Confusion and learning in the voluntary contributions game

Abstract

We use a limited information environment to assess the role of confusion in the repeated voluntary contributions game. A comparison with play in a standard version of the game suggests, that the common claim that decision errors due to confused subjects biases estimates of cooperation upwards, is not necessarily correct. Furthermore, we find that simple learning cannot generate the kind of contribution dynamics commonly attributed to the existence of conditional cooperators. We conclude that cooperative behavior and its decay observed in public goods games is not a pure artefact of confusion and learning.

Appendix

1.1 A.1 Details of the learning model

So we arrive at an extremely simple learning model. Define the set of players as I={1,2,3,4} and the action space as C={0,1,…,20}. Denote the contribution of person i ∈I in period t∈{1,2,…,20} as $c_{t}^{i}\in C$. The player uses the payoffs p and the own choices of the last two periods to determine the contribution in the current period (if possible). The attraction of choosing a certain contribution $A(c_{t}^{i})$ is therefore a function of the two past contributions and the payoffs in the two last periods:

$$ A\bigl(c_{t}^{i}\bigr)=f\bigl(c_{t-1}^{i},c_{t-2}^{i},p_{t-1}^{i},p_{t-2}^{i} \bigr). $$

(1)

After having observed the two last outcomes given the choices made, for the next round individuals only consider choices which are closer to the choice that resulted in a higher payoff.²³ Suppose $c_{t-1}^{i}$ was greater than $c_{t-2}^{i}$ and the payoff in period t−1 was greater than in period t−2, then the individual only chooses values in the interval from the midpoint between the two previous choices to the maximum choice (20). For equal profits in periods t−1 and t−2 the support is [0,20], as then the history contains no information about in which direction to go. Moreover, the support will also be the whole spectrum of possible choices if the previous two choices were identical.

To find the region of choices (the support) that satisfies these conditions given the history, define the changes in choices and payoffs between periods t−1 and t−2 as

(2)

(3)

Then we can introduce a variable $d_{t}^{i}$ that tells us whether the player wants to choose a number closer to the higher ($d_{t}^{i}=1$) or the lower of the previous choices ($d_{t}^{i}=-1$):

$$ d_{t}^{i}=\operatorname {sign}\bigl(\Delta p_{t}^{i} \cdot \Delta c_{t}^{i}\bigr). $$

(4)

Note that if either the profits or the previous choices have not changed between periods t−2 and t−1 then we have $d_{t}^{i}=0$. Denoting the admissible support for period t as $C_{t}^{i}$ we have:

$$ C_{t}^{i}=\left \{\begin{array}{l@{\quad }l} \{c\in C:c \leq (c_{t-1}^{i}+c_{t-2}^{i})/2 \} & \mathrm{if}\ d_{t}^{i}=-1 \\ \noalign {\vspace {3pt}} \{c\in C:c \geq (c_{t-1}^{i}+c_{t-2}^{i})/2 \} & \mathrm{if}\ d_{t}^{i}=1 \\ \noalign {\vspace {3pt}} \{c\in C \} & \mathrm{if} \ d_{t}^{i}=0\end{array} \right . $$

(5)

Next, we have to specify which point within the admissible range will be chosen. The simplest assumption is that subjects are equally likely to choose any element of $C_{t}^{i}$.²⁴ To implement this we set the attraction for a choice in $C_{t}^{i}$ equal to one, while the attraction of a contribution outside of $C_{t}^{i}$ is set to zero:

$$ A\bigl(c_{t}^{i}\bigr)=\left \{\begin{array}{l@{\quad }l} 1 & \mathrm{if}\ c_{t}^{i} \in C_{t}^{i} \\ \noalign {\vspace {3pt}} 0 & \mathrm{if} \ c_{t}^{i} \not \in C_{t}^{i}\\ \end{array} \right . $$

(6)

To arrive at the desired uniform distribution over the support $C_{t}^{i}$ we transform attractions into probabilities using the following rule:

$$ g\bigl(c_{t}^{i}\bigr)=\frac{A(c_{t}^{i})}{\sum_{c_{t}^{i}\in C_{t}^{i}}{A(c_{t}^{i})}} $$

(7)

1.2 A.2 Dynamic-panel estimation

We estimated a dynamic panel, which allows for contributions to depend on past own contributions and on past contributions of other group members. By design any unobserved panel-level effects are correlated with the lagged own contributions. For this reason we used the Arellano-Bond-Bover GMM estimator with additional moment conditions developed in Blundell and Bond (1998) that can handle this endogeneity problem. Table 3 reports the results.

Table 3 Estimation of contributions in the Standard Condition (by subject type) and in the Learning Condition