# Deterrence by Collective Punishment May Work against Criminals but Never against Freedom Fighters

## Abstract

**:**

## 1. Introduction

#### 1.1. The Fallacy of the Higher Effectiveness of Higher Punishment

#### 1.2. Substitute or Amplification of Individual Punishment

#### 1.3. Moral and Legal Issues

#### 1.4. Selected Literature

**Experiments:**Behavioral economics teach us that behavior derived from theoretical models often does not match behavior in the field or in the laboratory. For example, we may conclude from Wang et al.’s [8] Prisoner’s Dilemma experiments that adding an (irrelevant) alternative to resisting or complying with the authority may considerably change behavior. Decker et al. [9] find increasing contributions to a public good after the introduction of CP for insufficient aggregate contributions. Dickson [10] finds that trying to prevent the production of a public bad by CP is counterproductive. Neither does he confirm Decker et al.’s [9] result for the positive frame, at least not in the long-run. Gao et al. [11] find that collective punishment is more effective in promoting cooperation (producing a public good) than collective rewards. As we have emphasized above, CP will be more effective if the wrongdoer has an empathic connection with the punished group. [Complementary to this expectation, Pereira and van Prooijen [12] show that people are less reluctant to carry out CP against an “entitative” group (for example, a biker gang), because members of such groups are perceived to be more similar and interchangeable. Falomir-Pichastor et al. [13] and Pereira et al. [14] connect the readiness to carry out CP between groups with the decision structure in these groups. The effectiveness of extant CPs has rarely been investigated. We will come back to this question in the conclusion. Related research. There is a loose connection with discussions of the effectiveness of individual punishment (theory starting with the seminal work of Becker [15]; experiments with an environment of producing a public good starting with and Fehr and Gächter [16], in particular when there are strong external effects of asocial behavior. Even without CP, such investigations have a certain relation to this investigation if individual punishment relies on or induces network effects. For example, punishment may need the backing by the group (in principle [17,18,19], or as at least some consensus [20], or has consequences for network reciprocity [21].

#### 1.5. CP against the Costly Production of a Public Good

#### 1.6. CP against the Profitable Production of a Public Bad

## 2. The CP Game and the Equilibria of CP(x) Subgames

**Definition**

**1.**

- In Stage 1, the principal determines his rules of behavior (regime), the trigger, and the height $\mathrm{x}\in \left[0,\mathrm{M}\right]$ of CP. We do not specify the regime and we assume that the trigger of CP is one resisting agent. Hence, only $\mathrm{x}$ has to be determined.
- In Stage 2, all of the agents simultaneously decide whether to comply with the regime or resist against it. The subgames of stage 2 are called CP(x).

- If $j=0$, the utility of all agents is ${U}_{i}=0$.
- If $0<j<k$, then the group suffers from CP with utilities ${U}_{i}=-c-x$ if $i$ had resisted and ${U}_{i}=-x$ if $i$ had complied.
- If $j\ge k$, all agents enjoy the additional utility $G$, i.e., ${U}_{i}=G-c$ if $i$ had resisted and ${U}_{i}=G$ if $i$ had complied.

- The principal wants to maximize the probability of compliance.
- Agent $i$ wants to maximize her expected utility$${U}_{i}=G\ast prob\left(j\ge k\right)-x\ast prob\left(0<j<k\right)-c\ast {p}_{i},$$

**Lemma**

**1.**

**Proof.**

**Assumption**

**1.**

**Lemma**

**2.**

- If$G<c$and$c>0$, then$\pi =0$is a dominant strategy, independent of$x$.
- If$c<G$and$c<-x$, then$\pi =1$is a dominant strategy.

**Proof.**

**Definition**

**2.**

**Lemma**

**3.**

**Proof.**

**Proposition**

**1.**

- In the CPG game,$\pi =0$is always an equilibrium and$\pi =1$never. In the CPB game,$\pi =1$is always an equilibrium and$\pi =0$if and only if$x+c>0$.
- In the CPG game,${U}_{i}^{\prime}\left(\pi \right)$is a unimodal function. Either${U}_{i}^{\prime}\left(\pi \right)=0$has no solution and${U}_{i}^{\prime}\left(\pi \right)<0$for all$\pi $or it has two solutions${\pi}^{1}\le {\pi}^{2}$and${U}_{i}^{\prime}\left(\pi \right)>0$for${\pi}^{1}<\pi <{\pi}^{2}$and${U}_{i}^{\prime}\left(\pi \right)\le 0$otherwise.
- In the CPB game, there are three different cases.

- If$x<-c$, then${U}_{i}^{\prime}\left(\pi \right)$is a unimodal function.${U}_{i}^{\prime}\left(\pi \right)=0$has either no solution with${U}_{i}^{\prime}\left(\pi \right)>0$for all$\pi $or it has two solutions${\pi}^{1}\le {\pi}^{2}$with${U}_{i}^{\prime}\left(\pi \right)<0$for${\pi}^{1}<\pi <{\pi}^{2}$and${U}_{i}^{\prime}\left(\pi \right)\ge 0$otherwise.
- For$-c<x<-G$, ${U}_{i}^{\prime}\left(\pi \right)$has either no or two local extrema.${U}_{i}^{\prime}\left(\pi \right)=0$has either one solution${\pi}^{*}$with${U}_{i}^{\prime}\left(\pi \right)<0$for$\pi <{\pi}^{*}$and${U}_{i}^{\prime}\left(\pi \right)\ge 0$otherwise; or, it has three solutions${\pi}^{1}<{\pi}^{*}<{\pi}^{2}$with${U}_{i}^{\prime}\left(\pi \right)<0$for$\pi <{\pi}^{1}$and${\pi}^{*}<\pi <{\pi}^{2}$and${U}_{i}^{\prime}\left(\pi \right)\ge 0$otherwise.
- If$x>-G$, then${U}_{i}^{\prime}\left(\pi \right)=0$has exactly one solution${\pi}^{*}$with${U}_{i}^{\prime}\left(\pi \right)<0$for$\pi <{\pi}^{*}$and${U}_{i}^{\prime}\left(\pi \right)>0$for$\pi <{\pi}^{*}$.

**Proof.**

## 3. Equilibrium Selection in the CP(x) Game

#### 3.1. The Limit of Logistic Quantal Response Equilibria

**Proposition**

**2.**

- Let us assume adjacent mixed strategy equilibria$0\le {\pi}^{1}<{\pi}^{2}\le 1$and$\frac{1}{2}\in ({\pi}^{1},{\pi}^{2})$. If${{U}_{i}}^{\prime}\left(\pi \right)$is negative in the interval$\left({\pi}^{1},{\pi}^{2}\right)$, then${\pi}^{1}$is selected, if${{U}_{i}}^{\prime}\left(\pi \right)$is positive in this interval, then${\pi}^{2}$is selected.
- If, including pure strategy equilibria, the smallest equilibrium is${\pi}^{1}>0$and$\frac{1}{2}\in (0,{\pi}^{1})$, then${\pi}^{1}$is selected. If the largest equilibrium is${\pi}^{2}<1$and$\frac{1}{2}\in \left({\pi}^{2},1\right)$, then${\pi}^{2}$is selected.

**Proof.**

**Conclusion**

**1.**

**Conclusion**

**2.**

#### 3.2. The Linear Tracing Procedure

**Proposition**

**3.**

- If$Q\left(0\right)<0$and$\pi =0$is an equilibrium, i.e.,${U}^{\prime}\left(0\right)0$, then$\pi =0$is selected.
- If$Q\left(0\right)>0$and$\pi =1$is an equilibrium, i.e.,${U}^{\prime}\left(1\right)0$, then$\pi =1$is selected.
- If$Q\left(0\right)\langle 0,{U}^{\prime}\left(0\right)\rangle 0$and${U}_{i}^{\prime}$is non-increasing in the interval$\left(0,{\pi}^{1}\right)$, then$\pi ={\pi}^{1}$is selected.
- If$Q\left(0\right)>0,{U}^{\prime}\left(1\right)0$and${U}_{i}^{\prime}$is non-increasing in the interval$\left({\pi}^{2},1\right)$, then${\pi}^{2}$is selected.

**Proof.**

**Conclusion**

**3.**

**Conclusion**

**4.**

#### 3.3. Comparison and Summery

## 4. Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Lemma**

**3.**

**Proof**

**of**

**Proposition**

**1.**

- follows from (A6). For 2 and 3 we have to investigate the function ${U}_{i}^{\prime}\left(\pi \right)$ more closely. We compute$$\frac{\partial {U}_{i}^{\prime}\left(\pi \right)}{\partial \pi}={\left(1-\pi \right)}^{n-2}\left(x+G\right)\ast \left[h\left(\rho \right)+\left(n-1\right)\frac{x}{x+G}\right],$$$$h\left(\rho \right):=\left(\begin{array}{c}n-1\\ k-1\end{array}\right)\ast {\rho}^{k-2}\ast \left[\left(k-1\right)-\left(n-k\right)\rho \right],$$$$\rho =\frac{\pi}{1-\pi},$$
- Because of 0 < c < G, we have no or two mixed strategy equilibria. In the case without a mixed strategy equilibrium, π = 0 is the only equilibrium. In the case with two equilibria, ${U}_{i}^{\prime}\left(\pi \right)$ increases from π = 0 to the lower equilibrium ${\pi}^{1}$ and it decreases from the higher equilibrium ${\pi}^{2}$ to π = 1.
- For G + x < 0 and c + x < 0, we have zero or two local maxima of ${U}_{i}^{\prime}$. Considering the values for π = 0 and π = 1 and the derivative at π = 0 we find either no or two equilibria, together with the described signs of ${U}_{i}^{\prime}$. For G + x < 0 and c + x > 0, the conditions at π = 0 and π = 1 allow one or three equilibria together with the described signs of ${U}_{i}^{\prime}$. Exactly one equilibrium exists for G + x > 0 and c + x > 0. Note that this case is different from 2. because of c < 0. □

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**Figure 1.**The equilibria of the CPB game as intersections of the black curve $q\left(\pi \right)$ and the colored curves $r\left(\pi ,x\right)$. The extreme cases are no CP, i.e., $x=0$, and CP with $x\to \infty $ (red curves). Blue curves are with intermediate $x$. Parameters of the figure: $n=5,k=3,c=-0.15,G=-1$.

**Figure 2.**QRE in the case ${\pi}^{1}\left(x\right)<\frac{1}{2}<{\pi}^{2}\left(x\right)$. Parameters of the figure: $\lambda =0.5,x=1,n=5,\text{}k=3,\text{}c=0.15,\text{}G=1$.

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Bolle, F.
Deterrence by Collective Punishment May Work against Criminals but Never against Freedom Fighters. *Games* **2021**, *12*, 41.
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Bolle F.
Deterrence by Collective Punishment May Work against Criminals but Never against Freedom Fighters. *Games*. 2021; 12(2):41.
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2021. "Deterrence by Collective Punishment May Work against Criminals but Never against Freedom Fighters" *Games* 12, no. 2: 41.
https://doi.org/10.3390/g12020041