Do economic inequalities affect long-run cooperation and prosperity?

Abstract

We explore if fairness and inequality motivations affect cooperation in indefinitely repeated games. Each round, we randomly divided experimental participants into donor–recipient pairs. Donors could make a gift to recipients, and ex-ante earnings are highest when all donors give. Roles were randomly reassigned every period, which induced inequality in ex-post earnings. Theoretically, income-maximizing players do not have to condition on this inequality because it is payoff-irrelevant. Empirically, payoff-irrelevant inequality affected participants’ ability to coordinate on efficient play: donors conditioned gifts on their own past roles and, with inequalities made visible, discriminated against those who were better off.

Appendix A

1.1 Procedural details

We recruited 256 subjects through announcements to the standing subject pool for the Behavioral Business Research Laboratory at the University of Arkansas. No subject had previous experience with this type of game in the lab.¹⁹ After giving informed consent, subjects were seated at private terminals. Neither communication nor eye contact was possible among subjects at any time during the session. The experimenter publicly read the paper instructions at the start of the experiment, a copy of which were then left on the subjects’ desks. The experiment was programmed and conducted with the software z-Tree (Fischbacher 2007). On average, a session lasted 94 rounds for a running time of about 120 minutes including instructions, a paid post-instruction comprehension quiz, and post-experiment payment. Average earnings were $27.00 per subject ($\min =\$6.50, \max = \$55.50$) excluding a $\$5$ fixed participation payment and an average of $\$2.10$ ($\min =\$.75, \max = \$2.50$) from providing correct answers to the comprehension quiz ($\$0.25$ for each correct answer out of 10 questions). Only one randomly selected supergame from the session was paid and subjects knew this fact in advance.

1.2 Proof of Proposition 1

This analysis is based on the existence of equilibrium proof in Camera et al. (2013). In each round $t=0,1,2\ldots$ individuals in the group are matched in pairs, with uniform probability of selection. In each pair, a computer randomly determines who is the donor and who is the recipient (with equal probability). If cooperation (=Help) is the outcome, then g is the payoff to the recipient and for generality let a denote the payoff to the donor. If defection (=Do nothing) is the outcome, then d is the payoff to the donor and $d-l$ to the recipient. Round payoffs are geometrically discounted at rate $\beta \in (0,1)$ starting from round $n \ge 0$.

The equilibrium payoff (=expected lifetime utility) at $t=0$ is

$$\begin{aligned} v(n) := (n+1)\times \frac{g+a}{2}+ \sum _{j=1}^{\infty }\beta ^j \times \frac{g+a}{2} \ = \ \frac{g+a}{2}\times \left( n+\frac{1}{1-\beta }\right) . \end{aligned}$$

A player is a donor or a recipient with equal probability in each round, hence expects to earn $\dfrac{g+a}{2}$ in each round. The payoff v(n) is increasing in n because payoffs are discounted by $\beta$ in rounds $t\ge n$.

The equilibrium payoff in the continuation game starting on any date $t\ge 0$, before any uncertainty is resolved, corresponds to

$$\begin{aligned} V_t = \left\{ \begin{array}{ll} v(n-t) &{} \quad \hbox {if } t < n \\ v^*:=\dfrac{g+a}{2 (1-\beta )} &{} \quad \hbox {if } t \ge n . \end{array} \right. \end{aligned}$$

The equilibrium payoff of a donor at the start of any date t is

$$\begin{aligned} V_{dt} = \left\{ \begin{array}{ll} a + v(n-t-1) &{} \quad \hbox {if } t < n \\ a + \beta v^* &{} \quad \hbox {if } t \ge n . \end{array} \right. \end{aligned}$$

We must check that in equilibrium donors have no incentive to defect; out of equilibrium, donors have no incentive to cooperate.

Defection is the dominant action off-equilibrium; i.e., it is always individually optimal to punish after a defection from equilibrium play is made public. To see this suppose a donor deviates by cooperating off equilibrium. She would earn a instead of d but her continuation payoff would not improve since everyone else keeps defecting—as prescribed by the rule of punishment. Since $d>a$, it is optimal to punish off equilibrium.

In equilibrium, cooperation is a best response in every round $t=0,1,\ldots$, if $V_{dt} \ge \hat{V}_{dt}$. The left-hand-side denotes the payoff to a donor who cooperates; the right-hand-side denotes the donor’s payoff when she moves off equilibrium under a one-time deviation. Such deviation is publicly observed, hence—when everyone follows the cooperative strategy—every donor will always defect in the future. The payoff to the deviator is thus

$$\begin{aligned} \hat{V}_{dt} = \left\{ \begin{array}{ll} \hat{v}(n-t): = d+ (n-t) \dfrac{2d-l}{2} + \beta \dfrac{2d-l}{2(1-\beta ) } &{} \quad \hbox {if } 1 \le t < n \\ \hat{v}^*:=d + \beta \dfrac{2d-l}{2(1-\beta )} &{} \quad \hbox {if } t \ge n \end{array} \right. \end{aligned}$$

Now define

$$\begin{aligned} \Delta _t=V_{dt}- \hat{V}_{dt} = a-d + \dfrac{g+a-2d+l}{2} \times \left\{ \begin{array}{ll} n - t +\dfrac{\beta }{1-\beta } &{}\quad \hbox {if } t < n \\ \dfrac{\beta }{1-\beta } &{}\quad \hbox {if } t \ge n \end{array} \right. \end{aligned}$$

The minimum value of $\Delta _t$ is achieved for $t \ge n$. The implication is that cooperation is individually optimal in all rounds t whenever

$$\begin{aligned} \beta \ge \beta ^*:= \dfrac{2(d-a)}{g+l-a}. \end{aligned}$$

1.3 The grim strategy is risk dominant

Our game is sequential, does not have a two-by-two structure, and admits more than two equilibria. Therefore, we consider a notion of risk dominance that departs from the standard formalization in Harsanyi and Reinhard (1988), but conforms to the heuristic argument that motivates it. A risk-dominant equilibrium maximizes the expected payoff when players have uniformly distributed second order beliefs on the best and worst equilibrium (full cooperation and full defection). We also add structure to beliefs about strategies play, by exploiting the dynamic structure of the game. In doing so, we employ the technique in Bigoni et al. (2018), to which we refer the reader.

Given random role alternation, the ex-ante expected payoff under full defection equilibrium is denoted

$$\begin{aligned} \hat{v}:= \dfrac{2d-l}{2(1-\beta )}. \end{aligned}$$

The ex-ante expected payoff under full cooperation is

$$\begin{aligned} v:= \dfrac{g}{2(1-\beta )}. \end{aligned}$$

Consider uncertainty over two competing strategies: “grim” (G) and “always defect” (AD). Initial donors select a strategy in round 1 and maintain it for the rest of the supergame. Initial recipients take no action in round 1, so we set them free to select G or AD in round 2. Given public monitoring, conjecture that those who choose strategy in round 2 coordinate their choices by best responding to the strategy play observed in round 1. If so, then all uncertainty about future play is resolved at the end of round 1 as if initial donors selected the equilibrium. G dominates AD after any history of play, for those who choose their strategy in round 2 (weakly dominates AD, if someone defected in round 1). Hence, if no-one (someone) defected in round 1, then every donor will cooperate (defect) in every future meeting.

Consider round 1. There is strategic uncertainty because an initial producer is not sure what strategy the other initial producers will select. Suppose that every initial producer believes that in round 1 there is probability p that C is the outcome in any given pair; D is the outcome with the complementary probability. A special case is $p=1/2$, which may be motivated by the “principle of insufficient reason” for a player who is unsure about what the others will do. That is, the individual believes that the other initial producer plays G with probability p, and AD otherwise. At the end of round 1 either C will be the outcome in every future meeting, or D will be the outcome in every future meeting.

Fix an initial donor. Let denote $V_G$ and $V_{AD}$ the expected utilities from choosing strategy G and AD where

$$\begin{aligned} V_G& = 0 + p \beta v + (1-p) \beta \hat{v} , \\ V_{AD}& = d + \beta \hat{v} . \end{aligned}$$

Consider $V_{AD}$: the initial donor defects so all future donors will defect whether or not they chose G or AD. Consider $V_G$: with probability p the other initial donor is also a grim player. In that case both cooperate and there is full cooperation forever so the continuation payoff is $\beta v$. With probability $1-p$ the other initial donor plays AD in which case the continuation payoff is $\beta \hat{v}$ because full defection ensues. We say that G is risk dominant if $V_G \ge V_{AD}$. If $p=0.5$, which is a standard consideration, this implies

$$\begin{aligned} \beta \ge \frac{4d}{g+2d+l} \qquad \Rightarrow \qquad \beta \ge 0.62 \end{aligned}$$

Therefore grim trigger is a risk-dominant strategy in our experiment since the discount factor is 0.75.