A population game model for the expansion of Airbnb in the city of Venice

: The emergence of Airbnb along with an increase in urban tourism has intensiﬁed the 1 pressure on urban areas while adding a new dimension in the dynamics of the housing distribution 2 especially in historic cities. These dynamics affect both local economies and alter signiﬁcantly the 3 characteristics of urban space arising the necessity to create not only policies that foster sustainable 4 tourism development but also to advance urban models that explore the relation between Airbnb 5 and the traditional rental and accommodation sector. Through the case of Venice, the present study 6 sheds light on the potential evolution of Airbnb housing in comparison to the traditional rental and homeowner market. In particular, it seeks to understand whether a potential equilibrium between 8 these uses exists and if so, at which point in regards to this equilibrium the historic center of Venice 9 is now. To tackle this question, methods deriving from the ﬁeld of game theory and speciﬁcally 10 evolutionary game theory are used. With the agents (players) being the housing units, the designed 11 theoretical model explores the population dynamics of the housing units in Venice given the three 12 options of homeownership or long-term rental (residential), short term rental over Airbnb (airbnb) or 13 no use (vacant). The ﬁndings of our theoretical population game model are validated and discussed 14 against a dataset describing the use patterns in the city of Venice during the past 20 years. A 15 veriﬁcation of the outcome through further case studies could eventually provide insights on future 16 behavior of tourism pressure in historic urban areas.

yet the literature in the field of urban management and planning is scarce, with the existing references 48 focusing on the decision-making process amongst different actors in urban design [12]. Through a 49 symmetric game given in normal form by a 3 × 3 payoff matrix, we explore the potential consequences 50 for a population to transform their housing units over peer to peer accommodation platforms, to make 51 it available for rent on a local rental platform (or ownership occupancy) or to leave it vacant, given 52 the choices of the other players. It is important to stress at this point that while in evolutionary game 53 models the agents range from biological organisms to economic institutions [13], in the given case 54 the agents are the housing units per se. And as Airbnb is the leading marketplace of peer to peer 55 accommodation for those seeking short-term housing options, the tourist pressure is measured in 56 terms of houses transformed into Airbnb. 57 The proposed model focuses on three possible choices for each agent: to convert the housing 58 unit into an Airbnb rental (a), to either inhabit or rent it on the traditional rental market (r) or leave 59 it vacant (v). Options such as hotel accommodation and bed and breakfast are not taken under the 60 consideration. This is based on the hypothesis that Airbnb is an emerging sector that could follow to this increase. 86 The paper is organized as follows; first, a brief introduction in regards to the concepts of 87 evolutionary game theory is provided and subsequently the characteristics of the urban population 88 model are described. In Sec.4 the dataset used to elaborate the case study model is explained. In 89 chapter 5 the results of the proposed model are analysed and discussed against the dataset describing 90 the use patterns in the city of Venice during the past 20 years. Finally, we consider limitations and 91 future prospects of the game theory approach to urban planning. 92 2. The evolutionary game theory 93 As introduced above, the article explores the possibility of exploiting the population game and 94 the evolutionary dynamics approach in urban management and planning. In particular, a model based 95 on the framework of evolutionary game theory is developed, a field of game theory that originated in 96 biological contexts and has become of interest amongst a broader spectrum of social and economic 97 sciences [18] [19]. The choice derives from the fact that "evolution" is not limited to biological systems 98 but can be a powerful paradigm to understand social systems [9] or complex systems such as the urban 99 context. Moreover, evolutionary game theory has become an important tool to model the interaction from economic models of evolutionary games is that here the payoff of each strategy, as well as the 106 urban quality, is not considered in mere economic terms, but is the result of a synthetic element which 107 concerns the morphological and environmental aspects of urban space, its intangible qualities (social 108 and historic values) as well as the economic values of these.

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The payoff is a quantity determined not only on the agent's strategy, but it depends on the 110 strategies chosen by the other agents. In other words, the game is between one agent and a 111 population.In our case a housing unit (the agent) can adopt a strategy amongst "resident", "vacant" and 112 "airbnb" (r, v, a) but the payoff of this decision depends on how the different strategies are distributed 113 on the overall population. Once the sets of strategies and the payoff of the agents are defined, we 114 analyse the agent's possible collective behaviors.

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The dominant idea in game theory is the Nash equilibrium, a state in which no agent can change 116 strategy without decreasing their payoff, and each currently selected strategy is the best response to 117 the strategy of the other players [20] [21]. In general, the Nash equilibrium is neither unique, nor a 118 Pareto optimum, and always exists for strategies played with a given probability (mixed strategies) [9].  In urban modelling, the need to switch from static models towards dynamic ones has often been 125 stressed, and the population game approach can add to the exploration of dynamic urban models where the housing units act as agents that choose amongst the three available strategies, can become a 128 useful way to understand the urban space's reaction in pressing issues and obtain indicators for future 129 scenarios. 131 In order to describe the dynamic evolution of a system of agents in terms of population game 132 it is necessary to have a large population and at the same time assume that the action of each 133 individual agent has a very small impact on the distribution of strategies in the overall population. The 134 participants in the game are in our case the proportions between inhabited, vacant and Airbnb housing 135 units which form part of the total population of Venice's units, here represented as x r (t), x v (t) and 136 x a (t) respectively. In general, these quantities tend to change over time giving rise to a state dynamics.

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The state x(t) of the population of Venice's housing units is therefore given by the distribution vector of the three possible uses of these units.
A population game is the set of functions that describe the temporal evolution of this state. In the 138 given case these functions describe a housing unit's transformation of use that adapts to the landscape 139 generated by the distribution of other uses, and, unlike the "classic" formulation of game theory, they 140 allow us to study the system far from the equilibrium. It is not known if the pressure of Airbnb in

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Venice has reached a state of equilibrium or not, but we do know that understanding the dynamics 142 beyond equilibrium is decisive from an urbanistic point of view as it will allow us to design urban 143 policy interventions which would shift the state of equilibrium.

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The questions that can be answered by analyzing the population game is whether there are equilibrium states for our system based on the three urban uses (r, v, a) and what the current state of the city is in regards to these equilibria. This information is summarized by the vector field determined by the population game equations on the simplex (2) which represents the set of all possible states of the considered population.

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Each agent has a finite strategy set S = (r, v, a) consisting of the following three options: to rent 146 the housing unit over the traditional rental market or leave it available for homeowner accommodation 147 (r), to leave it vacant (v) or to rent over the shared rental platform of Airbnb (a).

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The payoff of an agent is given by the matrix π rr π rv π ra π vr π vv π va π ar π av π aa    where π ij is the payoff of the agent adopting the strategy i when the opponent plays j. The matrix Π 149 represents a symmetrical game with two players and three strategies and describes the payoff of a 150 strategy depending on the strategy chosen by its opponent. It is supposed that π ji is the payoff of the 151 opponent so that the game is symmetric.

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The payoff of a strategy depends on what other agents do, and every agent does not need to have 153 a clear vision nor a global understanding of the game. In fact, it is assumed that an agent is chosen 154 randomly from the population and plays with an opponent always chosen randomly as well. In other 155 words, a housing unit decides to adopt the state (r, v, a) but the payoff of its choice depends on how 156 they are distributed (x r , x v , x a ) in the overall population.
Airbnb yet not socially sustainable on a long term, as this would lead to a museumization putting 164 at risk the viability of the city. The values of the elements of the matrix Π are therefore a balance 165 between economic, social, environmental and intangible payoffs. It is important to mention that the 166 current payoff matrix is designed taking into account urban areas where tourism is considered to be 167 one of the core activities of the city. If an agent is currently playing strategy i ∈ S and the opponent 168 is also playing the same strategy, then it can be assumed that π rr = π vv = π aa = 0, considering that 169 when the agents are using the same strategy their behavior does not affect the evolution of the overall 170 population. When the agent plays the strategy r against an agent playing v, the payoff π rv is positive 171 because we assume that in a touristic urban area empty units stimulate the expansion of r, but if the 172 opponent is a then the π ra is negative as in a touristic city where Airbnb is operating, the expansion 173 of r is hindered and eventually decreased. As stated in [29] "... while the total supply of housing is 174 not affected by the entry of Airbnb, Airbnb listings increase the supply of short-term rental units and 175 decrease the supply of long-term rental units".

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If an agent adopts the strategy v and the opponent the strategy r, the payoff π vr is negative, but bigger than π ra . This is based on the assumption that the strategy of a population of Airbnbs over an agent currently under residential use, tends to take over residential uses in a more "aggressive" way than the residential uses tend to take over the vacant ones hindering their expansion. However, if an agent is currently adopting the strategy v and the opponent chooses a, then the payoff π va is again negative, but bigger than π vr and π ra .
The hypothesis made here is that although use patterns of Airbnb would hinder the expansion of 177 vacant units, it is less likely that a vacant unit will be directly transformed into an Airbnb but more 178 likely to shift to homeownership or long-term rental before becoming an Airbnb. As per [30] it is more 179 probable that Airbnb will shift to long-term rentals than consider to sell a housing unit which could 180 mean shifting to any of the three strategies. This is reflected in the payoff of our strategies with π ar 181 being the most "aggressive" one.

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If the agent is playing a and the opponent plays r, the π ar is negative as in a population of r an  192 Since the agents are the housing units of Venice distributed in the three classes of strategies S = {r, v, a} and the total number of houses is large enough then x i ≥ 0, with i ∈ S, can be considered in an equivalent way as the proportion of houses that adopt a given strategy or the probability that the strategy is adopted and therefore x r (t) + x v (t) + x a (t) = 1 for all t. If this point of view is considered, then, making time dependence implicit for notational simplicity, the i strategy adopted by the population in the state x has the average payoff and if its average value is taking over all the strategies, the result is the average payoff of the entire population If it is assumed that a strategy spreads proportionally to its convenience Π i − Π, the replication equation can be written [31] which describes the evolution over time of the state of the population. Finally, the research introduces a coefficient of randomness µ ∈ [0, 1] which represents the probability that an agent will choose his strategy randomly. The equation (7) then becomes [13] [32]

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The dataset considered in our analysis focuses on the timeframe of the past 20 years. This makes it possible to observe the impact and evolution pattern of the Airbnb shared accommodation in the city of Venice as the platform started operating only after 2008 [33]. In order to study the model and the population's selections in regards to the three strategies (r, v, a), the number r u of housing units inhabited by residents (homeowners or tenants), the vacant housing units v u and the housing units rented over the Airbnb platform a u are measured. The number N u represents the non-vacant housing units, which include units occupied by both residents r u and Airbnb tenants a u . T u refers to the total number of available housing units in the historic center of Venice The non-vacant housing units N u is the sum between residentially occupied ones and the Airbnb ones N u = a u + r u so that we have When elaborating the data following assumption was made; the housing units offering a room 225 and not the entire unit over Airbnb were calculated under the Airbnb category a u and not under the 226 category of residential units r u . This assumption was made on the basis of "testing" the scenario where 227 a maximum number of Airbnb rentals is considered. If a housing unit appeared more than once in 228 Airbnb, meaning that the owner would rent the rooms into different visitors, these duplicates were 229 removed. That way every housing unit in Airbnb appears only once in our dataset independent of the 230 various rooms that might be available to different groups of visitors.

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The equation (8) determines the time evolution of the urban state vector x(t). The dynamic we have used is the replication dynamics, which is a sort of "survival of the fittest" [31] [34]. Replication dynamics do not allow for new strategies to emerge. However, in the given case the a strategy appears for the first time in 2008. Therefore, it can be assumed that the replication dynamic is valid from 2008 and onwards analysing how the a strategy invaded the population previously formed only by r and v.

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The dynamics of Airbnb in Venice seem to be well described by this class of models, as can 239 be observed below. Based on the interaction amongst the agent and the population as well as the 240 possibility of random choice, the agents are able to reevaluate their strategy. To test the evolutionary 241 game model, the notebook EvoDyn-3s, designed to analyse the dynamics and equilibria of 3 × 3 matrix 242 games is used [32]. Starting from µ = 0, the value of µ is being progressively increased until obtaining the minimum 255 distance between the theoretical curves and the data. The meaning of µ is that behind the dynamics of 256 the strategies (r, v, a) there is a complex social and urban dynamic that can give rise to non-rational, or 257 at least non-optimal behaviors.

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We found that for µ = 0.154, the solution starting on the segment r − v with initial condition   The point (x r = 0.839, x v = 0.0259, x a = 0.135) has the second eigenvalue with positive real part 267 and it is not stable. The closest theoretical equilibrium to the state of the population in 2020, which 268 is x 2020 = (x r = 0.935, x v = 0.018, x a = 0.045), is the point (x r = 0.883, x v = 0.0236, x a = 0.0938).

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Therefore it seems that, in this model, the state of the housing units in Venice is converging towards 270 an equilibrium where less of 1% of the houses will be transformed into Airbnb and the 0.23% will be 271 uninhabited, as can be seen in Fig.4, which is an enlargement of Fig.3. A further equilibrium is at the if µ < 0.16 the city can "contain" the expansion of Airbnb.

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The µ parameter can be controlled by an urban planning policy that reduces the randomness rate 290 in strategic choices. This means that although the randomness is governed by personal situations that can lead to non-optimal choices, an urban management policy that "stabilizes" the quality of life of the 292 inhabitants of Venice can reduce the value of µ and consequently limit the tourist pressure. This means 293 that in order to control the growth of Airbnb, it is not necessary to impede such activities in urban 294 areas but to provide incentives that would support local inhabitants. Adding to that, finding ways to 295 minimize the factor of uncertainty for the agents choices could help mitigating phenomena such as the 296 uncontrolled expansion of Airbnb.

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The main purpose of this work is to explore the possibility to use some of the ideas of population 299 dynamics and game theory to model the urban dynamics in regards to the Airbnb pressure. Using the 300 houses as agents, it is possible to determine a dynamic that reproduces the qualitative behavior of the 301 real data and offers an overall picture of the possible states of equilibrium.

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The model remains rather abstract and could not be considered realistic in the sense that the 303 µ parameter should be calculated through a much more detailed dataset than the one we had the 304 opportunity to examine. However, the theoretical implications of this simple model are important as 305 they represent possible consequences of a competitive mechanism in urban dynamics that takes into 306 account not only the economic value of the housing units but also their urban use.

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In order to further explore its capacity, the proposed class of models could be compared with the