Prisoners’ other Dilemma

Abstract

We introduce a measure for the riskiness of cooperation in the infinitely repeated discounted Prisoner’s Dilemma and use it to explore how players cooperate once cooperation is an equilibrium. Riskiness of a cooperative equilibrium is based on a pairwise comparison between this equilibrium and the uniquely safe all defect equilibrium. It is a strategic concept heuristically related to Harsanyi and Selten’s risk dominance. Riskiness 0 defines the same critical discount factor $\delta ^{*}$ that was derived with an axiomatic approach for equilibrium selection in Blonski et al. (Am Econ J 3:164–192, 2011). Our theory predicts that the less risky cooperation is the more forgiving can parties afford to be if a deviator needs to be punished. Further, we provide sufficient conditions for cooperation equilibria to be risk perfect, i.e. not to be risky in any subgame, and we extend the theory to asymmetric settings.

Appendix

1.1 The bicentric prior and the tracing procedure in the infinitely repeated Prisoner’s Dilemma

Harsanyi and Selten’s more general definition of risk dominance is based on the so called bicentric prior and the tracing procedure and is difficult to check for games with many strategies. This might explain why it rarely found its way into the applied literature with the exception of $2\times 2$-games with two strict equilibrium points (see Harsanyi and Selten (1988), Chap. 3). While the bicentric prior and the tracing procedure have ever been controversial among game theorists they offer no clue in our context since HS defined them only for finite games. The game theoretic literature has not yet offered a generalization of risk dominance to infinite games. We learned from Klaus Ritzberger that a fundamental mathematical difficulty does not allow a direct generalization of the tracing procedure to infinite games. The idea of the tracing procedure relies on connecting strategy profiles by a generically unique continuous path representing continuous adjustment of beliefs. The problem is that this continuous path is neither unique nor even defined if equilibria are not topologically isolated which happens to be the case in infinite games. In finite games, in contrast, equilibria can shown to be generically isolated.

The bicentric prior—loosely speaking—is a mixed strategy profile reflecting players’ initial beliefs on which of two equilibria under comparison should be played. More precisely, say, player $-i$ attaches subjective probability $\left( z,1-z\right)$ to strategies $\varphi _{-i},\omega _{-i}$. Since player $i$ has no idea of player $-i$’s subjective probability HS invoke the principle of insufficient reason and let player $i$’s bicentric prior $p_{i},1-p_{i}$ be defined as a best response on the mixture $\left( Z,1-Z\right)$ where $Z$ is a random variable uniformly distributed over $\left[ 0,1\right]$. Since generally the bicentric prior is not an equilibrium in itself HS define the tracing procedure such as to readjust beliefs to obtain the risk-dominant equilibrium. To do this precisely, define the payoffs of a parametrized family of auxiliary games denoted by $\Gamma ^{t}\left( \delta ,p\right)$ as follows. Consider strategy profile $\xi =\left( \xi _{1},\xi _{2}\right)$ of $\Gamma \left( \delta \right)$. Let player $i$ assume that player $-i$ plays his bicentric prior abbreviated by $p_{-i}$ with probability $\left( 1-t\right)$ and $\xi _{-i}$ with probability $t$ such that player $i$ obtains

$$\begin{aligned} U_{i}^{t}\left( \xi \right) =\left( 1-t\right) U_{i}\left( \xi _{i},p_{-i}\right) +tU_{i}\left( \xi _{i},\xi _{-i}\right) . \end{aligned}$$

The so defined game $\Gamma ^{t}\left( \delta ,p\right)$ can be interpreted as a convex combination of the original game $\Gamma \left( \delta \right)$ and the trivial game where each player plays against his bicentric prior. Denote by $G=\left\{ \left( t,\xi \right) \left| \xi \text { is an equilibrium for }\Gamma ^{t}\left( \delta ,p\right) \right. \right\}$ the graph of the equilibrium correspondence for the $t$-parametrized family of games $\Gamma ^{t}\left( \delta ,p\right)$.

For generic finite games it can be shown that $G$ contains a unique distinguished curve connecting the best reply (which is the equilibrium) of $\Gamma ^{0}\left( \delta ,p\right)$ equilibrium with one of the two given equilibria (either $\varphi$ or $\omega$) in $\Gamma ^{1}\left( \delta ,p\right)$ for almost any prior $p$. Unfortunately, this is not anymore the case for infinite games. Therefore, for the purpose of the present paper we cannot follow the tracing procedure in order to define risk dominance.

Our criteria for an appropriate model of strategic risk in our context are (i) it must be defined for the discounted infinitely repeated Prisoner’s Dilemma, (ii) it should capture the intuitive observation that payoff $a$ enters the propensity to cooperate, (iii) be as simple as possible (Ockham’s razor). HS’s theory at least fails on criteria (i) and (iii) and clearly any extension of it would fail to a larger extent on (iii). It also fails on criterion (ii) since at least in their famous book HS impose priority for payoff-dominance over risk-dominance whenever the two concepts contradict. The cooperative equilibrium payoff-dominates the all-defect equilibrium whenever it exists. Hence, parameter $a$ does not infuence equilibrium selection with respect to the HS theory.

As already mentioned, it is known that for finite games with more than two strategies risk dominance within $\Gamma _{\varphi }$ and risk dominance based on the bicentric prior and the tracing procedure can differ (see van Damme and Hurkens 1998, 1999; and Carlsson and van Damme 1993b). For example, it may be the case that the best reply against the bicentric prior puts some positive weight on a third strategy which is not among the two equilibria under comparison and thereby tilts the bicentric prior tracing procedure comparison to another direction than the direct comparison does. As we have seen this concern is not well defined within the repeated PD. To the contrary, we note that here we do not compare two arbitrary equilibria of the repeated PD game. Rather, we compare a cooperation equilibrium $\varphi$ with the uniquely “safe” all-defection equilibrium $\omega$. For this comparison there is a best reply to the bicentric prior that is independent of the auxiliary game parameter $t$. One such best reply against different mixtures parametrized by $t$ is always either all defect $\omega$ or grim trigger denoted by $\psi$. There is no potential gain in playing any other cooperation strategy since grim trigger is payoff-equivalent with all other cooperation strategies against another cooperation strategy ¹⁹ but is better than any other cooperation strategy against all defect while all defect is a best response against all defect.

To see that the direct risk comparison in this paper is in line with the belief adjustment according to this constant best response to the bicentric prior (instead of the tracing procedure which is not well defined) we compute player $i$’s bicentric prior by defining expected payoffs of responding to the joint mixture $z\varphi _{-i}+\left( 1-z\right) \omega _{-i}$ by either of the pure strategies $\psi _{i}$ or $\omega _{i}$:

$$\begin{aligned} \tilde{c}_{i}\left( z\right) :&= U\left( \psi _{i},z\varphi _{-i}+\left( 1-z\right) \omega _{-i}\right) =\frac{zc}{1-\delta }+\left( 1-z\right) \left( a+\delta \frac{d}{1-\delta }\right) . \\ \tilde{d}_{i}\left( z\right) :&= U\left( \omega _{i},z\varphi _{-i}+\left( 1-z\right) \omega _{-i}\right) =z\left( b+\delta V_{i\omega }\right) +\left( 1-z\right) \frac{d}{1-\delta } \end{aligned}$$

Now compare these expected payoffs making use of $u_{i}\left( \varphi \right) =\frac{c}{1-\delta }-b-\delta V_{i\omega }$ and $v_{i}\left( \psi \right) =\frac{d}{1-\delta }-a-\delta \frac{d}{1-\delta }=d-a:$

$$\begin{aligned} \tilde{c}_{i}\left( z\right)&\ge \tilde{d}_{i}\left( z\right) \Leftrightarrow \\ z\left( u_{i}\left( \varphi \right) +v_{i}\left( \psi \right) \right)&\ge v_{i}\left( \psi \right) \Leftrightarrow \\ z&\ge \frac{v_{i}\left( \psi \right) }{u_{i}\left( \varphi \right) +v_{i}\left( \psi \right) }. \end{aligned}$$

Player $i$’s bicentric prior probabilities are given by the lengths of the subintervals $\left[ z,1\right]$ and $\left[ 0,z\right]$:

$$\begin{aligned} p_{i}\left( \psi _{i}\right)&= \frac{u_{i}\left( \varphi \right) }{ u_{i}\left( \varphi \right) +v_{i}\left( \psi \right) }\text { and } \\ p_{i}\left( \omega _{i}\right)&= \frac{v_{i}\left( \psi \right) }{ u_{i}\left( \varphi \right) +v_{i}\left( \psi \right) }. \end{aligned}$$

Since the tracing procedure is not well defined we now turn to a constant (in $t$) best response to the bicentric prior. At the starting point $t=0$ player $i$ picks a best response to his bicentric prior $p_{-i}:=p_{-i}\psi _{-i}+\left( 1-p_{-i}\right) \omega _{-i}$. For any $t\in \left( 0,1\right)$ responding to $p_{-i}$ by playing $\varphi _{i}$ or $\omega _{i}$ yields the payoff comparison

$$\begin{aligned} U_{i}\left( \omega _{i},p_{-i}\right)&> U_{i}\left( \varphi _{i},p_{-i}\right) \Leftrightarrow \\ \frac{v_{i}\left( \varphi \right) v_{-i}\left( \varphi \right) }{ u_{-i}\left( \varphi \right) +v_{-i}\left( \psi \right) }&> \frac{ u_{i}\left( \varphi \right) u_{-i}\left( \varphi \right) }{u_{-i}\left( \varphi \right) +v_{-i}\left( \psi \right) }\Leftrightarrow \\ v_{i}\left( \varphi \right) v_{-i}\left( \varphi \right)&> u_{i}\left( \varphi \right) u_{-i}\left( \varphi \right) \end{aligned}$$

This shows in the language of HS that $U_{i}\left( \omega _{i},p_{-i}\right) >U_{i}\left( \varphi _{i},p_{-i}\right)$ for $i=1,2$ iff $\omega$’s Nash product $v_{i}\left( \varphi \right) v_{-i}\left( \varphi \right)$ is strictly larger than $\varphi$’s Nash product $u_{i}\left( \varphi \right) u_{-i}\left( \varphi \right)$. In this case an equilibrium in $\Gamma ^{t}\left( s,p\right)$ contains no positive weight on $\varphi$ for any $t\in \left( 0,1\right)$ since $\omega$ outperforms $\varphi$ against the bicentric prior $p$ and the constant best response $p_{-i}$. The converse holds for $U_{i}\left( \varphi _{i},p_{-i}\right) >U_{i}\left( \omega _{i},p_{-i}\right)$. This shows that this form of belief adjustment corresponds exactly to our definition of risk dominance within the $2\times 2$-game$\ \Gamma _{\varphi }$.