Agent-Based Modeling and Simulation for Two-Dimensional Spatial Competition Masashi Miura Hidetoshi Shiroishi Tottori University 2018.06.20. KES-AMSTA-18 

Abstract • Applying agent-based approach to spatial competition. • Constructing the agent-based model and simulation for 2-dementinal Hotelling model Contents Background Agent-based modeling Simulation results Summary & Future 

Hotelling Model (Hotelling, 1929)  Two shops decide their location in the game situation.  Consumers are distributed uniformly in the line.  A consumer decides to purchase a product from the shop closer to him. ShopＡ ShopＢ Consumers choose the closer shop 

Principle of minimum differentiation share ofＡ share of B share ofＡ share of B share ofＡ share of B share ofＡ share of B Ａ Ａ Ａ Ａ B B B B 

Hotelling can be used to explain:  Retail shops who are located relatively close to each other.  Similar products in the same category.  Similar policies by two political parties. 

Introducing “Price” The cost for -th consumer: • ShopＡ: • ShopＢ: Choose the shop with the lower cost price: price: d′Aspremont et al.（1979） 

Extended Models Shop numbers – 2 shops → 3 shops（Eaton and Lipsey, 1975） Spatial competition +α – Spatial/Price competition （d'Aspremont, Gabszewicz & Thisse, 1979） – Spatial/Quantity competition （Anderson & Neven, 1991） Dimension – Two dimension（Tabuchi, 1994） 

x market space Shop 1 location︓(x 2, y 2 ) price ︓ p 2 Shop 1 location︓(x 1, y 1 ) price ︓ p 1 dxdy y y t x x t p y y t x x t p y x D            } ) ( ) ( ) ( ) ( ) , {( 2 2 2 2 2 2 1 2 1 1 D p1 1   ) 1 ( 2 2 D p    , Market share: Profit:  Consumers are distributed continuously and uniformly  A consumer choose the shop with the lower sum of the price and the transportation cost  The transportation cost is: Tabuchi Model ① 

Tabuchi Model ② Equilibrium state :  Maximum differentiation in the major axis  Minimum differentiation in the minor axis 

By introducing agent-based approach into spatial competition  consumers are assumed to be distributed continuously and uniformly. Analytical approaches Agent-based approach  Enable to deal with the discrete and non-uniform consumer distributions  Enable to describe situations closer to the actual and more complex. 

Agent-based Modeling Behaviors of Consumer Agents -th consumer ) , ( c i c i y x Purchase a product from the shop with lower cost. Total cost = Price ＋ Transportation cost = ＋ ︓Transportation cost function Shop B location︓(x B, y B ) price ︓ p B Shop A location︓(x A, y A ) price ︓ p A 

Agent-based Modeling Behaviors of Shop Agents  choose the location and set the price  act strategically to maximize each profit function Profit = Price × Number of consumers = N A consumers choose A N B consumers choose B Shop A location︓(x A, y A ) price ︓ p A Shop B location︓(x B, y B ) price ︓ p B 

Agent-based Modeling Backward Induction Method A pair of location A pair of price Solution for the sub game After finding the optimal price for each location pair, choose the location. 

 Assuming shops alternately update each price  Narrowing down the candidates for the update price to the reservation prices for each consumer agent. Agent-based Modeling Ideas for Practical Computation 2 1 2 1 2 2 2 2 2 1 ) ( ) ( ) ( ) ( y y t x x t y y t x x t p p c i c i c i c i i          

Candidates for the update price: the reservation prices : , ,…, = the upper limit price for -th consumer then he choose shop A = + − Agent-based Modeling Reservation Price -th consumer ) , ( c i c i y x Shop B location︓(x B, y B ) price ︓ p B Shop A location︓(x A, y A ) price ︓ p A 

If the shop agent chooses some price which is not equal to the reservation price for any consumer agent, then the shop can increase the price up to the nearest reservation price without changing the number of consumers Price can be raised up to ̂ without change of number of cunsumer Agent-based Modeling Reservation Prices are candidates 

Agent-based Modeling Flow of pricing stage 

Agent-based Modeling Location choice Shop B location︓m –th candidate location Candidate locations are set in the market space in a lattice Shop A location︓n –th candidate location Shop A chooses the location while fixing the location of shop B. For all candidate locations, the process of pricing stage is carried out. 

Agent-based Modeling Flow of the entire process of competition 

Graphical Simulator  Constructed with artisoc  graphical interface with which you can see results of the simulation as animation  graphical control panel to set the parameters 

Simulation Result ①: Model Validation Price location number of consumers 200 consumer distribution uniformly in the line transportation cost corresponding to the model of (d'Aspremont 1979) 

Simulation Result ②: Model Validation Price location number of consumers 200 consumer distribution in the grid pattern transportation cost corresponding to the model of (Tabuchi 1994) 

Simulation Result ③ Price location number of consumers 200 consumer distribution in the grid pattern transportation cost 

Simulation Result ④ Price location number of consumers 200 consumer distribution uniformly at random transportation cost 

Simulation Result ⑤ Price location number of consumers 200 consumer distribution randomly and condensed transportation cost 

Simulation Result ③ Price location number of consumers 200 consumer distribution biased to both sides transportation cost 

Summary  We constructed the agent-based model for two-dimensional spatial competition.  It can describe the discrete and non- uniform consumer distributions.  We developed the graphical simulator for that model and showed simulation results in some cases.  The model validation was achieved by comparing the results of our simulation with those of the previous researches. 

Future Works  Investigating spatial competition under different and complex conditions using our model and simulator.  Situations in which decision making of consumers (and shops) are affected by the results of their own previous behavior. e.g) the situation with the switching