Fairness considerations in joint venture formation

Abstract

Using a series of laboratory experiments in the context of bilateral bargaining over whether and how to engage in a joint venture, this paper shows that fairness concerns result in failures to undertake profitable joint production opportunities. We find that framing an opportunity as an employment relationship rather than as a partnership significantly reduces these inefficiencies and increases subjects’ welfare. Consistent with the theoretical model developed in the paper, text analysis and a follow-up experiment demonstrate that the lower likelihood of an efficient outcome in the partnership frame is driven primarily by a concern for fairness generated by the perceived social relationship associated with partnerships, and not by differences in the economic structure, cognition, subject motivation, or changes in relative bargaining power.

Appendix: Proofs

Below we present the proofs for the propositions presented in Sect. 3 and two related lemmas. Without loss of any generality, we assume that $w_{1} \ge w_{2}$.

Lemma 1

We can characterize the outcomes under three different cases as following:

• (1) When the size of pie is smaller than the sum of outside options then the players will always choose the outside options.

• (2) When $\pi >w_{1}+w_{2}$ and $\frac{\pi }{2}>w_{i}$ for both $i\in \left\{ 1,2\right\}$, then there is a Pareto improving division of $\pi$ for any $\alpha$, if either both players or neither player incorporate outside options in fairness consideration. However, if one player incorporates outside options in fairness consideration and the other player does not, then there is a Pareto improving division of $\pi$ only if $\alpha$ is low enough.

• (3) When $\pi >w_{1}+w_{2}$ and $\frac{\pi }{2}<w_{i}$ for some $i\in \left\{ 1,2\right\}$, then there is a Pareto improving division of $\pi$ for any $\alpha$, if both players incorporate outside options in fairness consideration. However, if at least one player does not incorporate outside options in fairness consideration, then there may not be a Pareto improving division of $\pi$ if $\alpha$ is high enough.

Proof

First we consider case 1, where $\pi <w_{1}+w_{2}$. Note that, for joint production, $u_{i}\left( x_{1},x_{2},w_{1},w_{2}\right) \le x_{i}$ independent of whether player i incorporates outside options to define fairness. As $x_{1}+x_{2}<w_{1}+w_{2}$ for any allocation of the pie, both cannot be (weakly) better-off by sharing the pie. Hence, the players should choose the outside options in the bargaining game.

For case 2, where $\pi >w_{1}+w_{2}$ and $\frac{\pi }{2}>w_{i}$ for both $i\in \left\{ 1,2\right\}$, $u_{i}\left( \frac{\pi }{2},\frac{\pi }{2} ,w_{1},w_{2}\right) =\frac{\pi }{2}>w_{i}$ if neither player incorporates outside options in fairness and $u_{i}\left( w_{1}+\frac{\pi -w_{1}-w_{2}}{2},w_{2}+\frac{\pi -w_{1}-w_{2}}{2} ,w_{1},w_{2}\right) =w_{i}+\frac{\pi -w_{1}-w_{2}}{2}>w_{i}$ if both players do. Thus, there is some allocation of the pie that makes both players better off when they both define fair allocation the same way. However, if the two players differ in how they define fair allocation, then it is possible that there is no division of $\pi$ that makes both better off if $\alpha$ is large enough and $w_{1}\ne w_{2}$.

Now consider case 3, where $\pi >w_{1}+w_{2}$ and $w_{1}>\frac{\pi }{2} >w_{2}$. If both players incorporate outside options in fairness, $u_{i}\left( \frac{\pi +w_{1}-w_{2}}{2},\frac{\pi -w_{1}+w_{2}}{2},w_{1},w_{2}\right) = \frac{\pi +w_{i}-w_{j}}{2}>w_{i}.$ Thus, there will be a Pareto improving division of the pie in that case. When at least one player does not incorporate the outside options in fairness, then if $\alpha$ or $w_{1}- w_{2}$ is large enough, there might not be any division of $\pi$ that is Pareto improving, leading to bargaining failure. Note that even when both players believe that equal division is the fair allocation, when the outside options are very different, giving player 1 just $w_{1}$ would already make player 2 worse off than her outside options. $\square$

Lemma 2

Suppose the two players differ in how they define fair allocation. If $\alpha$ is such that successful bargaining outcome is feasible for some outside option combination $\left( w_{1},w_{2}\right)$ where $w_{1}> \frac{\pi }{2}>w_{2}$, for a given $\pi$, then successful bargaining outcome will be feasible for any outside option combination $\left( w_{1}-\epsilon ,w_{2}+\epsilon \right)$ such that $\frac{w_{1}-w_{2}}{2}\ge \epsilon >0$.

Proof

The restriction on $\epsilon$ implies that $w_{1}-\epsilon \ge w_{2}+\epsilon$. First, we consider the case where ${\mathbf {1}}_{O_{1}}=0$ and ${\mathbf {1}}_{O_{2}}=1$ and joint production is feasible. Hence, there are $x_{1},x_{2}$ such that $x_{1}+x_{2}=\pi$ and

$$\begin{aligned} u_{1}\left( x_{1},x_{2},w_{1},w_{2}\right) =x_{1}-\alpha \left| x_{1}-x_{2}\right| \ge w_{1}{\text { and}}\text { }\text { }u_{2}\left( x_{1},x_{2},w_{1},w_{2}\right) =x_{2}-\alpha |x_{1}-w_{1}-\left( x_{2}-w_{2}\right) |\ge w_{2}. \end{aligned}$$

If we offer the two players $x_{1}-\epsilon$ and $x_{2}+\epsilon$ when the outside options are $\left( w_{1}-\epsilon ,w_{2}+\epsilon \right)$, then player 1’s utility from agreeing to accept $x_{1}-\epsilon$ is

$$\begin{aligned} u_{1}\left( x_{1}-\epsilon ,x_{2}+\epsilon ,w_{1},w_{2}\right) =x_{1}-\epsilon -\alpha \left| x_{1}-\epsilon -\left( x_{2}+ \epsilon \right) \right| =x_{1} -\alpha \left| x_{1}-x_{2}-2\epsilon \right| -\epsilon \ge w_{1} - \epsilon . \end{aligned}$$

Moreover, player 2’s utility from bargaining agreement is

$$\begin{aligned} u_{2}\left( x_{1}-\epsilon ,x_{2}+\epsilon ,w_{1},w_{2}\right)&=x_{2}+\epsilon -\alpha \left| x_{1}-\epsilon -w_{1}+\epsilon -\left( x_{2}+\epsilon -w_{2}-\epsilon \right) \right| \\&=x_{2}-\alpha \left| x_{1}-w_{1}-\left( x_{2}-w_{2}\right) \right| +\epsilon \ge w_{2}+\epsilon . \end{aligned}$$

Hence, bargaining agreement will be feasible.

Similarly, when ${\mathbf {1}}_{O_{1}}=1$ and ${\mathbf {1}}_{O_{2}}=0$, bargaining success under case 3 implies,

$$\begin{aligned}&u_{1}\left( x_{1},x_{2},w_{1},w_{2}\right) =x_{1}-\alpha |x_{1}-w_{1}-\left( x_{2}-w_{2}\right) |\ge w_{1}{\text { and}} \\&u_{2}\left( x_{1},x_{2},w_{1},w_{2}\right) =x_{2}-\alpha \left( x_{1}-x_{2}\right) \ge w_{2}. \end{aligned}$$

Using arguments similar to the ones above, we can show that for a different set of outside options $\left( w_{1}-\epsilon ,w_{2}+\epsilon \right)$ such that $\frac{w_{1}-w_{2}}{2}\ge \epsilon >0$, there will be a share of pie which makes both players better off. Note that the restriction on $\epsilon$ covers all possible case 2 situations as well as case 3 situations with less lop-sided outside options. Thus, the existence of feasible bargaining solution in a case 3 scenario involving two players whose fairness definitions differ, would suggest that there will be feasible bargaining solution in any case 2 scenario involving those two players. $\square$

Proposition 1

Bargaining failure is more likely in case 3 than in case 2.

Proof

Lemma 1 characterizes scenarios where bargaining failure may occur under cases 2 and 3. It suggests that it will suffice to focus on the case where the players differ in their definition of a fair outcome. Lemma 2 shows that, in that scenario, fixing $\pi$ and the sum of outside options, if the parameters ($\alpha ,w_{1},w_{2}$) are such that there is a Pareto-improving allocation of $\pi$ when $w_{1}>\frac{\pi }{2}>w_{2}$, then there must be a Pareto-improving allocation of $\pi$ when $w_{1}$ is decreased and $w_{2}$ is increased by the same amount in a way that $\frac{ \pi }{2}\ge w_{i}$ for $i\in \left\{ 1,2\right\}$. Hence, while bargaining failure can happen in both cases 2 and 3, it would happen more frequently under case 3. $\square$

Proposition 2

Suppose more players believe that fair allocation is equal division of surplus under frame A than under frame B. Then, the probability of bargaining success is higher under frame A for both cases 2 and 3.

Proof

We will prove this result by considering different scenarios. First, consider case 2. When both players define fairness the same way, they will successfully bargain to share the pie. If the two players define fairness differently, if $\alpha$ is small enough given $\left( \pi ,w_{1},w_{2}\right)$ then the players will bargain successfully. Therefore, we need to compare bargaining success probability under the two frames when bargaining failure occurs if the two players define fairness differently. In that case, the probability of bargaining success under frame $j \in {A,B}$ is $p_{j}^2+\left( 1-p_{j} \right) ^2$ for j. The difference in the success rate between frames A and B is $2p_{A}^2-2p_{A}-2p_{B}^2+2p_{B} =2\left( p_{A}+p_{B}-1 \right) \left( p_{A}-p_{B} \right)$. This difference is strictly positive if $p_{A}+p_{B}>1$. Therefore, the probability of bargaining success in case 2 will be higher for employment frame. For case 3, first consider $\left( \pi ,w_{1},w_{2}\right)$ combinations for which bargaining failure occurs only when the players differ in their definition of fairness. The above argument shows that the employment frame will increase probability of bargaining success when $p_{A}>p_{B}$ and $p_{B}+p_{A}>1$. If $\left( \pi ,w_{1},w_{2}\right)$ is such that bargaining success occurs only when both players define fairness with respect to outside options, then the employment frame will increase bargaining success as $p_{A}>p_{B}$. Therefore, frame A will lead to a higher likelihood of bargaining success for both cases 2 and 3. $\square$