The value of intermediation

Martin Stancsics

Introduction

Motivation

  • Bargaining between participants is rather overlooked in platform settings
    • Most papers implicitly assume take-it-or-leave-it offers
  • The use of cooperative bargaining concepts is not widespread in IO
    • Although they are well established in certain subfields, such as upstream-downstream markets
  • Most papers only consider finite-player games
    • Modeling the small players as a continuum simplifies analysis

Main ideas

  • Use the Shapley-value instead of an extensive-form bargaining model
  • Focus on the “few large players – continuum of small players” case (oceanic games)
    • Good approximation of the finite player case even for not too many players
    • Results have a nice geometric interpretation
  • Balance generality and having useful results

Preview of results

  • The Shapley-value seems to be a very tractable way of modeling bargaining in certain important contexts
    • One/two-sided platforms
    • Upstream-downstream setups
  • Comparative statistics give support to the bargaining interpretation
  • Extensions to include some important cases are relatively straightforward
  • Even a simple application yields interesting results about hybrid platforms

The Shapley-value

The Shapley-value

An example game in coalitional form

Imagine a game with three players:

  • Two producers: \(A_1, A_2\)
    • can each make $1 on their own
  • A platform: \(P\)
    • Triples the firms’ profits
Coalition Profits
\(\{P\}\) 0
\(\{A_i\}\) 1
\(\{A_1, A_2\}\) 2
\(\{P, A_i\}\) 3
\(\{P, A_1, A_2\}\) 6

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

Why do we care about it?

Same idea as with the Nash bargaining game/solution

  • Axiomatic reasoning
    • The Shapley-value is the only sharing rule that satisfies a nice set of axioms (efficiency, symmetry, linearity, dummy player axiom) (Shapley 1953)
  • Bargaining foundations
  • A tractable and sensible-looking gain-sharing rule
    • Comparative statics show that it behaves as one would expect

Main results

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

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

Main proposition

Proposition 1 Let \(f\) be continuous on \([0, 1]\). Then

\[ \lim_{n \to \infty} \varphi_P^n(f) = \int_0^1 f(t) \, \mathrm{d}t \]

Even as firms become infinitesimally small, the platform cannot take all the created value

Simple comparative statics

The platform gets a larger slice when

  1. the firms are substitutes to each other
  1. the platform has intrinsic value
  1. the fringe firms have no outside option

Example application

The model

  • Intentionally simple model
  • Most elements borrowed from (Anderson and Bedre-Defolie 2021)
  • Main changes:
    • Lump-sum entry fees
    • Entry fees are not set bilaterally (bargaining)

Market setup

  • Measure 1 of consumers (à la Anderson and Bedre-Defolie 2021)
    • Each of them buy a unit of products
    • Logit-type utility function (\(u_i = v_i - p_i + \mu \epsilon_i\))
  • Continuum of firms (\(a(t), t \in \mathbb{R}\)) with one differentiated product each
  • Measure 1 of outside options
  • A platform
    • necessary for the firms to sell their products
    • May or may not have its own products

Hybrid platforms with lump-sum fees

  1. Platforms set lump-sum fee \(F\)
  1. Firms decide whether to enter at cost \(i + F\)
  1. Firms and platform set product prices, consumers buy products

Outcomes

  • Constant additive markup \(\implies\) profit \(\propto\) market share
  • Per unit fringe profit \[ \frac{\mu B}{N_P B + N_F B + 1} \quad \text{where } B = \frac{v - c - \mu}{\mu} \]
  • Free entry \[ N_F^* = \frac{\mu B}{F + c} - N_P - \frac{1}{B} \]

Outcomes - the platform’s decision

  • Platform’s profit: own sales plus entry fees \[ \pi_P = F N_F^* + \frac{\mu N_P B}{N_P B + N_F^* B + 1} \]
  • In a pure marketplace the second term is zero
  • Here, second term is increasing in \(N_P\)
  • \(\implies\) higher fee and lower entry in the hybrid case

Model with bargaining

  1. Firms decide whether to invest at cost \(i\)
  1. Platform entry fees and decisions are negotiated
  1. Firms and platform set product prices, consumers buy products

Idea: stage 2 bargaining is over final allocations

Results

  • Assuming no collusion in the pricing game, total profits as a function of entering firms are \[ v(N_F) = \mu \frac{N_P B + N_F B}{N_P B + N_F B + 1} \quad \text{ where } B = e^\frac{v - c - \mu}{\mu} \]
  • If entry fees were fixed, \(N_P\) would not influence product variety / consumer welfare
  • However, a higher \(N_P\) leads to a better bargaining position for the platform and lower number of entrants

Results graphically

Conclusion

Conclusions

Contributions

  • A tractable way of modeling bargaining applicable to important contexts
  • Interesting insights into hybrid platforms

Future work

  • The example application could be turned into a proper, full-fledged model
  • A lab experiment about the outcomes of group bargaining

References

Anderson, Simon P, and Özlem Bedre-Defolie. 2021. “Hybrid Platform Model.”
Billera, Louis J, David C Heath, and Joseph Raanan. 1978. “Internal Telephone Billing Rates—a Novel Application of Non-Atomic Game Theory.” Operations Research 26 (6): 956–65.
Fogelman, Francoise, and Martine Quinzii. 1980. “Asymptotic Value of Mixed Games.” Mathematics of Operations Research 5 (1): 86–93.
Gul, Faruk. 1989. “Bargaining Foundations of Shapley Value.” Econometrica: Journal of the Econometric Society, 81–95.
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, Sergiu. 1973. “Values of Mixed Games.” International Journal of Game Theory 2 (1): 69–85.
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.
Segal, Ilya. 2003. “Collusion, Exclusion, and Inclusion in Random-Order Bargaining.” The Review of Economic Studies 70 (2): 439–60.
Shapley, Lloyd S. 1953. Additive and Non-Additive Set Functions. Princeton University.
Stole, Lars A, and Jeffrey Zwiebel. 1996. “Intra-Firm Bargaining Under Non-Binding Contracts.” The Review of Economic Studies 63 (3): 375–410.
Winter, Eyal. 1994. “The Demand Commitment Bargaining and Snowballing Cooperation.” Economic Theory 4 (2): 255–73.

Appendix

Multiple platforms

Imagine that

  • There are \(m\) platforms
  • They are perfect substitutes

That is, \[ v(S) = \begin{cases} 0 \; \text{ if } P_i \notin S \; \forall i \\ f\left(\frac{\#_A(S)}{n}\right) \end{cases} \]

Proposition 2 (Multiple platforms) The limit of the Shapley-value (as \(n \to \infty\)) of each player of type \(P\) is \[ \varphi_{P_i}^{\infty, m} = \int_0^1 (1-t) ^ {m-1} f(t) \mathrm{d}t . \]

Weighted value

  • Endow the game with a weight system
    • The platform has weight \(\lambda\)
    • The firms have weight \(1\)
  • Weights can thoughts of as innate bargaining power (Shapley 1953; S. Hart 1973)

Proposition 3 (Weighted values) Let \(f(t)\) be continuous on \([0, 1]\). Then \[ \varphi_P^\infty(f, \lambda) = \int_0^1 \underbrace{\lambda t^{\lambda - 1}}_{g(t)} f(t) \mathrm{d}t \]

Two-sided bargaining

Imagine that there are two fringe types (\(A_i\) and \(B_i\))

\[ v(S) = \begin{cases} 0 \; \text{ if } P \notin S \\ f\left(\frac{\#_A(S)}{n}, \frac{\#_B(S)}{n}\right). \end{cases} \]

Proposition 4 (Two-sided bargaining) Let \(f(a, b)\) continuous on \([0, 1] \times [0, 1]\). Then \[ f(1, 1) = \underbrace{\int_0^1 f(t, t) \mathrm{d}t}_{\varphi_P^\infty(f)} + \underbrace{\int_0^1 t \frac{\partial f(t, t)}{\partial a} \mathrm{d}t}_{\varphi_A^\infty(f)} + \underbrace{\int_0^1 t \frac{\partial f(t, t)}{\partial b} \mathrm{d}t}_{\varphi_B^\infty(f)} \]