Supplementary Material for

An Experimental Investigation of ‘Pledge and Review’ in Climate Negotiations

Scott Barrett & Astrid Dannenberg

This file contains Supplementary Text,Supplementary Tables S1-S5, and additional references.

  1. Theory

Our underlying game-theoretic model assumes that there are N symmetric countries, each i of which may reduce its emissions using a low-cost technology denoted B for “Black” and a high-cost technology denoted R for “Red” (the reason for these labels will become clear later), with and Aggregate abatement, denoted Q, is thus with The per-unit costs of reducing emissions are assumed to be constant, withReductions in emissions lessen “gradual” climate change, yielding each country a benefit, b, per unit of emission reduced, whether the reduction was achieved using the Red or the Black technology. Reductions in emissions can also decrease if not eliminate the chance of crossing a “dangerous” climate threshold. We assume that the threshold, , is a random variable distributed uniformly over the supports and such that the probability of avoiding danger is 0 for forand 1 forWe further assume and restrict parameters so that when countries cooperate fully they abate collectively, eliminating the risk of crossing the threshold, and when countries choose their abatement levels non-cooperatively, they do nothing to limit their emissions, making it inevitable that the threshold will be crossed. Our experiment assumes that the impact of crossing the threshold, X, is certain.In a previous paper (S1) we showed that, consistent with theory (S2), impact uncertainty has no effect on behavior.

Our experimental game was played by groups of five players (N = 5). At the start of the game, every subject was given 5 black poker chips worth €.10 each and 15 red poker chips worth €1.00 each. A contribution of either type of chip by any player gave every player a benefit equal to €.05 (that is, b = 0.05). These values implyWe also assume In other words, to reduce the risk of “dangerous” climate change, at least some players must contribute a sizable number of their expensive chips. To eliminate the risk of “dangerous” climate change, every player must contribute all of his or her chips.

To complete the model, we must also choose a value for X. In our experiment we set X = €20. This ensures that the game of avoiding “catastrophic” climate change is also a prisoners’ dilemma.

An important feature of this game is that cooperation promises a high reward, if the other players cooperate and Nature chooses a value for that is less than the total amount contributed. However, cooperation is also risky. If the other players don’t cooperate, or Nature chooses a high value for , a cooperative player could go home with very little money. Actual payouts in our experiment covered almost the full range of feasible outcomes, from €1 to €37. This large range is consistent with the view that climate change is a high-risk game. The smallest payout in the first five groups was €0 (as is convention in experiments we cut the payoffs at zero when a player had earned a negative profit). After increasing the endowment fund, negative profits were no longer possible.

  1. Choice of X

X must be neither “too small” nor “too large.” If X were too small, avoiding a “catastrophic” outcome would not be optimal for the group. (Arguably, in this case, exceeding the threshold would not be truly “catastrophic.”) If X were too big, avoiding a “catastrophic” outcome would result in a coordination game—meaning that avoiding the threshold would be a Nash equilibrium.

Under our assumptions, it will always pay the players collectively to contribute all of their black chips. Once at least chips have been contributed, it will also pay all of the players together to contribute all of their remaining red chips. However, when fewer thanchips have been contributed in total, contributing a single red chip loses the group money. Hence, it will either pay a group to contribute all of their red chips or none of them. If a group contributes all of their chips, each member will get If a group contributes only its black chips, each member will get Full cooperation thus requires contributing all chips if and only if which for our parameter values implies

For contributing all red chips (in addition to all black chips) not to be a Nash equilibrium, it must be the case that, if all other players contribute all of their chips, the remaining player is better off contributing one fewer than all of his chips. Given that the other players contribute all of their chips, the remaining player gets if he contributes all of his remaining chips. If he contributes one fewer red chip, he gets For contributing all red chips not to be a Nash equilibrium, it must therefore be the case that Using our parameter values, this condition implies

Combining both results, to have a prisoners’ dilemma we must use a value of X satisfying Our chosen value of X = €20 clearly satisfies this condition.

  1. The effect of contributing in stages

In our previous experiments (S1, S3), players were allowed to choose their contributions in a single stage. In this experiment, we allow the players to spread their contributions over two stages. In theory, the difference shouldn’t matter; in both cases, the game is “one shot.” However, in our game, there are at least two reasons for believing that cooperation will be higher in the two-stage contribution game than in the one-stage game. Other differences between this experiment and our previous experiments are the number of players (N = 5 versus N = 10) and the number of cheap and expensive chips ( and versus and ).

First, in a two-stage game, conditional cooperators can condition their second-stage contributions on the contributions they actually observe in the first stage. By contrast, in a one-stage game, conditional cooperators can only condition their contributions on the amounts they expect their co-players to contribute. In a one-stage game, each conditional cooperator may hold back, fearing that her co-players will contribute very little, whereas in a two-stage game, each such player may contribute some red chips in the first stage, and then see how many their co-players contribute before deciding whether to contribute any more red chips in the second stage. This illustrates a basic insight (S4) that if players are able to break up their contributions into smaller units, and contribute in stages, then they may be more willing to contribute, as each player “can try the other’s good faith for a small price,” and “no one ever may need risk more than one small contribution at a time.”

Second, in our game, the marginal (expected) private return to contributing a chip increases once fifty chips have been (or are expected to be) contributed. When only a “small” number of chips have been contributed, a player gains just €.05 for each additional chip contributed, whereas when a “large” number of chips have been contributed, a player gains €.05 + €20/(100 – 50) = €.45 in expected terms for each additional chip contributed. Observing that a large number of chips have been contributed in a first stage thus strengthens the motivation to contribute in the second stage. Moreover, an expectation about this behavior may in turn increase the incentive to contribute in the first stage. In a one-stage game, by contrast, contributing chips seems more risky.

  1. Experimental design

The games were played in stages. The stages were identical for each treatment, with the exception of the grading stage. Ignoring grading, the stages were as follows:

  1. Subjects proposed a collective target (the sum of contributions to be made over the two stages by all players, a number between 0 and 100), after which every player’s proposal was displayed to all the players in the group. The median value was chosen as the group’s collective target. This target was non-binding.
  2. Subjects pledged a total individual contribution for the game (the sum of the own contributions to be made over the two stages, a number between 0 and 20), after which every player’s pledge was displayed to all the players in the group. Pledges were non-binding.
  3. Subjects were asked to estimate the average contribution level of the other group members. Players knew that they would get one euro for a correct guess (that is, for guesses within 1 of the actual number). Guesses were not displayed to the group.
  4. Subjects played two contribution stages. In each stage they could contribute any number of black chips and red chips in their possession. Over the two stages, subjects could not contribute more chips in total than they had to start with—five black and 15 red chips. Contribution decisions were binding. After each contribution stage, the contributions made by every player and the group total were displayed to the group. After the second stage, subjects were also shown the probability of the loss and the payoffs they would get depending on whether or not they avoided the loss.

This concluded play of the game.

The experiment just described is for the No-Review treatment. For the other treatments, a grading stage was inserted in the above sequence. In this stage, subjects were asked to assign grades to the other players and also to themselves (1 = very good, 2 = good, 3 = satisfactory, 4 = fair, 5 = poor, 6 = insufficient; this grading scheme is used in German schools and is thus familiar to the students). Each player’s average grade (excluding the grade which the player gave to him or herself) was then displayed to the group. In the Ex-Ante-Review treatment, this grading round was inserted between stages two and three above. In the Mid-Point-Review treatment, the grading round was inserted between the first and second contribution stage. In the Ex-Post-Review, the grading round was inserted after the second and final contribution stage.

Decisions in every stage of the game were made simultaneously and independently. All the rules of the game were common knowledge. Players also knew that, before the real game started, there would be three trial rounds, which were not relevant for their payoff. The groups were reshuffled after each trial round. After play of the game was over, subjects completed a short questionnaire that elicited their emotions and motivation while playing the game; see below.

Upon completing the questionnaire, “Nature” chose the threshold in order for the groups contributing 50 or more chips to know their payoff. For people to understand the implications of uncertainty for their decision-making, probabilities must be communicated very carefully (S5). In our experiment, the threshold was determined by asking a student chosen at random within each session to activate a spinning arrow on a computer wheel. The wheel represented a uniform distribution, with 12 o’clock indicating the “ends” of the distribution ([50, 100]) and with intermediate values being demarcated in 0.01 increments. Within this range, the arrow could move freely. After being activated, the momentum of the arrow would slow until the arrow finally came to rest at some point. This was the point chosen by “Nature” from the distribution. All students could observe the arrow being spun and learn their “fate” live on their own screen. To conclude the session, earnings were paid out in cash at the end of the experiment.

Table S1 summarizes the key features of the experimental design.

  1. Experimental instructions

The sample instructions reproduced below are from the Ex-Ante-Review treatment, translated from German. The instructions for the other treatments are similar.

Instructions

Welcome to this experiment!

1. General information

In our experiment you can earn money. How much you earn depends on how you and your co-players play the game. You will receive a lump sum participation fee of €19. Note that a gain during the experiment will be added to this amount while a loss will be deducted from that amount. For a successful run of this experiment, it is essential that you do not talk to other participants. Now, read the following rules of the game carefully. If you have any questions, please give us a hand signal.

2. Game rules

There are 5 players in the group, meaning you and four other players. Each player faces the same decision problem. All decisions in the experiment are anonymous. For the purpose of anonymity, you will be identified by a letter (between A and E), which you will see later in the lower left corner of your display.

At the beginning of the game, you will receive 20 poker chips: 5 black chips and 15 red chips. During the game, you can use these poker chips to contribute to a joint account or you can keep them. Black chips are cheap; they are worth €0.10 each. Red chips are more expensive; they are worth €1.00 each. Overall you can contribute any integer amount of chips between 0 and 20 to the joint account: at most 5 black chips and at most 15 red chips.

Three rules determine your payoff:

First, you will receive the value of the chips you have not contributed to the joint account. That is, you will get €0.10 for every black chip you keep and €1.00 for every red chip you keep.

Second, you will get €0.05 for every poker chip contributed to the joint account, irrespective of which player contributed the chip and whether it was a black chip or a red chip.

Third, if the total number of chips contributed by your group is smaller than a threshold, every player will lose €20. If the group contribution is equal to or greater than the threshold, no player will lose any money. The threshold is some number between 50 and 100, but you will not know the exact value of the threshold until after the game is played. The exact value of the threshold will be determined at the end of the experiment by the spinning wheel. The wheel is programmed so that each value between 50 and 100 has the same probability of being selected.

Players decide whether and by how much to contribute to the joint account in two stages. A player’s overall contribution is the sum of his or her first-stage contribution and his or her second-stage contribution. So, for example, if a player contributes 3 chips in the first stage and 2 chips in the second stage, then his or her overall contribution is 5 chips. After the first stage, the first-stage contributions made by all players will be revealed to the group and after the second stage the second-stage contributions as well as the overall contributions by all players will be revealed. Players will see only the number of chips contributed, not whether those chips were red or black.

Before choosing their contributions to the joint account, players will have the opportunity to propose and announce a few things.

First, each player will make a proposal for how many chips he or she thinks the group as a whole should contribute to the joint account. After that, all the proposals made will be displayed to the group. The median of all proposals will then become the final group target. So, for example, if the five players propose 14, 45, 60, 77, and 98, then 60 chips will become the final group target (since two of the proposals are above 60 and two below 60). The group target is non-binding.

Second, each player will make a pledge for how many chips he or she intends to contribute overall to the joint account (in both stages together). All pledges will be displayed to the group. These pledges are also non-binding.

Third, after the pledges have been displayed, players will assess the decisions made so far. In particular, every player will assign a grade from 1 (very good) to 6 (insufficient) to the other co-players and also to himself or herself. The grades given by each player will be private and will not be shown to the group. However, players will learn the average grade given to them by their co-players. These average values, one for each player, will also be revealed to the group.

To summarize, the experiment involves the following steps:

-Proposals for the group target

-Pledges for intended contributions

-Grading of decisions

-Choice of first-stage contributions

-Choice of second-stage contributions

-Determination of the threshold by the spinning wheel

-Payment

3. Determining the threshold with the spinning wheel

Whether or not the loss occurs depends on the overall group contribution and the threshold which will be determined by the spinning wheel. If the group as a whole contributes less than 50 chips, the loss occurs with certainty. Recall that the threshold will be between 50 and 100. But what happens if the group contributes more than 50 chips? Consider the following numerical examples.

Suppose the group contributes 63 chips. Then the loss occurs if the threshold exceeds 63 (the red area in the pie chart shown on the right). The loss does not occur if the threshold is 63 or less (the blue area). You can see that the loss occurs with probability 74% (= (100 − 63)/(100 − 50)).