutes a consistent SMPE. By Lemma
A2, p̂2 (x) = 0 ∀x ∈ X and thus, by Lemma A7, A(x|σ̂) = [−|x|, |x|].
Fix i ∈ {1, 3} and x0 ∈ X such that p̂i (x0 ) ∈ (−|x0 |, |x0 |), which implies
|p̂i (x0 )| < |x0 | so that [|p̂i (x0 )|, |x0 |] is well defined. Fix, towards a contradiction, x ∈ [|p̂i (x0 )|, |x0 |] such that p̂i (x) 6= p̂i (x0 ). Since x ∈ [|p̂i (x0 )|, |x0 |],
A(|p̂i (x0 )| |σ̂) ⊆ A(x|σ̂) ⊆ A(x0 |σ̂). Clearly p̂i (x0 ) ∈ A(|p̂i (x0 )| |σ̂) and thus
p̂i (x0 ) ∈ A(x|σ̂). However, p̂i (x) ∈ A(x|σ̂) implies p̂i (x) ∈ A(x0 |σ̂), which,
along with consistency and p̂i (x) 6= p̂i (x0 ), implies p̂i (x0 ) ∈
/ A(x|σ̂).
We proceed via a series of claims. First, we claim p̂1 (x1 ) = maxx∈X p̂1 (x)
and p̂3 (x3 ) = minx∈X p̂3 (x). We prove the claim for i = 1. The argument
for i = 3 is analogous and omitted. We start by showing that maxx∈X p̂1 (x)
exists. When X ( R, X = [−z, z] for some z ≥ x1 and thus, since p̂1 is nondecreasing on [0, z] and symmetric about 0, maxx∈X p̂1 (x) = p̂1 (z). When
X = R, if maxx∈X p̂1 (x) does not exist, for any x ∈ X, there exists y ∈ X
such that p̂1 (x) < p̂1 (y). In particular, there exists y > x1 such that p̂1 (x1 ) <
p̂1 (y). Since p̂1 (x1 ) 6= p̂1 (y) and p̂1 (x1 ) ∈ A(y|σ̂), by consistency, p̂1 (y) ∈
/
A(x1 |σ̂) = [−x1 , x1 ] and thus p̂1 (y) > x1 , a contradiction to supx∈X p̂1 (x) <
x1 . We now prove p̂1 (x1 ) = maxx∈X p̂1 (x). If X = [−x1 , x1 ], the claim is
immediate, since p̂1 is non-decreasing on [0, x1 ] and symmetric about 0. If
X ) [−x1 , x1 ], suppose, towards a contradiction, p̂1 (x1 ) < maxx∈X p̂1 (x).
Then there exists y ∈ X such that y > x1 and p̂1 (y) > p̂(x1 ), which, by
consistency, yields a contradiction to supx∈X p̂(x) < x1 .
39
Second, we claim p̂i (x) = p̂i (xi ) ∀x ∈ (−∞, −|p̂i (xi )|] ∪ [|p̂i (xi )|, ∞) and
∀i ∈ N . For i = 2 the claim writes p̂2 (x) = 0 ∀x ∈ X and holds by Lemma
A2. We prove the claim for i = 1. The argument for i = 3 is analogous
and omitted. By symmetry of p̂1 about 0, it suffices to show p̂1 (x) = p̂1 (x1 )
∀x ∈ [p̂1 (x1 ), ∞). That p̂1 (x) = p̂1 (x1 ) ∀x ∈ [p̂1 (x1 ), x1 ] follows by Lemma
A9 since we have p̂1 (x1 ) = maxx∈X p̂1 (x) = supx∈X p̂1 (x) < x1 . That
p̂1 (x) = p̂1 (x1 ) ∀x ∈ [x1 , ∞) follows since p̂1 is non-decreasing on [0, ∞) and
p̂1 (x1 ) = maxx∈X p̂1 (x).
Suppose now, without loss of generality, that |p̂3 (x3 )| ≤ p̂1 (x1 ) and
denote Xc = X \ (−p̂1 (x1 ), p̂1 (x1 )). By the second claim, the proposal
strategies in σ̂ are constant on Xc . Hence, third, we observe V1 (x|σ̂) =
V1 (p̂1 (x1 )|σ̂) ∀x ∈ Xc .
Fourth, we claim that player 1 has a profitable deviation. Since V1 (x|σ̂)
is constant on Xc 3 p̂1 (x1 ) and p̂1 (x1 ) < x1 , there exists ¯ > 0 such that
∀ ∈ (0, ¯), U1 (p̂1 (x1 ) + |σ̂) > U1 (p̂1 (x1 )|σ̂) and hence p̂1 (p̂1 (x1 ) + ) =
p̂1 (x1 ) ∈
/ arg maxs∈[−p̂1 (x1 )−,p̂1 (x1 )+] U1 (s|σ̂).
Remark 1: The sole purpose of the first two requirements in Proposition
A4, i) ∀x ∈ X, p̂i (−|x|) = p̂i (|x|) for i ∈ {1, 3} and ii) ∀x ∈ X and ∀y ∈ X
such that 0 ≤ x ≤ y, 0 ≤ p̂1 (x) ≤ p̂1 (y) and 0 ≥ p̂3 (x) ≥ p̂3 (y), is to
ensure that A(x|σ̂) = [−|x|, |x|] ∀x ∈ X. It is straightforward to show that,
assuming A(x|σ̂) = [−|x|, |x|] ∀x ∈ X, consistency implies both i) and ii)
and hence, Proposition A4 continues to hold if i) and ii) are replaced by
A(x|σ̂) = [−|x|, |x|] ∀x ∈ X.
Remark 2:
Proposition A4 continues to hold with a general number of
players. Extend G to n ≥ 3 players, n odd, by setting the profile of bliss
points to x = {x1 , . . . , xn } and the profile of recognition probabilities to r =
{r1 , . . . , rn }. The assumptions we made about x and r extend in a natural
way. The SMPE definition extends naturally by setting N = {1, . . . , n}.
Index players such that xi > xi+1 ∀i ∈ N \ {n}. The median is player m =
n+1
2
with xm = 0. Let N+ = {1, . . . , m−1} and N− = {m+1, . . . , n}. Given
a model with n players, straightforward modification of the argument leading
to Proposition A4 can be used to show the following. (As for Proposition A4,
the first two requirements can be replaced by A(x|σ̂) = [−|x|, |x|] ∀x ∈ X.)
40
Proposition A5. There does not exist a consistent SMPE σ̂ = ((p̂i , v̂i ))i∈N
such that i) ∀x ∈ X, p̂i (−|x|) = p̂i (|x|) for i ∈ N \ {m}; ii) ∀x ∈ X and
∀y ∈ X such that 0 ≤ x ≤ y, 0 ≤ p̂i (x) ≤ p̂i (y) ∀i ∈ N+ and 0 ≥ p̂i (x) ≥
p̂i (y) ∀i ∈ N− ; and iii) supx∈X p̂i (x) < x1 ∀i ∈ N+ and inf x∈X p̂i (x) > xn
∀i ∈ N− .
41