Some variations of the Hotelling game Marco Scarsini 1 1 LUISS based - PowerPoint PPT Presentation

Model Equilibria Infinite games Conclusion Poisson games Types Some variations of the Hotelling game Marco Scarsini 1 1 LUISS based on joint work with Matías Nuñez and Gaëtan Fournier . ADGO, Santiago, Chile, January 2016

Model Equilibria Infinite games Conclusion Poisson games Types Hotelling • In the classical Hotelling model consumers are distributed uniformly on the interval r 0 , 1 s . • Retailers can choose any location in r 0 , 1 s where to set up a shop. • Consumers shop at one of the closest retailers. • This defines a (one-shot) game where the players are the retailers, the action set is r 0 , 1 s and the payoff is the amount of consumers that a retailer attracts. • Depending on the number of retailers, equilibria may or may not exist and may or may not be unique.

Model Equilibria Infinite games Conclusion Poisson games Types Generalizations • Various generalizations have been considered. • The space where consumers are distributed can be different from r 0 , 1 s . • The distribution could be non-uniform. • Retailers could compete not only on location but also on prices.

Model Equilibria Infinite games Conclusion Poisson games Types The model • Consumers are distributed according to a measure λ on a compact Borel metric space p S , d q . • S could be a compact subset of R 2 or a compact subset of a 2-sphere, but it could also be a (properly metrized) network. • A finite set N n : “ t 1 , . . . , n u of retailers have to decide where to set shop, knowing that each consumer chooses one of his closest retailers. • Each retailer wants to maximize her market share. • The action set of each retailer is a finite subset of S . For instance retailers can set shop only in one of the existing shopping malls in town.

Model Equilibria Infinite games Conclusion Poisson games Types Tessellation • K “ t 1 , . . . , k u • X K : “ t x 1 , . . . , x k u Ă S is a finite collection of points in S . These are the points where retailers can open a store. • For every J Ă K call X J : “ t x j : j P J u . • V p X J q is the Voronoi tessellation of S induced by X J . • For each x j P X J the Voronoi cell of x j is v J p x j q : “ t y P S : d p y , x j q ď d p y , x ℓ q for all x ℓ P X J u . • The cell v J p x j q contains all points whose distance from x j is not larger than the distance from the other points in X J . • Call V p X J q : “ p v J p x j qq j P J the set of all Voronoi cells v J p x j q . • For J Ă L Ă K we have v J p x j q Ą v L p x j q for every j P J .

Model Equilibria Infinite games Conclusion Poisson games Types x 1 x 3 x 2 x 10 x 7 x 4 x 6 x 5 x 8 x 9 Figure: X K Ă r 0 , 1 s 2 , K “ t 1 , . . . , 10 u .

Model Equilibria Infinite games Conclusion Poisson games Types Figure: V p X J q , J “ t 1 , 2 u .

Model Equilibria Infinite games Conclusion Poisson games Types Figure: V p X J q , J “ t 3 , 4 , 5 u .

Model Equilibria Infinite games Conclusion Poisson games Types Figure: V p X J q , J “ t 3 , 4 , 5 , 6 u .

Model Equilibria Infinite games Conclusion Poisson games Types Figure: V p X J q , J “ t 1 , 2 , 7 , 8 , 9 , 10 u .

Model Equilibria Infinite games Conclusion Poisson games Types Figure: V p X J q , J “ K .

Model Equilibria Infinite games Conclusion Poisson games Types • λ p v J p x j qq is the mass of consumers who are weakly closer to x j than to any other point in X J . • If price is homogeneous, these consumers will prefer to shop at location x j rather than at other locations in X J . • Consumers that belong to r different Voronoi cells v J p x j 1 q , . . . , v J p x j r q , are equally likely to shop at any of the locations x j 1 , . . . , x j r . • λ is absolutely continuous with respect to the Lebesgue measure on this space and λ p v K p x j qq ą 0 for all x j P X K . • More general situations can be considered but they require more care in handling ties.

Model Equilibria Infinite games Conclusion Poisson games Types The game • N n : “ t 1 , . . . , n u is the set of players. • a i P X K is the action of player i . • a : “ p a i q i P N n is the profile of actions. • a ´ i : “ p a h q h P N n zt i u is the profile of actions of all the players different from i . • a “ p a i , a ´ i q . • a : “ p a 1 , . . . , a n q « X J if for all locations x j P X J there exists a player i P N n such that a i “ x j and for all players i P N n there exists a location x j P X J such that a i “ x j .

Model Equilibria Infinite games Conclusion Poisson games Types The payoff • The payoff of player i is 1 ÿ u i p a q “ λ p v J p a i qq 1 p a « X J q , card t h : a h “ a i u J Ă K i.e., the measure of the consumers that are closer to the location that she chooses than to any other location chosen by any other player, divided by the number of retailers that choose the same action as i . • Some locations may not be chosen by any player, this is why, for every J Ă K , we have to consider the Voronoi tessellation V p X J q with a « X J rather than the finer tessellation V p X K q .

Model Equilibria Infinite games Conclusion Poisson games Types Example S “ r 0 , 1 s , λ is the Lebesgue measure on r 0 , 1 s , X K “ t 0 , 1 { 2 , 1 u . $ r 0 , 1 s if X J “ t 0 u , ’ & v J p 0 q “ r 0 , 1 { 2 s if X J “ t 0 , 1 u , ’ r 0 , 1 { 4 s if X J “ X K or X J “ t 0 , 1 { 2 u . % $ r 0 , 1 s if X J “ t 1 { 2 u , ’ ’ ’ ’ r 1 { 4 , 1 s if X J “ t 0 , 1 { 2 u & v J p 1 { 2 q “ r 0 , 3 { 4 s if X J “ t 1 { 2 , 1 u , ’ ’ ’ ’ r 1 { 4 , 3 { 4 s if X J “ X K . % $ r 0 , 1 s if X J “ t 1 u , ’ & v J p 1 q “ r 1 { 2 , 1 s if X J “ t 0 , 1 u , ’ r 3 { 4 , 1 s if X J “ X K or X J “ t 1 { 2 , 1 u . %

Model Equilibria Infinite games Conclusion Poisson games Types Example, continued $ 1 if X J “ t 0 u , ’ & λ p v J p 0 qq “ 1 { 2 if X J “ t 0 , 1 u , ’ 1 { 4 if X J “ X K or X J “ t 0 , 1 { 2 u . % $ 1 if X J “ t 1 { 2 u , ’ & λ p v J p 1 { 2 qq “ 3 { 4 if X J “ t 0 , 1 { 2 u or X J “ t 1 { 2 , 1 u , ’ 1 { 2 if X J “ X K . % $ 1 if X J “ t 1 u , ’ & λ p v J p 1 qq “ 1 { 2 if X J “ t 0 , 1 u , ’ 1 { 4 if X J “ X K or X J “ t 1 { 2 , 1 u . %

Model Equilibria Infinite games Conclusion Poisson games Types Example, continued Therefore the payoff for player i , if she chooses location 0 when the rest of the players’ pure actions are a ´ i is 1 u i p 0 , a ´ i q “ card t h : a h “ a i u φ p a ´ i q , where $ if a « t 0 u , 1 ’ & 1 φ p a ´ i q “ if a « t 0 , 1 u , 2 ’ 1 if a « X K or a « t 0 , 1 { 2 u . % 4 The payoffs when she chooses either 1 { 2 or 1 can be similarly computed.

Model Equilibria Infinite games Conclusion Poisson games Types 0 0 . 5 1 v J p 0 q , X J “ t 0 u v J p 0 q , X J “ t 0 , 1 u v J p 0 q , X J “ t 0 , 0 . 5 , 1 u v J p 0 q , X J “ t 0 , 0 . 5 u

Model Equilibria Infinite games Conclusion Poisson games Types • We have defined a game G n “ x S , λ, N n , X K , p u i qy . • With an abuse of notation, we use the same symbol G n for the mixed extension of the game, where, for a mixed strategy profile σ “ p σ 1 , . . . , σ n q , the expected payoff of player i is ÿ ÿ U i p σ q “ u i p a q σ 1 p a 1 q . . . σ n p a n q . . . . a 1 P X K a n P X K

Model Equilibria Infinite games Conclusion Poisson games Types Equilibria We consider a sequence t G n u of games, all of which have the same parameters S , λ, X K . Example (A game without pure equilibria) G n with n “ 3, S “ r 0 , 1 s , λ the Lebesgue measure, and X K “ t i { 100 : i “ 0 , . . . , 100 u . The game does not have a pure equilibrium. This case is very similar to the classical Hotelling game on r 0 , 1 s with three players.

Model Equilibria Infinite games Conclusion Poisson games Types Dominated strategies Example (Weakly dominated locations) Consider a game G n with n “ 2, S “ r 0 , 1 s , λ the Lebesgue measure, and X K “ t 0 . 45 , 0 . 5 , 0 . 55 u . Then both 0 . 45 and 0 . 55 are weakly dominated by 0 . 5. Proposition Consider a sequence of games t G n u n P N . There exists ¯ n such that for all n ě ¯ n no location in X K is weakly dominated.

Model Equilibria Infinite games Conclusion Poisson games Types Pure equilibria Theorem Consider a sequence of games t G n u n P N . There exists ¯ n such n the game G n admits a pure equilibrium a ˚ . that for all n ě ¯ Moreover, for all n ě ¯ n, any pure equilibrium is such that for every j , ℓ P K n j p a ˚ q λ p v K p x ℓ qq ď n j p a ˚ q ` 1 n ℓ p a ˚ q ` 1 ď λ p v K p x j qq . n ℓ p a ˚ q

Model Equilibria Infinite games Conclusion Poisson games Types Mixed equilibria Theorem For every n P N the game G n admits a symmetric mixed equilibrium γ p n q “ p γ p n q , . . . , γ p n q q such that n Ñ8 γ p n q “ γ, lim with γ p x j q “ λ p v K p x j qq for all j P K . λ p S q