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Intransitiveness: From Games to Random Walks^{ †}

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## Abstract

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## 1. Introduction

## 2. Transitivity

## 3. Intransitiveness in Markov Chains

## 4. Applications

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Analysis of the Penney Game with $\ell =3$, fair coins and fully absorbing traps. (

**a**) Transition graph, where every node is a possibly winning sequence, the links go from one sequence to the sequences which can be obtained with a coin toss. Each link weights 1/2. (

**b**) Weighted adjacency matrix (Markov matrix) $\mathit{M}$ of the system, where the sequences (indexes) are read as base-two numbers. (

**c**) Victory matrix and (

**d**) victory graph, where arrows mark which sequence wins with the largest probability over the given one.

**Figure 4.**(

**a**) Intransitiveness index $\sigma $ vs absorbency $\u03f5$ for a scale-free network with 100 nodes as a function of the absorbency parameter $\epsilon $. There is intransitive behaviour for intermediate values of this parameter. (

**b**) The small subnet presumably responsible for the the intransitive region $\sigma >0$ (

**c**) Intransitiveness index $\sigma $ vs absorbency $\u03f5$ for this subnet.

**Figure 5.**(

**a**) Example index matrix for hierarchical matrices (communities ${l}_{1}=2,{l}_{2}=6,{l}_{3}=2$): for an entry $k={I}_{ij}$, the corresponding adjacency matrix has ones with probability ${p}_{k}$. (

**b**) Average intransitiveness index $\sigma $ vs absorbency $\u03f5$ for hierarchical communities (${l}_{1}={l}_{2}={l}_{3}=3$, average over 1000 simulations).

**Figure 6.**(

**a**) A “square” city, where streets are alternating one-way ($p=0$). (

**b**) Average intransitiveness index $\sigma $ vs absorbency $\u03f5$ for a “square” city, where p denotes the probability of having two-way streets instead of alternating one-way (city with $4\times 4$ streets, average over 1000 simulations; the results are largely independent on the city size).

**Figure 7.**(

**a**) “Square” city with periodic boundary conditions and different absorbancy (${\u03f5}_{1}=1$). For large enough asymmetry p, there is an intransitive region near (${\u03f5}_{2}=1$). (

**b**) Same city but with reflecting boundary conditions, here one observes an intransitive region for intermediate values of ${\u03f5}_{2}$. The curve with $p=0$ is different from other since in this case one doe not have to average over different realizations (1000) of the choice of one-way streets.

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**MDPI and ACS Style**

Baldi, A.; Bagnoli, F.
Intransitiveness: From Games to Random Walks. *Future Internet* **2020**, *12*, 151.
https://doi.org/10.3390/fi12090151

**AMA Style**

Baldi A, Bagnoli F.
Intransitiveness: From Games to Random Walks. *Future Internet*. 2020; 12(9):151.
https://doi.org/10.3390/fi12090151

**Chicago/Turabian Style**

Baldi, Alberto, and Franco Bagnoli.
2020. "Intransitiveness: From Games to Random Walks" *Future Internet* 12, no. 9: 151.
https://doi.org/10.3390/fi12090151