1 Introduction

The heterogeneity in decision-making behaviour observed in both field settings and their laboratory counterparts is by turns a great joy and a great frustration to practitioners of behavioural economics. The richness in the variety of individual behaviour is evidence that people are indeed different, and approach the same economic decision-making task in a variety of ways. However, parsimonious, practical, and tractable economic models try to capture the commonalities in behaviour. Extracting those commonalities from the embarrassment of riches offered by the data is an important challenge in the development of behavioural economics and game theory.

One approach is to group behaviour into a small number of distinct types, which we refer to as a typology. In this paper, we will focus on the case of public goods voluntary contribution games (VCGs), for which Fischbacher et al. (Reference Fischbacher, Gächter and Fehr2001) (FGF) have proposed one such typology, which groups participants into four types. We choose this as an interesting setting, because the P-experiment protocol introduced by FGF, based on the linear VCG (Ledyard Reference Ledyard, Kagel and Roth1997), has been employed as a standard methodology by many studies conducted in various languages and locations (Kocher et al. Reference Kocher, Cherry, Kroll, Netzer and Sutter2008). The analysis we conduct in this paper benefits from being able to re-use data from a number of studies using a sufficiently similar protocol.

Although a number of papers have used variants of the FGF typology, the literature in experimental economics has not employed a framework for defining or evaluating candidate typologies. To address this, we introduce techniques from machine learning, in which exactly these types of classification problems have been studied in depth. Ideally, a typology represents the data well when the behaviours of two participants classified as the same type are similar, while the behaviours of two participants classified as different types are dissimilar. Machine learning provides methods for evaluating the trade-offs between within-type similarity and across-type dissimilarity and for constructing classifications which are optimal according to some criterion with respect to these trade-offs. Machine learning is commonly associated with data sets with large numbers of observations, a problem experimental economists rarely face. However, it also studies the organisation of multi-dimensional data. In the data we analyse, a participant’s type is determined based on a 21-dimensional conditional contribution strategy elicited by the P-experiment protocol.

We use data from six previous studies using the P-experiment protocol to construct alternative typologies using hierarchical cluster analysis (Kaufman and Rousseeuw Reference Kaufman and Rousseeuw1990). Our typologies differ from FGF in the organisation of conditionally cooperative participants. FGF propose to categorise these participants primarily into conditional cooperators and non-monotonic “hump-shaped” contributors. In contrast, cluster analysis identifies a group of strong conditional cooperators, centred on participants who match group contributions on a one-for-one basis, and a group of weak conditional cooperators, centred on those who match group contributions at approximately a one-for-two rate.

Machine learning offers tools for visualising the properties of classifications of high-dimensional data, such as our behavioural typologies. We use silhouette analysis (Rousseeuw Reference Rousseeuw1987) to assess the cohesion of types using both approaches, and illustrate that, in the FGF typology, participants grouped in the same type exhibit behaviours with heterogeneous consequences in the VCG.

To be useful in understanding economic and strategic behaviour, the classifications in a typology should correlate with choices made by the same participants which are not used in the classification process. In the P-experiment, participants make two types of choices: conditional contributions, which are used in the classification, and unconditional contributions, which are not. Across our data set, FGF’s conditional cooperators and hump-shaped contributors do not differ in their unconditional contributions. In contrast, participants classified as strong conditional cooperators make generally higher unconditional contributions than those classified as weak conditional cooperators. This supports the strong/weak conditional cooperator distinction as being a more insightful description of the data and that the underpinnings of the behaviour of weak conditional cooperators may be distinct from those of strong conditional cooperators.

2 The game

The experiments used in our analysis involve one-shot interaction among participants in a VCG. Participants are anonymously placed into groups with M members. Each participant receives G tokens. She can allocate any number of tokens between 0 and G to a group account; tokens not allocated to the group account are kept in her private account. We refer to the tokens allocated to the group account as her contribution. The participant receives a point for each token kept in her private account. Each token contributed to the group account yields $P>1$ points, which are then split equally among the group members. The parameters P and M are chosen, so that the marginal per-capita return (MPCR), P/M, is less than one. With these parameters, a participant who cares only about maximising her own earnings has a strictly dominant strategy, which is to contribute no tokens. In contrast, the strategy profile that maximises total earnings of the group is for each member to contribute all G tokens.

In the P-experiment protocol, contributions are made in two stages. In Stage 1, $M-1$ members make their contributions. The remaining member learns the average contribution of other members, and then decides on her contribution. A participant does not know whether she will make her contribution in Stage 1 or Stage 2, nor, if she is to be the Stage 2 contributor, what the average contribution of the other members in Stage 1 will turn out to be. Decisions are, therefore, elicited using the strategy method (Selten Reference Selten and Sauerman1967). Each participant i states what her contribution will be if she is chosen to contribute in Stage 1; we write the unconditional contribution of participant i as $u^i$ . She also states her contribution in Stage 2, for each possible realisation of the average contribution of the other members of her group.Footnote ¹ We call these Stage 2 contributions the contribution strategy. We write the contribution strategy of i as a vector $c^i$ . The component $c_g^i$ is the contribution of participant i in Stage 2 if the other members contribute g tokens on average in Stage 1. The contribution strategy is the basis for identifying behavioural types.

3 Typologies

Let $N$ denote the set of participants, and $C={\left\{(i,c^i)\right\}}_{i\in N}$ be the set of all participants paired with their contribution strategies. We define a typology T as a partition of $C$ into equivalence classes. Each equivalence class is interpreted as a distinct behavioural type. We write T(i) as the type of participant i in typology T.

The existing state-of-the-art in the literature is the typology based on Fischbacher et al. (Reference Fischbacher, Gächter and Fehr2001), which we will call $T^F$ . $T^F$ partitions participants into one of four types.

• • Free riders (FR) always maximise individual earnings by keeping all tokens in the private account, irrespective of the outcome of the first stage.

• • Conditional cooperators (CC) increase their contributions to the group account based on higher contributions by others in the first stage. A participant i is deemed a conditional cooperator by testing whether the Spearman’s $\rho$ correlation coefficient between the vector $[0,1,\dots ,G]$ of possible average contributions g and the participant’s strategy $\left(c_0^i,,,c_1^i,,,\dots ,,,c_G^i\right)$ is significantly positive at significance level $\le 0.001$ . We separately tabulate exact conditional contributors (XC), who match exactly one-for-one, labeling other CC as inexact conditional contributors (IC).

• • Hump-shaped (HS) contributors are identified based on visual classification of contribution strategies, in which $c_0^i$ and $c_G^i$ are small, but $c_g^i$ is larger for some intermediate values $0<g<G$ ; these strategies often have a triangular shape when plotted.

• • Others (OT) is the residual type, comprised of participants, whose contribution strategies do not satisfy the criteria defining the other types.

The $T^F$ procedure is implemented by defining a stereotypical behaviour, combined with a formal or informal criterion for deciding when a given contribution strategy is “similar enough” to the stereotype. This similarity is a matter of judgment; alternative proposals for inclusion criteria have been made in subsequent papers (e.g., Rustagi et al. Reference Rustagi, Engel and Kosfeld2010; Fischbacher et al. Reference Fischbacher, Gächter and Quercia2012). By adjusting the classification criteria, one can make the residual “other” group smaller, but with the possibility that a participant’s contribution strategy might satisfy the criteria for more than one other type. The most recent refinement of the criteria by Thöni and Volk (Reference Thöni and Volk2018) encounters this problem, requiring a further criterion for assigning contribution strategies that satisfy their versions of both the CC and HS criteria.

The stereotypical behaviours in $T^F$ are chosen based on an ad-hoc combination of theoretical models and inspection of the data. We are interested first in assessing the performance of this classification in identifying coherent types.

Question 2

Given the distribution of contribution strategies in the data, what is an appropriate number of types to include in a typology?

Ward’s method proposes a partition for each C, which has the property that the partition $T^H\left(C\right)$ can be computed efficiently given $T^H(C+1)$ by combining together the two “most similar” elements of $T^H(C+1)$ . The trade-off in having more (resp., fewer) types is that the variability within a type will be less (resp., more). For example, there is a trivial, but unsatisfying, clustering which assigns each contribution strategy to its own distinct type. The resulting types are by definition perfectly coherent, having zero variability, but fail to capture that there may be many strategies which differ, for example, by only one token in one contingency.

There are several approaches in the literature to analysing this trade-off. Recall that solutions $T^H\left(C\right)$ and $T^H(C+1)$ differ in that one cluster in T(C) is divided into two in $T^H(C+1)$ . There are exactly two members $t_1,t_2\in T^H(C+1)$ , such that $t_1\ne t_2$ and $t_1\cup t_2\in T^H\left(C\right)$ . Let W(t) denote the sum of squared errors in cluster t. Duda and Hart (Reference Duda and Hart1973) define the index

(1) $\begin{array}{c}\mathrm{Je}\left(2\right)/\mathrm{Je}\left(1\right)=\frac{W\left(t_1\right)+W\left(t_2\right)}{W(t_1\cup t_2)}.\end{array}$

Because Ward’s method minimises the within-cluster sum of squared errors, $\mathrm{Je}\left(2\right)/\mathrm{Je}\left(1\right)\le 1$ . This is considered in conjunction with the value of a pseudo- $T^2$ statistic:

(2) $\begin{array}{c}{\text{PT}}^2=\left(\frac{1}{\mathrm{Je}\left(2\right)/\mathrm{Je}\left(1\right)},-,1\right)\times \left(|,t_1,|+|,t_2,|-2\right)=\left(\frac{W(t_1\cup t_2)}{W\left(t_1\right)+W\left(t_2\right)},-,1\right)\times \left(|,t_1,|+|,t_2,|-2\right),\end{array}$

where |t| is the number of members of cluster t. Duda and Hart recommend preferring clusterings with relatively high $\mathrm{Je}\left(2\right)/\mathrm{Je}\left(1\right)$ and relatively low ${\text{PT}}^2$ values.

The criteria of Duda and Hart refer specifically to the output of hierarchical clustering. Another measurement of type coherence, which can be applied to any typology T, is silhouette analysis (Rousseeuw Reference Rousseeuw1987). For any participant i, the average distance from i’s contribution strategy to the contribution strategies of other participants of a given type $t\in T$ is

(3) $\begin{array}{c}a(i,t)=\frac{{\sum }_{j\ne i:T\left(j\right)=t}d(c^i,c^j)}{{\sum }_{j\ne i:T\left(j\right)=t}1}.\end{array}$

For i, the distance to the “closest” type which is different from the type to which i is assigned is

(4) $\begin{array}{c}b\left(i\right)=\underset{t\ne T\left(i\right)}{min}a(i,t).\end{array}$

The participant’s silhouette index is then defined as

(5) $\begin{array}{c}s\left(i\right)=\frac{b\left(i\right)-a(i,T(i\left)\right)}{max\left\{b\right(i),a(i,T\left(i\right)\left)\right\}}.\end{array}$

The silhouette index ranges from − 1 to + 1. Values greater than zero indicate that the members of i’s type are closer, on average, than the members of the next closest type.

In the trivial typology that assigns each distinct strategy to its own cluster, the silhouette index is + 1 for all strategies. Taken to the other extreme, fixing a small number C of groups and assigning strategies at random to the groups leads to silhouette indices distributed with a median near zero and small absolute values. Although hierarchical clustering does not construct its solution for C groups at random, but by combining two similar groups from its solution for $C+1$ groups, any grouping of heterogeneous strategies under one type necessarily decreases the silhouette index. Kaufman and Rousseeuw (Reference Kaufman and Rousseeuw1990) suggest selecting an appropriate number of clusters C by analysing the levels and distributions of silhouette indices as an indicator of the trade-off between within-cluster similarity and across-cluster dissimilarity.