Endogenous claims and collective production: an experimental study on the timing of profit-sharing negotiations and production

Abstract

We explore the efficiency and distributive implications of a multilateral bargaining model with endogenous production of the surplus under two different timings: ex ante and ex post bargaining. Both timings are commonly observed in business partnerships and alliance formations. The theoretical predictions confirm an intuitive economic tenet: in ex post bargaining, effort is considered sunk and opportunistic bargaining behavior will dissuade players from producing. On the other hand, ex ante bargaining entails an allocation of ownership shares that induces at least certain members to invest in the common fund because their return is guaranteed. Experiments show opposite results: ex post bargaining yields almost fully efficient outcomes while the reverse timing entails near zero efficiency. The psychological theory of inequity is useful in reconciling these divergent results.

Appendices

Appendix A: Proof of Proposition 2

Starting in stage 2 after an equity agreement has been reached in round $\tau$ we have that a member finds it optimal to invest if and only if the return of doing so is larger than the cost. Formally, the equilibrium investment strategy in any subgame is

$$\begin{aligned} c_{i}^{*}({\mathbf {s}}^{\tau })=\left\{ \begin{array}[c]{ccc} 0 &{}\quad \text { if } &{} \alpha s_{i}^{\tau }<1\\ E &{}\quad \text {if} &{} \alpha s_{i}^{\tau }\ge 1 \end{array} \right. . \end{aligned}$$

(5)

Now we partition the set of subgames into three possible disjoint sets and show that equilibrium can only occur in one of those subsets. We define

$$\begin{aligned} S_{k}:=\left\{ (s^{1},\ldots ,s^{n})\in [0,1]^{n}:\sum _{i=1}^{n}c_{i}^{*}=kE\right\} \end{aligned}$$

(6)

where the $S_{k}$ determines all equity agreements in which k members receive a share that induces investments. It is straight forward to show that the sets $S_{k}$ are pair-wise disjoint and that $S_{0}\cup S_{1}\cup S_{2}=[0,1]^{n}$. Moreover, $S_{k}=\emptyset$ for $k>2$. If $\alpha <2$, then $S_{2}=\emptyset$ also. This is a consequence of the equity constraint which requires that the sum of shares must add to 1.

Lemma 3

In equilibrium,$\mathbf {s\notin }S_{0}$.

Proof

By the contrary, assume that ${{\tilde{\mathbf{s }}}\in }S_{0}$ is an equilibrium. Then, $u({{\tilde{\mathbf{s }}}})=1$ for each player which implies that the ex ante value of the game is equal to 1 for every player. As such, a voter will vote yes for any share, because it will yield at least a payoff of 1. A proposing member can deviate to $(1,0,\ldots ,0)$ where the first entry is the proposer’s share (without loss of generality). Every player would earn $u({\tilde{\mathbf{s }}})=$$u(1,0,\ldots ,0)=1$ which means they would vote in favor. The proposer who would find it profitable to invest and earn $\alpha >1$. $\square$

Lemma 4

In equilibrium,$\mathbf {s\notin }S_{2}$.

Proof

By the contrary, assume that ${{\tilde{\mathbf{s }}}\in }S_{2}$. This means that exactly 2 players are offered a share of $s_{Inv}=0.5$. Without loss of generality, the payoff vector that arises will be $(\alpha ,\alpha ,1,\ldots ,1)$. This is, two players will earn $\alpha$ and the $n-2$ will earn only their endowment. Since ex ante any player can attain each payoff with equal chances, the equilibrium expected value of the game is $V=\frac{2}{n}\alpha +\frac{n-2}{n}>1$. Here, we are imposing symmetry in the sense that what player i offers j is what j offers i. Recall that a member votes in favor if and only if her payoff resulting from the share received is greater than or equal to V. However, this proposal will never be approved, because only two players exhaust all the available equity and the proposal does not receive a majority vote. $\square$

Proof of Proposition 2

We now focus on the case of $\mathbf {s\in }S_{1}$ and will show that there exists an equilibrium with one productive member. Without loss of generality, we assume that the proposer assigns herself a share $s^{\text {prop}}$ greater than or equal to 1 / 2 which means that she invests in equilibrium. We denote by $s^{\text {vote}}$ the positive share offered to m coalition partners. The equity feasibility constraint is given by

$$\begin{aligned} s^{\text {prop}}+m\cdot s^{\text {vote}}=1. \end{aligned}$$

(7)

With probability $\frac{m}{n}$ a member is an included voter who earns a share of the fund $s^{\text {vote}}$ and keeps his endowment. With probability $1-\left( \frac{1+m}{n}\right)$ a member receives zero equity and only keeps his endowment. Hence, the ex ante value of the game is given by

$$\begin{aligned} V=\frac{\alpha s^{\text {prop}}}{n}+\frac{m}{n}(\alpha s^{\text {vote} }+1)+1-\left( \frac{1+m}{n}\right) \end{aligned}$$

(8)

which is simply the payoff associated to each role weighed by its probability. Imposing the equity constraint (7) we obtain that $V=\frac{\alpha -1}{n}+1$. As such, the proposer faces the following problem:

$$\begin{aligned}&\max \nolimits _{s^{\text {prop}},s^{\text {vote}},m\in \{1,\ldots ,n-1\}}\alpha \cdot s^{\text {prop}}\text { subject to} \end{aligned}$$

(9)

$$\begin{aligned}&s^{\text {prop}}\in [1/2,1] \end{aligned}$$

(10)

$$\begin{aligned}&s^{\text {prop}}\alpha \ge V \end{aligned}$$

(11)

$$\begin{aligned}&s^{\text {vote}}\in [0,1/2) \end{aligned}$$

(12)

$$\begin{aligned}&s^{\text {vote}}\alpha +1\ge V \end{aligned}$$

(13)

$$\begin{aligned}&m\ge \frac{n-1}{2}. \end{aligned}$$

(14)

The objective function states that the proposer is seeking to maximize his own payoff and must choose a share for himself, a share offerd to coalition partners $s^{\text {vote}}$, the number of shares $s^{\text {vote}}$ that she will offer (m). Constraints (10) and (12) simply guarantee that the chosen equity scheme ${\mathbf {s}}$ is in $S_{1}$. Conditions (11) and (14) are ensure that the proposal receives at least the required number of votes in favor. The equity constraint (7) specifies that the sum of shares have to be feasible, i.e., less than one from which we obtain that $s^{\text {prop} }=1-m\cdot v^{\text {vote}}$. This implies that (14) binds at $m=\frac{n-1}{2}$, namely, that minimum winning coalitions are optimal. To obtain $s^{\text {vote}}$ we use constraint (13) which clearly binds as well because if the share was higher the proposer could decrease it and still obtain a favorable vote. Thus, we have $s^{\text {vote}}=\frac{V-1}{\alpha }=\frac{\alpha -1}{\alpha n}$ and $s^{\text {prop}}=1-\frac{n-1}{2}\frac{\alpha -1}{\alpha n}$. With simple computations one verifies that (11) and (10) hold. $\square$

Appendix B: Supplementary figures

See Figs. 3, 4, 5 and 6.

Fig. 3

Investments and approved shares in pre-distributive bargaining

Fig. 4

Average investments in pre-distribution by session

Fig. 5

Average investments in redistribution by session

Fig. 6

Average contributions and total fund in pre-distribution by group size