# A Population Game Model for the Expansion of Airbnb in the City of Venice

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Evolutionary Game Theory

#### 2.2. The Urban Population Game

#### 2.3. The Venice Data

## 3. Results

#### 3.1. The Case $\mu =0$.

#### 3.2. The Case $\mu \ne 0$.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- UNWTO United Nations World Tourism Organization. Available online: https://www.unwto.org (accessed on 30 March 2021).
- Gutiérrez, J.; García-Palomares, J.C.; Romanillos, G.; Salas-Olmedo, M.H. The eruption of Airbnb in tourist cities: Comparing spatial patterns of hotels and peer-to-peer accommodation in Barcelona. Tour. Manag.
**2017**, 62, 278–291. [Google Scholar] [CrossRef][Green Version] - Pasquinelli, C.; Trunfio, M. Overtouristified cities: An online news media narrative analysis. J. Sustain. Tour.
**2020**, 28, 1805–1824. [Google Scholar] [CrossRef] - Poczta, J.; Dąbrowska, A.; Kazimierczak, M.; Gravelle, F.; Malchrowicz-Mośko, E. Overtourism and Medium Scale Sporting Events Organisations—The Perception of Negative Externalities by Host Residents. Sustainability
**2020**, 12, 2827. [Google Scholar] [CrossRef][Green Version] - Fedyk, W.; Sołtysik, M.; Olearnik, J.; Barwicka, K.; Mucha, A. How Overtourism Threatens Large Urban Areas: A Case Study of the City of Wrocław, Poland. Sustainability
**2020**, 12, 1783. [Google Scholar] [CrossRef][Green Version] - De la Calle-Vaquero, M.; García-hernández, M.; de Miguel, S.M. Urban planning regulations for tourism in the context of overtourism. Applications in historic centres. Sustainability
**2021**, 13, 70. [Google Scholar] [CrossRef] - Van der Borg, J.; Costa, P.; Gotti, G. Tourism in European heritage cities. Ann. Tour. Res.
**1996**, 23, 306–321. [Google Scholar] [CrossRef] - Neuts, B.; Nijkamp, P. Tourist crowding perception and acceptability in cities: An Applied Modelling Study on Bruges. Ann. Tour. Res.
**2012**, 39, 2133–2153. [Google Scholar] [CrossRef] - De Luca, G.; Dastgerdi, A.S.; Francini, C.; Liberatore, G. Sustainable cultural heritage planning and management of overtourism in art cities: Lessons from atlas world heritage. Sustainability
**2020**, 12, 3929. [Google Scholar] [CrossRef] - Garcia-Lépez, M.A.; Jofre-Monseny, J.; Martínez-Mazza, R.; Segú, M. Do short-term rental platforms affect housing markets? Evidence from Airbnb in Barcelona. J. Urban Econ.
**2020**, 119, 103278. [Google Scholar] [CrossRef] - Guttentag, D. Airbnb: Disruptive innovation and the rise of an informal tourism accommodation sector. Curr. Issues Tour.
**2015**, 18, 1192–1217. [Google Scholar] [CrossRef] - Amore, A.; Falk, M.; Adie, B. One visitor too many: Assessing the degree of overtourism in established European urban destinations. Int. J. Tour. Cities
**2020**, 6, 117–137. [Google Scholar] [CrossRef] - Zmyślony, P.; Leszczyński, G.; Waligóra, A.; Alejziak, W. The Sharing Economy and Sustainability of Urban Destinations in the (Over)tourism Context: The Social Capital Theory Perspective. Sustainability
**2020**, 12, 2310. [Google Scholar] [CrossRef][Green Version] - Hall, C.M. Constructing sustainable tourism development: The 2030 agenda and the managerial ecology of sustainable tourism. J. Sustain. Tour.
**2019**, 27, 1044–1060. [Google Scholar] [CrossRef] - Capocchi, A.; Vallone, C.; Pierotti, M.; Amaduzzi, A. Overtourism: A Literature Review to Assess Implications and Future Perspectives. Sustainability
**2019**, 11, 3303. [Google Scholar] [CrossRef][Green Version] - Fan, D.X.; Liu, A.; Qiu, R.T. Revisiting the relationship between host attitudes and tourism development: A utility maximization approach. Tour. Econ.
**2019**, 25, 171–188. [Google Scholar] [CrossRef] - Bimonte, S.; Punzo, L.F. The evolutionary game between tourist and resident populations and Tourist Carrying Capacity. Int. J. Technol. Glob.
**2007**, 3, 73. [Google Scholar] [CrossRef] - Vinogradov, E.; Leick, B.; Kivedal, B.K. An agent-based modelling approach to housing market regulations and Airbnb-induced tourism. Tour. Manag.
**2020**, 77, 104004. [Google Scholar] [CrossRef] - Cheng, F.; Su, F.; Chen, M.; Wang, Q.; Jiang, H.; Wang, X. An evolving assessment model for environmental carrying capacity: A case study of coral reef islands. J. Environ. Manag.
**2019**, 233, 543–552. [Google Scholar] [CrossRef] - Weibull, J.W. Evolutionary Game Theory; MIT Press: Cambridge, MA, USA, 1997. [Google Scholar]
- Smith, J.M. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982. [Google Scholar] [CrossRef][Green Version]
- Kreps, D.; Press, O.U. Game Theory and Economic Modelling; Clarendon Lectures in Economics; Oxford University Press: Oxford, UK, 1990. [Google Scholar]
- Batty, S.E. Game-Theoretic Approaches to Urban Planning and Design. Environ. Plan. B Plan. Des.
**1977**, 4, 211–239. [Google Scholar] [CrossRef] - He, P.; He, Y.; Xu, F. Evolutionary analysis of sustainable tourism. Ann. Tour. Res.
**2018**, 69, 76–89. [Google Scholar] [CrossRef] - Sandholm, W.H. Population Games and Evolutionary Dynamics; MIT Press: Cambridge, MA, USA, 2010. [Google Scholar]
- Volgger, M.; Taplin, R.; Pforr, C. The evolution of “Airbnb-tourism”: Demand-side dynamics around international use of peer-to-peer accommodation in Australia. Ann. Tour. Res.
**2019**, 75, 322–337. [Google Scholar] [CrossRef] - Bertocchi, D.; Visentin, F. ‘The Overwhelmed City’: Physical and Social Over-Capacities of Global Tourism in Venice. Sustainability
**2019**, 11, 6937. [Google Scholar] [CrossRef][Green Version] - Città di Venezia–Servizio Statistica e Ricerca: Popolazione: Dati e Studi. 2020. Available online: https://www.comune.venezia.it/it/content/statistica-statistiche-popolazione-0 (accessed on 31 March 2021).
- Cristiano, S.; Gonella, F. ‘Kill Venice’: A systems thinking conceptualisation of urban life, economy, and resilience in tourist cities. Palgrave Commun.
**2020**, 7, 1–13. [Google Scholar] [CrossRef] - Bertocchi, D.; Camatti, N.; Giove, S.; van der Borg, J. Venice and Overtourism: Simulating Sustainable Development Scenarios through a Tourism Carrying Capacity Model. Sustainability
**2020**, 12, 512. [Google Scholar] [CrossRef][Green Version] - Maynard Smith, J.; Price, G.R. The Logic of Animal Conflict. Nature
**1973**, 246, 15–18. [Google Scholar] [CrossRef] - Friedman, D. On economic applications of evolutionary game theory. J. Evol. Econ.
**1998**, 8, 15–43. [Google Scholar] [CrossRef][Green Version] - Nash, J.F. Equilibrium points in n-person games. Proc. Natl. Acad. Sci. USA
**1950**, 36, 48–49. [Google Scholar] [CrossRef][Green Version] - Nash, J. Non-Cooperative Games. Ann. Math.
**1951**, 54, 286–295. [Google Scholar] [CrossRef] - Simmonds, D.; Waddell, P.; Wegener, M. Equilibrium versus Dynamics in Urban Modelling. Environ. Plan. Plan. B Des.
**2013**, 40, 1051–1070. [Google Scholar] [CrossRef] - Nash, J. Non-Cooperative Games. Unpublished Ph.D. Thesis, University of Princeton, Princeton, NJ, USA, 1950. [Google Scholar]
- Sandholm, W.H.; Dokumaci, E.; Franchetti, F. Dynamo: Diagrams for Evolutionary Game Dynamics. Available online: https://www.ssc.wisc.edu/~whs/dynamo/ (accessed on 30 March 2021).
- Batty, M. Modelling Cities as Dynamic Systems. Nature
**1971**, 231, 425–428. [Google Scholar] [CrossRef] - Batty, M. Classifying urban models. Environ. Plan. Plan. Des.
**2016**, 43, 251–256. [Google Scholar] [CrossRef][Green Version] - Zeeman, E.C. Population dynamics from game theory. In Global Theory of Dynamical Systems; Nitecki, Z., Robinson, C., Eds.; Springer: Berlin/Heidelberg, Germany, 1980; pp. 471–497. [Google Scholar]
- Programme, U.N.H.S. Unpacking the Value of Sustainable Urbanization. In World Cities Report 2020; United Nations: New York, NY, USA, 2020; pp. 43–74. Available online: https://www.un-ilibrary.org/content/books/9789210054386c006 (accessed on 30 March 2021).
- Barron, K.; Kung, E.; Proserpio, D. The Sharing Economy and Housing Affordability: Evidence from Airbnb. In Proceedings of the 2018 ACM Conference on Economics and Computation, Ithaca, NY, USA, 18–22 June 2018; Association for Computing Machinery: New York, NY, USA, 2018; p. 5. [Google Scholar] [CrossRef]
- Taylor, P.D.; Jonker, L.B. Evolutionary stable strategies and game dynamics. Math. Biosci.
**1978**, 40, 145–156. [Google Scholar] [CrossRef] - Izquierdo, L.R.; Izquierdo, S.S.; Sandholm, W.H. EvoDyn-3s: A Mathematica computable document to analyze evolutionary dynamics in 3-strategy games. SoftwareX
**2018**, 7, 226–233. [Google Scholar] [CrossRef] - Núñez-Tabales, J.M.; Solano-Sánchez, M.Á.; y-López-del Río, L.C. Ten Years of Airbnb Phenomenon Research: A Bibliometric Approach (2010–2019). Sustainability
**2020**, 12, 6205. [Google Scholar] [CrossRef] - Hofbauer, J.; Sigmund, K. Evolutionary Games and Population Dynamics; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar] [CrossRef][Green Version]
- Häggström, O. Finite Markov Chains and Algorithmic Applications; London Mathematical Society Student Texts, Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]

**Figure 1.**The evolution of $({x}_{r}\left(t\right),{x}_{v}\left(t\right),{x}_{a}\left(t\right))$ for the historic center of Venice with $t\in [2001,2020]$. The three graphical scales are different and represent the interpolated data. While both uses–residential and vacant–decreased, the number of units transformed into Airbnb grew, as residential and vacant units were being transformed into tourist accommodation.

**Figure 2.**The geometrical representation of dynamics and the equilibrium states without random choice probability $\mu =0.0$. The three vertices represent pure populations, composed only by one type of use. The center of the triangle represents a population composed equally by residential, vacant and Airbnb housing units. The arrows indicate the direction towards which the distribution of the three populations is evolving till reaching an equilibrium point. The gradient blue–red color scale specifies the speed according to which the populations approximate the equilibrium points.

**Figure 3.**The dynamics on the simplex when the probability of random choice is $\mu =0.154$. This figure is indicative of the fact that the probability of random choice is a determinant of including the strategy of vacancy in the configuration of the equilibrium; otherwise this strategy tends to disappear.

**Figure 4.**The $\mu =0.154$ dynamics for the simplex subset ${x}_{r}\ge 0.85$. The real trajectory of the $({x}_{r}\left(t\right),{x}_{v}\left(t\right),{x}_{a}\left(t\right))$ dynamics for the city of Venice from 2008 to 2020 is represented in red and the theoretical curves are represented in black. This is the scaled up version of the upper part of the triangle in Figure 3. The real trajectory, based on the data represented in Figure 1, corresponds well to the one of the model. According to the model, the process of transformation of housing units into Airbnbs has not been yet completed and these will continue to occur till representing nearly the 10% of the housing units in Venice.

**Figure 5.**The population distribution $({\overline{x}}_{r},{\overline{x}}_{v},{\overline{x}}_{a})$ according to different values of the $\mu $ random choice parameter. The x-axis indicates the randomness rate affecting the choice of the agents to transform the units. In case the choice is based on merely rational choices; the randomness rate is zero. It is important to stress that when this rate overcomes the value of 0.16, nearly all residential units are being transformed into Airbnbs. When the randomness rate reaches 1 (100% of the population choosing based on random criteria) the distribution shows a city equally occupied by residential, vacant and Airbnb uses.

${\mathit{x}}_{\mathit{r}}$ | ${\mathit{x}}_{\mathit{v}}$ | ${\mathit{x}}_{\mathit{a}}$ | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ |
---|---|---|---|---|

0.0167 | 0.0512 | 0.932 | −2.95 | −1.03 |

0.839 | 0.0259 | 0.135 | −1.99 | 0.142 |

0.883 | 0.0236 | 0.0938 | −2.19 | −0.149 |

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**MDPI and ACS Style**

Arbara, S.; D’Autilia, R. A Population Game Model for the Expansion of Airbnb in the City of Venice. *Sustainability* **2021**, *13*, 3829.
https://doi.org/10.3390/su13073829

**AMA Style**

Arbara S, D’Autilia R. A Population Game Model for the Expansion of Airbnb in the City of Venice. *Sustainability*. 2021; 13(7):3829.
https://doi.org/10.3390/su13073829

**Chicago/Turabian Style**

Arbara, Sophia, and Roberto D’Autilia. 2021. "A Population Game Model for the Expansion of Airbnb in the City of Venice" *Sustainability* 13, no. 7: 3829.
https://doi.org/10.3390/su13073829