Cooperation among unrelated individuals in social-dilemma-type situations is a key topic in social and biological sciences. It has been shown that, without suitable mechanisms, high levels of cooperation/contributions in repeated public goods games are not stable in the long run. Reputation, as a driver of indirect reciprocity, is often proposed as a mechanism that leads to cooperation. A simple and prominent reputation dynamic function through scoring: contributing behaviour increases one's score, non-contributing reduces it. Indeed, many experiments have established that scoring can sustain cooperation in two-player prisoner's dilemmas and donation games. However, these prior studies focused on pairwise interactions, with no experiment studying reputation mechanisms in more general group interactions. In this paper, we focus on groups and scores, proposing and testing several scoring rules that could apply to multi-player prisoners' dilemmas played in groups, which we test in a laboratory experiment. Results are unambiguously negative: we observe a steady decline of cooperation for every tested scoring mechanism. All scoring systems suffer from it in much the same way. We conclude that the positive results obtained by scoring in pairwise interactions do not apply to multi-player prisoner's dilemmas, and that alternative mechanisms are needed.

Voluntary contribution games are a classic social dilemma in which the individually dominant strategies result in a poor performance of the population. However, the negative zero-contribution predictions from these types of social dilemma situations give way to more positive (near-)efficient ones when assortativity, instead of random mixing, governs the matching process in the population. Under assortative matching, agents contribute more than what would otherwise be strategically rational in order to be matched with others doing likewise. An open question has been the robustness of such predictions when heterogeneity in budgets amongst individuals is allowed. Here, we show analytically that the consequences of permitting heterogeneity depend crucially on the exact nature of the underlying public-good provision efficacy, but generally are rather devastating. Using computational methods, we quantify the loss resulting from heterogeneity vis-a-vis the homogeneous case as a function of (i) the public-good provision efficacy and (ii) the population inequality.

Many real-world mechanisms are "noisy" or "fuzzy", that is the institutions in place to implement them operate with non-negligible degrees of imprecision and error. This observation raises the more general question of whether mechanisms that work in theory are also robust to more realistic assumptions such as noise. In this paper, in the context of voluntary contribution games, we focus on a mechanism known as "contribution-based competitive grouping". First, we analyze how the mechanism works under noise and what happens when other assumptions such as population homogeneity are relaxed. Second, we investigate the welfare properties of the mechanism, interpreting noise as a policy instrument, and we use logit dynamic simulations to formulate mechanism design recommendations.Keywords: voluntary contributions; behavioral economics; noise; heterogeneity; mechanism design; welfare; efficiency; equality JEL Classification: C73; D02; D03; D63 MotivationTypically, individual decisions in social-dilemma interactions are not perfectly observable in the real world. Applied mechanism designers should keep this in mind when implementing mechanisms, in particular in the context of interactions that have the strategic nature of voluntary contribution games where imperfect observability is ubiquitous. 1 Real-world institutions are usually "noisy" or "fuzzy", operating with non-negligible degree of imprecision. By contrast, theory investigations of mechanisms for the most part study perfect mechanisms. In this paper, as a first step towards performing robustness checks of mechanisms under relaxed assumptions more generally, we investigate the mechanism of contribution-based competitive grouping (as introduced by [5]): we relax some assumptions and investigate what happens when there is (i) noise, (ii) heterogeneity and (iii) different action spaces.The paper is structured as follows. Next, we introduce our model. In Section 3, we first analyze how the existence of Nash equilibria depends on (iii) the strategy space of the game and on (ii) the underlying population homogeneity/heterogeneity in terms of contribution budgets. Then, we assess their robustness when we allow more and more (i) monitoring noise. In Section 4, we investigate how a social planner, interpreting monitoring noise and/or the other model characteristics as policy instruments, would trade-off efficiency and equality to maximize social welfare. Finally, we use agent-based simulations to study the logit dynamics of the game in order to quantify the effect of noisy monitoring. 1 For other social-dilemma contexts, see, for example, [1,2] for imperfect public monitoring, [3] for noisy prisoners' dilemmas and [4] for team-production games with group-level information. Our results are summarized as follows. In terms of the existence of Nash equilibria, the zerocontribution outcome is always a Nash equilibrium and becomes the unique one under too much noise. High-contribution equilibria exist if three things come together: the rate of return of the underlyin...

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