Confusion and learning in the voluntary contributions game


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.

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  1. 1.

    The study of behavior in limited information environments is common in experiments on learning; see, Mookherjee and Sopher (1994), Van Huyck et al. (2007), and Chen and Khoroshilov (2003).

  2. 2.

    The choices of subjects who are confused may also be more likely to be influenced by objectively irrelevant contextual cues in language and other merely procedural details of the experiment. Ferraro and Vossler (2010), for example, report that “many subjects believe they are playing a sort of stock market game” (p. 24) and they conjecture that this might be caused by the “investment language” used in many voluntary contributions experiments. Fosgaard, Hansen and Wengström (2011) find that more subjects are able to identify the dominant strategy when the game is presented in a “take” as compared to a “give frame.”

  3. 3.

    In contrast to repetition, which represents a sequence of decision rounds within the same group of subjects, the term ‘experience’ means that subjects play the game again with a different group.

  4. 4.

    The term ‘warm glow’ was introduced by Andreoni (1993). Based on a similar thought, some authors have proposed games with an interior equilibrium prediction to test economic theory. This literature typically reports a lower level of excess cooperation. See, e.g., Ziegelmeyer and Willinger (2001) and the references they cite.

  5. 5.

    Gintis et al. (2003) explain similar dynamics with an evolutionary approach.

  6. 6.

    Besides potential interaction effects between heterogeneous social preferences and decision errors, there are other arguments why subjects may learn differently between strategic and individual choice situations. See, for example, Duersch et al. (2010), who explore how subjects learn to play a Cournot-Duopoly game against computers that are programmed to follow one of various learning algorithms.

  7. 7.

    For studies on the comparative power of alternative learning models see, among others, Gale et al. (1995), Erev and Roth (1998), Chen and Tang (1998), Feltovich (2000), Janssen and Ahn (2006). Camerer and Ho (1999) propose a weighted model with choice reinforcement and belief-based (fictitious play) learning as two special cases. Using data from a large class of experimental games, they show that learning is best explained by a combination of both.

  8. 8.

    These authors use a large sample from the general population in Denmark. They also run a follow-up experiment with standard student subjects in which even fewer subjects correctly answer this question.

  9. 9.

    Ferraro and Vossler (2010) suggest that subjects may use the actions of others as an indication of profit-maximizing behavior. Note that such “herding” is ruled out by our design.

  10. 10.

    We are grateful to an anonymous referee who suggested this treatment as a robustness check.

  11. 11.

    Contributions in the Standard Condition are very similar to contributions in the experiments where the standard VCM was played in a second phase of the Learning Condition (the averages are 7.24 in the Standard Condition and 8.25 and 7.73 after the minimum and limited information condition, respectively). A Mann-Whitney U-test (two tailed, applied on group averages) is insignificant at p>0.28 (p>0.80) if we compare the contributions in the Standard Condition with the results in the standard VCM played in a second phase after the Minimum (Limited) Information Condition. We therefore pool the data of all experiments that involve the standard VCM. Note that this result is consistent with our finding, that a reduction of confusion (which may happen in this case because of learning with limited information), does not necessarily lead to lower contributions in the Standard Condition.

  12. 12.

    This findings is in line with Ferraro and Vossler (2010), who also observe that contributions decrease faster in the standard VCM game as compared to their learning condition involving virtual players.

  13. 13.

    A Kolmogorov-Smirnov test to test the null hypothesis of identical distributions is insignificant (p>0.19).

  14. 14.

    The typical argument claims that, since a rational selfish player chooses the lowest possible contribution (i.e. zero), any confusion would lead to a positive and therefore higher contribution.

  15. 15.

    Comparing the over-all average group contributions across the conditions reveals that there are no significant differences (p>0.35, Mann-Whitney U-test). The average group contribution per period is 39.1 percent in the Standard Condition and 40.0 percent in the Learning Condition.

  16. 16.

    See Camerer (2003, Chap. 6) and the references therein for a nice overview. Our approach is similar to the reinforcement learning model by Ferraro and Vossler (2010) and Mason and Philips (1997).

  17. 17.

    Initially, we planed to refine the distribution allowing for more mass on the past choice. Analyzing the data, we found that the median of choices for both experimental conditions is approximately in the middle of the support, which is consistent with assuming choices with equal probabilities. So we decided to stick with this simple formulation.

  18. 18.

    We also simulated the learning model with different initial choices. Even when starting with extreme values (only 0 or 20) simulated behavior quickly converges to that generated by starting values drawn from the empirical distributions.

  19. 19.

    The average mean square error of the simulation is almost five times larger in the Standard Condition (3.21 vs. 0.66 points).

  20. 20.

    Fischbacher, Gächter and Fehr (2001) report 50 percent conditional cooperators.

  21. 21.

    For the full regression results see Appendix A.2.

  22. 22.

    An important difference of our approach to agent-based modeling is that we study the behavior of human subjects rather than the outcomes generated by computerized agents.

  23. 23.

    This specific restriction could be interpreted as an extreme similarity function in a similarity augmented learning model (Sarin and Vahid 2004) or as an element taken from directional learning (Selten and Stöcker 1986).

  24. 24.

    This assumption differs slightly from a traditional reinforcement-learning model in that it allows for “strategy similarity” (Sarin and Vahid 2004). In our formulation admissible strategies are not only seen as similar but even identical by the subjects.


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We are grateful for financial support by the Austrian Science Fund (FWF) under Projects No. P17029 and S10307-G14 as well as by the Faculty of Profession Research Grant Scheme of the University of Adelaide. We thank the editor Jordi Brandts, Simon Gächter, Martin Sefton and three anonymous referees for helpful comments and suggestions.

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Correspondence to Ralph-C. Bayer.

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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 iI 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:

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

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.Footnote 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


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 (\(d_{t}^{i}=-1\)):

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

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:

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

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 \(C_{t}^{i}\).Footnote 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:

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

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

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

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

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Bayer, RC., Renner, E. & Sausgruber, R. Confusion and learning in the voluntary contributions game. Exp Econ 16, 478–496 (2013).

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  • Voluntary contribution mechanism
  • Public goods experiments
  • Learning
  • Limited information
  • Confusion
  • Conditional cooperation

JEL Classification

  • C90
  • D83
  • H41