Large Spatial Competition

Abstract

We consider spatial competition when consumers are arbitrarily distributed on a compact metric space. Retailers can choose one of finitely many locations in this space. We focus on symmetric mixed equilibria which exist for any number of retailers. We prove that the distribution of retailers tends to agree with the distribution of the consumers when the number of competitors is large enough. The results are shown to be robust to the introduction of (1) randomness in the number of retailers and (2) different ability of the retailers to attract consumers.

Appendix: Proofs

1.1 Proofs of Sect. 3

The proof of Theorem 2 requires some preliminary results.

Lemma 1.

Consider a sequence of games $\{\mathcal{G}_{n}\}_{n\in \mathbb{N}}$. There exists $\bar{n}$ such that for all $n \geq \bar{ n}$, if $\boldsymbol{\gamma }^{(n)}$ is a symmetric equilibrium of $\mathcal{G}_{n}$, then $\boldsymbol{\gamma }^{(n)}$ is completely mixed, i.e.,

$$\displaystyle{ \gamma ^{(n)}(x_{ j}) > 0\quad \text{for all }x_{j} \in X_{K}. }$$

Proof.

Assume by contradiction that for every $n \in \mathbb{N}$ there exists some x _j  ∈ X _K and a symmetric equilibrium $\boldsymbol{\gamma }^{(n)}$ of $\mathcal{G}_{n}$ such that γ ⁽ⁿ⁾(x _j ) = 0. Given that λ(S) < ∞, we have that for all i ∈ N _n ,

$$\displaystyle{ U_{i}(\boldsymbol{\gamma }^{(n)}) = \frac{\lambda (S)} {n}. }$$

If player i deviates and plays the pure action a _i  = x _j , then she obtains a payoff

$$\displaystyle{ U_{i}(a_{i},\boldsymbol{\gamma }_{-i}^{(n)}) \geq \lambda (v_{ K}(x_{j})) > \frac{\lambda (S)} {n}, }$$

where the strict inequality holds for n large enough. This contradicts the assumption that $\boldsymbol{\gamma }^{(n)}$ is an equilibrium. □ 

Lemma 2.

Let (Y ₁ ,…,Y _k ) be a random vector distributed according to a multinomial distribution with parameters (n − 1;γ ₁ ⁽ⁿ⁾ ,…,γ _k ⁽ⁿ⁾ ), with δ < γ _j ⁽ⁿ⁾ < 1 −δ, for some 0 < δ < 1 and for all j ∈ K. Then

$$\displaystyle{ \lim _{n\rightarrow \infty }\frac{\mathbb{E}\left [ \frac{1} {Y _{j} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{j}))\mathbb{1}(Y _{h} = 0\text{ for }h\not\in J)\right ]} {\mathbb{E}\left [ \frac{1} {Y _{\ell} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{\ell}))\mathbb{1}(Y _{h} = 0\text{ for }h\not\in J)\right ]} = 1,\quad \text{for all }j,\ell\in K }$$

(11)

iff

$$\displaystyle{ \lim _{n\rightarrow \infty }\gamma _{j}^{(n)} =\gamma (x_{ j}) = \frac{\lambda (v_{K}(x_{j}))} {\lambda (S)} \quad \text{for all }j \in K. }$$

(12)

Proof.

Given j ∈ K, consider all J ⊂ K such that j ∈ J and the family $\mathcal{V}_{j}$ of all corresponding Voronoi tessellations V (X _J ). Call $\widetilde{V }_{j}$ the finest partition of S generated by $\mathcal{V}_{j}$, that is, the set of all possible intersections of cells v _J (x _j ) ∈ V (X _J ) for $V (X_{J}) \in \mathcal{V}_{j}$. It is clear that $v_{K}(x_{j}) \in \widetilde{ V }_{j}$.

For $A \in \widetilde{ V }_{j}$, call $\widetilde{V }_{j}(A)$ the class of all cells in $\widetilde{V }_{j}$ whose intersection with A is nonempty.

Then

$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\left [ \frac{1} {Y _{j} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{j}))\mathbb{1}(Y _{h} = 0\text{ for }h\not\in J)\right ] {}\\ & & \quad = \mathbb{E}\left [\frac{\lambda (v_{K}(x_{j}))} {Y _{j} + 1} \right ] {}\\ & & \quad \quad + \mathbb{E}\left [ \frac{1} {Y _{j} + 1}\sum _{A\in \widetilde{V }_{j}}\lambda (A)\mathbb{1}\left (Y _{h} = 0\text{ if }v_{K}(x_{j}) \cap A\neq \varnothing \right )\right ] {}\\ & & \quad \leq \mathbb{E}\left [\frac{\lambda (v_{K}(x_{j}))} {Y _{j} + 1} \right ] {}\\ & & \quad \quad +\sum _{A\in \widetilde{V }_{j}}\lambda (A)\mathbb{P}\left (Y _{h} = 0\text{ if }v_{K}(x_{j}) \cap A\neq \varnothing \right ) {}\\ & & = \mathbb{E}\left [\frac{\lambda (v_{K}(x_{j}))} {Y _{j} + 1} \right ] + o(1/n)\quad \text{for }n \rightarrow \infty, {}\\ \end{array}$$

since $\mathbb{P}(Y _{i} = 0) = (1 -\gamma _{i}^{(n)})^{n} = o(1/n)$ for n → ∞. Therefore

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\frac{\mathbb{E}\left [ \frac{1} {Y _{j} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{j}))\mathbb{1}(Y _{h} = 0\text{ for }h\not\in J)\right ]} {\mathbb{E}\left [ \frac{1} {Y _{\ell} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{\ell}))\mathbb{1}(Y _{h} = 0\text{ for }h\not\in J)\right ]} & =& \lim _{n\rightarrow \infty }\frac{\mathbb{E}\left [\frac{\lambda (v_{K}(x_{j}))} {Y _{j} + 1} \right ]} {\mathbb{E}\left [\frac{\lambda (v_{K}(x_{\ell}))} {Y _{\ell} + 1} \right ]} \\ & =& \lim _{n\rightarrow \infty }\frac{\lambda (v_{K}(x_{j}))} {\lambda (v_{K}(x_{\ell}))} \frac{\gamma _{\ell}^{(n)}} {\gamma _{j}^{(n)}} \\ & =& \frac{\lambda (v_{K}(x_{j}))} {\lambda (v_{K}(x_{\ell}))} \frac{\gamma (x_{\ell})} {\gamma (x_{j})} {}\end{array}$$

(13)

Given that ∑ _j = 1 ^k γ(x _j ) = 1, (13) holds if and only if (12) does. □ 

Proof (Proof of Theorem 2).

The game $\mathcal{G}_{n}$ is finite and symmetric, so it admits a symmetric mixed Nash equilibrium $\boldsymbol{\gamma }^{(n)} = (\gamma ^{(n)},\ldots,\gamma ^{(n)})$. Then, given Lemma 1, for all j, ℓ ∈ K,

$$\displaystyle{ U_{i}(x_{j},\boldsymbol{\gamma }_{-i}^{(n)}) = U_{ i}(x_{\ell},\boldsymbol{\gamma }_{-i}^{(n)}). }$$

(14)

Using (2) we obtain

$$\displaystyle\begin{array}{rcl} U_{i}(x_{j},\boldsymbol{\gamma }_{-i}^{(n)})& =& \sum _{ a_{1}\in X_{K}}\ldots \sum _{a_{n}\in X_{K}}u_{i}(a_{1},\ldots,a_{i-1},x_{j},a_{i+1},\ldots,a_{n}) {}\\ & & \qquad \gamma ^{(n)}(x_{ 1})^{n_{1}(\boldsymbol{a}_{-i})}\ldots \gamma ^{(n)}(x_{ j})^{n_{j}(\boldsymbol{a}_{-i})+1}\ldots \gamma ^{(n)}(x_{ k})^{n_{k}(\boldsymbol{a}_{-i})} {}\\ & =& \mathbb{E}\left [ \frac{1} {Y _{j} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{j}))\mathbb{1}(Y _{h} = 0\text{ for }h\not\in J)\right ], {}\\ \end{array}$$

where (Y ₁, …, Y _k ) has a multinomial distribution with parameters (n − 1; γ ⁽ⁿ⁾(x ₁), …, γ ⁽ⁿ⁾(x _k )). Notice that $\boldsymbol{a} \approx X_{J}$ is equivalent to Y _h  = 0 for all h ∉ J.

Therefore (14) holds if and only if

$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\left [ \frac{1} {Y _{j} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{j}))\mathbb{1}(Y _{h} = 0\text{ for }h\not\in J)\right ] {}\\ & & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \mathbb{E}\left [ \frac{1} {Y _{\ell} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{\ell}))\mathbb{1}(Y _{h} = 0\text{ for }h\not\in J)\right ], {}\\ \end{array}$$

which implies (11). Lemma 2 provides the result. □ 

1.2 Proofs of Sect. 4

The next two lemmata are similar to Lemmata 1 and 2, respectively.

Lemma 3.

Consider a sequence of games $\{\mathcal{P}_{n}\}_{n\in \mathbb{N}}$ . There exists $\bar{n}$ such that for all $n \geq \bar{ n}$ , if $\boldsymbol{\gamma }^{(n)}$ is a symmetric equilibrium of $\mathcal{P}_{n}$ , then $\boldsymbol{\gamma }^{(n)}$ is completely mixed, i.e.,

$$\displaystyle{ \gamma ^{(n)}(x_{ j}) > 0\quad \text{for all }x_{j} \in X_{K}. }$$

Proof.

Assume by contradiction that for every $n \in \mathbb{N}$ there exists some x _j  ∈ X _K and a symmetric equilibrium $\boldsymbol{\gamma }^{(n)}$ of $\mathcal{P}_{n}$ such that γ ⁽ⁿ⁾(x _j ) = 0. Given that λ(S) < ∞, we have that for each player i

$$\displaystyle{ U_{i}(\boldsymbol{\gamma }^{(n)}) = \mathbb{E}\left [\frac{\lambda (S)} {\varXi _{n}} \right ], }$$

where Ξ _n has a Poisson distribution with parameter n. If player i deviates and plays the pure action a _i  = x _j , then she obtains a payoff

$$\displaystyle{ U_{i}(a_{i},\boldsymbol{\gamma }_{-i}^{(n)}) \geq \lambda (v_{ K}(x_{j})) > \mathbb{E}\left [\frac{\lambda (S)} {\varXi _{n}} \right ], }$$

where the strict inequality holds for n large enough. This contradicts the assumption that $\boldsymbol{\gamma }^{(n)}$ is an equilibrium. □ 

Lemma 4.

Let (Ξ ₁ ,…,Ξ _k ) be a random vector of independent random variables where Ξ _j has a Poisson distribution with parameter nγ _j ⁽ⁿ⁾ , with δ < γ _j ⁽ⁿ⁾ < 1 −δ, for some 0 < δ < 1 and for all j ∈ K. Then

$$\displaystyle{ \lim _{n\rightarrow \infty }\frac{\mathbb{E}\left [ \frac{1} {\varXi _{j} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{j}))\mathbb{1}(\varXi _{h} = 0\text{ for }h\not\in J)\right ]} {\mathbb{E}\left [ \frac{1} {\varXi _{\ell} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{\ell}))\mathbb{1}(\varXi _{h} = 0\text{ for }h\not\in J)\right ]} = 1,\quad \text{for all }j,\ell\in K }$$

(15)

iff

$$\displaystyle{ \lim _{n\rightarrow \infty }\gamma _{j}^{(n)} =\gamma (x_{ j}) = \frac{\lambda (v_{K}(x_{j}))} {\lambda (S)} \quad \text{for all }j \in K. }$$

(16)

Proof.

Given j ∈ K, consider all J ⊂ K such that j ∈ J and the family $\mathcal{V}_{j}$ of all corresponding Voronoi tessellations V (X _J ). Call $\widetilde{V }_{j}$ the finest partition of S generated by $\mathcal{V}_{j}$, that is, the set of all possible intersections of cells v _J (x _j ) ∈ V (X _J ) for $V (X_{J}) \in \mathcal{V}_{j}$. It is clear that $v_{K}(x_{j}) \in \widetilde{ V }_{j}$.

For $A \in \widetilde{ V }_{j}$, call $\widetilde{V }_{j}(A)$ the class of all cells in $\widetilde{V }_{j}$ whose intersection with A is nonempty.

Then

$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\left [ \frac{1} {\varXi _{j} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{j}))\mathbb{1}(\varXi _{h} = 0\text{ for }h\not\in J)\right ] {}\\ & & \quad = \mathbb{E}\left [\frac{\lambda (v_{K}(x_{j}))} {\varXi _{j} + 1} \right ] {}\\ & & \quad \quad + \mathbb{E}\left [ \frac{1} {\varXi _{j} + 1}\sum _{A\in \widetilde{V }_{j}}\lambda (A)\mathbb{1}\left (\varXi _{h} = 0\text{ if }v_{K}(x_{j}) \cap A\neq \varnothing \right )\right ] {}\\ & & \quad \leq \mathbb{E}\left [\frac{\lambda (v_{K}(x_{j}))} {\varXi _{j} + 1} \right ] {}\\ & & \quad \quad +\sum _{A\in \widetilde{V }_{j}}\lambda (A)\mathbb{P}\left (\varXi _{h} = 0\text{ if }v_{K}(x_{j}) \cap A\neq \varnothing \right ) {}\\ & & \quad = \mathbb{E}\left [\frac{\lambda (v_{K}(x_{j}))} {\varXi _{j} + 1} \right ] + o(1/n)\quad \text{for }n \rightarrow \infty, {}\\ \end{array}$$

since $\mathbb{P}(\varXi _{i} = 0) =\mathop{ \mathrm{e}}\nolimits ^{-n} = o(1/n)$ for n → ∞. Therefore

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\frac{\mathbb{E}\left [ \frac{1} {\varXi _{j} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{j}))\mathbb{1}(\varXi _{h} = 0\text{ for }h\not\in J)\right ]} {\mathbb{E}\left [ \frac{1} {\varXi _{\ell} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{\ell}))\mathbb{1}(\varXi _{h} = 0\text{ for }h\not\in J)\right ]} & =& \lim _{n\rightarrow \infty }\frac{\mathbb{E}\left [\frac{\lambda (v_{K}(x_{j}))} {\varXi _{j} + 1} \right ]} {\mathbb{E}\left [\frac{\lambda (v_{K}(x_{\ell}))} {\varXi _{\ell} + 1} \right ]} \\ & =& \lim _{n\rightarrow \infty }\frac{\lambda (v_{K}(x_{j}))} {\lambda (v_{K}(x_{\ell}))} \frac{\gamma _{\ell}^{(n)}} {\gamma _{j}^{(n)}} \\ & =& \frac{\lambda (v_{K}(x_{j}))} {\lambda (v_{K}(x_{\ell}))} \frac{\gamma (x_{\ell})} {\gamma (x_{j})} {}\end{array}$$

(17)

Given that ∑ _j = 1 ^k γ(x _j ) = 1, (17) holds if and only if (16) does. □ 

Proof (Proof of Theorem 3).

Since the number of types and actions is finite, [23, Theorem  3] implies that the Poisson game $\mathcal{P}_{n}$ admits a symmetric equilibrium $\boldsymbol{\gamma }^{(n)}$. Given Lemma 3, for all j, ℓ ∈ K,

$$\displaystyle{ U_{i}(x_{j},\boldsymbol{\gamma }_{-i}^{(n)}) = U_{ i}(x_{\ell},\boldsymbol{\gamma }_{-i}^{(n)}). }$$

(18)

For j ∈ K call $n_{j}(\boldsymbol{a},\xi )$ the number of players who choose x _j under strategy $\boldsymbol{a}$ when the total number of players in the game is ξ. Using (2) we obtain

$$\displaystyle\begin{array}{rcl} U_{i}(x_{j},\boldsymbol{\gamma }_{-i}^{(n)})& =& \sum _{\xi =1}^{\infty }\Bigg[\sum _{ a_{1}\in X_{K}}\ldots \sum _{a_{\xi }\in X_{K}}u_{i}(a_{1},\ldots,a_{i-1},x_{j},a_{i+1},\ldots,a_{\xi }) {}\\ & & \gamma ^{(n)}(x_{ 1})^{n_{1}(\boldsymbol{a}_{-i},\xi )}\ldots \gamma ^{(n)}(x_{ j})^{n_{j}(\boldsymbol{a}_{-i},\xi )+1}\ldots \gamma ^{(n)}(x_{ k})^{n_{k}(\boldsymbol{a}_{-i},\xi )}\Bigg] {}\\ & & \cdot \frac{\mathop{\mathrm{e}}\nolimits ^{-n}n^{\xi }} {\xi !} {}\\ & =& \mathbb{E}\left [ \frac{1} {\varXi _{j} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{j}))\mathbb{1}(\varXi _{h} = 0\text{ for }h\not\in J)\right ], {}\\ \end{array}$$

where (Ξ ₁, …, Ξ _k ) are independent random variables such that Ξ _j has a Poisson distribution with parameter n γ ⁽ⁿ⁾(x _j ). Notice that $\boldsymbol{a} \approx X_{J}$ is equivalent to Ξ _h  = 0 for all h ∉ J.

Therefore (18) holds if and only if

$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\left [ \frac{1} {\varXi _{j} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{j}))\mathbb{1}(\varXi _{h} = 0\text{ for }h\not\in J)\right ] {}\\ & & \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \mathbb{E}\left [ \frac{1} {\varXi _{\ell} + 1}\sum _{J\subset K}\lambda (v_{J}(x_{\ell}))\mathbb{1}(\varXi _{h} = 0\text{ for }h\not\in J)\right ], {}\\ \end{array}$$

which implies (15). Lemma 4 provides the result. □ 

1.3 Proofs of Sect. 5

Lemma 5.

Consider a sequence of games $\{\mathcal{D}_{n}\}_{n\in \mathbb{N}}$ . There exists $\bar{n}^{A}$ such that for all $n^{A} \geq \bar{ n}^{A}$ , if $(\boldsymbol{\gamma }^{A,n},\boldsymbol{\gamma }^{D,n})$ is an (A,D)-symmetric equilibrium of $\mathcal{D}_{n}$ , then γ ^A,n is completely mixed, i.e.,

$$\displaystyle{ \gamma ^{A,n}(x_{ j}) > 0\quad \text{for all }x_{j} \in X_{K}. }$$

Proof.

Assume by contradiction that for every $n \in \mathbb{N}$ there exists some x _j  ∈ X _K and an (A, D)-symmetric equilibrium $(\boldsymbol{\gamma }^{A,n},\boldsymbol{\gamma }^{D,n})$ of $\mathcal{D}_{n}$, such that γ ^A, n(x _j ) = 0. Given that λ(S) < ∞, we have that for i ∈ N _n ^A

$$\displaystyle{ U_{i}^{A}(\boldsymbol{\gamma }^{A,n},\boldsymbol{\gamma }^{D,n}) \leq \frac{\lambda (S)} {n^{A}}. }$$

If player i ∈ N _n ^A deviates and plays the pure action a _i  = x _j , then she obtains a payoff

$$\displaystyle{ U_{i}^{A}(a_{ i},\boldsymbol{\gamma }_{-i}^{A,n},\boldsymbol{\gamma }^{D,n}) \geq \lambda (v_{ K}(x_{j})) \geq \frac{\lambda (S)} {n^{A}}, }$$

for n ^A large enough. Indeed, note that even if some D-players choose x _j in γ ^D, n, the A player attracts all the consumers from x _j . Therefore $(\boldsymbol{\gamma }^{A,n},\boldsymbol{\gamma }^{D,n})$ is not an equilibrium for n ^A large enough. □ 

Lemma 6.

Let (Y ₁ ,…,Y _k ) be a random vector distributed according to a multinomial distribution with parameters (n;γ ₁ ⁽ⁿ⁾ ,…,γ _k ⁽ⁿ⁾ ), with δ < γ _j ⁽ⁿ⁾ < 1 −δ, for some 0 < δ < 1 and for all j ∈ K. Then

$$\displaystyle{ \lim _{n\rightarrow \infty }\mathbb{P}(Y _{j} = 0) = 0\quad \text{for all }j \in K. }$$

Proof.

The result is obvious, since

$$\displaystyle{ \qquad \qquad \qquad \qquad \mathbb{P}(Y _{j} = 0) = (1 -\gamma _{j}^{(n)})^{n} \leq (1-\delta )^{n} \rightarrow 0.\qquad \qquad \qquad \qquad }$$

 □ 

Proof (Proof of Theorem 4).

Whenever a location x _j is occupied by an advantaged player, any disadvantaged player choosing x _j gets a payoff equal to zero. Therefore (10) is an immediate consequence of Lemmata 5 and 6. Moreover, asymptotically, the actions of disadvantaged players do not affect the payoff of advantaged players. Therefore an application of Lemma 2 with n ^A replacing n provides (9). □