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
  • Analytically more tractable
  • Has bargaining interpretation (e.g. S. Hart and Mas-Colell 1996; Gul 1989)
  • Has precedents in the IO literature (e.g. O. Hart and Moore 1990; Montez 2007)
• 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

Related literature

• Shapley-value as a bargaining outcome
  • (Gul 1989; Winter 1994; S. Hart and Mas-Colell 1996; Stole and Zwiebel 1996)
• Cooperative bargaining in IO models
  • (Billera, Heath, and Raanan 1978; O. Hart and Moore 1990; Levy and Shapley 1997; Segal 2003; Inderst and Wey 2003; Montez 2007)
• Values of oceanic games
  • (S. Hart 1973; Fogelman and Quinzii 1980)
• Hybrid platforms
  • (Anderson and Bedre-Defolie 2021; Hagiu, Teh, and Wright 2022)

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

An example game in coalitional form

The Shapley-value

Each player gets their expected marginal contribution

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
  • The value arises as the subgame-perfect equilibrium of various alternating-offer bargaining games (e.g. Gul 1989; Winter 1994; S. Hart and Mas-Colell 1996; Stole and Zwiebel 1996; Inderst and Wey 2003)
• A tractable and sensible-looking gain-sharing rule
  • Comparative statics show that it behaves as one would expect

Baseline model

Value of the platform

Main proposition

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

Simple comparative statics

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 ($u_i=v_i-p_i+\mu {ϵ}_i$)
• 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

Outcomes

• Constant additive markup $⟹$ profit $\propto$ market share
• Per unit fringe profit $$\frac{\mu B}{N_{P}B+N_{F}B+1}\text{where }B=\frac{v-c-\mu }{\mu }$$
• Free entry $$N_F^{\ast }=\frac{\mu B}{F+c}-N_P-\frac{1}{B}$$

Outcomes - the platform’s decision

• Platform’s profit: own sales plus entry fees $${\pi }_P=F{N}_F^{\ast }+\frac{\mu N_{P}B}{N_{P}B+N_F^{\ast }B+1}$$
• In a pure marketplace the second term is zero
• Here, second term is increasing in $N_P$
• $⟹$ higher fee and lower entry in the hybrid case

Model with bargaining

Results

• Assuming no collusion in the pricing game, total profits as a function of entering firms are $$v(N_F)=\mu \frac{N_{P}B+N_{F}B}{N_{P}B+N_{F}B+1}\text{ where }B=e^{\frac{v-c-\mu }{\mu }}$$
• 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

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

Multiple platforms

Weighted value

Two-sided bargaining