When Public Information Backfires: Heterogeneous Groups (Un)coordinate on a Fairness Norm

A.1 The Environment

We choose a classical environment for the experimental investigation of cooperation: the public-goods game (PGG).^[14] The one-shot PGG G consists of an arbitrary finite number of players N = {1, …, n}. Each player i contributes a _i ∈ A _i simultaneously to a public good. A denotes the set of all possible action vectors, where a typical element is a = (a₁, …, a _n ). For the set of all possible action vectors of all players but i, A\{a _i }, we define A_−i with generic element a_−i. To model the heterogeneity of players, we introduce different initial endowments θ _i ∈ {θ ^L , θ ^H } with θ ^L  < θ ^H . Each players’ possible endowment must lie between 0 and the player’s maximum endowment, i.e., player i’s action set A _i is bounded above by the initial endowment and consists only of positive integers. We assume that the distribution over initial endowments is uniform and perfectly correlated. The monetary payoff each player obtains from the public good is π _i (a _i , a_−i) = θ _i  − a _i  + α∑_j∈Na _j with α ∈ ( 1 n , 1 ) . The unique equilibrium of G is in dominant strategies and given by the strategy vector a* = (0, …, 0). This reflects the collective action problem: each player’s dominant strategy is a contribution of zero, which means that the public good will not be produced.

We now introduce identity theory, as proposed by Akerlof and Kranton (2000), into our model by expanding the monetary payoff by identity preferences of the players. A player’s utility no longer depends on monetary concerns alone; they also derive utility from behaving according to certain norms that are perceived to be prevalent in their group. For simplicity in our analysis, we restrict identity utility to be based on a match between one’s own behavior (action chosen) and the other players’ behavior. A player’s utility function u _i (a, c_−i) thus consists of the monetary payoff π _i (a) and the identity utility I _i (a, c_−i). We assume that these utilities are additive, where the latter is weighted by γ ≥ 0. The correspondence c_−i: A _i → A_−i is player i’s normative expectations, i.e., applied fairness norm, towards other players given action a _i .

We consider two potential fairness norms: equal relative contribution (for all j ∈ N\{i}: c j = a i θ i θ j ) – everyone contributes a certain percentage of their endowment – and equal absolute contribution (for all j ∈ N\{i}: c _j  = a _i if a _i  ≤ θ _j and c _j  = θ _j otherwise) – everyone contributes the same amount regardless of their endowment. For the analysis we assume that contributions sufficiently close to the normative expectation lead to the same identity utility, i.e., symmetric peakedness around the ideal point.^[15] We use q _i (a_−i) to denote the ex-ante belief of player i that action vector a_−i is played, i.e., the probability that a_−i is the state of the world. The random variable is then denoted by a ̃ − i and expectations are taken with respect to all random variables in the squared brackets. Given action a _i , the ex-ante expected utility is given by

Before entering the PGG, each player receives a correlated signal δ _i  ∈ Δ _i which recommends an action by stressing one of the fairness norms, i.e., f: Δ → A with Δ = ∏_i∈NΔ _i . The set of all possible signal combinations is denoted by F = F _i  × F_−i. The signal transforms the PGG into the sequential game G + : nature sends the correlated signal to each player, who then update his or her belief about the actions played by other players to q(a_−i|f _i (δ _i )). Finally, the players choose their action considering whether to follow the signal or not. This leads to the interim expected utility

We use the correlated equilibrium concept by Aumann (1974, 1987) to analyze G + . Formally, we define that

In other words, the equilibrium concept demands that every player prefers to follow the signal over deviating from it, given that all other players also follow their respective signal.

A.2 Analysis

For the theoretical analysis, we focus on the existence of a full-contribution correlated equilibrium, i.e., the correlated signal recommends that everyone contribute an equal percentage of their endowment, in this case 100 %. The equilibrium hypothesis assumes that all players but i follow their signals, so that their respective signals are equivalent to the action played (f_−i = a_−i). A second necessity for the existence of the equilibrium is that indeed each player i ∈ N receives the signal to contribute the full endowment to the public good. For an arbitrary player i, we can then focus on the behavior after the full-contribution signal f _i  = θ _i has been received.

In the analysis, we first introduce public knowledge on signals. In a second step, we add identity preferences to the utility function. Finally, we investigate interactive effects of identity preferences and signal announcement.

B.1 Instructions

B.2 The Puzzle

In order to increase participants’ time spent in their respective group and thus foster group identity, we altered the puzzle treatment of Eckel and Grossman (2005). To the physical puzzle that the group members have to assemble, we added a decoding task. For each group there is one puzzle containing instructions, one with a symbol-to-letter conversion table, and two with symbols (all in the respective group color). Participants assemble the puzzles, decode the words, put them together to form the composite German word FEUERWERK, and enter this word once they are back at their respective work stations. The puzzles are shown in Figure 3.

Figure 3:

Puzzles for the group identity formation task.

B.3 Belief Elicitation

We asked participants the following questions to gauge their beliefs about other participants’ contribution behavior and also in order to check the effectiveness of private and public information. In the case of public information, the answers to 1) and 2) should be the same. When there is only private information, the answers to 1) and 2) may differ depending on how sure the participant is that their recommendation is the same as everyone else’s.

Question 1: Please indicate what you believe is the probability with which all your group members contributed their entire endowment.

“Frage 1: Bitte geben Sie an, mit welcher Wahrscheinlichkeit (0 bis 100) Sie glauben, dass alle anderen Gruppenmitglieder ihre gesamte Anfangsausstattung beigetragen haben.”

Question 2: Please indicate what you believe is the probability with which all your group members followed their recommendation.

“Frage 2: Bitte geben Sie an mit welcher Wahrscheinlichkeit (0 bis 100) Sie glauben, dass alle anderen Gruppenmitglieder ihrer Empfehlung gefolgt sind.”

Question 3: Please indicate what you believe is the probability with which all your group members with an initial endowment of 10 contributed their entire endowment.

“Frage 3: Bitte geben Sie an, mit welcher Wahrscheinlichkeit (0 bis 100) Sie glauben, dass alle anderen Gruppenmitglieder mit einer Anfangsausstattung von 10 ihre gesamte Anfangsaussattung beigetragen haben.”

Question 4: Please indicate what you believe is the probability with which all your group members with an initial endowment of 20 contributed their entire endowment.

“Frage 4: Bitte geben Sie an, mit welcher Wahrscheinlichkeit (0 bis 100) Sie glauben, dass alle anderen Gruppenmitglieder mit einer Anfangsausstattung von 20 ihre gesamte Anfangsausstattung beigetragen haben.”

B.4 NS-5 Scale

Participants are asked to distribute a total budget of 100 points between themselves and a second person with whom they have not interacted with before. This version of the dictator game is meant to gauge the fairness of the participant’s behavior and their moral convictions. At random, either the participant’s own answers to the questions or their partner’s answers are chosen and paid out with a conversion rate of 1:100.

Question 1: Out of 100 – what do you believe is the morally right amount to give to the other person?

“Frage 1: Ich halte es für moralisch geboten, dass von der 100 Punkten der folgende Betrag an die andere Person abgegeben wird.”

Question 2: What do you think did your partner answer in Question 1?

“Frage 2: was hat die andere Person auf Frage 1 geantwortet?”

Question 3: I give the following amount to the other person:

“Frage 3: Ich gebe den folgenden Betrag an die andere Person:”

Question 4: What was the other person’s answer to Question 3?

“Frage 4: Was hat die andere Person auf Frage 3 geantwortet?”

Question 5: How did the other person answer Question 4?

“Was hat die andere Person auf Frage 4 geantwortet?”

Questions are repeated on the screen, so that the participant need not be able to remember the first question when answering the second, etc. However, answers are recorded in sequence and cannot be altered after they have been given.

B.5 SVO Measure

Social Value Orientation (SVO), i.e., the magnitude of concern that participants have for others, is measured according to the method developed in Murphy et al. (2011). The participant becomes a decision-maker in an allocation situation, deciding which payoffs they would like to see realized between themselves and another randomly chosen participant. We thus obtain the complete ranking of participants’ social preferences in order to understand their decision-making behavior during the public-goods game more fully.

Figure 4 shows the six primary SVO sliders as seen by the participants. Each slider gives participants the choice of different allocations between them and the other participant and asks them to repeat the chosen allocation in the end. Each slider measures a different social preference. Figure 5 shows the underlying SVO Ring measure. Each slider in Figure 4 represents the allocation choice that a participant can make on each of the blue lines in the SVO Ring measure, showing their social preferences in terms of competitiveness, altruism, prosociality and individualism.

Figure 4:

SVO measurement sliders as seen by the participants (Murphy et al. 2011, p.772).

Figure 5:

Underlying SVO ring-measure (Murphy et al. 2011, p.773).

The individual measures can be compiled into a single index of SVO containing the mean allocation to self, A ̄ S , and mean allocation for the other, A ̄ O , in order to compare SVO measures easily across participants:

This SVO measure is also incentivized with one round of allocation decisions randomly chosen and paid out to both participants.

B.6 Group Manipulation Check

In order to check whether the group manipulation prior to the public-goods game was successful, we administered a manipulation check that is based on Aron et al. (1992). Their Inclusion of Other in the Self (IOS) Scale is a single item, pictorial measure of closeness between the participant and their group, which has been shown to be test-retest reliable and compatible with a range of other closeness tests between individuals and between individuals and groups. We ask participants to circle the image that best represents their relationship to their group from Figure 6. This measure has proven inconclusive in the experiment.

Figure 6:

Closeness measure adapted from Aron et al. (1992).

Table 3 displays the frequency distribution of responses (on a 7-point scale) for the Inclusion of Others in the Self (IOS) scale across the four treatment conditions. In all conditions, option 5 is the most frequently chosen, indicating high closeness within the groups. Notably, in the treatment with group identity and public information, option 6 is the most common response, suggesting increased closeness. However, the overall distribution is not significantly different (χ² = 17.54, df = 18, p = 0.48).

B.7 Questionnaire

The questionnaire consists of six demographic questions that ask participants for the (1) age, (2) gender, (3) level of education, (4) parents’ level of education, as well as (5) their field of study if they are students.

Answers are recorded in free form, except for question 2, about the participants’ gender, where they may choose from male, female, or may choose not to answer.