In this paper the standard alternating offer model is analysed. An alternative proof is suggested to show that the system of equations that characterises the stationary SPE proposals admits a unique solution. Furthermore, this proof is more general than the existing proofs.</p

Houba, H.

Harold Houba

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Economics Letters

An alternative proof of uniqueness in non-cooperative bargaining

Houba, Harold

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VU Research PortalAn alternative proof of uniqueness in non-cooperative bargainingHouba, H.1993document versionPublisher's PDF, also known as Version of recordLink to publication in VU Research Portalcitation for published version (APA)Houba, H. (1993). An alternative proof of uniqueness in non-cooperative bargaining. (Serie ResearchMemoranda; No. 1993-15). 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May. 2021aculteit der Economische Wetenschappen en Econometrie ET -akgroep Econometrie 05348 Serie Research Memoranda An Alternative Proof of Uniqueness in Non-Cooperative Bargaining Harold Houba Research-Memorandum 1993-15 Maren 1993 vrije Universiteit amsterdam - 1 -An Alternative Proof of Uniqueness in Non-Cooperative Bargaining* Harold Houba Department of Econometrics, Free University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. A b s t r a c t In this paper the Standard alternating offer model is analysed. An al-ternative proof is suggested to show that the system of equations that characterises the stationary SPE proposals admits a unique solution. Fur-thermore, this proof is more general than the existing proofs. The author thanks Hans Haller, Gerard van der Laan and Aart de Zeeuw. JEL-code: C78 - 9 -1 Intro duet ion The bargaining model in this note is the alternating offer model (Rubinstein 1982) in which players bargain over a set of feasible payoff vectors. Furthermore, it is assumed that the players face an exogenously given probability of a terminal break down after each bargaining round. If the bargaining breaks down while the players have not agreed in the past, then the players get their exogenous disagreement payoffs. The bargaining problem in this particular setting has been analysed in van Damme (1991, chapter 7) and Okada (1991).1 In Okada (1991) it is assumed that the Pareto frontier is concave and piecewise linear and the proof is quite long. In van Damme (1991) it is assumed that the disagreement payoffs are equal to 0 and that the Pareto frontier is differentiable, concave and strictly decreasing. In both papers the Standard approach to prove uniqueness of the subgame perfect equilibrium (SPE) is foliowed. This is to establish existence of a stationary SPE first, then prove that the system of equations that characterises stationary SPE's admits at most one so-lution and, finally, apply the method in Shaked and Sutton (1984) to prove that no non-stationary equilibria exist [see, e.g., van Damme (1991) and Osborne and Rubinstein (1990)]. The aim of this paper is to suggest an alternative proof to show that the system of equations that characterises the stationary SPE proposals admits at most one solution. Furthermore, this proof requires weaker conditions than the conditions imposed in van Damme (1991) and Okada (1991), namely that the Pareto frontier is continuous, concave and strictly decreasing. 1In Okada (1991) the players bargaining over a stream of payoffs and the disagreement payoffs are endogenuously determined by the players. However, it should be acknowledged that Okada's arguments are also valid in the bargaining model analysed in this note. - 3 -2 Results The bargaining problem is defined as a couple (S,d), where S £ R 2 denotes the non-empty set of feasible payoffs and d £ S the disagreement payoff vector. It is assumed that the Pareto efficiënt frontier of {x 6 S\x > d} (x £ S is Pareto efficiënt if fiy £ S s.t. yi > x,-, i = 1,2) can be described by a strictly decreasing, concave and continuous function. If the set S is closed, convex and strongly comprehensive (strongly comprehensive means that if x £ 'S and y < x, then y £ S) and {x £ 5|x > d} is compact, then such a function exists. This function is denoted by fi(zj), i,j = 1,2, i ^ j , and specifies player i's maximum payoff in 5 given the payoff Zj for player j . Furthermore, f2 is the inverse function of / x and visa versa. It is also assumed that d £ S is not Pareto efficiënt, that is da < f\{d2), to rule out trivial bargaining problems. Finally, 8 £ [0,1) is defined as the exogenous probability that the bargaining process continues for one more bargaining round after the proposal in the current bargaining round is rejected and (1 — 8) is the probability that the bargaining breaks down after the proposal in the current bargaining round is declined by the responding player. The bargaining procedure is the same as in the standard alternating offer model (Rubinstein 1982). In every odd bargaining round player 1 proposes a vector of payoffs x £ S to player 2. Player 2 either accepts or rejects this proposal. If player 2 accepts then the game ends. However, if player 2 rejects, then this player proposes a vector of payoffs y £ S in the next (even) bargaining round (provided there is one). Player 1 can either accept or reject player 2's proposal. Again, if player 1 accepts then the game ends, but if player 1 rejects then this player has the initiative to propose in the next (odd) bargaining round (provided there is one). As long as the two players do not agree the bargaining process continues, unless the bargaining breaks down and d £ S is the terminal payoff vector to the players. As mentioned in the introduction, the standard approach to prove uniqueness of the subgame perfect equilibrium is foliowed. For every stationary SPE it holds that a responding player is indifferent between accepting or rejecting the current proposal - 4 -and the proposing player obtains the remainder [see, e.g., van Damme (1991) and Osborne and Rubinstein (1990)]. This leads to the following system of equations that completely characterises the stationary SPE proposal x of player 1 and the stationary SPE proposal y of player 2, namely xi = /i(-(l - 8)d2 + Sy2), x2 = (1 - 6)d2 + 6y2 and yj = (1 - 6)dx + Sxu 2/2 = /2((1 - S)di + Sx-i). Each pair (x,y) corresponds to one stationary SPE and visa versa. In order to prove that the couple (x, y) is unique the function p : S —» S is defined as •Mv) = (! - 6)d2 + Sy2, pi(y) = / i(p2(y)), and the function q : S —> S is defined as qi{x) = (1 - ê)dl + ^ ! , q2(x) = f2(qi(x)). Brouwer's fixed point theorem can be applied to prove that pxq: SxS—^SxS has a fixed point (x,y) = (p(y),q(x)). This establishes the existence of at least one stationary SPE. It is easy to show that each pair (x, y) possesses the properties known from the Standard alternating offer model (Rubinstein 1982). By construction of the functions p and q it follows that the proposals x and y are Pareto efficiënt. The concavity of / j and f2 can be used to show that x and y are individually rational with respect to d, e.g. xi = A ( ( l - S)d2 + 8y2) > (1 - 8)h{d2) + 8h{y2) > (1 - ê)d1 + S[(l - 8)d1 + 8x,} and, therefore, xx > dx and also yx = qi(x) = (1 — 8)d1 + 8x\ G {dux{). Moreover, the latter result implies that x lies to the right of y on the Pareto frontier, a property that is better known as the first mover advantage. Both in van Damme (1991) and Okada (1991) it is first proved that tbe couple (x, y) is unique and then it is shown that the corresponding x and y have the same - ö -Nash product. This latter result is used to prove that x and y converge to the Nash bargaining solution with threat vector d as 6 goes to 1. The proofs in these pub-lications make use of different mathematical techniques to establish the uniqueness of (x,y). However, lemma 2.1 states that, even before uniqueness is established, for every fixed point (x, y) of p x q it holds that x and y have the same Nash product. This insight is then used to construct an alternative proof, which requires less strin-gent conditions than the conditions in van Damme (1991) and Okada (1991). This alternative proof exploits the geometry of the bargaining problem and is the main result presented in this note. Lemma 2.1 If (x,y) is a fixed point of p x q, then x and y have the same Nash product with threat vector d. proof. The points x and y have the same Nash product with threat vector d, because X2 = p2{y) and yi = qi{x) imply that (xj - d1)(x2 - d2) = S(xx - d1)(y2 - d2) and (j/i - dx){y2 - d2) = 8(x1 - di)(y2 - d2). This completes the proof. • Proposi t ion 2.1 The function p x q has a unique fixed point (x, y). Proof. Convexity of the set S implies that each curve of Nash products (this is a curve of points in R2 that all have the same Nash product and the same threat vector) inter-sects the Pareto frontier of S at most twice. Suppose (x,y) and (x,y) are both fixed points of p x g, then x (y) and x (y) must lie on distinct curves of Nash products (if not, either the curve would intersect the Pareto frontier four times, which is a contradiction, or x = x and y — y). Therefore, their Nash products differ. Without - 6 -loss of generality assume that the Nash product of x (y) is strictly larger than x (y). This implies that x lies to the right of x and y lies to the left of y (all four points lie on the Pareto frontier of S). Thus, x\ > x\ and y2 > 2/2- However, X\ > x\ implies that 2/i = qi{£) > q\{x) = 2/1 and, hence, y2 = f2{yi) < Mvi) = 2/2, which contradicts J/2 > 2/2 above. Hence, (x,y) is unique. ü The final step in proving that the unique stationary SPE is the only SPE is based upon the method in Shaked and Sutton (1984). Define xM G S as the best SPE payoff vector for player 1 whenever this player has the initiative to propose and ym G S as the worst SPE payoff vector for player 2 whenever this player has the initiative to propose. Using the arguments in van Damme (1991) it follows that xM and ym satisfy the same relations as x and y, namely x? = M(l- 8)d2 + 8y?), x? = (l- 8)d2 + 6y? and y? = (1-6)^ +8.x?, y? = M(l-8)d1 + 8x?). Therefore, (xM,ym) is a fixed point of p x q. Hence, (xM,ym) = (x,y), where (x,y) denotes the unique fixed point of proposition 2.1. Similar definitions and arguments imply (xm.,yM) = (x,y). Hence, the stationary SPE is the unique SPE. Finally, it is also easy to prove that x and y converge to the Nash bargaining solution with threat vector d £ S as 8 goes to 1. Firstly, x and y converge to the same limit vector, because 0 < *i - ï/i = (1 - W i ( ( l - 8)d2 + 8y2) - d,} < (1 - 8)\fi(d2) - d1] -* 0, as 8 goes to 1. This latter argument is also simpler than the arguments in van Damme (1991) and Okada (1991). Finally, following van Damme (1991), for every 8 G [0,1) it holds that the points x and y are two distinct points that lie on the same Nash product curve and that have a common limit vector, which must be the unique Pareto efficiënt point where the Nash product curve {zx — di){z2 — d2) is tangent to the Pareto frontier, i.e. the limit vector is the Nash bargaining solution with threat point d. - i -3 References van Damme, E., 1991, Stability and perfection of Nash equilibria, second edition (Springer-Verlag, New York/Berlin). Okada, A., 1991, A non-cooperative approach to the Nash bargaining problem, in: R. Selten, ed., Game theoretic equilibrium models III: strategie bargaining (Springer-Verlag, New York/Berlin), 7-33. Osborne, M., and A. Rubinstein, 1990, Bargaining and markets, (Aca-demie Press, San Diego). Rubinstein, A., 1982, Perfect equilibrium in a bargaining model, Econo-metrica 50, 97-109. Shaked, A., and J. Sutton, 1984, Involuntary unemployment as a perfect equilibrium in a bargaining model, Econometrica 52, 1351-1364. 

English

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aculteit der Economische Wetenschappen en Econometrie 
ET -akgroep Econometrie 
05348 
Serie Research Memoranda 
An Alternative Proof of Uniqueness in Non-Cooperative 
Bargaining 
Harold Houba 
Research-Memorandum 1993-15 
Maren 1993 
vrije Universiteit amsterdam 
- 1 -
An Alternative Proof of Uniqueness in 
Non-Cooperative Bargaining* 
Harold Houba 
Department of Econometrics, 
Free University, 
De Boelelaan 1105, 
1081 HV Amsterdam, 
The Netherlands. 
A b s t r a c t 
In this paper the Standard alternating offer model is analysed. An al-
ternative proof is suggested to show that the system of equations that 
characterises the stationary SPE proposals admits a unique solution. Fur-
thermore, this proof is more general than the existing proofs. 
The author thanks Hans Haller, Gerard van der Laan and Aart de Zeeuw. 
JEL-code: C78 
- 9 -
1 Intro duet ion 
The bargaining model in this note is the alternating offer model (Rubinstein 
1982) in which players bargain over a set of feasible payoff vectors. Furthermore, 
it is assumed that the players face an exogenously given probability of a terminal 
break down after each bargaining round. If the bargaining breaks down while the 
players have not agreed in the past, then the players get their exogenous disagreement 
payoffs. 
The bargaining problem in this particular setting has been analysed in van Damme 
(1991, chapter 7) and Okada (1991).1 In Okada (1991) it is assumed that the 
Pareto frontier is concave and piecewise linear and the proof is quite long. In van 
Damme (1991) it is assumed that the disagreement payoffs are equal to 0 and that 
the Pareto frontier is differentiable, concave and strictly decreasing. In both papers 
the Standard approach to prove uniqueness of the subgame perfect equilibrium (SPE) 
is foliowed. This is to establish existence of a stationary SPE first, then prove that 
the system of equations that characterises stationary SPE's admits at most one so-
lution and, finally, apply the method in Shaked and Sutton (1984) to prove that 
no non-stationary equilibria exist [see, e.g., van Damme (1991) and Osborne and 
Rubinstein (1990)]. 
The aim of this paper is to suggest an alternative proof to show that the system 
of equations that characterises the stationary SPE proposals admits at most one 
solution. Furthermore, this proof requires weaker conditions than the conditions 
imposed in van Damme (1991) and Okada (1991), namely that the Pareto frontier is 
continuous, concave and strictly decreasing. 
1In Okada (1991) the players bargaining over a stream of payoffs and the disagreement payoffs 
are endogenuously determined by the players. However, it should be acknowledged that Okada's 
arguments are also valid in the bargaining model analysed in this note. 
- 3 -
2 Results 
The bargaining problem is defined as a couple (S,d), where S £ R 2 denotes the 
non-empty set of feasible payoffs and d £ S the disagreement payoff vector. It is 
assumed that the Pareto efficiënt frontier of {x 6 S\x > d} (x £ S is Pareto efficiënt 
if fiy £ S s.t. yi > x,-, i = 1,2) can be described by a strictly decreasing, concave 
and continuous function. If the set S is closed, convex and strongly comprehensive 
(strongly comprehensive means that if x £ 'S and y < x, then y £ S) and {x £ 
5|x > d} is compact, then such a function exists. This function is denoted by fi(zj), 
i,j = 1,2, i ^ j , and specifies player i's maximum payoff in 5 given the payoff Zj 
for player j . Furthermore, f2 is the inverse function of / x and visa versa. It is also 
assumed that d £ S is not Pareto efficiënt, that is da < f\{d2), to rule out trivial 
bargaining problems. Finally, 8 £ [0,1) is defined as the exogenous probability that 
the bargaining process continues for one more bargaining round after the proposal 
in the current bargaining round is rejected and (1 — 8) is the probability that the 
bargaining breaks down after the proposal in the current bargaining round is declined 
by the responding player. 
The bargaining procedure is the same as in the standard alternating offer model 
(Rubinstein 1982). In every odd bargaining round player 1 proposes a vector of 
payoffs x £ S to player 2. Player 2 either accepts or rejects this proposal. If player 2 
accepts then the game ends. However, if player 2 rejects, then this player proposes a 
vector of payoffs y £ S in the next (even) bargaining round (provided there is one). 
Player 1 can either accept or reject player 2's proposal. Again, if player 1 accepts then 
the game ends, but if player 1 rejects then this player has the initiative to propose in 
the next (odd) bargaining round (provided there is one). As long as the two players 
do not agree the bargaining process continues, unless the bargaining breaks down 
and d £ S is the terminal payoff vector to the players. 
As mentioned in the introduction, the standard approach to prove uniqueness of 
the subgame perfect equilibrium is foliowed. For every stationary SPE it holds that 
a responding player is indifferent between accepting or rejecting the current proposal 
- 4 -
and the proposing player obtains the remainder [see, e.g., van Damme (1991) and 
Osborne and Rubinstein (1990)]. This leads to the following system of equations 
that completely characterises the stationary SPE proposal x of player 1 and the 
stationary SPE proposal y of player 2, namely 
xi = /i(-(l - 8)d2 + Sy2), x2 = (1 - 6)d2 + 6y2 
and 
yj = (1 - 6)dx + Sxu 2/2 = /2((1 - S)di + Sx-i). 
Each pair (x,y) corresponds to one stationary SPE and visa versa. 
In order to prove that the couple (x, y) is unique the function p : S —» S is defined 
as 
•Mv) = (! - 6)d2 + Sy2, pi(y) = / i(p2(y)), 
and the function q : S —> S is defined as 
qi{x) = (1 - ê)dl + ^ ! , q2(x) = f2(qi(x)). 
Brouwer's fixed point theorem can be applied to prove that pxq: SxS—^SxS 
has a fixed point (x,y) = (p(y),q(x)). This establishes the existence of at least one 
stationary SPE. 
It is easy to show that each pair (x, y) possesses the properties known from the 
Standard alternating offer model (Rubinstein 1982). By construction of the functions 
p and q it follows that the proposals x and y are Pareto efficiënt. The concavity of / j 
and f2 can be used to show that x and y are individually rational with respect to d, 
e.g. 
xi = A ( ( l - S)d2 + 8y2) > (1 - 8)h{d2) + 8h{y2) > (1 - ê)d1 + S[(l - 8)d1 + 8x,} 
and, therefore, xx > dx and also yx = qi(x) = (1 — 8)d1 + 8x\ G {dux{). Moreover, 
the latter result implies that x lies to the right of y on the Pareto frontier, a property 
that is better known as the first mover advantage. 
Both in van Damme (1991) and Okada (1991) it is first proved that tbe couple 
(x, y) is unique and then it is shown that the corresponding x and y have the same 
- ö -
Nash product. This latter result is used to prove that x and y converge to the Nash 
bargaining solution with threat vector d as 6 goes to 1. The proofs in these pub-
lications make use of different mathematical techniques to establish the uniqueness 
of (x,y). However, lemma 2.1 states that, even before uniqueness is established, for 
every fixed point (x, y) of p x q it holds that x and y have the same Nash product. 
This insight is then used to construct an alternative proof, which requires less strin-
gent conditions than the conditions in van Damme (1991) and Okada (1991). This 
alternative proof exploits the geometry of the bargaining problem and is the main 
result presented in this note. 
Lemma 2.1 If (x,y) is a fixed point of p x q, then x and y have the same Nash 
product with threat vector d. 
proof. 
The points x and y have the same Nash product with threat vector d, because X2 = 
p2{y) and yi = qi{x) imply that 
(xj - d1)(x2 - d2) = S(xx - d1)(y2 - d2) 
and 
(j/i - dx){y2 - d2) = 8(x1 - di)(y2 - d2). 
This completes the proof. • 
Proposi t ion 2.1 The function p x q has a unique fixed point (x, y). 
Proof. 
Convexity of the set S implies that each curve of Nash products (this is a curve of 
points in R2 that all have the same Nash product and the same threat vector) inter-
sects the Pareto frontier of S at most twice. Suppose (x,y) and (x,y) are both fixed 
points of p x g, then x (y) and x (y) must lie on distinct curves of Nash products 
(if not, either the curve would intersect the Pareto frontier four times, which is a 
contradiction, or x = x and y — y). Therefore, their Nash products differ. Without 
- 6 -
loss of generality assume that the Nash product of x (y) is strictly larger than x (y). 
This implies that x lies to the right of x and y lies to the left of y (all four points lie on 
the Pareto frontier of S). Thus, x\ > x\ and y2 > 2/2- However, X\ > x\ implies that 
2/i = qi{£) > q\{x) = 2/1 and, hence, y2 = f2{yi) < Mvi) = 2/2, which contradicts 
J/2 > 2/2 above. Hence, (x,y) is unique. ü 
The final step in proving that the unique stationary SPE is the only SPE is based 
upon the method in Shaked and Sutton (1984). Define xM G S as the best SPE payoff 
vector for player 1 whenever this player has the initiative to propose and ym G S as 
the worst SPE payoff vector for player 2 whenever this player has the initiative to 
propose. Using the arguments in van Damme (1991) it follows that xM and ym satisfy 
the same relations as x and y, namely 
x? = M(l- 8)d2 + 8y?), x? = (l- 8)d2 + 6y? 
and 
y? = (1-6)^ +8.x?, y? = M(l-8)d1 + 8x?). 
Therefore, (xM,ym) is a fixed point of p x q. Hence, (xM,ym) = (x,y), where (x,y) 
denotes the unique fixed point of proposition 2.1. Similar definitions and arguments 
imply (xm.,yM) = (x,y). Hence, the stationary SPE is the unique SPE. 
Finally, it is also easy to prove that x and y converge to the Nash bargaining 
solution with threat vector d £ S as 8 goes to 1. Firstly, x and y converge to the 
same limit vector, because 
0 < *i - ï/i = (1 - W i ( ( l - 8)d2 + 8y2) - d,} < (1 - 8)\fi(d2) - d1] -* 0, 
as 8 goes to 1. This latter argument is also simpler than the arguments in van Damme 
(1991) and Okada (1991). Finally, following van Damme (1991), for every 8 G [0,1) 
it holds that the points x and y are two distinct points that lie on the same Nash 
product curve and that have a common limit vector, which must be the unique Pareto 
efficiënt point where the Nash product curve {zx — di){z2 — d2) is tangent to the Pareto 
frontier, i.e. the limit vector is the Nash bargaining solution with threat point d. 
- i -
3 References 
van Damme, E., 1991, Stability and perfection of Nash equilibria, second 
edition (Springer-Verlag, New York/Berlin). 
Okada, A., 1991, A non-cooperative approach to the Nash bargaining 
problem, in: R. Selten, ed., Game theoretic equilibrium models III: 
strategie bargaining (Springer-Verlag, New York/Berlin), 7-33. 
Osborne, M., and A. Rubinstein, 1990, Bargaining and markets, (Aca-
demie Press, San Diego). 
Rubinstein, A., 1982, Perfect equilibrium in a bargaining model, Econo-
metrica 50, 97-109. 
Shaked, A., and J. Sutton, 1984, Involuntary unemployment as a perfect 
equilibrium in a bargaining model, Econometrica 52, 1351-1364. 


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An alternative proof of uniqueness in non-cooperative bargaining

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