# Contribution-Based Grouping under Noise

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Motivation

## 2. The Mechanism

#### 2.1. The Model

## 3. Nash Equilibria

#### 3.1. Game 1: ‘Baseline’

#### 3.2. Game 2: ‘Heterogeneity Extension’ (Extension 1)

#### 3.2.1. Heterogeneity and Continuous Action Space

#### 3.2.2. Binary Action Space

#### 3.3. Game 3: ‘Noise Extension’ (Extension 2)

#### 3.4. Remark: Mixed-Strategy Nash Equilibria

## 4. Welfare Comparison

You can’t have your cake and eat it, too, is a good candidate for the fundamental theorem of economic analysis. We can’t have our cake of market efficiency and share it equally.[24], p. 1.

#### 4.1. Homogeneous Endowments

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

#### 4.2. Heterogeneous Endowments

## 5. Logit Dynamics

Algorithm 1: Logit choice dynamics. | |

_{1} | Initialization: Assign the initial endowments to each player sampling them from a Gaussian distribution with mean ${W}_{0}$ and standard deviation $\sigma $. If some endowments are zero or negative, the sampling is repeated.^{a} |

_{2} | Initialization: Set initial strategy ${\alpha}_{i}$ to 0 $\forall i$ (start the simulation in the fully-defective state^{b}), and set time t to zero. |

repeat | |

until $t<T$; |

^{a}- The standard deviation is chosen for this to be an unlikely event.
^{b}- Different initial starting conditions have been explored, and they have been observed to have no effect on the final outcome of the simulation.
^{c}- Inertia was added to ensure the convergence to the pure-strategy Nash equilibrium (if existing). Without inertia, the best-response dynamics could oscillate around such an equilibrium.
^{d}- For a definition, see, e.g., [34].

## 6. Summary

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Pure Strategy Nash Equilibria

#### Appendix A.1. Perfect Meritocracy

- We have a population of n players with $N=\left(\right)open="\{"\; close="\}">1,2,\dots ,n$.
- We have g groups, $\left(\right)$, of size s.
- ${\varphi}_{i}\left(\right)open="("\; close=")">{c}_{i}\mid {\mathit{c}}_{-\mathit{i}}$ is the payoff of agent i with ${\mathit{c}}_{-\mathit{i}}\equiv \mathit{c}\backslash \left(\right)open="\{"\; close="\}">{c}_{i}$.

**Proposition**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

**Proposition**

**A5.**

**Proof.**

**Lemma**

**A6.**

**Proof.**

#### Appendix A.2. Fuzzy Mechanism

**Proposition**

**A7.**

**Proposition**

**A8.**

**Proof.**

**Remark**

**A9.**

## Appendix B. Mixed-Strategy Nash Equilibria

**Proposition**

**A10.**

**Proof.**

**Claim**

**A11.**

**Figure A1.**Expected payoffs of contributing versus free-riding.The expected values of ${\varphi}_{i}(0|{1}_{-i}^{p})$ and ${\varphi}_{i}(1|{1}_{-i}^{p})$ are plotted as functions of the probability p for some fixed $\beta >\beta $. The two planes intersect at the bifurcating symmetric mixed-strategy Nash equilibrium-values of $\overline{p}$ and p (see Proposition A10). The relative slopes of the two curves illustrate the proposition. Note that this figure is a slice through Figure A2 along a value of $\beta >\beta $.

**Figure A2.**The figure depicts the expected payoffs of contributing versus free-riding for the economy with $n=16$, $s=4$, $r=1.6$. The expected values of ${\varphi}_{i}(0|{1}_{-i}^{p})$ and ${\varphi}_{i}(1|{1}_{-i}^{p})$ are plotted as functions of contribution probability p and meritocratic matching fidelity $\beta $. The two planes intersect at the bifurcating symmetric mixed-strategy Nash equilibrium-values of $\overline{p}$ and p (see Proposition A10). Notice that the expected values of both actions increase linearly in p when the meritocratic matching fidelity is zero, but turn increasingly S-shaped for larger values, until they intersect at $\overline{p}$ and p.

**Remark**

**A12.**

**Corollary**

**A13.**

**Corollary**

**A14.**

**Proof.**

**Proposition**

**A15.**

**Proof.**

**Remark**

**A16.**

## Appendix C. Properties of the Fuzzy Ranking

- For $\beta =1$, the contributions are perfectly observable, and the ranking follows a perfect ordering.
- For $\beta =0$, contributions do not matter, and the ordering of the players is determined completely at random. This follows from the property of the normal distribution for $\sigma \to \infty $ ($\beta \to 0$).
- For $0<\beta <1$, the expected ranking for every player has the following properties:$$\frac{\partial \mathbf{E}\left(\right)open="["\; close="]">{\overline{k}}_{i}^{\beta}\left(\right)open="("\; close=")">{c}_{i}}{}0$$$$\mathbf{E}\left(\right)open="["\; close="]">{\overline{k}}_{i}^{\beta}\left(\right)open="("\; close=")">{c}_{L}$$$$\frac{\partial \mathbf{E}\left(\right)open="["\; close="]">{\overline{k}}_{i}^{\beta}\left(\right)open="("\; close=")">{c}_{L}}{-}\partial \beta $$

#### Proofs

**Lemma**

**A17.**

**Proof.**

**Lemma**

**A18.**

**Proof.**

**Lemma**

**A19.**

**Proof.**

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1 | |

2 | With a slight abuse of notation, from now on, we will write $\beta \in \left(\right)open="["\; close="]">0,1$ and $\beta =0$ to indicate the “no meritocracy” case where $\sigma \to \infty $, and the ranking, and consequentially, the matching, of the players is (uniformly) randomly selected. |

3 | Meritocracy $\beta $ guides smoothly from (i) no meritocracy (grouping is random as in [6]) to (ii) full meritocracy (the case of perfect contribution-based grouping as in [5]). Note that many public goods experiments use variants of Andreoni’s random (re-)matching implementation (e.g., [6,7,8,9,10,11,12,13,14,15,16]); see [17,18] for reviews. |

4 | Thus, full-contribution is collectively efficient, and zero-contribution, despite collectively inefficient, is the unique Nash equilibrium under no meritocracy. |

5 | See [5], Theorem 1. |

6 | In fact, recent evidence suggests that players may endogenously implement variants of contribution-based competitive grouping over time and then converge to the high-contribution equilibria [21]. |

7 | For a marginal per capita rate of return $R>s$, the game is not a social dilemma anymore: “cooperate” becomes a dominant strategy, and the only existing equilibrium is for everybody to contribute everything. |

8 | Note that, whenever pure strategy highly-efficient equilibria exist, there could also exist several asymmetric Nash equilibria whose characterization is not easily obtained. |

9 | Note that Harsanyi’s social welfare approach [25], by contrast, would consider ex ante payoffs, that is expected payoffs. His social welfare function is ${W}_{H}(\varphi )=\frac{1}{n}{\sum}_{i\in N}\mathbf{E}\left[{\varphi}_{i}\right]$. See, for example, [26] for a discussion of ex ante versus ex post approaches. |

10 | The work in [27] defines that outcome $\varphi $ payoff-dominates ${\varphi}^{\prime}$ if ${\varphi}_{i}\ge {\varphi}_{i}^{\prime}$ for all i, and there exists a j such that ${\varphi}_{j}>{\varphi}_{j}^{\prime}$. |

11 | See, for example, [29] for a discussion of this generalization. |

12 | With $e=10.3$, ${W}_{e}$ requires efficiency gains of more than twice the amount lost by any player to compensate for the additional inequality. |

13 | For continuous action space, the only equilibrium is non-contribution by all, and therefore, there is no social welfare analysis to be made. |

14 | |

15 | |

16 | Using the terminology of [46], our paper therefore studies a ‘system’ rather than moral ‘acts’ or ‘intentions’. In our mechanism, the system assorts contributions, i.e., actions, as other evolutionary biology mechanisms that lead to cooperation as, for example, kin selection (e.g., [47,48,49]), local interaction and/or assortative matching of preferences [50,51,52,53,54]. |

17 | It is easy to check that ${\mathrm{lim}}_{n\to \infty}mpcr=1/s$. |

18 | See Appendix C. |

19 | Details concerning the use of the law of large numbers can be followed based on the proof in [55]. |

20 | See, e.g., [56] |

**Figure 1.**The figure illustrates the three dimensions that we vary to assess the robustness of the mechanism. Two are binary (indicated by the blue dotted lines): For one, players have either the same or different initial endowments; further, they either choose a proportional or an all-or-nothing contribution. One dimension is varied continuously (indicated by the blue continuous line): contributions are observed with different degrees of noise.

**Figure 2.**The figure summarizes the existence of the “high-efficiency” equilibria described in this section. For homogeneous endowments, regardless of the action space, there exist high contribution equilibria for high enough marginal rates of return. In the case of heterogeneous endowments, only a binary action space allows for contributive equilibria. Whenever high contribution equilibria exist under full meritocracy, there exists a marginal rate of return (higher than the one for $\beta =1$) such that the high equilibria exist also with a fuzzy mechanism.

**Figure 3.**The figure summarizes the average level of contributions in continuous and binary action spaces. The average is shown as a function of time for the voluntary contribution game with meritocratic matching with homogeneous endowments in four different cases: perfect and imperfect observations for continuous (

**a**) and binary (

**b**) action spaces. When the matching mechanism is implemented without noise ($\beta =1$, blue lines in (

**a**,

**b**)), we observe high levels of cooperation in both cases. On the other hand, when the matching mechanism is implemented with partially-noisy observations $\beta =0.7$, orange lines in (

**a**,

**b**), the result depends on the strategy space of the game: for a continuous action space (

**a**), we observe a very low percentage of average contributions, while for a binary action space (

**b**), we see a higher level of cooperation. The blurred areas around the lines represent the $95\%$ confidence interval. The simulations were obtained for the following set of parameters: $s=4$, $R=0.5$, $EN=200$, $p=0.99$. The simulations for the binary action space are obtained with a lower value for the rationality parameter in the logit function in order to speed up convergence and with a higher number of agents to reduce the error.

**Figure 4.**The figure depicts the average percentage of contributions as a function of the rate of return R and the level of noise in the matching mechanism $\beta $. For values of $R\le \frac{1}{s}=0.25$ and/or too low values of $\beta $, we observe no cooperation. Otherwise, as predicted, the average level of cooperation weakly increases with increasing R and $\beta $, and we observe almost complete cooperation when approaching one. The contour lines indicate when the average contribution is above $25\%$, $50\%$ and $75\%$ of the whole population, respectively. The simulation was obtained for the following set of parameters: $N=80$, $s=4$, $\lambda =30$, $EN=100$, $p=0.99$, $T=2000$.

**Table 1.**Stem-and-leaf plot of individual payoffs for the non-contribution equilibrium (valid for any $\beta \in [0,1]$) and for the high-contribution equilibrium (when evaluated at $\beta =1$). Parameter values are $n=16$, $s=4$, $r=1.6$ and $\beta =1$.

High-Contribution Equilibrium | Payoff | Non-Contribution Equilibrium |
---|---|---|

When ${\beta}=\mathbf{1}$ | When $\mathit{\beta}=\mathbf{0}$ | |

0 | 0.0 | 0 |

0 | 0.2 | 0 |

0 | 0.4 | 0 |

0 | 0.6 | 0 |

13 14 (${c}_{i}=1$) 2 | 0.8 | 0 |

0 | 1.0 | 16 (${c}_{i}=0$) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 |

0 | 1.2 | 0 |

0 | 1.4 | 0 |

1 2 3 4 5 6 7 8 9 10 11 12 (${c}_{i}=1$) 12 | 1.6 | 0 |

15 16 (${c}_{i}=0$) 2 | 1.8 | 0 |

24.4 | efficiency | 16 |

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**MDPI and ACS Style**

Nax, H.H.; Murphy, R.O.; Duca, S.; Helbing, D.
Contribution-Based Grouping under Noise. *Games* **2017**, *8*, 50.
https://doi.org/10.3390/g8040050

**AMA Style**

Nax HH, Murphy RO, Duca S, Helbing D.
Contribution-Based Grouping under Noise. *Games*. 2017; 8(4):50.
https://doi.org/10.3390/g8040050

**Chicago/Turabian Style**

Nax, Heinrich H., Ryan O. Murphy, Stefano Duca, and Dirk Helbing.
2017. "Contribution-Based Grouping under Noise" *Games* 8, no. 4: 50.
https://doi.org/10.3390/g8040050