On the Provision of Public Goods with Probabilistic and Ambiguous Thresholds


Many natural systems involve thresholds that, once triggered, imply irreversible damages for the users. Although the existence of such thresholds is undisputed, their location is highly uncertain. We explore experimentally how threshold uncertainty affects collective action in a series of threshold public goods games. Whereas the public good is always provided when the exact value of the threshold is known, threshold uncertainty is generally detrimental for the public good provision as contributions become more erratic. The negative effect of threshold uncertainty is particularly severe when it takes the form of ambiguity, i.e. when players are not only unaware of the value of the threshold, but also of its probability distribution. Early and credible commitment helps groups to cope with uncertainty.

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  1. 1.

    Our paper is also related to the experimental literature on linear public goods games involving uncertainty about the marginal benefits of the public good (Gangadharan and Nemes 2009; Levati et al. 2009; Levati and Morone 2013). Most of these experiments report negative effects of risky or ambiguous marginal benefits on public goods contributions as compared to certain benefits.

  2. 2.

    For the effects of uncertainty in the context of climate change mitigation versus adaptation, see Alpizar et al. (2011), Hasson et al. (2010, 2012).

  3. 3.

    Experimental investigations have shown that the fourfold pattern of risk attitude (risk aversion for gains and risk seeking for losses at high probability, and risk seeking for gains and risk aversion for losses at low probability) also extends to ambiguity (Di Mauro and Maffioletti 2004).

  4. 4.

    The experimental literature has shown that the restricted and anonymous kind of communication used in our games can improve coordination but works much less reliably for cooperation (for reviews see e.g. Balliet 2010; Croson and Marks 2000).

  5. 5.

    For different methods to implement uncertainty in the experimental lab, see e.g. Hey et al. (2010), Levati and Morone (2013), Morone and Ozdemir (2012). Note that in our experiment there was no information asymmetry between experimenters and subjects, meaning that the former were also ignorant of the probability distribution. This is an important feature of our design because decision makers perceive ambiguity differently when there is somebody else (the experimenter) who has more information than they do (Chow and Sarin 2002). Threshold uncertainty that revolves around ecological tipping points is typically one of the unknowable types, as nobody has nor could obtain additional information. Ecological validity concerns thus imposed to implement a procedure in which subjects and experimenters had the same information regarding the threshold distribution. Moreover, this setup makes our test of ambiguity effects a particularly conservative one with respect to the potential hampering effects of ambiguity, as information asymmetries have been shown to boost ambiguity aversion.

  6. 6.

    Note that, while in Certainty \(F_I (T)=0\), if \(I<120\) and \(F_I (T)=1\), if \(I\ge 120\), in Risk \(F_I (T)>0\) for each investment level (i.e. there is a positive provision probability even for \(I=0\)). At the other end of the spectrum, only \(I=240\) guarantees provision in Risk, which would leave each player with \(w-\sum _{t=1}^r {c_i^t} =0\).

  7. 7.

    Note that, while \(I=120\) is payoff-dominant with respect to the free-riding equilibrium, it is also unstable: should there be a “tremble” by one player (e.g. switching from \(C_i =2\) to \(C_i =0\) at a given round), the remaining players’ best response may be to also switch.

  8. 8.

    Additionally, if all players are (sufficiently) risk averse, some higher provision equilibria obtain. For instance, if we drop risk neutrality and assume \(u_i (x_i)=x_i^{1/6}\) where the argument is the take home money (either \(w-c_i \) when the group is successful or \((w-c_i)d\) when unsuccessful), \(I=60\) is a symmetric Nash equilibrium. That is, given \(c_{-i} =50, u_i (c_i =10)>u_i (c_i =0)\). While the next achievable symmetric equilibrium \(I=120\) is not equilibrium, contributing \(c_i =16\cong 100/6\) when \(c_{-i} =84\) is a dominant strategy. Hence \(I=100\) is also attainable under risk aversion and the bit of asymmetry required to split a burden of 100.

  9. 9.

    In fact, there are \(\sum _{t=0}^5 {{5!}/{t!^{2}(10-2t)!}} \) profiles consistent with \(c_i =20\).

  10. 10.

    For instance, take the point of view of a player \(i\) who has follow ed the free-riding strategy \(C_i =0\) for the first nine rounds. Should the other \(j\ne i\) players have contributed \(\sum _{t=1}^9 {\sum _{j=1}^5 {c_j^t}} =96\) collectively, player \(i\)’s best response is to provide enough to reach a higher threshold (and no player has an incentive to deviate). In this case, \(i \)would optimally contribute \(C_i =4\) in the last round, a pivotal contribution in reaching \(I=100\). Similarly, a selfish individual would be willing to switch from \(C_i =0\) to \(C_i =4\) in the last round if instrumental in reaching \(I=120\).

  11. 11.

    Statistical tests are based on group averages as units of observation. If not stated otherwise, the reported tests are two-sided throughout the paper. Note also that the differences between Certainty and the other treatments are robust to multiple comparison corrections.

  12. 12.

    All the results on the correlation between variables do also hold if we employ the Spearman’s rank correlation test.

  13. 13.

    In the Certainty treatment, the correlation between first round contributions and subsequent contributions is also significant but negative (\(\rho = -0.84, p = 0.00\)), reflecting the presence of groups that had a slow start but ultimately strived and managed to reach the threshold.

  14. 14.

    We do not include the second proposals in the regression models because they were elicited during the game and therefore are likely to be endogenous. We did not find significant relationships between the variables we elicited in our ex post questionnaire and the behavior in the game.


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Correspondence to Astrid Dannenberg.



Experimental Instructions (Risk treatment, translated from German)

Welcome to our experiment!

  1. 1.

    General information

In our experiment, you can earn money. How much you earn depends on the gameplay, or more precisely on the decisions you and your fellow players make. Regardless of the gameplay, you will receive €2 for your participation. For a successful run of this experiment, it is absolutely necessary that you do not talk to other participants or do not communicate in any other way. Now read the following rules of the game carefully. If you have any questions, please give us a hand signal. It is important that you read up to the STOP sign only. Please wait when you get there, as we will give you a brief oral explanation before we continue.

  1. 2.

    Game rules

There are six players in the game, meaning you and five other players. Each player is faced with the same decision problem. In the beginning of the experiment, you receive a starting capital of €40, which is credited to your personal account. During the experiment, you can use the money in your account or let it be. In the end, your current account balance is paid to you in cash. Your decisions are anonymous. For the purpose of anonymity, you will be allocated a pseudonym which will be used for the whole duration of the game. The pseudonyms are chosen from the names of moons in the Solar System (Ananke, Telesto, Despina, Japetus, Kallisto or Metis). You can see your pseudonym in the lower left corner of your display.

The experiment has exactly ten rounds. In each round, you can invest your money in order to try and prevent damage. The damage will have a considerable negative financial impact on all players. In each round of the game, all six players are asked the following question at the same time:

‘How much do you want to invest to prevent damage?

You can answer with €0, €2 or €4. After each player has made her or his decision, the six decisions are displayed at the same time. After that, all money paid by the players is booked to a special account for damage prevention.

At the end of the game (after exactly ten rounds), the computer calculates the total investments made by all players. If the investments have reached a certain minimum, the damage is prevented. In this case, each player is paid the money remaining in her or his account, meaning the €40 starting capital minus the money the player has invested in preventing damage over the course of the game. However, if the total investments are lower than the minimum, the damage occurs: All players lose 90 % of the remaining money in their personal accounts. The minimum to be reached in order to prevent damage will be drawn randomly. We will draw the minimum after the game in your presence. The draw goes like this: The minimum can take the values 0, 20, 40, 60 etc. up to 240 (always in steps of 20). For each of these 13 values, a certain number of balls in different colors is put into a bag. One ball is drawn from the bag and the value shown on the ball is the minimum value for the game. The following figure shows the distribution of the different balls. There are 52 balls altogether. These balls are put into a bag, and one is drawn randomly.


For each possible value, four balls are put into a bag. The probability of being drawn is thus equal for every value and comes to 4/52 (\(\approx \)8 %). Assuming that a light blue ball with the value 100 was drawn, all players together must have invested at least €100 in order to prevent damage. If a single player has invested, say, a total of €10 in damage prevention after ten rounds, he or she has a credit of €30 on his or her personal account. If the group of players as a whole has invested €100 or more in damage prevention, the damage will not occur and this player will receive €30 from the game. However, if the group has invested less than €100, the damage will occur and the player will receive €3 (10 % of €30) from the game.

Please note the following feature of the game: Before the players decide how much they want to invest into preventing damage, they exchange non-binding suggestions for their common investment goal. Each player makes a suggestion of how much the group as a whole should invest into preventing damage over the total of ten rounds. After that, the suggestions made by all players and an average value from all suggestions are shown on the monitor. After round 5, all players can make a new suggestion for the total investments to be made by the group over the ten rounds. After that, the suggestions made by each player and an average value for all suggestions are shown on the monitor.

  1. 3.


Here, you can see an example of the decisions made by the six players in one round (round 3).


The right column shows the investments made in the current round (round 3). The players Ananke and Kallisto have invested €2 each, the players Telesto and Japetus have invested €4 each and Despina and Metis have not made any investments. In total, €12 were invested in this round. The middle column shows the cumulative investments made by each player from the first to the current round (rounds 1–3). The players Ananke and Telesto have each invested €6 in the first three rounds. Despina, Kallisto and Metis have each invested €4 and Japetus has invested €10 in the first three rounds. In total, €34 were invested in the first three rounds.

The left column shows the suggestions made by each player as to how much the group as a whole should invest into preventing damage over the ten rounds in total. For example, Metis suggests that the group should invest €140. The average of all suggestions is €108. In the game, you will see this information after each round.

“STOP sign” (oral explanation of the game)

  1. 4.

    Control questions

Please answer the following control questions.


Please give us a hand signal after you have answered all control questions. We will come to you and check the answers. The game will begin after we have checked the answers of all players and answered any questions you may have. Good luck!

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Dannenberg, A., Löschel, A., Paolacci, G. et al. On the Provision of Public Goods with Probabilistic and Ambiguous Thresholds. Environ Resource Econ 61, 365–383 (2015). https://doi.org/10.1007/s10640-014-9796-6

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  • Cooperation
  • Experiment
  • Public good
  • Threshold uncertainty
  • Ambiguity

JEL Classification

  • C72
  • C92
  • H41
  • Q54