Equilibria in sequential bargaining games as solutions to systems of equations

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Equilibria in sequential bargaining games

as solutions to systems of equations

Tasos Kalandrakis*

Department of Political Science, Yale University, and Wallis Institute, University of Rochester, USA Received 7 December 2003; accepted 10 March 2004

Available online 7 June 2004

Abstract

No-delay, stationary equilibrium points of sequential bargaining games with history-dependent random recognition rules and general agreement rules are characterized via a finite number of equalities and inequalities. Existence of equilibrium is established using Brouwer’s fixed point theorem.

Keywords:Sequential bargaining; Equilibrium existence JEL classification:C72

We characterize no-delay, stationary equilibrium points in a general class of sequential bargaining games via a finite number of equalities and inequalities. We then adopt arguments ofNash (1951), to establish existence of equilibrium using Brouwer’s theorem. Existence has been obtained for a general class of discounted games with stationary random recognition rules by Banks and Duggan (2000), using Glicksberg’s theorem. Jackson and

Moselle (2002)analyze an application of a related majority rule game and establish existence using Kakutani’s

theorem.

The arguments we present below further simplify the existence proof. We allow for both discounting and fixed delay costs. We also study more general recognition rules to allow as special cases both alternating offers models and stationary random recognition rules. Thus, when it comes to institutions, we encompass models analyzed by

Rubinstein (1982),Binmore (1987),Baron and Ferejohn (1989),Baron (1991),Banks and Duggan (2000), etc.

Besides its simplicity, we believe the characterization we present is enlightening as to the structure of the equilibrium set. It is used inKalandrakis (2003) to show generic determinacy of pure strategy equilibria for a

* W. Allen Wallis Institute of Political Economy, 107 Harkness Hall, University of Rochester, Rochester, NY 14627-0158, USA. Tel.: 1-585-273-4902; fax: +1-585-271-3900.

E-mail address:[email protected] (T. Kalandrakis).

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subclass of these games. It is also possible that this formulation may facilitate the construction of algorithms for the computation of equilibrium.

We start our analysis by describing the bargaining environment. Consider a set ofnz2 playersN={1,. . .,n}.

They convene in periodst= 1,2,. . .to reach an agreementxdrawn from a setX.Xis a convex, compact subset of

Rd, dz1. A decision requires the approval of a winning coalition, CpN. The set of winning coalitions is

determined by the underlying voting rule and is denoted byDo2N\ Ø,Dp Ø. For example, if all players have one vote and the voting rule is simple majority,Dconsists of all coalitions with more thann/2 members. As inBanks and Duggan (2000)the only restriction on the voting rule ismonotonicity, so that for any two coalitionsA,Bwith

ApBpN, we haveAaDZBaD.

In each period t= 1,2,. . ., one of the players is recognized to make a proposal zaX. Having observed the proposal, players voteyesorno. If a winning coalition voteyes, then the game ends withzbeing implemented, else the game moves to the next period. Proposers in each period are recognized probabilistically and probabilities of recognition may depend on the identity of the proposer in the last period. Thus, if the game reaches periodt> 1 and playerjaNwas the proposer in periodt1, then playeriaNis recognized with probabilityp_ij_z0,Pn

h¼1p j h ¼1, for eachjaN.

Player iaN derives von Neuman – Morgenstern stage utility u_i:X!R from the agreement x. We assume throughoutuiis continuous and add further assumptions as necessary. To complete the description of payoffs, we entertain two standard possibilities in the literature:

Assumption A1 (discounting):

Players discount the future by a factor

d

i

a

[0,1], i

a

N, and u

_i

(

x

)

z

0, for

all x

a

X, all i

a

N

.

Thus, the payoff of playerifrom a decisionxaXreached in periodt_z1 is given byd_it1u_i(x), and it is zero in the case of perpetual disagreement. The second possibility is:

Assumption A2 (fixed delay cost):

Players incur a delay cost c

i

>0, i

a

N.

Under this assumption, the payoff of player i from a decision xaX reached in period t_z1 is given by

ui(x)(t1)ci, and it is l in the case of perpetual disagreement.

As is standard in the literature, we focus on stationary subgame perfect (SSP) equilibria and require that in every structurally identical subgame players behavior is ex ante (prior to any randomization) identical. A stationary proposal strategy for playeriaNis an element l_iaP[X], where P[X] is the set of Borel probability measures over X. We say that an SSP involves no delay if every proposal in the support of players’ proposal strategies is approved.

Before we describe voting strategies, we define thecontinuation value of players as their expected utility if the game moves in the next period. With no-delay proposal strategies the continuation value of player i, mij, when the proposer in the current period is j is given by

m

_ij

¼

X

n h¼1

p

_hj

Z

X

u

i

ð

x

Þ

l

h

*ðd

x

Þ

;

for all

i;

j

a

N

ð1Þ

Thus, the space of all possible continuation values,VoRn, in no-delay equilibria is obtained as the image of a mapping v:P[X]!Rn where the ith coordinate is defined as miðlÞum_Xu_iðxÞlðdxÞ. Clearly, Vuv(P[X]) is

convex and it is also compact as the continuous image of compact set P[X] (Aliprantis and Border, 1999,

Theorem 14.11).

We specify stationary voting strategies for playeriby acceptance setsAijoX,jaN.A_ijcontains the proposals by player j which i approves. We restrict voting strategies in order to rule out equilibria where undesirable agreements are approved or desirable agreements are rejected solely because each of the players is not pivotal

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and hence is indifferent between her voting actions. Thus, in equilibrium we require that Aij:VOX is a correspondence that satisfies

x

a

A

j i

ð

v

j

Þ

Z

u

_i

ð

x

Þ

z

d

_i

m

j i

;

under A1

ð2Þ

x

a

A

j i

ð

v

j

Þ

Z

u

_i

ð

x

Þ

z

m

j i

c

i

;

under A2

ð3Þ

for all j,iaN. Following Baron and Kalai (1993) we call such voting strategies stage-undominated.

Consider the subset of winning coalitions that include playerjand are minimum winning in the sense that if any player ipj is removed from the coalition, the coalition ceases to be winning. Call this set of coalitions

t_joD, defined as t_j_u{CaD:jaC, C\ {i}gD,ipj}. Let the number of coalitions in t_j be n_j_uAt_jA. By non-emptiness and monotonicity of the agreement rule, D, we are guaranteed that njz1 for all jaN.

We now define the agenda setting plan, fxj, of proposer j and coalition Cxat_j, x= 1,. . .,n_j, as a correspondence fxj:VOX, given by

f

_xj

ð

v

j

Þ

u

arg max

x

u

j

ð

x

Þ

:

x

a

u

haC_x

A

_hj

ð

v

j

Þ

:

In certain modal cases, as we establish in Lemma 1 at the end of our analysis,fxjis a non-empty, single-valued function for all jaN, x= 1,. . .,n_j.

When all agenda setting plans,fxj, are functions we are afforded the following reduction in the description of stationary proposal strategies. Instead of a measureljaP[X] we specify the proposal strategy of player j as a choice among the finite number of coalitions int_j. Thus, for continuation valuesvjaV, a proposal strategy for player j is an element, mjaDnj1, where Dnj1is the (n_j1)-dimensional unit simplex in Rnj. The coalition

mixing probability mxj denotes the probability that coalition Cxat_j is chosen. Implicitly, when the chosen

coalition isCxat_j, the proposed agreement is given by f_xj(vj)aX.

We now have the following characterization of no-delay SSP equilibrium for the entire class of games we consider:

Theorem 1.

If f

xj

is a function for all j

a

N,

x

= 1,

. . .

,

n

_j

, then no-delay SSP equilibria in

stage-undominated voting strategies under either A1 or A2 are characterized by coalition mixing probabilities

m

j

*

a

D

s_j 1

, and continuation values

v

j

*

a

V, j

a

N such that:

m

_ij

*

¼

X

n h¼1

p

_hj

X

nh c¼1

m

h_c

*

u

i

ð

f

ch

ð

v

h

*

ÞÞ

;

for all

i;

j

a

N

ð4Þ

m

j_x

*

>

0

Z

u

j

ð

f

_xj

ð

v

j

*

ÞÞ

z

u

j

ð

f

_xVj

ð

v

j

*

ÞÞ

;

x

V

¼

1

;

. . .

;

n

j

:

ð5Þ

Proof.

Every winning coalition

C

g

t

_j

with

j

a

C

is a superset of some coalition

C

V

at

j. Thus, when

continuation values are given by some

v

j

a

V

, if there exist optimal, no-delay proposals for

j

that are

acceptable by members of

C

, they are also acceptable by coalition

C

V

at

_{j, i.e. optimal proposals must}

be drawn from {

f

xj

(

v

j

)}

nxj= 1

and conditions (4) and (5) are necessary in equilibrium. Note that Eqs. (4)

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and (5) ensure there do not exist profitable one-stage deviations at the proposal stage. Also, by

definition, agenda setting plans

f

xj

respect condition (2) or (3), hence there are no profitable one-stage

deviations at the voting stage either with stage-undominated voting strategies. Thus, there are no

profitable finite period deviations. Now, since

u

i(

x

)

z

0,

x

a

X

, there do not exist profitable infinite

deviations under A1.

c

i

>0 ensures the same under A2, and we have established sufficiency.

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The above characterization invites a proof of existence of equilibrium using Brouwer’s fixed point theorem. We provide such a proof in the next theorem:

Theorem 2.

Consider a game under A1 or A2 for which f

xj

is a continuous function for all j

a

N,

x

= 1,

. . .

,

n

j

. A no-delay SSP equilibrium exists.

Proof.

Let

M

=

D

nj1

j= 1n

. Define a function

F

:

V

n

M

!

V

n

M

so that the first

n

2

coordinates (in

V

n

) correspond to the

n

vectors of continuation values,

v

=(

v

1

,

. . .

,

v

n

), and the

P

n_j_¼₁

n

j

remaining

coordinates in

M

correspond to coalition mixing probabilities. Set the coordinate that corresponds to the

continuation value of the

i

th player when the proposer is

j

as

F

_ð_mj

iÞ

ð

v;

m

Þ

u

P

n h¼1

p

h

Pn

h x¼1

m

h x

u

i

ð

f

xh

ð

v

h

ÞÞ

.

To specify coordinates corresponding to the mixing probabilities

m

xj

*, we shall use

Nash’s (1951)

analogous formulation for finite games in normal form.

For given vjaV and by using a particular mixing, mj, among coalitions, proposer j’s expected utility is Pnj

x¼1mjxujðfxjðvjÞÞ. If instead j proposes only to coalition Cxat_j her utility is u_j(f_xj(vj)). The potential improvement injVs payoff,uxj(v,mj), obtained by proposing exclusively to coalitionCxinstead of mixing among coalitions according to mj, is defined as uj

xðvj;mjÞumax 0;ujðfxjðvjÞÞ Pnj

x¼1mxjujðfxjðvjÞÞ

n o

. This is a continuous function by the continuity of fxj, uj, by the theorem of the Maximum. Then, we define the xth coordinate of the mixing vector of proposerjas F(mx

j

)(v,m)umxj+uxj(vj,mj)/1+P nj

x¼1uxj(vj,mj).

Importantly,Fis a continuous function of (v,m)aVnMas a result of the continuity off_xj,u_j, andu_xj. Also,VnM

VnM is convex and compact since both M and Vare convex and compact. Thus, Brouwer’s fixed point theorem applies and F has a fixed point. It is trivial to see that (v, m) is a fixed point of F if an only if it

satisfies conditions (4) and (5). 5

Theorems 1 and 2 could be void if the conditions on the agenda setting plans fxj are not met. In the next lemma we provide a pair of sufficient conditions:

Lemma 1.

Assume that for all i

a

N either: (A3) u

_i

strictly concave, or (A4) X =

D

n1

, u

_i

(

x

) = m

_i

(x

_i

), with

m

i

(0) = 0, m

i

V

>0, and m

i

W

V

0. Then, f

xj

is a continuous function for all j

a

N,

x

= 1,

. . .

,

n

_j

under A1 or A2.

Proof.

Under either A3 or A4,

A

Cj

:

V

O

X

defined as

\h

aC

A

h

j

(

v

j

) for

C

at

_j

is a non-empty, continuous

correspondence (e.g.,

Banks and Duggan, 2000). Also,

\

haC

A

h

j

(

v

j

) is convex valued as the intersection

of convex sets. Thus, A3 implies there is a unique maximizer, and continuity of

f

xj

follows from Berge’s

theorem of the Maximum. Also, under A4, the inverse

m

i 1

exists, hence the agenda setting plan is

obtained as the following continuous function:

f

_xj

ð

v

j

Þ ¼

0

if

h

g

Cx

m

1 i

ðmaxf0

;

m

h

gÞ

if

h

a

C

_x

q

f

j

g

1

X

laC_xqfjg

m

_l 1

ðmaxf0

;

m

l

gÞ

if

h

¼

j

8

>

<

>

:

ð6Þ

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Condition A3 covers political environments with spatial or ideological spaces, while A4 covers standard divide-the-dollar environments. While in combination, these sufficient conditions cover most applications in the literature, we note that the same existence arguments apply if fxj are correspondences that admit continuous selections. Alternatively, if we relax A3 to ‘‘ui concave’’, instead of strictly concave, we can use Geanakoplos’ (2003) ‘‘satisficing principle’’ to establish existence via Brouwer’s theorem.

Lastly, an easy consequence of our formulation is a bound on the number of possible agreements which cannot be larger thanPn_i_¼₁n_iin any equilibrium. For oligarchic rules,ni= 1, and all SSP equilibria are in pure strategies, so that we extend Theorem 2, Part (ii) ofBanks and Duggan (2000)to non-stationary recognition rules and the case of fixed delay costs.

References

Aliprantis, C.D., Border, K., 1999. Infinite Dimensional Analysis: a Hitchhiker’s Guide, 2nd ed. Springer-Verlag: Berlin. Banks, J.S., Duggan, J., 2000. A bargaining model of collective choice. American Political Science Review 94 (March),

73 – 88.

Baron, D., 1991. A spatial bargaining theory of government formation in parliamentary systems. American Political Science Review 85 (March), 137 – 164.

Baron, D.P., Ferejohn, J.A., 1989. Bargaining in legislatures. American Political Science Review 83 (December), 137 – 164. Baron, D.P., Kalai, E., 1993. The simplest equilibrium of a majority rule game. Journal of Economic Theory 61, 290 – 301. Binmore, K., 1987. Perfect Equilibria in Bargaining Models, Chapter 5. In: Binmore, K., Dasgupta, P. (Eds.), The Economics of

Bargaining. Blackwell, Oxford.

Geanakoplos, J., 2003. Nash and Walras equilibrium via Brouwer. Economic Theory 21 (2 – 3), 585 – 603.

Jackson, M., Moselle, B., 2002. Coalition and party formation in a legislative voting game. Journal of Economic Theory 103, 49 – 87.

Kalandrakis, T., 2003. Regularity of pure strategy stationary equilibria in a class of bargaining games. Mimeo, Yale University. Nash, J., 1951. Non-Cooperative games. The Annals of Mathematics, 2nd Ser. 54 (2), 286 – 295.