Fairness motivation in bargaining: a matter of principle

Abstract

In this paper, we study the role of fairness motivation in bargaining. We show that bargaining between two strongly fairness motivated individuals who have different views about what represents a fair division may end in disagreement. Further, by applying the Nash bargaining solution, we study the influence of fairness motivation on the bargaining outcome when an agreement is reached. In particular, we show that the bargaining outcome is sensitive to the fairness motivation of the two individuals, unless they both consider an equal division fair. We argue that our results accommodate existing experimental and field data on bargaining.

Appendices

Appendix 1: Analytical results

We here present analytical results referred to in the main text.

1.1 Reservation points

As referred to in footnote 11, we here show that for any fair share, $s_{i}$, there is a value $\hat{\beta }_{i}$ such that for any $0<\beta _{i}<\hat{\beta }_{i}$ there exists a unique reservation point, $x_{i}^{L}$, and for any $\beta _{i}\ge \hat{\beta _{i}}$ there exist two reservation points, $x_{i}^{L}$ and $x_{i}^{H}$. The utility function,

$$\begin{aligned} u_{i}(x_{i},s_{i},\beta _{i}) = x_{i} - \beta _{i} (x_{i} - s_{i})^{2}, \end{aligned}$$

is a quadratic equation,

$$\begin{aligned} a_{1}x^2 + a_{2}x + a_{3} = 0, \end{aligned}$$

where the coefficients are reduced to

$$\begin{aligned} a_{1}&= -\beta _{i}, \\ a_{2}&= (1+2\beta _{i} s_{i}), \\ a_{3}&= -\beta _{i} (s_{i})^2. \end{aligned}$$

The discriminant, $a_{2}^{2}-4a_{1}a_{3} = 1+4\beta _{i}s_{i}$, is positive and hence the utility function has two real, distinct roots. The quadratic formula gives the two solutions:

$$\begin{aligned} x_{i}^{L}= \frac{1 + 2 \beta _{i} s_{i} - \sqrt{1+4\beta _{i}s_{i}}}{2\beta _{i}}, \qquad x_{i}^{H}= \frac{1 + 2 \beta _{i} s_{i} + \sqrt{1+4\beta _{i}s_{i}}}{2\beta _{i}}. \end{aligned}$$

These solutions are not defined for $\beta _{i}=0$. By definition, a fair share, $s_{i}$, can have values in the interval $0\le s_{i}\le 1$. We observe that if $s_{i}=0$, then $x_{i}^{L}=0$. Differentiate $x_{i}^{L}$ with respect to $\beta _{i}$:

$$\begin{aligned} \frac{\hbox {d} x_{i}^{L}}{\hbox {d}\beta _{i}} = \frac{1+2\beta _{i}s_{i}-\sqrt{1+4\beta _{i}s_{i}}}{2\beta _{i}^{2} \sqrt{1+4\beta _{i}s_{i}}}. \end{aligned}$$

For the numerator to be positive, $1+2\beta _{i}s_{i}>\sqrt{1+4\beta _{i}s_{i}}$. By squaring both sides of the inequality we find that the numerator is always positive for $s_{i}>0$. Hence, $\frac{\mathrm{d}x_{i}^{L}}{\mathrm{d} \beta _{i}}>0$, and $x_{i}^{L}$ is strictly increasing in $\beta _{i}$ for $s_{i}>0$. We know from Sect. 3 that $\lim _{\beta _{i} \rightarrow \infty } x_{i}^{L} = s_{i}$. Thus, there always exists a lower reservation point, $x_{i}^{L}$, which attains values in the interval $0\le x_{i}^{L}\le s_{i}$.

We then consider the upper reservation point, $x_{i}^{H}$. We observe that if $s_{i}=1$, then $x_{i}^{H}>1$, which is outside of the domain of the utility function for argument $x_{i}$. We now differentiate $x_{i}^{H}$ with respect to $\beta _{i}$:

$$\begin{aligned} \frac{\hbox {d} x_{i}^{H}}{\hbox {d}\beta _{i}} = \frac{-1-2\beta _{i}s_{i}-\sqrt{1+4\beta _{i}s_{i}}}{2\beta _{i}^{2}\sqrt{1+4\beta _{i}s_{i}}}. \end{aligned}$$

We observe that $x_{i}^{H}$ is strictly decreasing in $\beta _{i}$, since $\frac{\mathrm{d} x_{i}^{H}}{\mathrm{d} \beta _{i}}<0$. If $s_{i}<1$, then $x_{i}^{H}$ may attain values in the interval $0< x_{i}^{H}\le 1$, depending on the relationship between $s_{i}$ and $\beta _{i}$. Define $\hat{\beta _{i}}$ such that $x^{H}=1$, which gives

$$\begin{aligned} \hat{\beta _{i}}=\frac{1}{(1-s_{i})^{2}}. \end{aligned}$$

Any $\beta _{i} \ge \hat{\beta _{i}}$ will give an upper reservation point in the interval $0<x_{i}^{H}\le 1$. Hence, for $\beta _{i}$ in the interval $0<\beta _{i}<\hat{\beta _{i}}$, there exists a unique reservation point, $x_{i}^{L}$, and for $\beta _{i}\ge \hat{\beta _{i}}$ there exist two reservation points, $x_{i}^{L}$, and $x_{i}^{H}$.

1.2 The Nash bargaining solution

As refereed to in Sect. 5, we here provide the analytical solution to the Nash bargaining problem. The Nash bargaining solution can be found by solving the optimization problem:

$$\begin{aligned} \begin{array}{ll} \underset{x_{1}}{\max } &{} (u_{1}(x_{1},s_{1},\beta _{1})- u_{1}(d,s_{1},\beta _{1}))(u_{2}(x_{2},s_{2},\beta _{2})- u_{2}(d,s_{2},\beta _{2}))\\ \text {s.t.} &{} x_{1} + x_{2} = 1. \\ \end{array} \end{aligned}$$

We assume that $u_{i}(d,s_{i},\beta _{i})=0$ and consider the utility function $u_{i}(x_{i},s_{i}, \beta _{i}) = x_{i} - \beta _{i} (x_{i} - s_{i})^{2}$. By substituting the constraint into the objective function, the optimization problem can be written as

$$\begin{aligned} \max f(x_1) =\Big (x_{1}-\beta _{1}(x_{1}-s_{1})^{2}\Big )\Big (1-x_{1}- \beta _{2}(1-x_{1}-s_{2})^{2}\Big ). \end{aligned}$$

Differentiating $f$ with respect to $x_1$ gives a cubic equation (the subscript on $x$ is suppressed):

$$\begin{aligned} a_{1}x^3 + a_{2}x^2 + a_{3}x + a_{4} = 0, \end{aligned}$$

where the coefficients are

$$\begin{aligned} a_{1}&= 4\beta _1 \beta _2, \\ a_{2}&= 3\beta _1-3 \beta _2-6 \beta _1 \beta _2-6 s_1 \beta _1 \beta _2+6 s_2 \beta _1 \beta _2, \\ a_{3}&= -2-2 \beta _1-4 s_1 \beta _1+4 \beta _2-4 s_2 \beta _2+2 \beta _1 \beta _2+ 8 s_1 \beta _1 \beta _2+2 (s_1)^2 \beta _1 \beta _2 \\&\quad -4 s_2 \beta _1 \beta _2-8 s_1 s_2 \beta _1 \beta _2+2 (s_2)^2 \beta _1 \beta _2, \\ a_{4}&= 1+2 s_1 \beta _1+(s_1)^2 \beta _1-\beta _2+2 s_2 \beta _2-(s_2)^2 \beta _2-2 s_1 \beta _1 \beta _2- 2 (s_1)^2 \beta _1 \beta _2\\&\quad +4 s_1 s_2 \beta _1 \beta _2+2 (s_1)^2 s_2 \beta _1 \beta _2-2 s_1 (s_2)^2 \beta _1 \beta _2. \end{aligned}$$

Define the following relationships:

$$\begin{aligned} Q&\equiv \frac{a_3}{3a_1}-\left( \frac{a_2}{3a_1}\right) ^2, \\ R&\equiv \frac{a_3a_2}{6(a_1)^2}-\frac{a_4}{2a_1}-\left( \frac{a_2}{3a_1}\right) ^3,\\ D&\equiv Q^3+R^2, \\ S&\equiv \left( R+\sqrt{D}\right) ^{\frac{1}{3}}, \\ T&\equiv \left( R-\sqrt{D}\right) ^{\frac{1}{3}}. \end{aligned}$$

$D$ is the discriminant that determines the nature of the roots of the equation. If $D>0$, there are one real root and two conjugate complex roots; if $D=0$, there are real roots of which at least two are equal; and if $D<0$, there are three distinct real roots. In this model, $D$ is negative and there are three distinct real roots. Cardano’s formulae for the roots are as follows:

$$\begin{aligned} \hbox {root}_1&= -\frac{a_2}{3a_1}+(S+T), \\ \hbox {root}_2&= -\frac{a_2}{3a_1}-\frac{1}{2}(S+T)+\frac{1}{2} i \sqrt{3}(S-T), \\ \hbox {root}_3&= -\frac{a_2}{3a_1}-\frac{1}{2}(S+T)-\frac{1}{2} i \sqrt{3}(S-T), \end{aligned}$$

where $i = \sqrt{-1}$. For this model, it turns out that $\hbox {root}_2 < \hbox {root}_3< \hbox {root}_1$. The solution is only defined for $\beta _{1}>0$ and $\beta _{2}>0$. To make sure the solution is the Nash bargaining solution, check that $(u_{1}(x_{1}^{N}),u_{2}(1-x_{1}^{N}))>(0,0)$. The optimal solution is $x^{N}_{1}=root_3$, and the Nash bargaining solution is $(x^{N}_{1},1-x^{N}_{1})$.

1.3 The Nash bargaining solution: a special case

We here provide the Nash bargaining solution for the special case where one individual is fairness motivated and the other is only motivated by material self-interest, as referred to in footnote 16. We assume

$$\begin{aligned} u_{1}&= x_{1}-\beta _{1}(x_{1}-s_{1})^{2},\\ u_{2}&= x_{2},\\ u_{i}(d,s_{i},\beta _{i})&= 0. \end{aligned}$$

By substituting the constraint into the objective function, the optimization problem can be written as

$$\begin{aligned} \max f(x_1) =\Big (x_{1}-\beta _{1}(x_{1}-s_{1})^{2}\Big )\Big (1-x_{1}\Big ). \end{aligned}$$

Differentiating $f$ with respect to $x_1$ gives a quadratic equation, where the root that satisfies the Nash bargaining solution is,

$$\begin{aligned} x_{1}^{N}=\frac{1+\beta _{1}+2\beta _{1}s_{1}-\sqrt{1-\beta _{1}+4\beta _{1}s_{1} +(\beta _{1})^{2}-2(\beta _{1)}^{2}s_{1}+(\beta _{1}s_{1})^{2}}}{3\beta _{1}}. \end{aligned}$$

The Nash bargaining solution has the property that it approaches the solution for two self-interested individuals when fairness motivation is weak, $\lim _{\beta _{1} \rightarrow 0} (x_{1})^{N} = \frac{1}{2}$. For a strongly fairness motivated individual, the solution approaches his fair share, $\lim _{\beta _{1} \rightarrow \infty } (x_{1})^{N} = s_{1}$.

The effect of a change in $\beta _{1}$ can be derived by differentiating the Nash bargaining solution for this special case,

$$\begin{aligned} \frac{\mathrm{d}x_{1}^{N}}{\mathrm{d}\beta _{1}}=\frac{2+\beta _{1}(4s_{1}-1)-2\sqrt{1+\beta _{1}(4s_{1}-1+\beta _{1} (s_{1}-1)^{2}}}{6(\beta _{1})^{2} \sqrt{1+\beta _{1}(4s_{1}-1+\beta _{1}(s_{1}-1)^{2}}}. \end{aligned}$$

This expression is zero for $s_{1}=0.5$, it is positive for any change in $\beta _{1}$ provided that $s_{1}>0.5$, and it is negative if $s_{1}<0.5$. Hence, when the fairness motivation of an individual increases, the Nash bargaining solution assigns a share that is closer to his fair share.

Appendix 2: Experiment instructions

1.1 General introduction

Welcome to this experiment. My name is (...) and I will guide you through the experiment. The results from the experiment will be used in a research project, and it is therefore important that you all stick to the rules that have been distributed:

• You should not talk to other participants.

• If you have questions or problems during the experiment, raise your hand and we will come to you.

• You should not open other web pages.

If you breach these rules, you will have to leave the room. There will be pauses during the experiment and it is important that you sit still and keep quiet during these.

You will be completely anonymous in the experiment. You will not at any time be asked about who you are. It will not be possible for us or the other participants to find out which choices you have made. You will be asked to make choices in several different situations in this experiment. For every situation, you will be randomly connected to another person in this room. Your actual payment will be determined as follows: we randomly draw one of the situations you were involved in and pay the amount of money you received in that situation. The choices that you make will not influence which situation is drawn; it will be an entirely random draw and there is an equal chance for all situations to be drawn. You should therefore think about each situation as if it is the one that determines how much you will earn.

When the experiment is finished, you will see a payment code on the screen. You are asked to write down this code on a form that will then be sent to the accounting department at (...). Employees at the accounting department will receive a list of codes and amounts from us and match these with the payment instructions from the forms. This is done so that nobody will know how much you have earned.

The experiment consists of four phases. I will now explain the main features of the experiment. I will stop before we start a new phase and explain in more detail what you should do in each phase. In the first phase of the experiment, you will be copying text in Word for 10 min. You will be paid a price for each correct word you have typed. In phase two of the experiment, you will be randomly matched with other persons in this room, and each person in each pair will choose how much of the combined production value to distribute to yourself and to the other person. You will be involved in four such situations of distribution.

In the third phase of the experiment, you will also be randomly matched with people in this room. You will then negotiate about the division of the combined production value by sending proposals to each other until one of you accepts the other’s proposal. The production value shrinks by 4 % every time one of you does not accept the other’s proposal. You will be involved in four such situations of negotiation. In the last phase of the experiment, you will be asked to answer a few questions about the types of situations that you have experienced.

1.2 Introduction phase 1

The first thing you will do is to copy text from an official report that is marked with either an A or a B, and which you will find in the folder on your desk. You will start copying the text into Word when I tell you. When 10 min have passed, I will let you know and everybody must then stop. You will be paid for each correct word you type. You may use the spellchecker in Word.

I remind you that you should raise your hand if you have any problems or questions, and then someone from the research group will come and help you. You can now open a new document in Word and we will soon start to type. You can start typing now.

Everybody must now stop typing. You should now highlight all the text typed and copy it to the window in the Mozilla browser, then click on the button marked “submit text.”

After having submitted the text, you will see a screen that shows how much you have produced and the value of your production. The production is rounded off to the nearest 50 words. Half of you have copied text marked A, which is an excerpt from an official report on the merger of the telecoms, IT, and media sectors. You will receive one krone and 50 oere for each correct word you have typed. The other half has copied text marked B, which is an excerpt from an official report about Norwegian performing art. You will receive 75 oere for each correct word you have typed. These prices are randomly determined by us. Finally, click on the button marked “continue.”

1.3 Introduction phase 2

You will now be randomly matched with other people in this room. In each situation of distribution, you will not know who the other person is and the other person will not know who you are. You will be informed about how many words he or she has produced and what price each of you has randomly been allocated. You will then choose a distribution of the combined production value between you and the other person. Remember that this is real money and that the way that you divide the money determines how much you earn and how much the other person earns. You will be asked to make decisions in two such situations of distribution. In two other situations of distribution, another person will decide how much he or she will distribute to you.

After you have registered the distribution, you will see a new screen where you are asked either to confirm the distribution or to go back and change the distribution. When you have confirmed your choices, you will receive a message that you have finished the second phase of the experiment. You should then quietly wait for all the other people in the room to finish making choices in their situations. On the computer, you will soon see a screen with the first situation and you can then start making choices.

1.4 Introduction phase 3

Everybody has finished the second phase and I shall now explain what you will be doing in the third phase of the experiment. You will this time also be randomly matched with other people in this room. In each situation of negotiation, you will not know who the other person is and the other person will not know who you are. You will be informed about how many words the other person has produced and what price he or she has randomly been allocated. One of you is randomly drawn to make the first proposal for division of your combined production value. The proposal will be sent to the other person and he or she has two choices: to accept your proposal or to make a new proposal for division. New proposals are sent back and forth until one of you chooses to accept the other’s proposal. Every time one of you does not accept the proposal for division, but comes up with a new proposal, the remaining production value will be reduced by 4 %. Everybody will be involved in four such negotiation situations.

In some situations, you will be asked what you think will be the final outcome of the negotiation. If your answer is within a deviation of plus or minus 20 kroner of the actual result, you will receive 20 kroner in extra payment, with one exception: if you guessed the negotiation result in a situation, and this particular result was randomly drawn, you will not receive the extra payment for a correct guess but only the payout in this situation. You will also be asked to state how certain you are about your guess. Your answer should be given in terms of a certainty percentage, that is, a number between 0 and 100. It is important that you write a high percentage if you are certain that this will be the final result, and a low percentage if you are uncertain if this will be the final result.

On the computer, you will soon see a new screen with the first situation and you can then start to negotiate. When the situation is accepted, you will automatically get a new situation to negotiate. When you have finished all the negotiation situations, you will be asked to wait until everybody has finished their choices.

1.5 Introduction phase 4

Everybody has finished and we will soon draw the situation that will decide your payment from this experiment. First, we ask you to answer a few questions. Soon you will see a new screen with information about the first question. Click on the button marked “go forward” when you have read the information and thereafter please answer all the questions.

1.6 Closing and payment

Everybody has now answered the questions. You will soon see a screen that informs you about which situation that has been drawn randomly, and how much you have earned in this situation. This screen will be open for 45 s. Thereafter, you will automatically be forwarded to a new screen, which only contains a payment code.

Everybody now has a screen with the payment code. Write down this payment code on the form that you find in the folder next to you. On the form also write your name and bank account details. Put the form in the envelope and place the envelope in the box by the door when you leave the room.

The experiment is now finished and, on behalf of the research team, I thank you again for your participation in this experiment.