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Contents
  1. Platform competition, tipping and interoperability: a selection-robust welfare threshold in a two-sided Hotelling model
  2. Abstract
  3. 1. Model
  4. 2. Equilibria and the threshold
  5. 3. Identification
  6. 4. Scope
  7. Completion Estimate
  8. References

Platform competition, tipping and interoperability: a selection-robust welfare threshold in a two-sided Hotelling model

Abstract

In a two-platform, two-sided Hotelling model with single-homing, full coverage, quasilinear utilities and an interoperability parameter $\phi\in[0,1]$, we prove the following. (i) Welfare depends only on the allocation of users and on $\phi$. Fees are transfers. (ii) Under full interoperability the participation equilibrium is unique and first-best, and tipping is impossible. (iii) Moving from $\phi=0$ to $\phi=1$ raises welfare for every equilibrium selection iff the incremental compatibility cost is at most $W^{\rm FB}-\max_{e\in\mathcal E(0)}W(e)$. For the two canonical equilibria this threshold equals $(\alpha_1+\alpha_2)/2$ against the split equilibrium and $t/2$ against tipping. (iv) Partial interoperability can lower welfare relative to $\phi=0$ under tipping selections. (v) The welfare-relevant primitives $(t,\alpha_1,\alpha_2)$ are identified from exogenous variation in opposite-side participation. Without such variation, the reflection problem leaves $\alpha$ unidentified.

1. Model

Sides $a\in\{1,2\}$ each have a unit mass of users uniform on $[0,1]$. Platform $1$ is at $0$ and platform $2$ at $1$. A side-$a$ user at $x$ on platform $k$ gets $v+\alpha_aN_k^{-a}(\phi)-p_k^a-t\,d(x,k)$, where $N_k^{-a}(\phi)=n_k^{-a}+\phi\,n_{-k}^{-a}$ is the accessible opposite side. Compatibility costs $\kappa(\phi)$, which is increasing. With platform owners counted at unit weight, total welfare at allocation $(n^1,n^2)$, where $n^a$ is side $a$'s mass on platform 1 (sorted by location), is \[ W(n;\phi)=2v+\sum_a\alpha_a\left[n^an^{-a}+(1-n^a)(1-n^{-a})+\phi\left(n^a(1-n^{-a})+(1-n^a)n^{-a}\right)\right] -\tfrac t2\sum_a\left[(n^a)^2+(1-n^a)^2\right]-\kappa(\phi). \]
Lemma: $\sup_nW(n;\phi)\le W^{\rm FB}:=2v+\alpha_1+\alpha_2-t/2-\kappa(\phi)$, with equality iff $\phi=1$ and $n^1=n^2=1/2$.
Proof: The bracket is at most $1$, with equality iff $\phi=1$ or the allocation is tipped. The transport term is minimised at $n^a=1/2$, with value $t/4$ per side. At $\phi<1$ the two maxima are attained at different allocations. ∎

2. Equilibria and the threshold

Proposition: At $\phi=1$, each user's network benefit is $\alpha_a$ whatever platform it joins. At symmetric fees, the unique participation equilibrium is $n^a=1/2$, so tipping cannot occur. At $\phi=0$ with symmetric fees, the split $n^a=1/2$ is an equilibrium. It is the unique one if $t>\sqrt{\alpha_1\alpha_2}$ (more precisely, if the best-response map is a contraction, for which $\max(\alpha_1,\alpha_2)<t$ suffices), and tipped allocations are equilibria if $\alpha_a\ge t$ for both sides.
Proof: At $\phi=1$, the indifferent user on side $a$ solves $-tx=-t(1-x)$. At $\phi=0$, with symmetric fees, the indifferent location is $x^a=\tfrac12+\tfrac{\alpha_a}{2t}(2n^{-a}-1)$. This map is a contraction when $\alpha_a<t$. If $\alpha_a\ge t$, then $n^{-a}=1$ gives $x^a\ge1$, so tipping is self-confirming. ∎
Theorem (Selection-robust threshold): Let $\mathcal E(0)$ be the set of participation allocations sustainable in equilibrium at $\phi=0$, under any pricing equilibrium selection. Then $W(e_1;1)\ge W(e_0;0)$ for all $e_0\in\mathcal E(0)$ iff \[ \kappa(1)-\kappa(0)\le\alpha_1+\alpha_2-\tfrac t2-\max_{e\in\mathcal E(0)}\left[W(e;0)+\kappa(0)-2v\right]. \] For the split equilibrium the right side is $(\alpha_1+\alpha_2)/2$; for a tipped equilibrium it is $t/2$.
Proof: By Proposition Proposition, $e_1$ is the split, and Lemma Lemma gives $W(e_1;1)=W^{\rm FB}$. Compute $W(\text{split};0)=2v+(\alpha_1+\alpha_2)/2-t/2-\kappa(0)$ and $W(\text{tipped};0)=2v+\alpha_1+\alpha_2-t-\kappa(0)$, then compare. ∎
Proposition (Partial interoperability can hurt): Suppose $\alpha_a\ge t$ and the selection picks tipping at $\phi=0$. Let $\phi\in(0,1)$ be small enough that tipping is no longer self-confirming, i.e. $(1-\phi)\alpha_a<t$, so that the split is selected. Then $W(\text{split};\phi)-W(\text{tipped};0)=-(1-\phi)(\alpha_1+\alpha_2)/2+t/2-[\kappa(\phi)-\kappa(0)]$. This is negative whenever $(1-\phi)(\alpha_1+\alpha_2)>t$.
Proof: At the split, the bracket equals $(1+\phi)/2$; substitute. ∎
So interoperability mandates should be either complete or accompanied by an argument that tipping was not efficient.

3. Identification

Proposition: From observed shares and fees in a symmetric single-market equilibrium, $\alpha_a$ is not identified. The demand equation $n^a=\tfrac12+\tfrac{\alpha_a(2n^{-a}-1)-(p_1^a-p_2^a)}{2t}$ involves the endogenous $n^{-a}$, and is observationally equivalent to a model with $\alpha_a=0$ and correlated fee shocks (reflection). With an exogenous shifter $z$ of side $-a$ participation, excluded from side-$a$ utility (for example a side-$(-a)$ subsidy), $(t,\alpha_a)$ are identified by IV: $t$ from own-fee variation, and $\alpha_a/t$ from the $z$-induced variation in $n^{-a}$.
Proof: The structural equation is linear in $(p_1^a-p_2^a,\,2n^{-a}-1)$, with coefficients $(-1/2t,\ \alpha_a/2t)$. Two instruments, the own-fee shifter and $z$, satisfy the rank condition. With only fees varying, $n^{-a}$ moves collinearly with them in symmetric equilibria, and the rank condition fails. ∎

4. Scope

Multi-homing, entry, quality investment and asymmetric platforms break the closed forms. Theorem Theorem (Selection-robust threshold)'s structure persists, however: full interoperability removes the network-size component from users' platform choice and therefore removes tipping-based multiplicity. Dynamic attraction basins and investment incentives (the statement's remaining items) require a dynamic model, which we leave open [rt,arm,weyl].

Completion Estimate

30%

This is a post hoc, heuristic estimate for the full catalogue target. The Hotelling participation calculations provide fee-cancelling welfare comparisons and conditional thresholds between split and tipped allocations. Joint platform pricing equilibrium, quality and entry responses, and dynamic attraction basins are not established, limiting the claimed robustness to the full target.

References

  • [rt] J.-C. Rochet, J. Tirole, Platform competition in two-sided markets, JEEA 1 (2003), 990--1029.
  • [arm] M. Armstrong, Competition in two-sided markets, RAND J. Econ. 37 (2006), 668--691.
  • [weyl] E. G. Weyl, A price theory of multi-sided platforms, AER 100 (2010), 1642--1672.