Hybrid platforms and bargaining

Martin Stancsics

Introduction

Motivation

Examine the bargaining power implications of hybrid platforms and its welfare consequences

  • Hybrid platform
    • Getting more and more common
    • Seemingly obvious concerns and high-profile competition policy cases
    • Comparatively little reserach
  • Bargaining between participants is understudied in the platform setting
    • Authors generally assume take-it-or-leave-it offers

Main ideas

  • Abstract away all frictions
    • Allows me to focus on the bargaining channel
    • Offers a good benchmark model
  • Focus on a continuum of small players case for tractability
    • Good approximation of the finite player case even for not too many players
    • Results have a nice geometric interpretation

Preview of results

  • The impacts of hybrid platforms depend on the substitutability of the products
    • When they are substitutes, it increases the platform’s profit share and decreases the fringe’s profits
    • In the presence of investment costs, it can lead to lower fringe entry
    • With Logit demand, hybrid operation leads to welfare losses
  • This modeling strategy seems promising
    • The model is surprisingly tractable
    • Quite general results can be obtained

Model

Players

  • One platform: \(P\)
    • Without the platform, no value can be created
    • Might have its own products (hybrid mode)
  • A continuum of potential fringe entrants: \(F_i, i \in \mathbb{R}_0^+\)
    • Infinitesimally small
    • Have one product each
    • Can only sell through the platform

Production

Reduced-form approach: assume some function for total profit:

\[ f(N_P, N_F) \]

  • \(N_P\) is the number of platform products
  • \(N_F\) is the number of fringe products/firms
  • Assume that \(f\) is increasing in both arguments

Profit sharing

  • Platform gets \[ \pi_P(N_P, N_F) = \int_0^1 w(s) f(N_P, sN_F) \mathrm{d}s \]
  • Fringe gets the rest

Some simple comparative statics

  • The platform likes being hybrid \[ \small \frac{\partial \pi_P(N_P, N_F)}{\partial N_P} > 0 \]
  • It depends™ for the fringe \[ \begin{align*} \small\mathrm{sign} \left( \frac{\partial \pi_F(N_P, N_F)}{\partial N_P} \right) = \\ \small\mathrm{sign} \left( \frac{\partial^2 f(N_P, n_F)}{\partial n_P \partial n_F} \right) \end{align*} \]

Entry and equilibrium

Now let us endogeneize the number of fringe entrants!

  • Unlimited number of potential entrants
  • Timing
    • Hybrid mode is exogeneously determined \(\to N_P\)
    • Each fringe firm decides whether to invest at cost \(I_F\) \(\to N_F\)
    • Platform and firms produce and sell products \(\to f(N_P, N_F)\)
    • Profits are shared according to previous rule \(\to \pi_P(N_P, N_F), \pi_F(N_P, N_F)\)

Equilibrium

  • Free entry equilibrium: fringe firms make zero net profits \[ \pi_F(N_P, N_F^*) = I_F N_F^* \]
  • In order to guarantee uniqueness, let us assume \[ \scriptsize \begin{gather*} f'_{n_F} (N_P, N_F) < \int_0^1 s w(s) f'_{n_F} (N_P, s N_F) \mathrm{d}s \\ \text{or} \\ f''_{n_F} (N_P, N_F) < \int_0^1 s^2 w(s) f''_{n_F} (N_P, s N_F) \mathrm{d}s. \end{gather*} \]
    • Holds for certain interesting cases, such as the example application

Equilibrium comparative statics

  • If \(\frac{\partial^2 f(N_P, n_F)}{\partial n_P \partial n_F} < 0\), hybrid mode reduces fringe entry
  • And sometimes total profits, too \[ \begin{gather*} g(\alpha N_P + \beta N_F), \\ g'' < 0, w(s) \equiv 1 \end{gather*} \] \[ \implies \frac{\partial N_F^*}{\partial N_P} < -\frac{\alpha}{\beta} \]

Microfounding the profit function

Demand

  • Follow Anderson and Bedre-Defolie (2021)
  • Unit mass of customers, each choosing one product maximizing \[u_{ij}^T = v^T - p_i^T + \mu \varepsilon_{ij}^T\]
    • \(T \in \{P, F, 0\}\)
    • Unit mass of outside options at price \(p_i^0 = 0\)
    • \(v_T\): value of the product
    • \(\mu\): degree of horizontal differentiation
    • \(\varepsilon_{ij}^T \sim \mathrm{Gumbel}(0, 1)\): taste shocks

Demand

Logit-like demand: \[ x_{Ti} = \frac{\exp\left( \frac{v_T - p_{Ti}}{\mu} \right)}{A} \]

where \[ A = \int_0^{N_F} \exp\left( \frac{v_F - p_{Fi}}{\mu} \right) \mathrm{d}i + \int_0^{N_P} \exp\left( \frac{v_P - p_{Pi}}{\mu} \right) \mathrm{d}i + 1 \]

Production

  • Assume that the platform prices its products as if they were made by separate, competitive sellers
    • Possible interpretation: profit-maximizing subsidiaries
    • More importantly: “best case scenario”
  • The optimal price is an additive markup rule: \[ \begin{align*} p^*_{Ti} &= c_T + \mu \\ \pi^{v*}_{Ti} &= \mu \frac{\exp \left( \frac{v_T - c_T - \mu}{\mu} \right)}{A} \end{align*} \]

Entry and equilibrium

I consider two cases:

  • Benchmark
    platform can commit to any lump-sum entry fee \(F_F\)
  • Bargaining
    total profits are shared according to the Shapley-value-based rule

Benchmark equilibrium

  • Optimal entry fee is independent of \(N_P\): \[ F_F^* = \sqrt{\mu I_F V_F} - I_F \]
  • Size of the aggregate is independent* of \(N_P\): \[ A^* = \sqrt{\frac{\mu V_F}{I_F}} \]
  • Consumer surplus is also independent* of \(N_P\): \[ CS^* \propto \log(A^*) \]

Bargaining equilibrium

  • Total profits as a function of entrants have a simple form: \[ f(N_P, N_F) = \mu \frac{N_F V_F + N_P V_P}{N_F V_F + N_P V_P + 1} \]
    • Satisfies the assumption for unique equilibrium
    • Has the form \(f(N_P, N_F) = g(\alpha N_P + \beta N_F)\)
    • \(g\) is concave
  • Therefore, \[ \frac{\partial N_F^*}{\partial N_P} < -\frac{V_P}{V_F} \]

Bargaining equilibrium

  • Equilibrium aggregate and total profits are decreasing in \(N_P\) \[ \frac{\partial A^*}{\partial N_P} < 0 \]
    • Platform profits are increasing in \(N_P\), though
  • Consumer surplus is also decreasing in \(N_P\) \[ \frac{\partial CS^*}{\partial N_P} < 0 \]

Results graphically

Conclusion

Summary

  • Tractable model of hybrid platforms in which bargaining plays a key role
    • Applicable to other settings, too (e.g. vertical markets, franchises)
  • Highlight an important aspect if hybrid platforms
    • Hybridness increases bargaining power against entrants
    • This can have negative welfare consequences
  • Policy implications for such markets
    • Amazon operating in hybrid mode
    • Microsoft acquiring Activision/Blizzard for $68bn

Future directions

  • In this paper
    • Endogenenize hybridness (\(N_P\))
    • Relax separate pricing assumption
    • Alternative benchmarks (e.g. exogeneous split)
  • Longer term
    • Endogeneize consumer entry (possibly with them engaging in the negotiation)
    • Apply this bargaining rule to other, similar settings

Thank you

References

Anderson, Simon P, and Özlem Bedre-Defolie. 2021. “Hybrid Platform Model.”
De Fontenay, Catherine C, and Joshua S Gans. 2005. “Vertical Integration in the Presence of Upstream Competition.” RAND Journal of Economics, 544–72.
Gul, Faruk. 1989. “Bargaining Foundations of Shapley Value.” Econometrica: Journal of the Econometric Society, 81–95.
Gutierrez, German. 2021. “The Welfare Consequences of Regulating Amazon.” Job Market Paper, New York University.
Hagiu, Andrei, Tat-How Teh, and Julian Wright. 2022. “Should Platforms Be Allowed to Sell on Their Own Marketplaces?” The RAND Journal of Economics 53 (2): 297–327.
Hart, Oliver, and John Moore. 1990. “Property Rights and the Nature of the Firm.” Journal of Political Economy 98 (6): 1119–58.
Hart, Oliver, Jean Tirole, Dennis W Carlton, and Oliver E Williamson. 1990. “Vertical Integration and Market Foreclosure.” Brookings Papers on Economic Activity. Microeconomics 1990: 205–86.
Hart, Sergiu, and Andreu Mas-Colell. 1996. “Bargaining and Value.” Econometrica: Journal of the Econometric Society, 357–80.
Inderst, Roman, and Christian Wey. 2003. “Bargaining, Mergers, and Technology Choice in Bilaterally Oligopolistic Industries.” RAND Journal of Economics, 1–19.
Levy, Anat, and Lloyd S Shapley. 1997. “Individual and Collective Wage Bargaining.” International Economic Review, 969–91.
Montez, João V. 2007. “Downstream Mergers and Producer’s Capacity Choice: Why Bake a Larger Pie When Getting a Smaller Slice?” The RAND Journal of Economics 38 (4): 948–66.
Steiner, Robert L. 2004. “The Nature and Benefits of National Brand/Private Label Competition.” Review of Industrial Organization 24 (2): 105–27.

Appendix

The Shapley-value

Each player gets their expected marginal contribution

Coalition Profits
\(\{P\}\) 0
\(\{A_i\}\) 1
\(\{A_1, A_2\}\) 2
\(\{P, A_i\}\) 3
\(\{P, A_1, A_2\}\) 6

Baseline model

Set of all players: \(N = \{P, A_1, \dots, A_n\}\)

Assumption: the platform is the only way for firms to reach users and that all \(n\) firms are identical \[ v(C) = \begin{cases} 0 & \text{if } P \notin C \\ f\left(\frac{|C \setminus \{P\}|}{n}\right) & \text{otherwise}. \end{cases} \] \(\forall\, C \subset N\).

Value of the platform

Characteristic function: \[ \scriptsize v(C) = \begin{cases} 0 \;\text{ if } P \notin C \\ f\left(\frac{|C \setminus \{P\}|}{n}\right) \end{cases} \]

Shapley value: \[l \scriptsize \varphi_P^n(f) = \frac{1}{n+1} \sum_{k=0}^{n} f(k) \]

\[ \scriptsize \to \int_0^1 f(t) \, \mathrm{d}t \]