Words are not Wind -- How Public Joint Commitment and Reputation Solve the Prisoner's Dilemma

TL;DR

This study tackles the enduring question of how cooperation can emerge in the Prisoner's Dilemma without external enforcement. It introduces a joint-commitment mechanism where cooperation is conditional on mutual commitment and private reputations, turning assessment into a function of arrangements rather than unconditional actions. Using an evolutionary game-theoretic framework with a Staying-like reputation norm and a nine-strategy space (including RA), the authors show that cooperation can be stable when the net benefit of cooperation exceeds the sum of arrangement and cooperation costs, $b-1>c_a$, and that RA typically dominates yet coexists with other strategies. The work provides a plausible avenue for understanding the ubiquity of commitments (public or private) in human societies, suggesting they can underpin cooperation long before robust legal institutions, and offers a theoretical bridge between joint commitments and reputation-based indirect reciprocity.

Abstract

To achieve common goals, we often use joint commitments. Our commitment helps us to coordinate with our partners and assures them that their cooperative efforts will benefit themselves. However, if one of us can exploit the other's cooperation (as in the Prisoner's Dilemma), our commitment appears less useful. It cannot remove the temptation for our partners to exploit us. Using methods from evolutionary game theory, we study the function of joint commitments in the Prisoner's Dilemma. We propose a reputation system akin to indirect reciprocity, wherein agents observe interactions even when not directly involved. They judge cooperation as good and defection as bad, but, crucially, only if the parties involved had committed to cooperate. This results in stable cooperation even though judgments are made privately, which had been a weakness in previous models of indirect reciprocity. Our work shows that joint commitments have utility beyond coordination problems, which could explain their prevalence. The proposed link between joint commitments and reputation could also explain why some joint commitments are pointedly public, like wedding vows. A reputation-based mechanism might have been particularly relevant in our distant past, in which no institutions existed to enforce commitments.

Figures (8)

• Figure 1: Principles of Joint Commitments. This schema shows all the steps of a strategy that uses joint commitment and reputations to enable cooperation. When playing the Prisoner's Dilemma (PD) with another individual, you first apply the commitment rule (top). If you think your partner is good (i.e. trustworthy), you are willing to enter a joint commitment. If you are both willing, a joint commitment (or “arrangement”) is made. In such an arrangement, you will cooperate. Everybody assesses actions that are made in an arrangement. The opinions formed will be used when they play the PD with this individual in the future. The symbol '$=>$' indicates 'if and only if', so that rules could be simplified (i.e. these rules were not listed: partner bad $=>$ not willing, no arrangement $=>$ defect, no arrangement + any action $=>$ no assessment.) For later reference, we indicated some steps as individual (blue) or global (grey). This distinction will be referred to in the Method section when we introduce the evolutionary setting.
• Figure 2: Strategies, i.e. combinations of commitment rules, $\alpha$, and cooperation rules, $\beta$. Strategies are named by abbreviation, such as RA for 'Reputation observant Arrangement upholder'. They broadly fall into three categories: the mentioned RA which applies all principles, 'nice' strategies (yellow, green and blue) which are either more generous (cooperate outside arrangement), more naive (enter arrangements with everybody) or both, and lastly 'mean' strategies (dark and red) that either never commit nor cooperate (AllD) or commit but defect anyway (fakers).
• Figure 3: Predictions of Reputations. On the right is an overlay of the three different predictions under Assumptions 1 and 2 for the nine strategies. On the left is a graph of reputation values. We predict that faker strategies should always have bad reputations ($r=0$), but that perception error increases their average reputation to $r=\epsilon$, as is indicated by the broken line. The case is similar for 'high' reputations.
• Figure 4: Evolution of cooperation as a function of the benefit of cooperation ($b$) and cost of commitment arrangement ($c_a$). Stereotypical case (centre) with Assumption 2.b, perception error rate $\epsilon=0.01$, imitation strength $s=1$ and mutation rate $\mu=0.01$. The matrix of graphs on the left shows different game parameters in the same evolutionary setting $s=1$, $\mu=0.01$. The matrix on the right shows different evolutionary parameters in the same game setting $\epsilon=0.01$, Assumption 2.b. Result 'Copop1': The frequency of cooperation in the population is very high for almost the entire area ($>0.7$), where the size of benefits minus the cost of cooperation exceeded the cost of arrangements (i.e. to the right of the diagonal white line $b-1=c_a$). If errors are frequent, selection weak or mutation rare, cooperation rates decrease, but cooperation never vanishes. Result 'Copop2': Cooperation tends to decrease for higher benefits and smaller costs of arrangements.
• Figure 5: Average frequency of all nine strategies over time. Benefit $b$ varies from left to right (low: $b=1.5$, medium: $5.5$, high: $9.5$). In all cases, we set $c_a=1$, $\epsilon=0.01$; in Assumption 2.b, we also set $s=1$, $\mu=0.01$. For low benefits (left), $b-1>c_a$ does not hold. Cooperation is only profitable if it would not require arrangements. But the only strategy that cooperates and does not enter arrangements, AllC (aka 0+), vanishes as well. The only prevailing strategies are AllD (0- and 0A) and observant faker (R-). What they have in common is that they do not waste resources on cooperation or arrangements (R- does not trust other fakers or defectors). For medium benefits (center), RA is by far the most common strategy (Result 'Strats1'), while there are also more naive and more generous cooperators (1A, 1+ and R+) present (Result 'Strats2'). AllD players are virtually non-existent, but a small share of R- remains (Result 'Strats3'). For high benefits (right), naive cooperator strategies become even more frequent, and so does R-, lowering the frequency of RA. Note, as Figure \ref{['fig:res1']} indicates, that increase in fakers' frequency seems to cause overall cooperation to decline, despite the increase in naive cooperators.