Risk aversion and equilibrium selection in a vertical contracting setting: an experiment

Abstract

The theoretical literature on vertical relationships usually assumes that beliefs about secret contracts take specific forms. In a recent paper, Eguia et al. (Games Econ Behav 109:465–483,2018) propose a new selection criterion that does not impose any restriction on beliefs. In this article, we extend their criterion by generalizing it to risk-averse retailers, and we show that risk aversion modifies the size of the belief subsets that support each equilibrium. We conduct an experiment which revisits that of Eguia et al. (Games Econ Behav 109:465–483,2018). We design a new treatment effect on equilibrium selection depending on the retailers’ risk sensitivity. Experimental results confirm the treatment effect: the more sensitivity there is towards risk, the more the equilibrium played is consistent with passive beliefs. In addition, extending Eguia et al.’s (Games Econ Behav 109:465–483,2018) criterion to risk-averse retailers improves its predictive power on the equilibria played, especially for a population of retailers with moderate to extreme risk aversion.

Appendices

Appendices

Proof of Lemma 1

To show equilibrium L and equilibrium H are Perfect Bayesian Equilibria in pure strategies (henceforth PBE), we show that no player deviates from equilibrium strategies given their beliefs and that these beliefs are consistent with equilibrium choices. We begin with the proof for equilibrium L and finish with the proof for equilibrium H.

$\bullet$ Proof equilibrium L

To prove the existence of the equilibrium L, we consider “passive beliefs”. We focus our proof on the case where $r=0$. As the level of r does not modify the preferences over the payoffs our proof then extends to all values of r considered in our analysis.¹⁸ In this equilibrium, the supplier sets $p^L=15$ to both retailers thinking each retailer rejects $p^H=36$ while buys 2 units at $p^L=15$. Each retailer rejects $p^H$ while buys 2 units at $p^L$ thinking the supplier sets $p^L$ to both retailers. It boils down to the following strategy vector $((p^L,p^L), (0,2), (0,2))$.

Retailers do not deviate (1) On-path and given the beliefs, the price offered is $p^L=15$, and the retailer i earns $\pi _i (q_i, 2)= e - 33 + P(q_i + 2) q_i - 15 q_i$ if $q_i>0$ and $\pi _i (q_i, 2)= e$ otherwise. We find $\pi _i(0, 2) = e$, $\pi _i(1, 2) = e - 2$, $\pi _i(2, 2) = e + 27$, and $\pi _i(3, 2) = e + 6$ which implies that $\pi _i(2, 2)$ is higher than the other profits for all admissible value of e. The retailer does not deviate on-path. (2) Off-path and given beliefs, the price offered is $p^H=36$, the retailer i earns $\pi _i (q_i, 2)= e - 33 + P(q_i + 2) q_i - 36 q_i$ if $q_i>0$ and $\pi _i (q_i, 2)= e$ otherwise. We find $\pi _i(0, 2) = e$, $\pi _i(1, 2) = e - 23$, $\pi _i(2, 2) = e -15$, and $\pi _i(3, 2) = e -57$ which implies that $\pi _i(0, 2)$ is higher than the other profits for all admissible value of e. The retailer does not deviate off-path.

Supplier does not deviate Given the beliefs, the suppliers earns $\pi _0(p^L,p^L)= 15 \times 2 + 15 \times 2 = 60$. If it deviates bilaterally or multilaterally, it, respectively, earns $\pi _0(p^H,p^L)= 36 \times 0 + 15 \times 2 = 30$ and $\pi _0(p^H,p^H)= 36 \times 0 + 36 \times 0 = 0$ which is less than the previous profit. The supplier does not deviate.

Belief consistency Given beliefs, we see that each retailer rejects $p^H$ and buys 2 units at $p^L$ while the supplier offers $p^L$ to both retailers. Therefore, beliefs are consistent; Equilibrium L is a PBE with passive beliefs.

$\bullet$ Proof equilibrium H

To prove the existence of equilibrium H, we consider “symmetric beliefs”. As previously, we focus our proof on the case where $r=0$. In this equilibrium, the supplier sets $p^H=36$ to both retailers thinking each retailer buys 1 unit at $p^H=36$ while buys 2 units at $p^L=15$. Each retailer buys 1 unit at $p^H$ while buys 2 units at $p^L$ thinking the supplier sets $p_i = p^H$ to both retailers. It boils down to the following strategy vector $((p^H,p^H), (1,2), (1,2))$.

Retailers do not deviate (i) On-path and given the beliefs, the price offered is $p^H=36$, and the retailer earns $\pi _i (q_i, 1)= e - 33 + P(q_i + 1) q_i - 36 q_i$ if $q_i>0$ and $\pi _i (q_i, 1)= e$ otherwise. We find $\pi _i(0, 1) = e$, $\pi _i(1, 1) = e + 31$, $\pi _i(2, 1) = e - 13$, and $\pi _i(3, 1) = e - 6$ which implies that $\pi _i(1, 1)$ is higher than the other profits for all admissible value of e. The retailer does not deviate on-path. (ii) Off-path and given the beliefs, the price offered is $p^L=15$, and the retailer earns $\pi _i (q_i, 2)= e - 33 + P(q_i + 2) q_i - 15 q_i$ if $q_i>0$ and $\pi _i (q_i, 2)= e$ otherwise. We find $\pi _i(0, 2) = e$, $\pi _i(1, 2) = e - 2$, $\pi _i(2, 2) = e + 27$, and $\pi _i(3, 2) = e + 6$ which implies that $\pi _i(2, 2)$ is higher than the other profits for all admissible value of e. The retailer does not deviate off-path.

Supplier does not deviate Given the beliefs, the supplier earns $\pi _0(p^H,p^H)= 36 \times 1 + 36 \times 1 = 72$. If it deviates bilaterally or multilaterally, it, respectively, earns $\pi _0(p^H,p^L)= 36 \times 1 + 15 \times 2 = 66$ and $\pi _0(p^L,p^L)= 15 \times 2 + 15 \times 2 = 60$ which is less than the previous profit. The supplier does not deviate.

Belief consistency Given beliefs, we see that each retailer buys 1 unit at $p^H$ and buys 2 units at $p^L$ while the supplier offers $p^H$ to both retailers. Therefore, beliefs are consistent; Equilibrium H is a PBE with symmetric beliefs.

Proof of Lemma 2

After observing the out-of-equilibrium offer $\bar{p_{i}}=p^H$, retailer i does not deviate from the equilibrium L by purchasing one unit rather than zero unit whenever:

$$\begin{aligned}&w(p_{-i}^{H}|\bar{p_{i}}=p^H) \cdot u(\pi _i (1 , 0 , 36)) +(1-w(p_{-i}^{H}|\bar{p_{i}}= p^H)) \cdot u(\pi _i (1 , 3 , 36)) \le u(e) \text { or } \\&w(p_{-i}^{H}|\bar{p_{i}}=p^H) \cdot u(e-33+103-36) +(1-w(p_{-i}^{H}|\bar{p_{i}}= p^H))\\&\quad \cdot u(e-33+46-36) \le u(e), \text { or } \\&w(p_{-i}^{H}|\bar{p_{i}}= p^H) \le w_{L_{1/0}} \equiv \frac{u(e)-u(e-23)}{u(e+34)-u(e-23)}. \end{aligned}$$

Similarly, purchasing two units (respectively, three units) is not a profitable deviation whenever $w(p_{-i}^{H}|\bar{p_{i}}= p^H) \le w_{L_{2/0}} \equiv \frac{u(e)-u(e -15)}{u(e+95) - u(e-15)}$ (respectively, $w(p_{-i}^{H}|\bar{p_{i}}= p^H) \le w_{L_{3/0}} \equiv \frac{u(e)-u(e -57)}{u(e-3)-u(e-57)} )$.

As a consequence, the subset of retailer i’s belief $w(p_{-i}^{H}|\bar{p_{i}}=p^L) \le w_H$, with $w_L =\min \{$ $w_{L_{1/0}}$, $w_{L_{2/0}}$, $w_{L_{3/0}}\}$, implies that purchasing either one, two or three units is not a profitable deviation from the equilibrium L after observing the out-of-equilibrium offer $\bar{p_{i}}=p^H$. This subset supports then the equilibrium L.

Proof of Lemma 3

After observing the out-of-equilibrium offer $\bar{p_{i}}=p^L$, retailer i does not deviate from the equilibrium H by purchasing three units rather than two units whenever:

$$\begin{aligned}&w(p_{-i}^{H}|\bar{p_{i}}=p^L) \cdot u(\pi _i (3 , 1, 15)) +(1-w(p_{-i}^{H}|\bar{p_{i}}= p^L)) \cdot u(\pi _i (3 , 2 , 15))\\&\quad \le w(p_{-i}^{H}|\bar{p_{i}}=p^L) \cdot u(\pi _i (2 , 1, 15))\\&\quad +(1-w(p_{-i}^{H}|\bar{p_{i}}= p^L)) \cdot u(\pi _i (2 , 2 , 15)) \text { or } \\&w(p_{-i}^{H}|\bar{p_{i}}= p^L) \cdot u(e-33+3(45-15))\\&\quad +(1-w(p_{-i}^{H}|\bar{p_{i}}= p^L)) \cdot u(e-33+3(28-15)) \\&\quad \le \ w(p_{-i}^{H}|\bar{p_{i}}= p^L) \cdot u(e-33+2(46-15))\\&\quad +(1-w(p_{-i}^{H}|\bar{p_{i}}= p^L)) \cdot u(e-33+2(45-15)) \text { or } \\&w(p_{-i}^{H}|\bar{p_{i}}= p^L) \le w_{H_{3/2}} \equiv \frac{u(e+ 27) -u(e + 6) }{u(e + 57)-u(e+6) +u(e+27)-u(e + 29)} . \end{aligned}$$

Similarly, purchasing on unit (respectively, zero unit) is not a profitable deviation whenever $w(p_{-i}^{H}|\bar{p_{i}}= p^L) \le w_{H_{1/2}} \equiv \frac{u(e + 27)-u(e-2)}{u(e+52)-u(e-2)+u(e+27)-u(e + 29)}$ (respectively, $w(p_{-i}^{H}|\bar{p_{i}}= p^L)\le w_{H_{0/2}} \equiv \frac{u(e + 27)-u(e)}{u(e+27)-u(e + 29)} )$.

As a consequence, the subset of retailer i’s belief $w(p_{-i}^{H}|\bar{p_{i}}=p^L) \le w_L$, with $w_L = \min \{$ $w_{H_{0/2}}$, $w_{H_{1/2}}$, $w_{H_{3/2}}\}$ implies that purchasing either zero, one, or three units is not a profitable deviation from the equilibrium H after observing the out-of-equilibrium offer $\bar{p_{i}}=p^L$. This subset supports then the equilibrium H.

Simulator of payoffs for treatment LE

See Fig. 5.

Fig. 5

Screenshot of simulator of players’ payoffs

Issues on belief-elicitation methods in the experiment

Beliefs and equilibrium actions In the experiment, we elicit both actions and beliefs, which raises questions as to whether eliciting beliefs might change the action or not. This issue is discussed in the literature in which incentives for truthful reporting are based on the realization of a random variable by means of Proper Scoring Rules (PSR). Basically, a scoring rule measures the accuracy of a probabilistic prediction and its form can be linear, quadratic, logarithmic, etc. The reported results in this literature are inconclusive. Whereas some papers, albeit few (e.g., Smith 2013 and Costa-Gomes et al. 2014), report preliminary evidence of a causal effect between elicited beliefs and equilibrium actions in subjects’ play, most papers state that there is no evidence of such an impact (Blanco et al. 2010; Schotter and Trevino 2014; Holt and Smith 2016). In our experiment, the presence of a simulator should limit this impact insofar as participants can already have a clear overview of the impact of their actions. The additional understanding brought by the belief elicitation should be minimized.

Hedging In the experiment, the payoff of the game divides into the payoffs from the action and the payoff from the belief elicitation task. As we pay both actions and beliefs, the validity of the elicitation might be challenged by hedging motives. Subjects might use stated beliefs to hedge against adverse outcomes in the rest of the experiment. While hedging can indeed be a problem in belief-elicitation experiments, it is less likely to be so when the hedging possibilities are not strong and prominent (Blanco et al. 2010; Armantier and Treich 2013; Schotter and Trevino 2014). In our experiment, the retailer’s payoff, if its stated belief is correct, is 40 ECU (2 €), which is a small fraction of its average payoff to the game. Thus, the risk of hedging is minimized.

Risk elicitation task

Table 7 provides an overview of our risk elicitation task. We depart from the test designed by Crosetto and Filippin (2015) in their “EG treatment” (in line with Eckel and Grossman (2002, 2008)’s test (EG test/task) procedure) and divide all the outcomes by one half (due to budget constraints). Given the CRRA utility this theoretically does not affect the answers of the participants (see Moffatt 2015). This test is a simple single-choice design where subjects are asked to choose one gamble from six different gambles where the probabilities of low and high outcomes are always 0.5 in each gamble. In an experiment, Dave et al. (2010) compared the behaviors in this task to those in the Holt and Laury (2002)’s task and found that subjects considered the former task to be more simple to understand. The EG task provided more reliable estimates of risk aversion for subjects with limited mathematical ability.

Table 7 The ordered lottery selection (Eckel and Grossman’s method)

Participants’ characteristics

This appendix gives an overview of the participants’ demography across the two treatments. Participants were randomly assigned a session. It was however still possible that their characteristics might be correlated with the type of treatment afterwards. To fully extract the impact of the initial endowment on the participants’ behaviors, we check that there is no correlation between the participants’ characteristics and the type of treatment they played.

Table 8 summarizes the distributions of the participants’ demographic characteristics. Let us remind that there are 12 subjects per sessions. In this table, $\mu$ denotes the mean and $\sigma$ denotes the standard deviation. We ran two Mann–Whitney tests (MW) and a Kolmogorov–Smirnov test (KS) to verify whether the two samples of subjects were identical. Statistical tests confirm that the participants’ characteristics are not significantly different between the two treatments: KS for age gives $p = 0.699$, the MW for gender gives $z=0.091$ and $p=0.928$, and the MW for field of study gives $z=-0.223$ and $p=0.823$.

Table 8 Participants’ demographic characteristics

We conclude that there is no evidence of correlation between the participants’ characteristics and their assigned treatment.