This paper derives the Nash-equilibrium degrees of commitment to a partnership where lack of full commitment fuels suspicion and increases potential losses for partners.Structural break, unit root test, Lebanon economy

Levy, Amnon

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
Partnership with Partial Commitment 
 
 
 
 
 
 
 
 
Amnon Levy 
 
 
 
 
 
 
 
 
 
 
WP 06-05 
 
April 2006 
University of Wollongong 
Economics Working Paper Series 
2006 
 
http://www.uow.edu.au/commerce/econ/wpapers.html PARTNERSHIP WITH PARTIAL COMMITMENT  
 
 
 
 
Amnon Levy 
 
Economics Discipline 
School of Economics and Information Systems 
University of Wollongong 
& 
Department of Economics 
City College of New York 
City University of New York  
 
 
 
 
Abstract 
 
This paper derives the Nash-equilibrium degrees of commitment to a partnership 
where lack of full commitment fuels suspicion and increases potential losses for 
partners.  
 
JEL classification:  C72, D01, D80  
 
 
  
 
 
 
 
 
 
 
 
Corresponding Author: Amnon Levy, Economics Discipline, School of Economics 
and Information Systems, University of Wollongong, Wollongong, NSW 2522, 
Australia.  
Tel: 61-2-42213658 
Fax: 61-2-42213725 
E-mail: amnon_levy@uow.edu.au   1
1. Introduction 
 
The term partial commitment is usually used in game-theoretic and bargaining 
studies to indicate a player’s revocable obligation to a position on a partition of a cake 
(e.g., Muthoo, 1996; Calem, 1997; and Henkel, 2002). In this paper it is used to 
describe a partner’s degree of devotion to her partnership and disregard of external 
options. As frequently happens in business, politics and interpersonal relationships, 
attractive external options are realized after a partnership is formed. The effects of 
such external options on the rational choice of degrees of commitment to a 
partnership are conceptually analyzed in this paper. In particular, the paper considers 
the case where the direct returns on the external options exceed the internal returns, 
but high pecuniary and non-pecuniary costs prevent the partner(s) with the external 
option(s) from quitting (separation) and render partial commitment, though exposable, 
a viable strategy. The paper suggests that the Nash-equilibrium commitment of each 
member of a one-shot partnership (or, more generally, a known finite number of times 
automatically renewed partnership) increases with her internal return, quitting cost 
and costs of being deserted and with her counterpart’s external return and intrinsic 
capacity to detect lack of full commitment, but diminishes with her external return 
and intrinsic capacity to detect her counterpart’s lack of full commitment and with her 
counterpart’s internal return, quitting cost and costs of being deserted. 
The paper is structured as follows. Section 2 describes the effects of partial 
commitment on the partners’ internal returns, external returns, potential losses and 
detection of each other’s lack of full commitment. Section 3 presents the partner’s 
payoffs and strategies. Section 4 derives the Nash-equilibrium degrees of commitment 
in a one-period partnership. Section 5 illustrates the effects of the model parameters   2
on the partners’ Nash-equilibrium degrees of commitment with numerical 
simulations.  
 
2. External and internal returns, potential losses and detection capacities 
 
I and J are partners. With similar notations for J, the degree of I’s commitment 
to her partnership with J is denoted by  1 0 ≤ < i θ , where  1 = i θ  indicates full 
commitment, 1 0 < < i θ  partial commitment, and  i θ − 1  I’s degree of devotion to her 
external option.  
Suppose, that the external return for I increases, linearly for simplicity, with 
i θ − 1:   
i i i z z ˆ ) 1 ( θ − =              ( 1 )  
 
where  i i z z
i 0 lim ˆ
→ =
θ  indicates the external return when I quits the partnership and is fully 
devoted to her external option. Suppose, similarly, that I’s gross internal return is 
proportional to her degree of commitment to the partnership: 
i i i y y ˆ θ =              ( 2 )  
 
where  i i y y
i 1 lim ˆ
→ =
θ  indicates the gross internal return for I when I is fully committed to 
the partnership.  
Internal return for a partner is depreciated by the exposure of her own lack of 
full commitment and by her realization of her counterpart’s lack of full commitment. 
Naturally, the higher I’s degree of commitment the lower the probability ( j p ) of I’s 
lack of full commitment being detected by J and also the lower the loss (
i
i L ) for I 
from J’s discovery of I’s lack of full commitment. This assumption is displayed, for 
simplicity, by a linear specification of the detection probability   3
  
j i j p p ˆ ) 1 ( θ − = ,   1 ˆ 0 ≤ < j p            ( 3 )  
 
and by a linear specification of the potential loss 
 
i
i i
i
i L L ˆ ) 1 ( θ − = ,   0 ˆ >
i
i L .          ( 4 )  
 
In this framework,  j j p p
i 0 lim ˆ
→ =
θ  and 
i
i
i
i L L
i 0 lim ˆ
→ =
θ  and hence can be interpreted as J’s 
intrinsic capacity to detect I’s lack of full commitment and the maximum potential 
loss for I when I quits (the quitting cost), respectively. Similarly, the probability ( i p ) 
of  J’s lack of full commitment being detected by I and the loss (
i
j L ) for I from 
discovering J’s lack of full commitment are taken to be linearly diminishing in J’s 
degree of commitment:  
i j i p p ˆ ) 1 ( θ − = ,   1 ˆ 0 ≤ < i p            ( 5 )  
 
and  
 
i
j j
i
j L L ˆ ) 1 ( θ − = ,   0 ˆ >
i
j L .          ( 6 )  
 
Here,   i i p p
i 0 lim ˆ
→ =
θ  is I’s intrinsic capacity to detect J’s lack of full commitment and 
i
j
i
j L L
i 0 lim ˆ
→ =
θ  is the loss for I when J quits – the costs for I from being deserted by J.  
  The partners’ intrinsic capacities to detect each other’s lack of full 
commitment are not necessarily equal:  i j p p ˆ ˆ >  means that J is less susceptible to 
deception than I, and vice versa. If I’s ego is strong and I’s integrity is weak her 
quitting cost is smaller than the costs of being deserted (
i
j
i
i L L ˆ ˆ < ) and, in view of the 
above assumptions, I’s loss from the exposure of her lack of full commitment is 
smaller than I’s loss from realizing an identical lack of commitment of her 
counterpart. 
   4
3. Payoffs and strategies 
 
In view of the uncertainty about being exposed as partially committed and the 
uncertainty about J’s degree of commitment, the net return for partner I ( i x ) is a 
random variable distributed as follows
1 
 
⎪
⎪
⎪
⎩
⎪ ⎪
⎪
⎨
⎧
− − − − − − − − +
− − − − +
− − − +
− − − − − − +
=
i j j i i j j i i i i i
i j
i
j j i i i i
j i
i
i i i i i
i j j i
i
j j
i
i i i i i
i
p p p p z y
p L z y
p L z y
p p L L z y
x
ˆ ) 1 ( ˆ ) 1 ( ˆ ) 1 ( ˆ ) 1 ( 1 ˆ ) 1 ( ˆ
ˆ ) 1 ( ˆ ) 1 ( ˆ ) 1 ( ˆ
ˆ ) 1 ( ) ˆ ˆ )( 1 ( ˆ
ˆ ) 1 ( ˆ ) 1 ( ˆ ) 1 ( ) ˆ ˆ )( 1 ( ˆ
θ θ θ θ θ θ
θ θ θ θ
θ θ θ
θ θ θ θ θ
           . ( 7 )  
 
The expected net returns for I and J are: 
 
i
j j i i j
i
i i j j i i i i i i L p p L p p z y x E ˆ ] 1 ˆ ) 1 [( ˆ ) 1 ( ˆ ] 1 ˆ ) 1 [( ˆ ) 1 ( ˆ ) 1 ( ˆ ) (
2 2 + − − − + − − − − + = θ θ θ θ θ θ    (8) 
 
and 
 
j
i i j j i
j
j j i i j j j j j j L p p L p p z y x E ˆ ] 1 ˆ ) 1 [( ˆ ) 1 ( ˆ ] 1 ˆ ) 1 [( ˆ ) 1 ( ˆ ) 1 ( ˆ ) (
2 2 + − − − + − − − − + = θ θ θ θ θ θ  
.(9) 
  
The possible strategies and expected payoffs for the partners are summarized 
in Table 1. 
Insert Table 1 here 
 
The top-left cell of the matrix displays the payoffs for the ethical behavior - mutual 
full commitment - which is also the rational strategy for both partners when the 
partnership is automatically renewed indefinite number of times (e.g., a Catholic 
marriage). In the case of a one-shot partnership, or a partnership that is automatically 
renewed a known finite number of times, each partner might resort to a non-ethical 
                                                 
1 The first line in equation 7 displays the return for I in the event that J discovers I’s lack of full 
commitment and I discovers J’s lack of full commitment. The second line indicates the return for I in 
the event that J discovers I’s lack of full commitment and I does not discover J’s lack of full 
commitment. The third line presents the return for I in the event that I discovers J’s lack of full 
commitment and J does not discover I’s lack of full commitment. The fourth line displays the return for 
I in the event that both I and J do not discover each other’s lack of full commitment. 
   5
behavior by rationally choosing partial commitment so long that her expected net 
external return exceeds her internal return: 
i
i
i j i i y L p z ˆ ˆ ˆ ) 1 ( ˆ > − − θ                             (10) 
and  
j
j
j i j j y L p z ˆ ˆ ˆ ) 1 ( ˆ > − − θ .                          (11)  
In this case, the entries in the bottom-right cell indicate the Nash-equilibrium 
payoffs.
2  
 
4. Nash-equilibrium partial commitment in a one-shot partnership 
Assuming, for simplicity, risk-neutrality,
3 the partners’ Nash-equilibrium 
degrees of commitment in a one-shot partnership game are those maximizing each 
partner expected net return simultaneously. Assuming that conditions 10 and 11 hold, 
the partners’ chosen partial degrees of commitment satisfy: 
 
0 ) 1 ( ˆ ] 1 ˆ ) 1 [( ˆ 2 ) ˆ ) 1 ( ˆ ˆ ˆ ˆ (
2 = − + − − − − −
o
i
i
i i j j
i
j j j i i i L p p L p p y z θ θ θ                (12) 
 
and  
 
0 ) 1 ( ˆ ) 1 ˆ ) 1 [( ˆ 2 ) ˆ ) 1 ( ˆ ˆ ˆ ˆ (
2 = − + − − − − −
o
j
j
j j i i
j
i i i j j j L p p L p p y z θ θ θ                (13) 
 
which imply that I’s reaction function is 
 
i
i i j j
i
j j j i i i o
i L p p
L p p y z
ˆ ] 1 ˆ ) 1 [( ˆ 2
ˆ ) 1 ( ˆ ˆ ˆ ˆ
1
2
+ −
− − −
− =
θ
θ
θ                        (14) 
 
                                                 
2  If J chooses to be fully committed, I chooses partial commitment so long that 
i
i
i j i i i i i y L p z y ˆ ˆ ˆ ) 1 ( ˆ ) 1 ( ˆ
2 > − − − + θ θ θ  which, by rearranging terms, is equivalently rendered by 
condition 10. If I chooses to be fully committed, J chooses partial commitment so long that 
j
j
j i j j j j j y L p z y ˆ ˆ ˆ ) 1 ( ˆ ) 1 ( ˆ
2 > − − − + θ θ θ  which, by rearranging terms, is equivalently expressed 
by condition 11. 
3 Risk aversion can be incorporated, with a huge computational cost, by displaying the decision 
problem of an  i R -type risk averse person as  )] ( ) ( [ max i i i x VAR R x E
i − θ  and using 
2 2 )) ( ( ) ( i i x E x E −  to compute  ) ( i x VAR .    6
and J’s reaction function is   
 
j
j j i i
j
i i i j j j o
j L p p
L p p y z
ˆ ] 1 ˆ ) 1 [( ˆ 2
ˆ ) 1 ( ˆ ˆ ˆ ˆ
1
2
+ −
− − −
− =
θ
θ
θ .                       (15) 
 
The equilibrium degrees of commitment (
e
j
e
i θ θ , ) simultaneously satisfy 
equations 14 and 15. Subsequently, 
 
i
i j
j j
e
i i
j
i
e
i i j j j
i j
i
j j
j j
e
i i
j
i
e
i i j j j
j i i i
e
i
L
L p p
L p p y z
p p
L
L p p
L p p y z
p p y z
ˆ } 1 ˆ ] 1 ˆ ) 1 [( ˆ 2
ˆ ) 1 ( ˆ ˆ ˆ ˆ
ˆ { ˆ 2
ˆ } ˆ ] 1 ˆ ) 1 [( ˆ 2
ˆ ) 1 ( ˆ ˆ ˆ ˆ
{ ˆ ˆ ˆ ˆ
1
2
2
2
+
+ −
− − −
+ −
− − −
− −
− =
θ
θ
θ
θ
θ                   (16) 
 
and 
 
j
j i
i i
e
j j
i
j
e
j i j i i
j i
j
i i
i i
e
j j
i
j
e
j i j i i
j i j j
e
j
L
L p p
L p p y z
p p
L
L p p
L p p y z
p p y z
ˆ } 1 ˆ ] 1 ˆ ) 1 [( ˆ 2
ˆ ) 1 ( ˆ ˆ ˆ ˆ
ˆ { ˆ 2
ˆ } ˆ ] 1 ˆ ) 1 [( ˆ 2
ˆ ) 1 ( ˆ ˆ ˆ ˆ
{ ˆ ˆ ˆ ˆ
1
2
2
2
+
+ −
− − −
+ −
− − −
− −
− =
θ
θ
θ
θ
θ .
4                   (17) 
 
 
 
5. Numerical simulations and conclusion 
Table 2 summarizes the simulation results of the model parameter effects on 
e
i θ  and 
e
j θ  obtained by using equations 16 and 17 with the benchmark set of 
parameter values indicated by the bold numbers in the central column. The 
                                                 
4 In the special case where losses are total, rather than proportional to the degree of lack of 
commitment, close-form solutions are obtained. In this case, the expected return for I is 
i
j j i i j
i
i i j j i i i i i i L p p L p p z y x E ˆ ] 1 ˆ ) 1 [( ˆ ) 1 ( ˆ ] 1 ˆ ) 1 [( ˆ ) 1 ( ˆ ) 1 ( ˆ ) ( + − − − + − − − − + = θ θ θ θ θ θ , the 
necessary condition for maximum is  ) ( i x E  is  0 ˆ ˆ ) ˆ ˆ )( 1 ( ˆ ˆ ˆ ˆ = − + − − −
i
i j
i
j
i
i
e
j j i i i L p L L p p y z θ  and 
the Nash-equilibrium degree of commitment is 
) ˆ ˆ ( ˆ ˆ
ˆ ˆ ˆ ˆ
1
i
j
i
i i j
i
i j i i e
j L L p p
L p y z
+
− −
− = θ  for J and, by 
symmetry, 
) ˆ ˆ ( ˆ ˆ
ˆ ˆ ˆ ˆ
1
j
i
j
j i j
j
j i j j e
i L L p p
L p y z
+
− −
− = θ  for I .   7
benchmark parameter values portray a symmetric case ( 2 1 ˆ 000 , 100 $ ˆ y y = = , 
2 1 ˆ 000 , 200 $ ˆ z z = = , 
2
2
1
1 ˆ 000 , 100 $ ˆ L L = = , 
2
1
1
2 ˆ 000 , 150 $ ˆ L L = =  and  2 1 ˆ 9 . 0 ˆ p p = = ) and 
a 0.5907 commitment of I and J to their partnership. The off-central column cells in 
each row report the simulation results obtained by changing the value of one 
parameter while holding the rest at the benchmark level. Similar results were obtained 
with different benchmark sets of parameters. 
Insert Table 2 here 
 
The simulations’ results suggest the following properties of the partners’ 
Nash-equilibrium degrees of commitment to their partnership. First, the commitment 
of each partner increases with her own internal return and diminishes with her 
counterpart’s internal return. Second, the commitment of each partner diminishes with 
her external return and increases with her counterpart’s external return. Third, the 
commitment of each partner increases with her own quitting cost and decreases with 
her counterpart’s quitting cost. Fourth, the commitment of each partner increases with 
her own costs of being deserted and decreases with her counterpart’s costs of being 
deserted. Fifth, the commitment of each partner decreases with her own intrinsic 
capacity to detect her counterpart’s lack of full commitment and increases with her 
counterpart’s intrinsic capacity to detect lack of full commitment.    8
References 
 
Calem, Paul S., 1997, “Durable Goods Monopoly with Limited Choice of 
Commitment Strategy: When Should the Monopolist Precommit?”, Southern 
Economic Journal 64 (1), 104-17. 
 
Henkel, Joachim, 2002, “The 1.5th Mover Advantage”, Rand Journal of Economics 
33 (1), 156-70. 
 
Muthoo, Abhinay, 1996, “A Bargaining Model Based on the Commitment Tactic”, 
Journal of Economic Theory 69, 134-152. 
 
 
   9
Table 1. The partners’ expected payoff matrix 
             1→ 
2 
↓ 
Full Commitment  Partial Commitment 
 
Full 
Commitment 
1 1 ˆ ) ( y x E =
 
 
2 2 ˆ ) ( y x E =  
1
1 2
2
1
1 1 1 1 1
ˆ ˆ ) 1 (
ˆ ) 1 ( ˆ ) (
L p
z y x E
θ
θ θ
− −
− + =
 
 
2
1 2
2
1 2 2 ˆ ˆ ) 1 ( ˆ ) ( L p y x E θ − − =  
 
 
 
 
Partial 
Commitment 
 
1
2 1
2
2 1 1 ˆ ˆ ) 1 ( ˆ ) ( L p y x E θ − − =  
 
 
 
 
2
2 1
2
2
2 2 2 2 2
ˆ ˆ ) 1 (
ˆ ) 1 ( ˆ ) (
L p
z y x E
θ
θ θ
− −
− + =
 
 
1
2 2 1 1
2
2
1
1 1 2 2
2
1
1 1 1 1 1
ˆ ) 1 ˆ ) 1 [( ˆ ) 1 (
ˆ ] 1 ˆ ) 1 [( ˆ ) 1 (
ˆ ) 1 ( ˆ ) (
L p p
L p p
z y x E
+ − − −
+ − − −
− + =
θ θ
θ θ
θ θ
 
 
2
1 1 2 2
2
1
2
2 2 1 1
2
2
2 2 2 2 2
ˆ ) 1 ˆ ) 1 [( ˆ ) 1 (
ˆ ] 1 ˆ ) 1 [( ˆ ) 1 (
ˆ ) 1 ( ˆ ) (
L p p
L p p
z y x E
+ − − −
+ − − −
− + =
θ θ
θ θ
θ θ
 
 
 
   10
Table 2. Numerical simulation results 
1 ˆ y  
e
e
2
1
θ
θ
 
60,000 
 
0.4115 
0.6833 
80,000 
 
0.5011 
0.6354 
100,000 
 
0.5907  
0.5907 
120,000 
 
0.6803 
0.5465 
140,000 
 
0.7699 
0.5038 
2 ˆ y  
e
e
2
1
θ
θ
 
60,000 
 
0.6833 
0.4115 
80,000 
 
0.6354 
0.5011 
100,000 
 
0.5907  
0.5907 
120,000 
 
0.5465 
0.6803 
140,000 
 
0.5038 
0.7699 
1 ˆ z  
e
e
2
1
θ
θ
 
160,000 
 
0.7699 
0.5038 
180,000 
 
0.6803 
0.5465 
200,000 
 
0.5907  
0.5907 
220,000 
 
0.5011 
0.6364 
240,000 
 
0.4115 
0.6833 
2 ˆ z  
e
e
2
1
θ
θ
 
160,000 
 
0.5038 
0.7699 
180,000 
 
0.5465 
0.6803 
200,000 
 
0.5907  
0.5907 
220,000 
 
0.6364 
0.5011 
240,000 
 
0.6833 
0.4115 
1
1 ˆ L  
e
e
2
1
θ
θ
 
50,000 
 
0.1815 
0.7545 
75,000 
 
0.4543 
0.6438 
100,000 
 
0.5907  
0.5907 
125,000 
 
0.6726 
0.5597 
150,000 
 
0.7272 
0.5395 
1
2 ˆ L  
e
e
2
1
θ
θ
 
100,000 
 
0.5788 
0.6137 
125,000 
 
0.5843 
0.6022 
150,000 
 
0.5907  
0.5907 
175,000 
 
0.5972 
0.5794 
200,000 
 
0.6036 
0.5681 
2
2 ˆ L  
e
e
2
1
θ
θ
 
50,000 
 
0.7545 
0.1815 
75,000 
 
0.6438 
0.4543 
100,000 
 
0.5907  
0.5907 
125,000 
 
0.5794 
0.5972 
150,000 
 
0.5681 
0.7272 
2
1 ˆ L  
e
e
2
1
θ
θ
 
100,000 
 
0.6137 
0.5788 
125,000 
 
0.6022 
0.5843 
150,000 
 
0.5907  
0.5907 
175,000 
 
0.5794 
0.5972 
200,000 
 
0.5681 
0.6036 
1 ˆ p  
e
e
2
1
θ
θ
 
0.8000 
 
0.6036 
0.5492 
0.8500 
 
0.5969 
0.5712 
0.9000 
 
0.5907  
0.5907 
0.9500 
 
0.5850 
0.6081 
0.9999 
 
0.5796 
0.6237 
2 ˆ p  
e
e
2
1
θ
θ
 
0.8000 
 
0.5492 
0.6036 
0.8500 
 
0.5712 
0.5969 
0.9000 
 
0.5907 
0.5907 
0.9500 
 
0.6081 
0.5850 
0.9999 
 
0.6237 
0.5796 
 
 
 
 

A Bargaining Model Based on the Commitment Tactic”,

Durable Goods Monopoly with Limited Choice of Commitment Strategy: When Should the Monopolist Precommit?”,

Partnership with Partial Commitment

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Partnership with Partial Commitment

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