Social approval, competition and cooperation

Abstract

Non-monetary rewards are frequently used to promote pro-social behaviors, and these behaviors often result in approval from one’s peers. Nevertheless, we know little about how peer-approval, and particularly competition for peer-approval, influences people’s decisions to cooperate. This paper provides experimental evidence suggesting that people in peer-approval competitions value social approval more when it leads to unique and durable rewards. Our evidence suggests that such rewards act as a signaling mechanism, thereby contributing to the value of approval. We show that this signaling mechanism generates cooperation at least as effectively as cash rewards. Our findings point to the potential value of develo** new mechanisms that rely on small non-monetary rewards to promote generosity in groups.

Appendices

Appendices

1.1 Appendix 1

See Fig. 5.

Fig. 5

The photo on the left is the Haagen–Dazs ice-cream bar and the mug on the right is one that we have specifically designed for distribution only in this experiment

1.2 Appendix 2: The framework of Holländer (1990)

An individual receives utility from private consumption, public goods, and social approval. As a result, an individual must make decisions regarding private consumption, contribution to the public good and amount of approval to dispense to others. In making these decisions, a typical agent is confronted with the respective behaviour of others, characterized by $w$, the approval for each unit of contribution (i.e., the unit approval rate) and $\bar{g}$, the average contribution of the society.²⁵ An individual responds to others’ behaviour by some rationally chosen contribution $g_{i}$, and what Holländer refers to as an “emotionally” determined approval rate, $v$, which he applies to others’ contributions. These are the key quantities underlying the hypotheses in the current paper, and the following two sub-sections explain how these two variables are determined in Holländer’s (1990) model.

1.2.1 Individual contribution $g_{i}$

Given that the unit approval rate of others is $w$, the absolute contribution of the individual is $g_{i} ,$ the approval assigned due to absolute contribution is $wg_{i}$, and approval assigned due to his contribution relative to the average contribution is $w*\left( {g_{i} - \bar{g}} \right)$. If we assume that the overall approval assigned to any individual i is a weighted average of approval assigned due to an individual’s absolute contribution level (weight $1 - \beta )$ and approval assigned due to an individual’s contribution relative to the average contribution (weight $\beta )$. we have

Approval assigned to individual $i$

$$ \left( {1 - \beta } \right)wg_{i} + \beta w.\left( {g_{i} - \bar{g}} \right) = w.(g_{i} - \beta \bar{g}) $$

(1)

Note that if one contributes the same as the average contribution $\bar{g}$, the approval assigned to him is:

$$ \left( {1 - \beta } \right)w\bar{g} $$

(2)

Holländer’s key departure from the traditional public goods literature is that he assumes people value receiving social approval. Intuitively, we not only want to receive approbation, but also care about how much approbation we receive in comparison to others. Therefore, assume that utility from approval received is determined by both the approval received by individual $i$, that is, Eq. (1) (weight $(1 - \alpha )$), and how much he receives relative to that received by the group average contribution, that is, $w(g_{i} - \bar{g})w*\left( {g_{i} - \beta \bar{g}} \right) - \left( {1 - \beta } \right)w\bar{g} = w*(g_{i} - \bar{g})$ (weight $\alpha$). Therefore, the total utility from received approval for individual $i$ is:

$$ \left( {1 - \alpha } \right)w*\left( {g_{i} - \beta \bar{g}} \right) + \alpha w*\left( {g_{i} - \bar{g}} \right) = wg_{i} - \left( {\alpha + \beta - \alpha .\beta } \right)w\bar{g},\quad {\text{with}}\;0 \le \alpha ,\;\beta \le 1 $$

Let $\sigma = \alpha + \beta - \alpha \beta$, so that we have utility from received approval for individual i to be equal to:

$$ w*(g_{i} - \sigma \bar{g}),\quad {\text{with}}\;0 \le \sigma \le 1 $$

(3)

Note that $\sigma$ increases with both $\alpha$ and $\beta$. $\beta$ is a weight parameter that captures how competition influences approval assignment while $\alpha$ captures the role of competition on the way approval is valued. An increasing $\beta$ indicates that approval assigned to a typical individual is based more on how much he contributes relative to the group average; an increasing $\alpha$ captures how much a typical individual cares about the relative approval received. Intuitively, a more competitive environment will increase $\alpha$ and $\beta$ and thus increase $\sigma$.

An individual’s total utility is characterized by additively separable preferences between private consumption, public goods, and approval received. Thus, the utility function for individual $i$ is:

$$ U_{i} = U_{p} \left( {\pi - g_{i} } \right) + U\left( {\bar{g}} \right) + U_{A} \left( {w*\left( {g_{i} - \sigma \bar{g}} \right)} \right) \quad 0 \le g_{i} \le \pi $$

(4)

Taking the f.o.c. condition of $U_{i}$ with respect to $g_{i}$, we find:

$$ U_{p}^{{\prime }} \left( {\pi - g_{i} } \right) = wU_{A}^{{\prime }} [w*(g_{i} - \sigma \bar{g})] $$

(5)

We focus exclusively on symmetric equilibria, where an individual’s contribution is equal to the average contribution so that $g_{i} = \bar{g}$. Substituting this equality into Eq. (5), we have $U_{p}^{{\prime }} \left( {\pi - \bar{g}} \right) = wU_{A}^{{\prime }} [w*(\bar{g} - \sigma \bar{g})]$, so that,

$$ w = \frac{{U_{p}^{{\prime }} \left( {\pi - \bar{g}} \right)}}{{U_{A}^{{\prime }} [w*\bar{g}(1 - \sigma )]}} $$

(6)

As noted, this is referred as the $g_{i} \bar{g}$ equilibrium in Holländer (1990).

Assuming all utility functions are concave, an increase in $\bar{g}$ leads to an increased ratio $\frac{{U_{p}^{{\prime }} \left( {\pi - \bar{g}} \right)}}{{U_{A}^{{\prime }} [w*\bar{g}(1 - \sigma )]}}$. That is, the $g\bar{g}$ curve is an upward slo** curve, with approval rate w increasing with the average contribution $\bar{g}$. This curve is referred as the supply curve in the main text.

1.2.2 The approval rate $v$

The approval rate is determined by the marginal rate of substitution between public goods and private consumption. Intuitively, the higher the pleasure one derives from contributions to the public goods relative to that of private consumption, the higher the approval rate should be, and vice versa.²⁶

$$ v = \frac{{U_{g}^{{\prime }} \left( {\bar{g}} \right) - \sigma wU_{A}^{{\prime }} \left[ {w*\left( {g_{i} - \sigma \bar{g}} \right)} \right]}}{{U_{p}^{{\prime }} \left( {\pi - g_{i} } \right)}} $$

(7)

In equilibrium, the individual approval rate must be equal to the approval rate of others, $v = w$. Given (5), we know that $\sigma wU_{A}^{{\prime }} [w*(g_{i} - \sigma \bar{g})] = \sigma U_{p}^{{\prime }} \left( {\pi - g_{i} } \right)$. Substituting this and $v = w$ together into (7), we obtain $w = \frac{{U^{{\prime }}_{g} \left( {\bar{g}} \right) - \sigma U_{p}^{{\prime }} \left( {\pi - g_{i} } \right)}}{{U_{p}^{{\prime }} \left( {\pi - g_{i} } \right)}}$, or

$$ w = \frac{{U_{{\bar{G}}}^{{\prime }} \left( {\bar{g}} \right)}}{{U_{p}^{{\prime }} (\pi - g_{i} )}} - \sigma $$

(8)

Equation (8) is referred as the VW equilibrium in Holländer’s paper. It occurs when the individual approval rate is equal to others’ approval rates. This is the structure of the demand curve. We characterize the demand curve in social exchange equilibrium below. See the discussion surrounding Proposition 1.

1.2.3 Social exchange equilibrium

Holländer (1990) denotes the simultaneous $VW$ and $g\bar{g}$ equilibrium as a social equilibrium. Substituting $g_{i} = \bar{g}$ into (8), we have

$$ w = \frac{{U_{{\bar{G}}}^{{\prime }} \left( {\bar{g}} \right)}}{{U_{p}^{{\prime }} (\pi - \bar{g})}} - \sigma $$

(9)

Proposition 1

An increase in the competitiveness of the environment ($\sigma$) leads to a downward shift of the $VW$ curve.

Assuming the utility function to be concave, based on Eq. (9), an increase in $\bar{g}$ leads to a decrease in the ratio of $\frac{{U_{{\bar{G}}}^{{\prime }} \left( {\bar{g}} \right)}}{{U_{p}^{{\prime }} (\pi - \bar{g})}}$, and thus the $VW$ curve is negatively sloped. This is the demand curve mentioned in the manuscript. Meanwhile, an exogenous increase in the competitiveness of the environment $\sigma$ leads to a downward shift of the $VW$ curve (see movement (1) in Fig. 6 below).

Fig. 6

The approval rate $w$ is on the vertical axis and the average contribution  $\bar{g}$ on the horizontal axis. The solid black curve is the VW cruve, also referred as the demand curve in the main text. The solid grey curve is the $g\bar{g}$ curve, also referred as supply curve in the main text. The black dashed line shows a downward shift of the VW curve [movement (1)] after an increase in the competitiveness of the environment. The dashed grey line shows a downward shift of the $g\bar{g}$ curve after an increase in the competitiveness of the environment. The equilibrium changes from $e_{1}^{*}$ to $e_{2}^{*}$. We can see that the overall effect is a decreased approval rate w, and an ambiguous change in equilibrium contribution level

For the $g\bar{g}$ equilibrium, we had Eq. (6) as:

$$ w = \frac{{U_{p}^{{\prime }} \left( {\pi - \bar{g}} \right)}}{{U_{A}^{{\prime }} [w*\bar{g}(1 - \sigma )]}} $$

Proposition 2

An increase in competitiveness of the environment $\sigma$ shifts down the $g\bar{g}$ curve, which is referred as the supply curve in the main text. In particular, observe that the $g\bar{g}$ curve is influenced by two variables: the endowment $\pi$ and the competitiveness of the environment $\sigma$. An increase in $\sigma$ leads to a decreased ratio of $\frac{{U_{p}^{{\prime }} \left( {\pi - \bar{g}} \right)}}{{U_{A}^{{\prime }} [w*\bar{g}(1 - \sigma )]}}$, thus a downward shift of the $g\bar{g}$ curve (see movement (2) in Fig. 6 below).

Proposition 3

When $\sigma < \frac{{U_{{\bar{G}}}^{{\prime }} \left( 0 \right)}}{{U_{p}^{{\prime }} (\pi )}} - \frac{{U_{p}^{{\prime }} \left( \pi \right)}}{{U_{A}^{{\prime }} (0)}}$, there exists a unique social equilibrium with positive $w^{*}$ and $\bar{g}^{*}$.

The intersection of the $VW$ curve with the vertical axis occurs where $\bar{g} = 0$. Substituting this into in (9), we have: $w = \frac{{U_{{\bar{G}}}^{{\prime }} \left( 0 \right)}}{{U_{p}^{{\prime }} (\pi )}} - \sigma$. The intersection of the $g\bar{g}$ curve with the vertical axis occurs where $\bar{g} = 0$, and substituting this into (6) we have $w = \frac{{U_{p}^{{\prime }} \left( \pi \right)}}{{U_{A}^{{\prime }} (0)}}$. Thus, if $\frac{{U_{{\bar{G}}}^{{\prime }} \left( 0 \right)}}{{U_{p}^{{\prime }} (\pi )}} - \sigma > \frac{{U_{p}^{{\prime }} \left( \pi \right)}}{{U_{A}^{{\prime }} (0)}}$, that is, $0 < \sigma < \frac{{U_{{\bar{G}}}^{{\prime }} \left( 0 \right)}}{{U_{p}^{{\prime }} (\pi )}} - \frac{{U_{p}^{{\prime }} \left( \pi \right)}}{{U_{A}^{{\prime }} (0)}}$, then there exists a unique social equilibrium with positive $w^{*}$ and $\bar{g}^{*}$.