# Direct Reciprocity and Model-Predictive Strategy Update Explain the Network Reciprocity Observed in Socioeconomic Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. Tit-for-tat vs. D

- her own strategy, T or D, to be used at round t (in Table 1, second column, Tn and Dn denote the fact that the agent changed strategy after round $t-1$); and
- for each neighbor j, the strategy, T or D, that i expects j to use at round t and the last actions, C or D, played by i and j at round $t-1$.

#### 2.2. Tit-for-two-tats vs. D

#### 2.3. New-tit-for-two-tats vs. D

## 3. Results

#### 3.1. Tit-for-tat vs. D

**Theorem**

**1.**

**Corollary**

**1.**

**Theorem**

**2.**

**Corollary**

**2.**

#### 3.2. Tit-for-two-tats vs. D

**Theorem**

**3.**

**Corollary**

**3.**

**Theorem**

**4.**

**Corollary**

**4.**

**Theorem**

**5.**

**Corollary**

**5.**

#### 3.3. New-tit-for-two-tats vs. D

**Theorem**

**6.**

**Corollary**

**6.**

**Theorem**

**7.**

**Corollary**

**7.**

## 4. Discussion and Conclusions

#### 4.1. Three Versions of the T-strategy

#### 4.2. Comparison with the T-strategy in Dercole et al. (2019)

#### 4.3. Impact on Applied Behavioral Science

#### 4.4. Future Directions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Theorems and Proofs of Results in Section 3

#### Appendix A.1. Tit-for-tat vs. D

**Proof of Theorem**

**1.**

**Proof of Theorem**

**2.**

#### Appendix A.2. Tit-for-two-tats vs. D

**Proof of Theorem**

**3.**

**Proof of Theorem**

**4.**

**Proof of Theorem**

**5.**

#### Appendix A.3. New-tit-for-two-tats vs. D

**Proof of Theorem**

**6.**

**Proof of Theorem**

**7.**

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**Figure 1.**Invasion and fixation of the tit-for-tat strategy against a population of unconditional defectors. Panels show the fraction of T-agents reached in ${10}^{4}$ game rounds starting from a single initial T on WS and BA networks (left and right panels) as a function of the PD return r (average degree $\langle k\rangle =4$ and 8 in top and bottoms panels). Solid lines show the average fraction over 100 random initializations (network generation and random placement of the initial T). Dashed lines show the average fraction of cooperative actions, saturating at fixation to 0.5 because of the alternate exploitation triggered by new T-agents with their T-neighbors. Dots show the outcomes of single simulations (T-fraction); transparency is used to show dots accumulation. Colors code the predictive horizon h, from 3 to 5, and the corresponding invasion threshold $\langle {\overline{r}}_{\mathrm{T}}^{h}\left({k}_{\mathrm{n}}^{max}\right)\rangle $, averaged over the 100 networks, is reported; the invasion threshold for degree-4 and -8 regular networks are indicated by the vertical dotted lines. The results were obtained for strategy update rate $\delta =0.05$.

**Figure 2.**Invasion and fixation of the tit-for-two-tats strategy against a population of unconditional defectors. Panels show the fraction of T-agents reached in ${10}^{4}$ game rounds starting from a single initial T in WS and BA networks (left and right panels) as a function of the PD game return r (average degree $\langle {k}_{i}\rangle =4$ and 8 in top and bottom panels). Solid lines show the average fraction over 100 random initializations (network generation and random placement of the initial T). Dots show the outcomes of single simulations; transparency is used to show dots accumulation. Colors code the predictive horizon h, from 3 to 5, and the corresponding invasion threshold $\langle {\overline{r}}_{{\mathrm{T}}_{2}}^{h}\rangle $, averaged over the 100 networks, is reported; the invasion threshold for degree-4 and -8 regular networks are indicated by the vertical dotted lines. The results were obtained for strategy update rate $\delta =0.05$.

**Figure 3.**Invasion and fixation of the new-tit-for-two-tats strategy against a population of unconditional defectors. Panels show the fraction of T-agents reached in ${10}^{4}$ game rounds starting from a single initial T in WS and BA networks (left and right panels) as a function of the PD game return r (average degree $\langle {k}_{i}\rangle =4$ and 8 in top and bottom panels). Solid lines show the average fraction over 100 random initializations (network generation and random placement of the initial T). Dots show the outcomes of single simulations; transparency is used to show dots accumulation. Colors code the predictive horizon h, from 2 to 5, and the corresponding invasion threshold $\langle {\overline{r}}_{{\mathrm{nT}}_{2}}^{h}\rangle $, averaged over the 100 networks, is reported; the invasion threshold for degree-4 and -8 regular networks are indicated by the vertical dotted lines. The results were obtained for strategy update rate $\delta =0.05$.

**Table 1.**

**Tit-for-tat vs. D.**Neighborhood classification update for agent i when playing the game round t. For each neighbor j, the class label includes the strategy, T or D, that j is expected to use at round t and the last actions, C or D, played by i and j, respectively. The strategy of agent i (second column) takes values in $\{\mathrm{T},\mathrm{Tn},\mathrm{D},\mathrm{Dn}\}$, where the ‘n’ stands for ‘new’ and indicates that the agent is new to the strategy. The first four classes in the first column are those that can occur after the first game round. The other classes are considered, top-to-bottom, in order of appearance in the last column.

At Round t | Class of $\mathit{j}$ after Round t | ||||
---|---|---|---|---|---|

Class of $\mathit{j}$ | Strat. of $\mathit{i}$ | Act. of $\mathit{i}$ | Act. of $\mathit{j}$ | Act. of $\mathit{j}$ | |

(Expected) | |||||

TCC | T | C | C | C | TCC |

${}^{1,2}$ | D | DCD | |||

Dn | D | C | TDC | ||

D | DDD | ||||

DCD | T,Dn | D | D | C | TDC |

${}^{1,2}$ | D | DDD | |||

TDC | T,Tn | C | D | D ${}^{3}$ | TCD |

${}^{1}$ | D,Dn | D | D ${}^{3}$ | TDD | |

DDD | T | D | D | C | TDC |

${}^{1}$ | D | DDD | |||

Tn | C | C | TCC | ||

D | DCD | ||||

D,Dn | D | C | TDC | ||

D | DDD | ||||

TCD | T,Dn | D | C | C | TDC |

${}^{2}$ | D | DDD | |||

TDD | Tn | C | D | C | TCC |

${}^{4}$ | D | TCD | |||

D | D | C | TDC | ||

D | TDD |

When Computing | Class of j Updated after Round t | Contribution per Neighbor per Round | Number of Neighbors | ||||
---|---|---|---|---|---|---|---|

1 | 2 | $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathit{\dots}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$ | $2\mathit{l}+1$ | $2\mathit{l}$ | |||

${\pi}_{\mathrm{TT},i}^{h}$ | TCC | $\phantom{\rule{0.166667em}{0ex}}r-1\phantom{\rule{0.166667em}{0ex}}$ | $\phantom{\rule{0.166667em}{0ex}}r-1\phantom{\rule{0.166667em}{0ex}}$ | … | $r-1$ | $r-1$ | ${k}_{i}^{\mathrm{TCC}}$ |

TCD | r | $-1$ | … | r | $-1$ | ${k}_{i}^{\mathrm{TCD}}$ | |

TDC | $-1$ | r | … | $-1$ | r | ${k}_{i}^{\mathrm{TDC}}$ | |

DCD, DDD | 0 | 0 | … | 0 | 0 | – | |

TDD | Revising Ts have no TDD-neigbors | 0 | |||||

${\pi}_{\mathrm{TD},i}^{h}$ | TCC, TCD | r | 0 | … | 0 | 0 | ${k}_{i}^{\mathrm{TCC}}+{k}_{i}^{\mathrm{TCD}}$ |

TDC, DCD, DDD | 0 | 0 | … | 0 | 0 | – | |

TDD | Revising Ts have no TDD-neigbors | 0 | |||||

${\pi}_{\mathrm{DD},i}^{h}$ | TDC, TDD, DDD | 0 | 0 | … | 0 | 0 | – |

TCC, TCD, DCD | Revising Ds have no such neighbors | 0 | |||||

${\pi}_{\mathrm{DT},i}^{h}$ | TDC, TDD | $-1$ | r | … | $-1$ | r | ${k}_{i}^{\mathrm{TDC}}+{k}_{i}^{\mathrm{TDD}}$ |

DDD | $-1$ | 0 | … | 0 | 0 | ${k}_{i}^{\mathrm{DDD}}$ | |

TCC, TCD, DCD | Revising Ds have no such neighbors | 0 | |||||

$\begin{array}{ll}{\pi}_{\mathrm{TT},i}^{h}=\phantom{\rule{-0.56905pt}{0ex}}\left\{\phantom{\rule{-4.83694pt}{0ex}}\begin{array}{ll}h{k}_{i}^{\mathrm{TCC}}(r\phantom{\rule{-1.42262pt}{0ex}}-\phantom{\rule{-1.70717pt}{0ex}}1)+\frac{{\scriptstyle h}}{{\scriptstyle 2}}({k}_{i}^{\mathrm{TCD}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDC}})r-\frac{{\scriptstyle h}}{{\scriptstyle 2}}({k}_{i}^{\mathrm{TCD}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDC}})& \mathrm{if}\phantom{\rule{0.277778em}{0ex}}h\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{even}\\ h{k}_{i}^{\mathrm{TCC}}(r\phantom{\rule{-1.42262pt}{0ex}}-\phantom{\rule{-1.70717pt}{0ex}}1)+\frac{{\scriptstyle h-1}}{{\scriptstyle 2}}({k}_{i}^{\mathrm{TDC}}r-{k}_{i}^{\mathrm{TCD}})+\frac{{\scriptstyle h+1}}{{\scriptstyle 2}}({k}_{i}^{\mathrm{TCD}}r-{k}_{i}^{\mathrm{TDC}})& \mathrm{if}\phantom{\rule{0.277778em}{0ex}}h\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{odd}\end{array}\right.& {\pi}_{\mathrm{TD},i}^{h}=({k}_{i}^{\mathrm{TCC}}+{k}_{i}^{\mathrm{TCD}})r\\ {\pi}_{\mathrm{DT},i}^{h}=\phantom{\rule{-0.56905pt}{0ex}}\left\{\phantom{\rule{-5.406pt}{0ex}}\begin{array}{ll}\phantom{\rule{5.69054pt}{0ex}}\frac{{\scriptstyle h}}{{\scriptstyle 2}}\phantom{\rule{4.2679pt}{0ex}}({k}_{i}^{\mathrm{TDC}}+{k}_{i}^{\mathrm{TDD}})r\phantom{\rule{0.28453pt}{0ex}}-\phantom{\rule{4.83694pt}{0ex}}\frac{{\scriptstyle h}}{{\scriptstyle 2}}\phantom{\rule{4.2679pt}{0ex}}({k}_{i}^{\mathrm{TDC}}+{k}_{i}^{\mathrm{TDD}})-{k}_{i}^{\mathrm{DDD}}& \phantom{\rule{29.30634pt}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}h\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{even}\\ \frac{{\scriptstyle h-1}}{{\scriptstyle 2}}({k}_{i}^{\mathrm{TDC}}+{k}_{i}^{\mathrm{TDD}})r-\frac{{\scriptstyle h+1}}{{\scriptstyle 2}}({k}_{i}^{\mathrm{TDC}}+{k}_{i}^{\mathrm{TDD}})-{k}_{i}^{\mathrm{DDD}}& \phantom{\rule{29.30634pt}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}h\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{odd}\end{array}\right.& {\pi}_{\mathrm{DD},i}^{h}0\\ \Delta {\pi}_{\mathrm{T},i}^{1}={\pi}_{\mathrm{TD},i}^{1}\phantom{\rule{-0.56905pt}{0ex}}-{\pi}_{\mathrm{TT},i}^{1}={k}_{i}^{\mathrm{TCC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDC}}\Delta {\pi}_{\mathrm{D},i}^{1}={\pi}_{\mathrm{DT},i}^{1}\phantom{\rule{-0.56905pt}{0ex}}-{\pi}_{\mathrm{DD},i}^{1}=-({k}_{i}^{\mathrm{TDC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDD}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{DDD}})& \end{array}$ |

**Table 3.**

**Tit-for-two-tats vs. D.**Neighborhood classification update for agent i when playing the game round t. For each neighbor j, the class label includes the strategy, T or D, that j is expected to use at round t and the last actions, C or D, played by i and j, respectively. The actions labels CD1/DC1 means that i/j has been exploited by j/i at round $t-1$ but the same did not happen at round $t-2$. The strategy of agent i (second column) takes values in $\{\mathrm{T},\mathrm{Tn},\mathrm{D},\mathrm{Dn}\}$, where the ‘n’ stands for ‘new’ and indicates that the agent is new to the strategy. The first four classes in the first column are those that can occur after the first game round. The other classes are considered, top-to-bottom, in order of appearance in the last column.

At Round t | Class of $\mathit{j}$ after Round t | ||||
---|---|---|---|---|---|

Class of $\mathit{j}$ | Strat. of $\mathit{i}$ | Act. of $\mathit{i}$ | Act. of $\mathit{j}$ | Act. of $\mathit{j}$ | |

(Expected) | |||||

TCC | T | C | C | C | TCC |

${}^{1,2}$ | D | DCD1 | |||

Dn | D | C | TDC1 | ||

D | DDD | ||||

DCD1 | T | C | D | C | TCC |

${}^{1,2}$ | D | DCD | |||

Dn | D | C | TDC1 | ||

D | DDD | ||||

TDC1 | T,Tn | C | C | C | TCC |

${}^{1}$ | D | DCD1 | |||

D,Dn | D | C | TDC | ||

D | DDD | ||||

DDD | T,D,Dn | D | D | C | TDC1 |

${}^{1}$ | D | DDD | |||

Tn | C | C | TCC | ||

D | DCD1 | ||||

DCD | T,Dn | D | D | C | TDC1 |

${}^{1,2}$ | D | DDD | |||

TDC | Tn | C | D | D ${}^{4}$ | TCD1 |

${}^{1,3}$ | D,Dn | D | D ${}^{4}$ | TDD | |

TCD1 | T | C | C | C | TCC |

${}^{2}$ | D | DCD | |||

Dn | D | C | TDC1 | ||

D | DDD | ||||

TDD | Tn | C | D | C | TCC |

${}^{5}$ | D | TCD1 | |||

D | D | C | TDC1 | ||

D | TDD |

When Computing | Class of j updated after Round t | Contribution per Neighbor per Round | Number of Neighbors | ||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathit{\dots}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$ | h | |||

${\pi}_{\mathrm{TT},i}^{h}$ | TCC, TCD1, TDC1 | $\phantom{\rule{0.166667em}{0ex}}r-1\phantom{\rule{0.166667em}{0ex}}$ | $\phantom{\rule{0.166667em}{0ex}}r-1\phantom{\rule{0.166667em}{0ex}}$ | $\phantom{\rule{0.166667em}{0ex}}r-1\phantom{\rule{0.166667em}{0ex}}$ | … | $r-1$ | ${k}_{i}^{\mathrm{TCC}}+{k}_{i}^{\mathrm{TDC}1}+{k}_{i}^{\mathrm{TCD}1}$ |

DCD1 | $-1$ | 0 | 0 | … | 0 | ${k}_{i}^{\mathrm{DCD}1}$ | |

DCD, DDD | 0 | 0 | 0 | … | 0 | – | |

TDC, TDD | Revising Ts have no TDD-neigbors | 0 | |||||

${\pi}_{\mathrm{TD},i}^{h}$ | TCC, TCD1 | r | r | 0 | … | 0 | ${k}_{i}^{\mathrm{TCC}}+{k}_{i}^{\mathrm{TCD}1}$ |

TDC1 | r | 0 | 0 | … | 0 | ${k}_{i}^{\mathrm{TDC}1}$ | |

DCD, DCD1, DDD | 0 | 0 | 0 | … | 0 | – | |

TDC, TDD | Revising Ts have no TDD-neigbors | 0 | |||||

${\pi}_{\mathrm{DD},i}^{h}$ | TDC1 | r | 0 | … | 0 | 0 | ${k}_{i}^{\mathrm{TDC}1}$ |

TDC, TDD, DDD | 0 | 0 | … | 0 | 0 | – | |

TCC, TCD1, DCD, DCD1 | Revising Ds have no such neighbors | 0 | |||||

${\pi}_{\mathrm{DT},i}^{h}$ | TDC1 | $r-1$ | $r-1$ | $r-1$ | … | $r-1$ | ${k}_{i}^{\mathrm{TDC}1}$ |

TDC, TDD | $-1$ | $r-1$ | $r-1$ | … | $r-1$ | ${k}_{i}^{\mathrm{TDC}}+{k}_{i}^{\mathrm{TDD}}$ | |

DDD | $-1$ | $-1$ | 0 | … | 0 | ${k}_{i}^{\mathrm{DDD}}$ | |

TCC, DCD1, DCD, TCD1 | Revising Ds have no such neighbors | 0 | |||||

$\begin{array}{ll}{\pi}_{\mathrm{TT},i}^{h}=h({k}_{i}^{\mathrm{TCC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TCD}1}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDC}1})(r\phantom{\rule{-1.42262pt}{0ex}}-\phantom{\rule{-1.70717pt}{0ex}}1)-{k}_{i}^{\mathrm{DCD}1}& {\pi}_{\mathrm{TD},i}^{h}=({k}_{i}^{\mathrm{TCC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TCD}1})2r\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDC}1}r\\ {\pi}_{\mathrm{DT},i}^{h}=h{k}_{i}^{\mathrm{TDC}1}(r\phantom{\rule{-1.42262pt}{0ex}}-\phantom{\rule{-1.70717pt}{0ex}}1)\phantom{\rule{-1.70717pt}{0ex}}+\phantom{\rule{-1.70717pt}{0ex}}(h\phantom{\rule{-1.42262pt}{0ex}}-\phantom{\rule{-1.70717pt}{0ex}}1)({k}_{i}^{\mathrm{TDC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDD}})(r\phantom{\rule{-1.42262pt}{0ex}}-\phantom{\rule{-1.70717pt}{0ex}}1)-({k}_{i}^{\mathrm{TDC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDD}})\phantom{\rule{-1.99168pt}{0ex}}-\phantom{\rule{-0.56905pt}{0ex}}2{k}_{i}^{\mathrm{DDD}}& {\pi}_{\mathrm{DD},i}^{h}={k}_{i}^{\mathrm{TDC}1}r\end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{ll}\Delta {\pi}_{\mathrm{T},i}^{1}={\pi}_{\mathrm{TD},i}^{1}\phantom{\rule{-0.56905pt}{0ex}}-{\pi}_{\mathrm{TT},i}^{1}={k}_{i}^{\mathrm{TCC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TCD}1}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDC}1}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{DCD}1}& \Delta {\pi}_{\mathrm{D},i}^{1}={\pi}_{\mathrm{DT},i}^{1}\phantom{\rule{-0.56905pt}{0ex}}-{\pi}_{\mathrm{DD},i}^{1}=-({k}_{i}^{\mathrm{TDC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDC}1}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDD}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{DDD}})\end{array}$ |

**Table 5.**

**New-tit-for-two-tats vs. D.**Neighbors’ classification update for agent i when playing the game round t. For each neighbor j, the class label includes the strategy, T or D, that j is expected to use at round t and the last actions, C or D, played by i and j, respectively. The action labels CD1/DC1 mean that i/j changed to T after round $t-2$ and she has been exploited by j/i at round $t-1$. The strategy of agent i (second column) takes values in $\{\mathrm{T},\mathrm{Tn},\mathrm{D},\mathrm{Dn}\}$, where the ‘n’ stands for ‘new’ and indicates that the agent is new to the strategy. The first four classes in the first column are those that can occur after the first game round. The other classes are considered, top-to-bottom, in order of appearance in the last column.

At Round t | Class of $\mathit{j}$ after Round t | ||||
---|---|---|---|---|---|

Class of $\mathit{j}$ | Strat. of $\mathit{i}$ | Act. of $\mathit{i}$ | Act. of $\mathit{j}$ | Act. of $\mathit{j}$ | |

(Expected) | |||||

TCC | T | C | C | C | TCC |

${}^{1,2}$ | D | DCD | |||

Dn | D | C | TDC | ||

D | DDD | ||||

DCD | T,Dn | D | D | C | TDC1 |

D | DDD | ||||

TDC | Tn | C | D | D ${}^{4}$ | TCD1 |

${}^{1,3}$ | D,Dn | D | D ${}^{4}$ | TDD | |

DDD | T,D,Dn | D | D | C | TDC1 |

${}^{1}$ | D | DDD | |||

Tn | C | C | TCC | ||

D | DCD1 | ||||

TDC1 | T,Tn | C | C ${}^{5}$ | C | TCC |

${}^{1}$ | D | DCD1 | |||

Tn | D ${}^{6}$ | D ${}^{7}$ | TCD1 | ||

D,Dn | D | C ${}^{8}$ | C | TDC | |

D | DDD | ||||

D | D | D | TDD | ||

TCD1 | T | C | C | C | TCC |

${}^{2}$ | D | DCD | |||

Dn | D | C | TDC | ||

D | DDD | ||||

TDD | Tn | C | D | C | TCC |

${}^{2,9}$ | D | TCD1 | |||

D | D | C | TDC1 | ||

D | TDD | ||||

DCD1 ${}^{1,2}\phantom{\rule{-0.166667em}{0ex}}$ | T,Dn | D | D | C | TDC1 |

D | DDD |

When Computing | Class of j Updated after Round t | Contribution per Neighbor per Round | Number of Neighbors | ||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathit{\dots}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$ | h | |||

${\pi}_{\mathrm{TT},i}^{h}$ | TCC, TCD1, TDC1 | $\phantom{\rule{0.166667em}{0ex}}r-1\phantom{\rule{0.166667em}{0ex}}$ | $\phantom{\rule{0.166667em}{0ex}}r-1\phantom{\rule{0.166667em}{0ex}}$ | $\phantom{\rule{0.166667em}{0ex}}r-1\phantom{\rule{0.166667em}{0ex}}$ | … | $\mathbf{r}-\mathbf{1}$ | ${\mathbf{k}}_{\mathbf{i}}^{\mathrm{TCC}}+{\mathbf{k}}_{\mathbf{i}}^{\mathrm{TDC}1}+{\mathbf{k}}_{\mathbf{i}}^{\mathrm{TCD}1}$ |

DCD, DCD1, DDD | 0 | 0 | 0 | … | 0 | – | |

TDC, TDD | Revising Ts have no TDD-neigbors | 0 | |||||

${\pi}_{\mathrm{TD},i}^{h}$ | TCC,TCD1,TDC1 | r | 0 | 0 | … | 0 | ${k}_{i}^{\mathrm{TCC}}+{k}_{i}^{\mathrm{TDC}1}+{k}_{i}^{\mathrm{TCD}1}$ |

DCD, DCD1, DDD | 0 | 0 | 0 | … | 0 | – | |

TDC, TDD | Revising Ts have no TDD-neigbors | 0 | |||||

${\pi}_{\mathrm{DD},i}^{h}$ | TDC1 | r | 0 | … | 0 | 0 | ${k}_{i}^{\mathrm{TDC}1}$ |

TDC, TDD, DDD | 0 | 0 | … | 0 | 0 | – | |

TCC, TCD1, DCD, DCD1 | Revising Ds have no such neighbors | 0 | |||||

${\pi}_{\mathrm{DT},i}^{h}$ | TDC1 | $r-1$ | $r-1$ | $r-1$ | … | $r-1$ | ${k}_{i}^{\mathrm{TDC}1}$ |

TDC, TDD | $-1$ | $r-1$ | $r-1$ | … | $r-1$ | ${k}_{i}^{\mathrm{TDC}}+{k}_{i}^{\mathrm{TDD}}$ | |

DDD | $-1$ | 0 | 0 | … | 0 | ${k}_{i}^{\mathrm{DDD}}$ | |

TCC, TCD1, DCD, DCD1 | Revising Ds have no such neighbors | 0 | |||||

$\begin{array}{ll}{\pi}_{\mathrm{TT},i}^{h}=h({k}_{i}^{\mathrm{TCC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TCD}1}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDC}1})r\phantom{\rule{-1.42262pt}{0ex}}-\phantom{\rule{-1.70717pt}{0ex}}1)& {\pi}_{\mathrm{TD},i}^{h}=({k}_{i}^{\mathrm{TCC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TCD}1}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDC}1})r\\ {\pi}_{\mathrm{DT},i}^{h}=h{k}_{i}^{\mathrm{TDC}1}(r\phantom{\rule{-1.42262pt}{0ex}}-\phantom{\rule{-1.70717pt}{0ex}}1)+(h\phantom{\rule{-1.42262pt}{0ex}}-\phantom{\rule{-1.70717pt}{0ex}}1)({k}_{i}^{\mathrm{TDC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDD}})(r\phantom{\rule{-1.42262pt}{0ex}}-\phantom{\rule{-1.70717pt}{0ex}}1)-({k}_{i}^{\mathrm{TDC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDD}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{DDD}})& {\pi}_{\mathrm{DD},i}^{h}={k}_{i}^{\mathrm{TDC}1}r\end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{ll}\Delta {\pi}_{\mathrm{T},i}^{1}={\pi}_{\mathrm{TD},i}^{1}\phantom{\rule{-0.56905pt}{0ex}}-{\pi}_{\mathrm{TT},i}^{1}={k}_{i}^{\mathrm{TCC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TCD}1}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDC}1}& \Delta {\pi}_{\mathrm{D},i}^{1}={\pi}_{\mathrm{DT},i}^{1}\phantom{\rule{-0.56905pt}{0ex}}-{\pi}_{\mathrm{DD},i}^{1}=-({k}_{i}^{\mathrm{TDC}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDC}1}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{TDD}}\phantom{\rule{-1.99168pt}{0ex}}+\phantom{\rule{-0.56905pt}{0ex}}{k}_{i}^{\mathrm{DDD}})\end{array}$ |

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**MDPI and ACS Style**

Della Rossa, F.; Dercole, F.; Di Meglio, A. Direct Reciprocity and Model-Predictive Strategy Update Explain the Network Reciprocity Observed in Socioeconomic Networks. *Games* **2020**, *11*, 16.
https://doi.org/10.3390/g11010016

**AMA Style**

Della Rossa F, Dercole F, Di Meglio A. Direct Reciprocity and Model-Predictive Strategy Update Explain the Network Reciprocity Observed in Socioeconomic Networks. *Games*. 2020; 11(1):16.
https://doi.org/10.3390/g11010016

**Chicago/Turabian Style**

Della Rossa, Fabio, Fabio Dercole, and Anna Di Meglio. 2020. "Direct Reciprocity and Model-Predictive Strategy Update Explain the Network Reciprocity Observed in Socioeconomic Networks" *Games* 11, no. 1: 16.
https://doi.org/10.3390/g11010016