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Representation in Multi-Issue Delegated Bargaining

  • Shourjo Chakravorty EMAIL logo
Veröffentlicht/Copyright: 18. April 2018
The B.E. Journal of Theoretical Economics
Aus der Zeitschrift The B.E. Journal of Theoretical Economics Band 19 Heft 1

Abstract

Delegated bargaining over multiple issues of varying sizes is a frequent occurrence and happens during such contexts as real estate sales and political representation. Past work has shown that while simultaneously bargaining over two issues, it is in the best interest of an individual to choose a representative with identical preferences as herself. By allowing the sizes of the two issues to vary, this paper uses a two-issue, two-player Rubinstein bargaining game to show that an individual is best served by selecting a representative whose preferences closest resemble her bargaining rival’s preferences. It is also found that in some instances depending on the size of the issues the individual will be indifferent when choosing between a representative with identical preferences and another with different preferences to bargain on her behalf.

Keywords: delegated bargaining; multi-issue bargaining; simultaneous bargaining
JEL Classification: C7 Game Theory and Bargaining Theory

Acknowledgements

I am grateful to Steven Slutsky, David Sappington, Caprice Knapp and Sanford Berg for the guidance and support they provided while writing this paper. I also thank Dave Brown, Michelle Phillips and seminar participants at the Department of Economics at Istanbul Technical University for helpful comments.

Appendix

A Proof of Lemma 1

There are four candidate solutions for Equilibrium Condition eq. (4):

  1. xa<X‾, ya=0 and xb<X‾, yb=0,

  2. xa<X‾, ya=0 and xb=X‾,

  3. xa=X‾ and xb<X‾, yb=0,

  4. xa=X‾ and xb=X‾.

To prove Lemma 1, I examine which of the four candidate solutions (i), (ii), (iii) and (iv) solve Equilibrium Condition eq. (4).

Candidate solution (i): xa<X‾, ya=0andxb<X‾, yb=0

Candidate solution (i) implies that the Equilibrium Condition eq. (4) becomes:

(15)(X‾−xa)+Y‾=δ(X‾−xb+Y‾),
(16)xb=δxa.

Solving eqs (15) and (16) for xa and xb:

(17)xa=X‾+Y‾1+δ,
(18)xb=δ(X‾+Y‾)1+δ.

For Solutions eqs (17) and (18) to be a valid, it must be that 0≤xa<X‾ and 0≤xb<X‾. Because δ∈(0,1) and X‾ and Y‾ are nonnegative numbers, it must be that xa≥0 and xb≥0. For xa<X‾, it must be that Y‾<δX‾, and for xb<X‾, it must be that Y‾<X‾δ. Because δX‾<X‾δ, Candidate solution (i) is a valid solution in the range Y‾<δX‾.

Candidate solution (ii): xa<X‾, ya=0 and xb=X‾

Candidate solution (ii) implies that the Equilibrium Condition eq. (4) becomes:

(19)(X‾−xa)+Y‾=δ(Y‾−yb),
(20)X‾+αSyb=δxa.

Solving eqs (19) and (20) for xa and yb:

(21)xa=(δ−αS)X‾−αS(1−δ)Y‾δ2−αS,
(22)yb=(1−δ)X‾−δ(1−δ)Y‾δ2−αS.

For Solutions eqs (21) and (22) to be valid, it must be that 0≤xa<X‾ and 0≤yb≤Y‾. To check this validity, we examine the two possible subcases: (a) δ2−αS≥0 and (b) δ2−αS<0.

Subcase (a) δ2−αS≥0. For xa<X‾, it must be Y‾>δX‾αS and for yb≥0, it must be Y‾≤X‾δ. Therefore, for Candidate solution (ii) to be a valid solution in Subcase (a), it must be δX‾αS<X‾δ. But that would imply δ2−αS<0. But in Subcase (a), δ2−αS≥0. Consequently, Candidate solution (ii) cannot exist in Subcase (a).

Subcase (b) δ2−αS<0. For xa<X‾, it must be Y‾<δX‾αS and for yb≥0, it must be Y‾≥X‾δ. Therefore, for Candidate solution (ii) to be a valid solution in Subcase (b), it must be δX‾αS>X‾δ. But that would imply δ2−αS>0. But in Subcase (b), δ2−αS<0. Consequently, Candidate solution (ii) cannot exist in Subcase (b).

This means Candidate solution (ii) cannot be a valid solution for Equilibrium Condition eq. (4).

Candidate solution (iii): xa=X‾ and xb<X‾, yb=0

Candidate solution (iii) implies that the Equilibrium Condition eq. (4) becomes:

(23)Y‾−ya=δ(X‾−xb+Y‾),
(24)xb=δ(X‾+αSya).

Solving eqs (23) and (24) for ya and xb:

(25)ya=(1−δ)(Y‾−δX‾)1−δ2αS,
(26)xb=δ[(1−δαS)X‾+αS(1−δ)Y‾]1−δ2αS.

For Solutions eqs (25) and (26) to be a valid, it must be that 0≤ya≤Y‾ and 0≤xb<X‾. δ∈(0,1) and 0≤αS<1 imply that the denominator of Solutions eqs (25) and (26), 1−δ2αS>0. By inspection, we also see that the numerator of Solution eq. (26) is nonnegative. This implies xb≥0. Looking at the numerator of Solution eq. (25), for ya≥0 it must be that Y‾≥δX‾. For ya≤Y‾, it must be that (1−δ)(Y‾−δX‾)1−δ2αS≤Y‾. This inequality simplifies to (1−δαS)Y‾+(1−δ)X‾≥0. By inspection, we can see that this latter inequality always holds. Therefore, ya≤Y‾. For xb<X‾, it must be that δ[(1−δαS)X‾+αS(1−δ)Y‾]1−δ2αS<X‾. This inequality simplifies to Y‾<X‾δαS. Hence, Candidate solution (iii) is a valid solution in the range δX‾≤Y‾<X‾δαS.

Candidate solution (iv): xa=X‾ and xb=X‾

Candidate solution (iv) implies that the Equilibrium Condition eq. (4) becomes:

(27)Y‾−ya=δ(Y‾−yb),
(28)X‾+αSyb=δ(X‾+αSya).

Solving eqs (27) and (28) for ya and yb:

(29)ya=αSY‾−δX‾αS(1+δ),
(30)yb=δαSY‾−X‾αS(1+δ).

For Solutions eqs (29) and (30) to be a valid, it must be that 0≤ya≤Y‾ and 0≤yb≤Y‾. ya≤Y‾ and yb≤Y‾ imply δ(αSY‾+X‾)≥0 and αSY‾+X‾≥0 respectively. Because δ∈(0,1) and 0≤αS<0, we know that these latter two inequalities always hold by inspection. Therefore, ya≤Y‾ and yb≤Y‾ always hold. For ya≥0 and yb≥0, it must be Y‾≥δX‾αS and Y‾≥X‾δαS respectively. Because X‾δαS≥δX‾αS, Candidate solution (iv) is a valid solution in the range Y‾≥X‾δαS.

This completes the proof of Lemma 1.

B Proof of Lemma 2

The proof of Lemma 2 proceeds just like the proof of Lemma 1. There are four candidate solutions for Equilibrium Condition eq. (4):

  1. xa=0, ya<Y‾ and xb=0, yb<Y‾,

  2. xa=0, ya<Y‾ and yb=Y‾,

  3. ya=Y‾ and xb=0, yb<Y‾,

  4. ya=Y‾ and yb=Y‾.

To prove Lemma 2, I examine which of the four candidate solutions (i), (ii), (iii) and (iv) solve Equilibrium Condition eq. (4).

Candidate solution (i): xa=0, ya<Y‾ and xb=0, yb<Y‾

Candidate solution (i) implies that the Equilibrium Condition eq. (4) becomes:

(31)X‾+(Y‾−ya)=δ(X‾+Y‾−yb),
(32)αLyb=δαLya.

Solving eqs (31) and (32) for ya and yb:

(33)ya=X‾+Y‾1+δ,
(34)yb=δ(X‾+Y‾)1+δ.

For Solutions eqs (33) and (34) to be a valid, it must be that 0≤ya<Y‾ and 0≤yb<Y‾. Because δ∈(0,1) and X‾ and Y‾ are nonnegative numbers, it must be that ya≥0 and yb≥0. For ya<Y‾, it must be that Y‾>X‾δ, and for yb<Y‾, it must be that Y‾>δX‾. Because δX‾≤X‾δ, Candidate solution (i) is a valid solution in the range Y‾>X‾δ.

Candidate solution (ii): xa=0, ya<Y‾ and yb=Y‾

Candidate solution (ii) implies that the Equilibrium Condition eq. (4) becomes:

(35)X‾+(Y‾−ya)=δ(X‾−xb),
(36)xb+αLY‾=δαLya.

Solving eqs (35) and (36) for ya and xb:

(37)ya=(1−δ)X‾+(1−δαL)Y‾1−δ2αL,
(38)xb=αL(1−δ)(δX‾−Y‾)1−δ2αL.

For Solutions eqs (37) and (38) to be valid, it must be that 0≤ya<Y‾ and 0≤xb≤X‾. To check this validity, we examine the two possible subcases: (a) 1−δ2αL≥0 and (b) 1−δ2αL<0.

Subcase (a) 1−δ2αL≥0. For ya<Y‾, it must be Y‾>X‾δαL and for xb≥0, it must be Y‾≤δX‾. Therefore, for Candidate solution (ii) to be a valid solution in Subcase (a), it must be δX‾>X‾δαL. But that would imply 1−δ2αL<0. But in Subcase (a), 1−δ2αL≥0. Consequently, Candidate solution (ii) cannot exist in Subcase (a).

Subcase (b) 1−δ2αL<0. For ya<Y‾, it must be Y‾<X‾δαL and for xb≥0, it must be Y‾≥δX‾. Therefore, for Candidate solution (ii) to be a valid solution in Subcase (b), it must be δX‾<X‾δαL. But that would imply 1−δ2αL>0. But in Subcase (b), 1−δ2αL<0. Consequently, Candidate solution (ii) cannot exist in Subcase (b).

This means Candidate solution (ii) cannot be a valid solution for Equilibrium Condition eq. (4).

Candidate solution (iii): ya=Y‾ and xb=0, yb<Y‾

Candidate solution (iii) implies that the Equilibrium Condition eq. (4) becomes:

(39)X‾−xa=δ(X‾+Y‾−yb),
(40)αLyb=δ(xa+αLY‾).

Solving eqs (39) and (40) for xa and yb:

(41)xa=αL(1−δ)(X‾−δY‾)αL−δ2,
(42)yb=δ[(1−δ)X‾+(αL−δ)Y‾]αL−δ2.

For Solutions eqs (41) and (42) to be a valid, it must be that 0≤xa≤X‾ and 0≤yb<Y‾. δ∈(0,1) and αL>1 imply that the denominator of Solutions eqs (41) and (42), αL−δ2>0. By inspection, we also see that the numerator of Solution eq. (42) is nonnegative. This implies yb≥0. Looking at the numerator of Solution eq. (41), for xa≥0 it must be that Y‾≤X‾δ. xa≤X‾ implies αL(1−δ)(X‾−δY‾)αL−δ2≤X‾. This inequality simplifies to (αL−δ)X‾≥−αL(1−δ)Y‾ which always holds. Therefore, xa≤X‾. yb<Y‾ implies δ[(1−δ)X‾+(αL−δ)Y‾]αL−δ2<Y‾. This latter inequality simplifies to Y‾>δX‾αL. Hence, Candidate solution (iii) is a valid solution in the range δX‾αL<Y‾≤X‾δ.

Candidate solution (iv): ya=Y‾ and yb=Y‾

Candidate solution (iv) implies that the Equilibrium Condition eq. (4) becomes:

(43)X‾−xa=δ(X‾−xb),
(44)xb+αLY‾=δ(xa+αLY‾).

Solving eqs (43) and (44) for xa and xb:

(45)xa=X‾−δαLY‾1+δ,
(46)xb=δX‾−αLY‾1+δ.

For Solutions eqs (45) and (46) to be a valid, it must be that 0≤xa≤X‾ and 0≤xb≤X‾. Both xa≤X‾ and xb≤X‾ imply αLY‾+X‾≥0. Because αL>1 and X‾ and Y‾ are nonnegative numbers, we know that this latter inequality always holds by inspection. Therefore, xa≤X‾ and xb≤X‾ always hold. For xa≥0 and xb≥0, it must be Y‾≤X‾δαL and Y‾≤δX‾αL respectively. Because δX‾αL≤X‾δαL, Candidate solution (iv) is a valid solution in the range Y‾≤δX‾αL.

This completes the proof of Lemma 2.

References

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Published Online: 2018-04-18

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Articles
  2. Privatizing Multi-subsidiary Public Firm in Location Model
  3. Efficient Combinatorial Allocations: Individual Rationality versus Stability
  4. On Decay Centrality
  5. Sequential Auctions with Decreasing Reserve Prices
  6. The core of a strategic game
  7. Targeted Advertising on Competing Platforms
  8. Representation in Multi-Issue Delegated Bargaining
  9. Endogenous Mergers in Markets with Vertically Differentiated Products
  10. Standards of Proof and Civil Litigation: A Game-Theoretic Analysis
  11. Retained Earnings, Interest Rates and Lending Relationship
  12. Uniform Price Auctions with Asymmetric Bidders
  13. Conformity and Influence
  14. Sellouts, Beliefs, and Bandwagon Behavior
  15. Notes
  16. Eco-Firms and the Sequential Adoption of Environmental Corporate Social Responsibility in the Managerial Delegation
  17. Vertical Contract and Competition Intensity in Hotelling’s Model
  18. Constrained Allocation of Projects to Heterogeneous Workers with Preferences over Peers
  19. Irrelevance of the Strategic Variable in the Case of Relative Performance Maximization
  20. Critical Efficiencies as Upward Pricing Pressure with Feedback Effects
  21. On the Openness of Unique Pure-Strategy Nash Equilibrium
  22. Forecast Dispersion in Finite-Player Forecasting Games
https://doi.org/10.1515/bejte-2017-0100
Schlagwörter für diesen Artikel
delegated bargaining; multi-issue bargaining; simultaneous bargaining

Artikel in diesem Heft

  1. Articles
  2. Privatizing Multi-subsidiary Public Firm in Location Model
  3. Efficient Combinatorial Allocations: Individual Rationality versus Stability
  4. On Decay Centrality
  5. Sequential Auctions with Decreasing Reserve Prices
  6. The core of a strategic game
  7. Targeted Advertising on Competing Platforms
  8. Representation in Multi-Issue Delegated Bargaining
  9. Endogenous Mergers in Markets with Vertically Differentiated Products
  10. Standards of Proof and Civil Litigation: A Game-Theoretic Analysis
  11. Retained Earnings, Interest Rates and Lending Relationship
  12. Uniform Price Auctions with Asymmetric Bidders
  13. Conformity and Influence
  14. Sellouts, Beliefs, and Bandwagon Behavior
  15. Notes
  16. Eco-Firms and the Sequential Adoption of Environmental Corporate Social Responsibility in the Managerial Delegation
  17. Vertical Contract and Competition Intensity in Hotelling’s Model
  18. Constrained Allocation of Projects to Heterogeneous Workers with Preferences over Peers
  19. Irrelevance of the Strategic Variable in the Case of Relative Performance Maximization
  20. Critical Efficiencies as Upward Pricing Pressure with Feedback Effects
  21. On the Openness of Unique Pure-Strategy Nash Equilibrium
  22. Forecast Dispersion in Finite-Player Forecasting Games
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