When is the risk of cooperation worth taking? The prisoner’s dilemma as a game of multiple motives

ABSTRACT

This experimental article helps to understand the motives behind cooperation in the prisoner’s dilemma. It manipulates the pay-off in case both players defect in a two-player, one-shot prisoner’s dilemma and explains the degree of cooperation by a combination of four motives: efficiency, conditional cooperation, fear and greed. All motives are significant but some become only significant if one controls for all remaining ones. This seems to be the reason why earlier attempts at explaining choices in the prisoner’s dilemma with personality have not been successful.

KEYWORDS:

• Prisoner’s dilemma
• efficiency
• belief
• risk aversion
• loss aversion
• risky dictator game

JEL CLASSIFICATION:

• C72
• C91
• D03
• H41

I. Introduction

Many experimental economists have become sceptical about models that assume individuals to hold stable social preferences. Seemingly, effects vary across situations. In this article, we revisit the issue, focusing on a particularly involved game (the prisoner’s dilemma) to put the claim to the hardest test. We show that experimental participants interpret the prisoner’s dilemma as a game of identifiable, multiple, interconnected motives. Seemingly, earlier attempts have failed since they have not controlled for idiosyncratic beliefs about the cooperativeness of others.

Specifically, we consider efficiency (we vary gains from cooperation by manipulating pay-offs), conditional cooperation (we measure beliefs about the cooperativeness of others), fear (we measure loss aversion: Gächter, Johnson, and Herrmann 2007 and risk aversion: Holt and Laury 2002) and greed (we measure generosity in a modified dictator game). When controlling for the remaining motives, all have a significant effect. In line with differential psychology, personality traits turn out consistent only conditional on situation (Ross and Nisbett 1991).

II. Earlier results

For decades, prisoner’s dilemma games have been tested in the lab (for an overview, see Colman 1995; Sally 1995). Many participants cooperate, and hence violate the theoretical prediction.

Several papers explored how behaviour reacts to changes in pay-offs. We use the labels introduced by Rapoport and Chammah (1965). In Table 1, R stands for the reward from successful cooperation, S is the sucker pay-off, T the temptation pay-off and P signifies punishment (for defection).

Table 1. Generalized representation of a two-person two-action prisoner’s dilemma.

	C	D
C	R,R	S,T
D	T,S	P,P

C, cooperation; D, defection.

The closest analogue to our game is Rapoport and Chammah (1965). They too vary P and find that there is the more cooperation, the lower P (see also Ahn et al. (2001), Steele and Tedeschi (1967), Vlaev and Chater (2006)).

Previous attempts at explaining choices in the prisoner’s dilemma with behaviour in other tests have not been too encouraging. Dolbear and Lave (1966) do not find a connection with participants’ risk preferences. In Swope et al. (2008), a personality test turns out insignificant. Boone, De Brabander, and Van Witteloostuijn (1999) do not find an effect of locus of control, self-monitoring, aggressiveness and sensation seeking. Brosig, Riechmann, and Weimann (2007), Blanco, Engelmann, and Normann (2011) and Dreber, Fudenberg, and Rand (2014) find little consistency between a prisoner’s dilemma and a dictator game. Yet, Kramer, McClintock, and Messick (1986), McClintock and Liebrand (1988) and Boone, Weimann, and Kiyonari (2010) significantly explain choices in the prisoner’s dilemma with social value orientation scores. Dariel and Nikiforakis (2014) find that cooperators in the public-good game tend to reciprocate higher wages in the gift-exchange game with higher levels of effort.

We believe that the rather sobering earlier findings are due to the fact that the prisoner’s dilemma is a game of multiple, interrelated motives. Specifically, we believe that conditional cooperation is key. Therefore, it is critical to also measure beliefs.

III. Method

Prisoner’s dilemma

Subjects play a simultaneous, two-person, one-shot prisoner’s dilemma game. We hold $R=5$, $S=0$ and $T=10$ constant. Using the strategy method (Selten 1967), we vary $P$, in the interval $\left[S,R\right]$, in 10 steps: $P=0.005x^2$, for $x\in \left\{0.1;\dots ;10\right\}$. Participants are randomly matched to a partner after they have made all choices. One problem is randomly selected to be pay-off-relevant.

We choose these parameters to test whether cooperativeness is disproportionately more pronounced if a cooperator has little to lose (small $P-S$), or if gains from cooperation are large (large $R-P$). If $S=P$, cooperation is ‘free of charge’ – but due to $T>R$, the defection incentive is present. Conversely, if $R=P$, there are no gains from cooperation, but a cooperative move entails the risk of exploitation.

Explanatory variables

Gains from cooperation $\left(2R-P\right)$ measure efficiency. The smaller the $P$, the more cooperation is efficient.

To measure optimism, participants are asked how many of the 24 participants in their session they believe to cooperate when $P=2.45$.¹ Correct guesses are rewarded $€2$; a guess that is the true number $\pm 2$ earns the participant $€1$. We translate this estimate into percentage.

To measure fear, we conduct the Gächter, Johnson, and Herrmann (2007) test² for the λ-value of prospect theory for loss aversion, and the Holt and Laury (2002) test³ for relative risk aversion.

We measure greed with a risky dictator game. Each participant chooses between two situations: in situation 1, both the dictator and the recipient get $€5$. In situation 2, the dictator has a chance of $0\le \alpha \le 1$ to get $€10$ rather than $€5$, but the recipient gets nothing. Using the strategy method (Selten 1967), we vary $\alpha \in \left[0,1\right]$ in 10 equal steps of $.1$. Ex post participants are randomly assigned to be dictators or recipients. One problem is randomly selected to be pay-off-relevant. The switching point⁴ is transformed into β-values of the Fehr and Schmidt (1999) model.

Procedure

The sequence was as follows: prisoner’s dilemma, belief elicitation, loss aversion test, dictator game and risk aversion test. There was no feedback before the end of the experiment. All parts of the experiment were pay-off-relevant. To guarantee positive earnings, we announced a minimum payment of $€5$. It became effective for 11 participants.

The experiment was programmed in z-Tree (Fischbacher 2007) and conducted at the Bonn Econ Lab. Sessions lasted approximately 90 minutes. Subjects were recruited with the software ORSEE (Greiner 2015). A total of 96 students of various majors participated. Mean age was 24.81 years; 61.46% were female. Participants on average earned $€10.50$. Instructions are available in the supplementary data.

IV. Results

Prisoner’s dilemma choices

The defection rate monotonically increases in P (Fig. 1(a)). Exploiting that 84 (87.5%) of our participants behave consistently, we compress results into one continuous variable, the switching point. It is higher the more a participant is cooperative (Fig. 1(b)).⁵

Figure 1. Choices in the prisoner’s dilemma. (a) Effect of safety pay-off P on defection; beliefs are elicited for the shaded pay-off ($P=2.45$). (b) Switching points in the prisoner’s dilemma.

Explaining choices

Since many switching points are at either extreme, a Tobit model is appropriate (Table 2).⁶

Table 2. Explaining switching points by motives.

Tobit, robust standard errors
Dependent variable: lowest punishment pay-off at which participant defects
	Model 1	Model 2	Model 3	Model 4	Model 5	Model 6	Model 7	Model 8
Gains from Cooperation	1.676*** (.290)					1.475*** (.124)	1.794*** (.124)	1.831*** (.118)
Optimism		8.816*** (1.612)				2.814*** (.511)	2.440*** (.317)	2.694*** (.329)
λ			.849 (1.197)			.349 (.456)	−.579* (.249)	−.654* (.261)
Relative riskaversion				.617 (.968)		.249 (.327)	−3.401*** (.425)	−4.115*** (.546)
λ × Relative riskaversion							2.841*** (.292)	3.426*** (.394)
β					−1.916 (3.628)	−4.112** (1.146)	−3.070** (.876)	−2.405** (.877)
Female								−.590* (.285)
Age								−.055^+ (.031)
Econ, math or technical								−.321 (.273)
Law								.095 (.303)
Humanity								.496 (.442)
Social science								.533 (.382)
Cons	−5.707** (1.696)	−3.039*** (1.042)	.922 (1.817)	1.816* (.689)	2.645* (1.287)	−5.886*** (1.618)	−6.906*** (.943)	−5.769*** (1.103)
N	83	83	83	81	83	81	81	81
Left censored	26	26	26	26	26	26	26	26
Right censored	18	18	18	17	18	17	17	17
p Model	<.001	<.001	.4803	.5253	.5988	<.001	<.001	<.001
Pseudo R^2	.4150	.1086	.0018	.0019	.0011	0.5071	.6077	.6335

Gains from cooperation: when switching to defection at a certain point; optimism: estimated percentage of cooperation; λ: degree of loss aversion; relative risk aversion: degree of relative risk aversion; β: degree of aversion against advantageous inequity; standard errors in parentheses.

***p < .001; **p < .01; * p < .05; ⁺p < .1.

Gains from cooperation have a significant positive effect (Model 1).⁷ Optimism increases cooperation significantly (Model 2).⁸ Yet neither loss aversion alone (Model 3),⁹ nor relative risk aversion alone (Model 4),¹⁰ nor inequity aversion alone (Model 5) are significant.

If we include all explanatory variables (Model 6), inequity aversion is significant as well. The more participants are generous in the dictator game (the higher their β), the less they cooperate in the prisoner’s dilemma. This might seem counterintuitive. Yet, for most individuals, $\alpha >\beta$ (cf., Blanco, Engelmann, and Normann 2011). They are averse not only against exploiting others ($\beta$), but also against being exploited themselves ($\alpha$), and both are correlated. Choosing the selfish option in the prisoner’s dilemma protects the player against exploitation. Thus, the negative coefficient of $\beta$ suggests subjects’ general sensitivity to inequity.

If we further interact loss and risk aversion (Model 7), both the main and the interaction effects become significant. This is due to the fact that risk and loss aversion are negatively correlated (Spearman’s rho: r_S = .3452, p = .0009) (cf., Novemsky and Kahneman 2005). Note that our final model explains more than 60% of the variance.

Adding demographic controls (Model 8) has little effect.

V. Conclusion

Since the 1960s, many researchers have experimentally tested in which way and in which magnitude manipulations of pay-offs in a prisoner’s dilemma affect behaviour. Our main contribution is to show why previous experiments have seemingly had such a hard time finding the behavioural causes. For conditional cooperators, a prisoner’s dilemma is a complicated game of multiple motives. Yet, if one simultaneously controls for all of them, one is able to identify this mixture. Our results suggest that participants sense a trade-off: cooperation may pay; but is the risk worth taking?

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental data for this article can be accessed here.

Notes

¹ From pre-tests, we find that this is approximately the empirical mean of switching points.

² Almost all participants are consistent.

³ Ninety-two (95.83%) participants are consistent.

⁴ One participant is inconsistent.

⁵ Only data from consistent participants are used. We drop one datapoint from a participant switching in the opposite direction.

⁶ The results look qualitatively similar if we estimate a random effects logit model that explains choices on individual problems.

⁷ Note that gains from cooperation are not endogenous. Switching points compress choices. We explain choices with (expected) gains from cooperation. At a participant’s switching point, the expected gain from cooperation is given by the sum of gains from all games up to this point, divided by 11.

⁸ 58% cooperate, the mean belief is 59.81%.

⁹ In the regressions, results look very similar if we do not transform variables, but directly work with the switching points for loss aversion, risk aversion and the dictator game.

¹⁰ We lose two observations for participants who are inconsistent on the test for risk aversion.