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Essays on Cooperative Bargaining

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PhD Defense

+ +
+
+
+Martin Stancsics +
+

+ + University of Zürich + +

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+
+ +

16 October 2024

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+

Introduction

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Motivation

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+
    +
  • How to distribute value is a fundamental question in economics +
      +
    • Wage bargaining
      • +
    • Sharing the costs of a road trip
      • +
    • +
+
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    +
  • Many of such situations are characterized by an imbalance of power +
      +
    • Employer vs. employees
      • +
    • Driver vs. passengers
      • +
    • +
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  • The one indispensable / many smaller plyers is an economically important special case
    • +
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History

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+
    +
  • Bargaining theory has a long history in economics +
      +
    • Zeuthen and Schumpeter (1930), Hicks (1932)
      • +
    • +
+
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    +
  • Cooperative (or axiomatic) bargaining theory in the 1950s +
      +
    • Seminal paper: Nash et al. (1950)
      • +
    • More general cooperative game theory: Shapley (1953a), Gillies (1959)
      • +
    • +
+
+
+
    +
  • The Nash-program: non-cooperative microfoundations (Nash 1953) +
      +
    • Most well-known is Rubinstein (1982), Harsanyi (1956) is an early example
      • +
    • Microfoundations for the Shapley value: Gul (1989), Winter (1994), S. Hart and Mas-Colell (1996), Stole and Zwiebel (1996)
      • +
    • +
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This thesis

+
+
    +
  • Focus on the case of +
      +
    • Predictions from cooperative game theory
      • +
    • One (few) central player(s) and many smaller ones
      • +
    • +
+
+
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    +
  • Three aspects +
      +
    • A general, abstract treatment of the problem
      • +
    • An application to hybrid platforms
      • +
    • A lab experiment to get a better understanding of bargaining behavior
      • +
    • +
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Cooperative game theory

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Coalitional form games

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    +
  • A game \(\mathcal{G} = (N, v)\) consists of +
      +
    • A set of players \(N\)
      • +
    • A characteristic function \(v: 2^N \to \mathbb{R}\)
      • +
    • +
+
+
    +
  • Example 3-player game +
      +
    • Two producers (\(A_1, A_2\)): can each make $1 on their own
      • +
    • A platform: (\(P\)): triples the firms’ profits
      • +
    • +
+
+
+ +
+

Players: \[N = \{P, A_1, A_2\}\]

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Characteristic function \(v(S)\):

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Coalition (\(S\))Profits
\(\{P\}\)0
\(\{A_i\}\)1
\(\{A_1, A_2\}\)2
\(\{P, A_i\}\)3
\(\{P, A_1, A_2\}\)6
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+ +
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The Shapley-value

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Each player gets their expected marginal contribution

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Players: \[N = \{P, A_1, A_2\}\]

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Characteristic function \(v(S)\):

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

Core: Section 5.14

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Nucleolus: Section 5.15

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Why do we care about it?

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Same idea as with the Nash bargaining 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 1953b)
      • +
    • +
  • Bargaining foundations +
    • +
  • A tractable and sensible-looking gain-sharing rule +
      +
    • Comparative statics show that it behaves as one would expect
      • +
    • +
+ + +
+
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+

Chapter 1 – Theory

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The value of being indispensable

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+
+

Motivation

+
+
    +
  • Using cooperative game theory to model bargaining +
      +
    • Has precedents in labor and IO literature
      • +
    • General solution concepts are flexible but often not very tractable
      • +
    • +
+
+
+
    +
  • Focus on one indispensable player / many small players +
      +
    • Economically relevant
      • +
    • Even general solution concepts can be tractable in this case
      • +
    • +
+
+
+
    +
  • Generalize existing results +
      +
    • Use random order values instead of the Shapley value or weighted values
      • +
    • Relax the indispensable player assumption
      • +
    • New results for heterogeneous small players
      • +
    • +
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Literature

+
+
    +
  • Oceanic games (theoretical) +
      +
    • Focus on fundamental questions, such as existence and uniqueness
      • +
    • E.g., Milnor and Shapley (1978), S. Hart (1973), Fogelman and Quinzii (1980)
      • +
    • This paper focuses on a more specific setting to derive more concrete results
      • +
    • +
+
+
+
    +
  • IO and labor economics (applied) +
      +
    • Intermediation, vertical integration, multi-sided markets, wage bargaining
      • +
    • Examples of oceanic games: Stole and Zwiebel (1996), Levy and Shapley (1997)
      • +
    • This paper generalizes those results in a more abstract setting
      • +
    • An example application is based on Armstrong (2006)
      • +
    • +
+
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Simplest model

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

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Assumptions

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    +
  • Big player (\(P\)) is indispensable
    • +
  • Small players (\(A_i\)) are symmetric
    • +
+
+
+

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

+
+ +
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Shapley value

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+
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Characteristic fct.: \[ +\scriptsize +v(S) = \begin{cases} + 0 & \text{if } P \notin S \\ + f\left(\frac{\#_A (S)}{n}\right) & \text{o/w}. +\end{cases} +\]

+
+

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

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

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Random order values

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Characteristic fct.: \[ +\scriptsize +v(S) = \begin{cases} + 0 & \text{if } P \notin S \\ + f\left(\frac{\#_A (S)}{n}\right) & \text{o/w}. +\end{cases} +\]

+
+

Random order value: \[ +\scriptsize +\varphi_P^n(f) = \sum_{k=0}^{n} {\color{RoyalBlue}\Pr(|\mathrm{prec}P| = k)} f(k) +\]

+
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\[ +\scriptsize +\to \int_0^1 f(t) \mathrm{d} {\color{RoyalBlue}G(t)} +\]

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+ +
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Theorem 1.1.

+
    +
  • Let \(f\) be continuous on \([0, 1]\).
    • +
  • Let \(X_n \coloneqq \frac{|\mathrm{prec}_P|}{n} \xrightarrow[]{d} X\) +
      +
    • with cdf \(G(t)\)
      • +
    • and, if exists, pdf \(g(t)\)
      • +
    • +
+

Then

+

\[ +\begin{aligned} + \varphi_P^\infty &\coloneqq \lim_{n \to \infty} \varphi_P^n = \int_0^1 f(t) \mathrm{d}G(t) \\ + &= \int_0^1 g(t) f(t) \mathrm{d}t. +\end{aligned} +\]

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Some comparative statics

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The big player gets a larger slice when

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  • The small players are substitutes to each other
    • +
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  • The big player can creates some value on its own
    • +
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  • The small players cannot create value on their own
    • +
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Special case: multiple big players

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  • Assume that +
      +
    • There are \(m\) big players
      • +
    • Only one of them is needed to create value
      • +
    • +
+
+
+
    +
  • It turns out that +
      +
    • It’s equivalent to a random order value with one big player
      • +
    • Integrate wrt. the number of firms before first big player
      • +
    • +
+
+
+ +
+
+

Theorem 1.2.

+
    +
  • Let \(X^j_n \coloneqq \frac{|\mathrm{prec}_{P_j}|}{n} \xrightarrow[]{d} X^j\) +
      +
    • with cdf \(G(t) \, \forall j\)
      • +
    • \(f\) continuous, \(X^j_n\) independent of each other
      • +
    • +
+

Then

+

\[ +\varphi_P^{\infty, m} = \int_0^1 f(t) \mathrm{d}H(t) +\] with \(H(t) = 1 - (1 - G(t))^m\).

+
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Weighted value: Section 5.16

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Illustration: Shapley value

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+
+
+
    +
  • Assume that +
      +
    • There are \(m\) big players
      • +
    • Only one of them is needed to create value
      • +
    • +
+
+
+
    +
  • It turns out that +
      +
    • It’s equivalent to a random order value with one big player
      • +
    • Integrate wrt. the number of firms before first big player
      • +
    • +
+
+
+ +
+

SV for \(m \in \{1, 2, 3\}\) big players

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+ + + + + 2024-10-06T15:12:15.217313 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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Weighted value: Section 5.16

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Heterogeneous small players

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  • Let us assume +
      +
    • One indispensable big player
      • +
    • \(L\) types of small players
      • +
    • +
+
+ +
+

\[ + v(S) = \begin{cases} + 0 \; \text{ if } P \notin S \\ + f\left(\frac{\#_{A_1}(S)}{n}, \dots, \frac{\#_{A_L}(S)}{n}\right). + \end{cases} +\]

+
+
+
+

Theorem. 1.4. Assume that

+
    +
  • \(f\) is continuous on \([0, 1]^L\)
    • +
  • \(X_n \coloneqq \left( \frac{n_{A_1}(\mathrm{prec})}{n}, \dots, \frac{n_{A_L}(\mathrm{prec})}{n} \right) \xrightarrow[]{d} X\) with cdf \(G(t_1, \dots, t_L)\)
    • +
+

\[ +\implies \varphi_P^\infty = \int_0^1 \dots \int_0^1 f(t_1, \dots, t_L) \mathrm{d}G(t_1, \dots t_L) +\]

+
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Heterogeneity – Shapley value

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  • Diagonal formula +
      +
    • Unequal proportions of small players is unlikely (LLN)
      • +
    • Only have to integrate over one dimension
      • +
    • +
+
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+
    +
  • As before, but +
      +
    • \(P\) gets area under the diagonal of \(f\)
      • +
    • For \(A_i^l\), marginal contributions ≈ partial derivatives
      • +
    • +
+
+
+ +
+
+

Proposition 1.7.

+

Let \(f\) be continuous on \([0, 1]^L\). Then, \[ +\varphi_P^\infty = \int_0^1 f(t, \dots, t) \mathrm{d}t +\] and \[ +\varphi_{A^l}^\infty = \int_0^1 t \partial_l f(t, \dots, t) \mathrm{d}t. +\]

+
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General idea: Section 5.18

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Heterogeneity – Shapley value

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    +
  • Diagonal formula +
      +
    • Unequal proportions of small players is unlikely (LLN)
      • +
    • Only have to integrate over one dimension
      • +
    • +
+
+
+
    +
  • As before, but +
      +
    • \(P\) gets area under the diagonal of \(f\)
      • +
    • For \(A_i^l\), marginal contributions ≈ partial derivatives
      • +
    • +
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Shapley value (diagonal formula)

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+ + + + + 2024-10-06T15:10:30.838357 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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General idea: Section 5.18

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Heterogeneity – Weighted value

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  • Let us add weights +
      +
    • \(P\) has weight \(\lambda_P\)
      • +
    • Type \(A^l\) has weight \(\lambda_l\)
      • +
    • +
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  • Similar to the diagonal formula, but +
      +
    • The relevant manifold is curved
      • +
    • The curvature depends on the weights
      • +
    • +
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Proposition 1.8.

+

Let \(f\) be continuous on \([0, 1]^L\). Then, \[ +\varphi_P^\infty = \int_0^1 \lambda_P t^{\lambda_P - 1} f(t^{\lambda_1}, \dots, t^{\lambda_L}) \mathrm{d}t +\] and \[ +\varphi_{A^l}^\infty = \int_0^1 t^{\lambda_P} \lambda_l t^{\lambda_l - 1} \partial_l f(t^{\lambda_1}, \dots, t^{\lambda_L}) \mathrm{d}t. +\]

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Heterogeneity – Weighted value

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+
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  • Let us add weights +
      +
    • \(P\) has weight \(\lambda_P\)
      • +
    • Type \(A^l\) has weight \(\lambda_l\)
      • +
    • +
+
+
+
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  • Similar to the diagonal formula, but +
      +
    • The relevant manifold is curved
      • +
    • The curvature depends on the weights
      • +
    • +
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Weighted value with \(\lambda_1 < \lambda_2\)

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+ + + + + 2024-10-06T15:08:39.501824 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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Summary

+
+
    +
  • Provide a tool-kit for modeling bargaining in settings with one (few) central player(s) and many small ones +
      +
    • Use random order values as a general framework
      • +
    • Relax the indispensable player assumption
      • +
    • Generalize existing results for the heterogeneous small player case
      • +
    • Simple example application to two-sided markets (omitted here)
      • +
    • +
+
+
+
    +
  • Random order values are a convenient tool for modeling bargaining +
      +
    • Analytically tractable
      • +
    • More flexible than the Shapley value or the we
      • +
    • Sensible in terms of comparative statics
      • +
    • +
+
+ +
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+

Chapter 2 – Application

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Hybrid platforms and bargaining power

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Motivation

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    +
  • Hybrid platforms are +
      +
    • Getting more and more common
      • +
    • Seemingly obvious concerns and high-profile competition policy cases
      • +
    • Despite this, relatively little reserach
      • +
    • +
+
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+
    +
  • Bargaining between participants is not well understood in the platform setting +
      +
    • Authors generally assume take-it-or-leave-it offers
      • +
    • +
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Examine the bargaining power implications of hybrid platforms and its welfare consequences

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Main ideas

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  • Abstract away all frictions +
      +
    • Focus on the bargaining channel
      • +
    • Offers a good benchmark model
      • +
    • +
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  • Consider a continuum of small players case for tractability +
      +
    • Good approximation of the finite player case even for not too many players
      • +
    • +
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Model – Players

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  • One platform: \(P\) +
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    • Without the platform, no value can be created
      • +
    • Might have its own products (hybrid mode)
      • +
    • +
+
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    +
  • 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
      • +
    • +
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Model – Timing and overview

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Benchmark model

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+

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+ + +G + + + +entry_fee + +T1 +: Platform unilaterally +sets + the entry fee + + + +entry_decision + +T2 +: Potential entrants decide to +invest + in a product and +enter + + + +entry_fee->entry_decision + + + + + +sales + +T3 +: Platform and fringe +set product prices + + + +entry_decision->sales + + + + + +final + +T4 +: Consumers make +consumption decisions + + + +sales->final + + + + + +
+

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Bargaining model

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+

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+ + +G + + + +entry_decision + +T1 +: Potential entrants decide to +invest + in a product + + + +entry_fee + +T2 +: Entry fees are +negotiated +between platform and fringe entrants + + + +entry_decision->entry_fee + + + + + +sales + +T3 +: Platform and fringe +set product prices + + + +entry_fee->sales + + + + + +final + +T4 +: Consumers make +consumption decisions + + + +sales->final + + + + + +
+

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Demand

+

Logit-like demand for each product \(T_i\)

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    +
  • \(T \in \{P, F\}\)
    • +
  • \(v\): value of the product
    • +
  • \(p\): price
    • +
+

\[ +x_{Ti} = \frac{\exp\left( \frac{v_T - p_{T_i}}{\mu} \right)}{A} +\]

+
+
+

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

+
+
+

Utility function: Section 5.19

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Production

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+
    +
  • 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”
      • +
    • +
+
+
+

Proposition 2.3.

+

The optimal price is an additive markup over marginal costs:

+
    +
  • \(p^*_{Ti} = c_T + \mu\)
    • +
  • \(\pi^{v*}_{T_i} = \mu \frac{\exp \left( \frac{v_T - c_T - \mu}{\mu} \right)}{A} \coloneqq \frac{V_T}{A}\)
    • +
+
+
+
+

Free enetry

+
+
    +
  • Potential entrants decide to enter if they can cover the +
      +
    • Investment cost \(I_F\)
      • +
    • Platform entry fee \(K_F\)
      • +
    • +
+
+
+

Proposition 2.4.

+

If \(I_F\) and \(K_F\) are low enough, the equilibrium size of the aggregate is \[ +A = \mu \frac{V_F}{K_F + I_F} +\]

+
+
+
    +
  • Does not directly depend on the platform’s product variety
    • +
+
+
+
+

Benchmark model

+
+

Theorem. 2.1 The optimal entry fee is given by

+

\[K_F^{opt} = \sqrt{\mu I_F V_F} - I_F\]

+
+
+

Theorem. 2.2 In the benchmark model under hybrid regime,

+
    +
  • The equilibrium number of fringe firms is decreasing in the platform’s product variety: \(\frac{\mathrm{d} N_F}{\mathrm{d} N_P} = -\frac{V_P}{V_F} < 0\).
    • +
  • The equilibrium size of the aggregate and consumer surplus are independent of the platform’s product variety: \(\frac{\mathrm{d} A}{\mathrm{d} N_P} = \frac{\mathrm{d} CS}{\mathrm{d} N_P} = 0\).
    • +
+
+
+
+

Benchmark model

+
+
+
+
+
+

+ + + + + 2024-10-06T15:08:33.985265 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
Entry fee
+
+
+
+
+
+
+

+ + + + + 2024-10-06T15:08:38.641556 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
Number of fringe entrants
+
+
+
+
+
+
+
+

Bargaining model – profit division

+
+
    +
  • Total profits (≈ characteristic function):
    • +
+

\[ +\Pi(N_P, N_F) = \mu \frac{N_F V_F + N_P V_P}{N_F V_F + N_P V_P + 1} +\]

+
+
+
    +
  • Everyone gets their Shapley-value:
    • +
+

\[ +\pi^t_P = \int_0^1 \Pi(N_P, sN_F) \mathrm{d}s, \quad \pi^t_{F_i} = \frac{\Pi(N_P, N_F) - \pi^t_P}{N_F} +\]

+
+
+
    +
  • Implied entry fee:
    • +
+

\[ +K_F^{impl} = \pi^v_{F_i} - \pi^t_{F_i} +\]

+
+
+
+

Bargaining model – fringe entry

+
+
+
+
    +
  • The fringe profits are hump-shaped
    • +
+
+
+
    +
  • Platform product variety reduces fringe entry
    • +
+
+
+
    +
  • In the bargaining case, more than proportionally
    • +
+
+
+
+
+

+
+
+

+
+
+

+
+
+

+
+
+
+
+
+
+

Bargaining model – outcomes

+
+

Proposition 2.9. In the hybrid regime, the implied entry fee is increasing in the platform’s product variety:

+

\[ +\frac{\partial K_F^{impl}(N_P)}{\partial N_P} > 0 +\]

+
+
+

Theorem 2.3. In the bargaining model under hybrid regime

+
    +
  • The equilibrium number of fringe firms is decreasing fast in the platform’s product variety: \(\frac{\mathrm{d} N_F}{\mathrm{d} N_P} < -\frac{V_P}{V_F}\).
    • +
  • The equilibrium size of the aggregate and consumer surplus are decreasing in the platform’s product variety: \(\frac{\mathrm{d} A}{\mathrm{d} N_P}, \frac{\mathrm{d} CS}{\mathrm{d} N_P} < 0\).
    • +
+
+
+
+

Bargaining model – outcomes

+
+
+
+
+
+

+ + + + + 2024-10-06T15:07:55.215852 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
(Implied) Entry fee
+
+
+
+
+
+
+

+ + + + + 2024-10-06T15:07:53.593733 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
Number of fringe entrants
+
+
+
+
+
+
+

Other case: Section 5.20

+

Platform’s choice of product variety: Section 5.22

+

Extensiions: Section 5.23

+
+
+
+

Bargaining model – outcomes

+
+
+
+
+
+

+ + + + + 2024-10-06T15:07:26.377938 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
Consumer surplus
+
+
+
+
+
+
+

+ + + + + 2024-10-06T15:07:08.171339 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
Platform profits
+
+
+
+
+
+
+

Other case: Section 5.20

+

Platform’s choice of product variety: Section 5.22

+

Extensiions: Section 5.23

+
+
+
+

Conclusion

+
+
    +
  • 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 of hybrid platforms +
      +
    • Hybrid mode increases bargaining power against entrants
      • +
    • This can have negative welfare consequences
      • +
    • +
+
+
+
    +
  • Policy implications for such markets +
      +
    • Apple, Google, etc. having their own apps
      • +
    • Microsoft acquiring Activision/Blizzard
      • +
    • +
+
+ +
+
+
+

Chapter 3 – Experiment

+

Characterizing multiplayer free-form bargaining

+

joint work with Mia Lu

+
+
+

Motivation

+
+
    +
  • Much work on bargaining in experimental economics (understatement)
    • +
+
+
+
    +
  • Much less work on free-form bargaining +
      +
    • especially between more than two players
      • +
    • +
+
+
+
    +
  • The one indispensable player / multiple small players setting has real-world relevance +
      +
    • Wage bargaining
      • +
    • An inventor with an idea and multiple investors
      • +
    • A band, where one member owns the PA system
      • +
    • +
+
+ +
+
+

Research question

+
+
    +
  • Problem: non-cooperative game theory cannot provide predictions without structure +
      +
    • E.g. timing of the game, who makes the offers
      • +
    • NCGT solution is alternating offer games, but a lot depends on minor details (S. Hart and Mas-Colell 1996)
      • +
    • +
+
+
+
    +
  • How does bargaining power affect bargaining outcomes?
    • +
  • How well do cooperative game theory solution concepts describe the outcomes?
    • +
+
+ +
+
+

What we do

+
+
    +
  • Free-form bargaining between three players +
      +
    • Almost no structure
      • +
    • Group-level unrestricted chat
      • +
    • An interface for proposing and accepting allocations
      • +
    • No binding decision until the very last second
      • +
    • +
+
+
+
    +
  • Vary the bargaining power of the indispensable player +
      +
    • How important it is to have all small players on board
      • +
    • +
+
+
+
    +
  • We test, whether: +
      +
    • Outcomes vary as we would expect, based on bargaining power
      • +
    • Certain CGT solution concepts provide good predictions
      • +
    • +
+
+ +
+
+

Literature

+
+
+
    +
  • Early unstructured bargaining papers (1950s-1990s): +
      +
    • E.g. Kalisch et al. (1952), Maschler (1965), Nydegger and Owen (1874), Rapoport and Kahan (1976), Murnighan and Roth (1977), Murnighan and Roth (1978), Michener et al. (1979), Michener and Potter (1981), Leopold-Wildburger (1992)
      • +
    • Face to face bargaining, different experimental standards
      • +
    • +
+
+
+
    +
  • Free-form bargaining +
      +
    • E.g. Galeotti, Montero, and Poulsen (2018), Hossain, Lyons, and Siow (2020), Navarro and Veszteg (2020)
      • +
    • Almost always bilateral
      • +
    • +
+
+
+
    +
  • Multi-lateral bargaining +
      +
    • E.g. Montero, Sefton, and Zhang (2008), Mitsutsune and Adachi (2014), Tremewan and Vanberg (2016), Chessa et al. (2023), Shinoda and Funaki (2022)
      • +
    • Structured or semi-structured
      • +
    • +
+
+
+
    +
  • Fairness views in bargaining +
      +
    • E.g. Luhan, Poulsen, and Roos (2019), Schwaninger (2022), De Clippel and Rozen (2022)
      • +
    • +
+
+
+ +
+
+

The game

+
+
    +
  • Players: \(N = \{A, B_1, B_2\}\)
    • +
+
+
+
    +
  • Value function: \(v: 2^N \to \mathbb{R}\) +
      +
    • No one can create any value alone: \(v(\{A\}) = v(\{B_i\}) = 0\)
      • +
    • Player \(A\) is indispensable: \(v(\{B_1, B_2\}) = 0\)
      • +
    • Small players contribute to the value: \(v(\{A, B_i\}) = Y \in [0, 100]\)
      • +
    • The more small players the better: \(v(\{A, B_1, B_2\}) = 100\)
      • +
    • +
+
+
+

How to divide the value between the players?

+
+ +
+
+

Solution concepts

+ + + + + + + 2024-10-06T15:07:06.567998 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+

Experimental setup

+
+
    +
  • 4 treatments/sessions and 144 subjects
    • +
+
+
+
    +
  • Timing: +
      +
    • Instructions with comprehension checks
      • +
    • Slider task
      • +
    • Trial round + 5 bargaining rounds (5 minutes each)
      • +
    • Survey (demographics, reasoning, axioms)
      • +
    • +
+
+
+
    +
  • Free-form bargaining via public chat and interface for submitting proposals and current acceptances +
      +
    • Unlimited number of messages and proposals
      • +
    • Acceptances are not binding and can be changed any time
      • +
    • +
+
+ +
+
+

Main results: average payoffs

+
+
+

+ + + + + 2024-10-06T15:10:32.335063 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
+
    +
  • Dummy player gets something
    • +
  • \(Y=10\) and \(Y=30\) treatments look similar
    • +
  • Player A gets more in \(Y=30\) treatment
    • +
  • Nucleolus better in terms of ‘shape’ but worse in terms of ‘distance’
    • +
  • Linear regression and Mann-Whitney test yield the same conclusion
    • +
+
+
+
+
+

A deeper look

+
+
+

+ + + + + 2024-10-06T15:09:48.280035 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
+
    +
  • In all treatments, equal(ish) split is a frequent outcome +
      +
    • Even in the dummy player treatment!
      • +
    • +
  • \(Y=10\) and \(Y=30\) treatments still look similar
    • +
  • In the \(Y=90\) treatment +
      +
    • \(A\) only achieves significantly higher by excluding one small player (Section 5.24)
      • +
    • Neither efficient nor stable
      • +
    • +
+
+
+
+
+

Testing the axioms

+
+
    +
  • Axioms charactetrizing the Shapley value and the core +
      +
    • Look at observed behavior
      • +
    • Survey at the end of the experiment
      • +
    • +
+
+
+
    +
  • Observed choices provide +
      +
    • Strong evidence against dummy player and stability axiom
      • +
    • Evidence against symmetry and efficiency axioms in \(Y=90\) groups
      • +
    • Evidence for symmetry and efficiency axioms in other treatments
      • +
    • Some evidence against linearity
      • +
    • +
+
+
+
    +
  • Stated preferences and actions disagree +
      +
    • Concerns about subjects understanding the questions
      • +
    • +
+
+
+

Details: Section 5.25

+
+
+
+

Chat topics (before final agreement)

+
+
+

+ + + + + 2024-10-06T15:10:53.064459 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
+
    +
  • Lots of small talk +
      +
    • Early agreement → boredom
      • +
    • Negotiation tactic for small players
      • +
    • +
  • Relatively few fairness-related arguments +
      +
    • Although more in treatments with high bargaining power disparity
      • +
    • +
  • Discussion about the experiment itself +
      +
    • Players can clarify the rules for others
      • +
    • Provides feeddback for the experimenters
      • +
    • +
+
+
+
+

All chat messages: Section 5.35

+
+
+
+

Main takeaways

+
+
    +
  • Lots of equal splits in all treatments, but considerable heterogeneity +
      +
    • Heterogeneity in fairness concepts?
      • +
    • Norms established in early rounds?
      • +
    • +
+
+
+
    +
  • Nucleolus gives qualitatively correct predictions +
      +
    • No bargaining power advantage when there is no profitable deviation
      • +
    • +
+
+
+
    +
  • Both overestimate big player’s share +
      +
    • They do not capture all relevant fairness considerations
      • +
    • Combining other-regarding preferences and CGT might be promising
      • +
    • +
+
+
+
    +
  • People’s stated preferences and actions disagree
    • +
+
+ +
+
+
+

Thank you

+ +
+
+

References

+
+
+Anderson, Simon P, and Özlem Bedre-Defolie. 2021. “Hybrid Platform Model.” +
+
+Armstrong, Mark. 2006. “Competition in Two-Sided Markets.” The RAND Journal of Economics 37 (3): 668–91. +
+
+Chessa, Michela, Nobuyuki Hanaki, Aymeric Lardon, and Takashi Yamada. 2023. “An Experiment on Demand Commitment Bargaining.” Dynamic Games and Applications 13 (2): 589–609. +
+
+De Clippel, Geoffroy, and Kareen Rozen. 2022. “Fairness Through the Lens of Cooperative Game Theory: An Experimental Approach.” American Economic Journal: Microeconomics 14 (3): 810–36. +
+
+De Fontenay, Catherine C, and Joshua S Gans. 2005. “Vertical Integration in the Presence of Upstream Competition.” RAND Journal of Economics, 544–72. +
+
+Fogelman, Francoise, and Martine Quinzii. 1980. “Asymptotic Value of Mixed Games.” Mathematics of Operations Research 5 (1): 86–93. +
+
+Galeotti, Fabio, Maria Montero, and Anders Poulsen. 2018. “Efficiency Versus Equality in Bargaining.” Journal of the European Economic Association 17 (6): 1941–70. +
+
+Gillies, Donald B. 1959. “Solutions to General Non-Zero-Sum Games.” Contributions to the Theory of Games 4 (40): 47–85. +
+
+Gul, Faruk. 1989. “Bargaining Foundations of Shapley Value.” Econometrica, 81–95. +
+
+Gutierrez, German. 2021. “The Welfare Consequences of Regulating Amazon.” Job Market Paper, New York University. +
+
+Hagiu, Andrei. 2004. “Optimal Pricing and Commitment in Two-Sided Markets.” Rand Journal of Economics 20: 658–70. +
+
+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. +
+
+Harsanyi, John C. 1956. “Approaches to the Bargaining Problem Before and After the Theory of Games: A Critical Discussion of Zeuthen’s, Hicks’, and Nash’s Theories.” Econometrica, Journal of the Econometric Society, 144–57. +
+
+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. 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, 357–80. +
+
+Hicks, J. 1932. The Theory of Wages. Macmillan. https://books.google.ch/books?id=P-1AAAAAIAAJ. +
+
+Hossain, T., E. Lyons, and A. Siow. 2020. “Fairness considerations in joint venture formation.” Experimental Economics 23 (August): 632–67. +
+
+Inderst, Roman, and Christian Wey. 2003. “Bargaining, Mergers, and Technology Choice in Bilaterally Oligopolistic Industries.” RAND Journal of Economics, 1–19. +
+
+Kalisch, G., John Willard Milnor, John F. Nash, and E. D. Nering. 1952. Some Experimental n-Person Games. RAND Corporation. +
+
+Leopold-Wildburger, Ulrike. 1992. “Payoff Divisions on Coalition Formation in a Three-Person Characteristic Function Experiment.” Journal of Economic Behavior & Organization 17 (1): 183–93. https://doi.org/https://doi.org/10.1016/0167-2681(92)90086-Q. +
+
+Levy, Anat, and Lloyd S Shapley. 1997. “Individual and Collective Wage Bargaining.” International Economic Review, 969–91. +
+
+Luhan, W. J., O. Poulsen, and M. W. M. Roos. 2019. “Money or Morality: Fairness Ideals in Unstructured Bargaining.” Social Choice and Welfare 53: 655–75. https://doi.org/https://doi.org/10.1007/s00355-019-01206-5. +
+
+Maschler, Michael. 1965. “Playing an n-Person Game, an Experiment.” +
+
+Michener, H., and Kathryn Potter. 1981. “Generalizability of Tests in n-Person Sidepayment Games.” Journal of Conflict Resolution, 733–49. +
+
+Michener, H., Melvin M. Sakurai, Kenneth Yuen, and Thomas J. Kasen. 1979. “A Competitive Test of the M1 (i) and M1 (Im) Bargaining Sets.” The Journal of Conflict Resolution 23 (1): 102–19. +
+
+Milnor, John Willard, and Lloyd S Shapley. 1978. “Values of Large Games II: Oceanic Games.” Mathematics of Operations Research 3 (4): 290–307. +
+
+Mitsutsune, Masanori, and Takanori Adachi. 2014. “Estimating Noncooperative and Cooperative Models of Bargaining: An Empirical Comparison.” Empirical Economics 47: 669–93. +
+
+Montero, Maria, Martin Sefton, and Ping Zhang. 2008. “Enlargement and the Balance of Power: An Experimental Study.” Social Choice and Welfare 30 (1): 69–87. +
+
+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. +
+
+Murnighan, J., and A. Roth. 1977. “The Effects of Communication and Information Availability in an Experimental Study of a Three-Person Game.” Management Science 23 (12): 1336–48. +
+
+———. 1978. “Large Group Bargaining in a Characteristic Function Game.” Journal of Conflict Resolution 22: 299--317. +
+
+Nash, John F et al. 1950. “The Bargaining Problem.” Econometrica 18 (2): 155–62. +
+
+Nash, John F. 1953. “Two-Person Cooperative Games.” Econometrica: Journal of the Econometric Society, 128–40. +
+
+Navarro, Noemí, and Róbert F. Veszteg. 2020. “On the Empirical Validity of Axioms in Unstructured Bargaining.” Games and Economic Behavior 121: 117–45. +
+
+Nydegger, R. V., and G. Owen. 1874. “Two-Person Bargaining: An Experimental Test of the Nash Axioms.” International Journal of Game Theory 3: 239–49. https://doi.org/https://doi.org/10.1007/BF01766877. +
+
+Rapoport, Amnon, and James P Kahan. 1976. “When Three Is Not Always Two Against One: Coalitions in Experimental Three-Person Cooperative Games.” Journal of Experimental Social Psychology 12 (3): 253–73. https://doi.org/https://doi.org/10.1016/0022-1031(76)90056-1. +
+
+Rochet, Jean-Charles, and Jean Tirole. 2003. “Platform Competition in Two-Sided Markets.” Journal of the European Economic Association 1 (4): 990–1029. +
+
+Rubinstein, Ariel. 1982. “Perfect Equilibrium in a Bargaining Model.” Econometrica, 97–109. +
+
+Schwaninger, Manuel. 2022. “Sharing with the Powerless Third: Other-Regarding Preferences in Dynamic Bargaining.” Journal of Economic Behavior & Organization 197: 341–55. https://doi.org/https://doi.org/10.1016/j.jebo.2022.03.002. +
+
+Shapley, Lloyd S. 1953a. “A Value for n-Person Games.” Contributions to the Theory of Games, 307–17. +
+
+———. 1953b. Additive and Non-Additive Set Functions. Princeton University. +
+
+Shinoda, T., and Y. Funaki. 2022. “The Core and the Equal Division Core in a Three-Person Unstructured Bargaining Experiment: The Weakest Coalition Is Ignored.” https://ssrn.com/abstract=4291591. +
+
+Steiner, Robert L. 2004. “The Nature and Benefits of National Brand/Private Label Competition.” Review of Industrial Organization 24 (2): 105–27. +
+
+Stole, Lars A, and Jeffrey Zwiebel. 1996. “Intra-Firm Bargaining Under Non-Binding Contracts.” The Review of Economic Studies 63 (3): 375–410. +
+
+Tremewan, James, and Christoph Vanberg. 2016. “The Dynamics of Coalition Formation–a Multilateral Bargaining Experiment with Free Timing of Moves.” Journal of Economic Behavior & Organization 130: 33–46. +
+
+Winter, Eyal. 1994. “The Demand Commitment Bargaining and Snowballing Cooperation.” Economic Theory 4 (2): 255–73. +
+
+Zeuthen, F. L. B., and J. A. Schumpeter. 1930. Problems of Monopoly and Economic Warfare. Kelley. https://books.google.ch/books?id=w8kN0AEACAAJ. +
+
+
+
+
+

Appendix – CGT

+ +
+
+

The core

+
+
+

There are no profitable deviations

+
+

\[ +\begin{aligned} +\pi_P &\geq 0 \\ +\pi_{A_i} &\geq 1 \\ +\pi_{A_1} + \pi_{A_2} &\geq 2 \\ +\pi_{P} + \pi_{A_i} &\geq 3 +\end{aligned} +\]

+
+
+

May be multi-valued, e.g.:

+ + + + + + + + + + + + + + + + + + + + + + + + + +
\(\pi_P\)\(\pi_{A_1}\)\(\pi_{A_2}\)
222
033
321
+
+
+ +
+

Players: \[N = \{P, A_1, A_2\}\]

+

Characteristic function \(v(S)\):

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

🔙 Section 2.2

+
+
+
+
+
+

The nucleolus

+
+
+

Maximize smallest excess

+
+
+

Excess: payoff – value \[ +\begin{aligned} +e({P}) &= \pi_P \mathrel{\phantom{=2}} \\ +e({A_i}) &= \pi_{A_i} - 1 \mathrel{\phantom{=1}} \\ +e({P, A_i}) &= \pi_P + \pi_{A_i} - 3 \mathrel{\phantom{=1}} \\ +e({A_1, A_2}) &= \pi_{A_1} + \pi_{A_2} - 2 \mathrel{\phantom{=2}} +\end{aligned} +\]

+
+
+

Excess: payoff – value \[ +\begin{aligned} +e({P}) &= \pi_P \mathrel{\color{RoyalBlue}{=2}} \\ +e({A_i}) &= \pi_{A_i} - 1 \mathrel{\color{RoyalBlue}{=1}} \\ +e({P, A_i}) &= \pi_P + \pi_{A_i} - 3 \mathrel{\color{RoyalBlue}{=1}} \\ +e({A_1, A_2}) &= \pi_{A_1} + \pi_{A_2} - 2 \mathrel{\color{RoyalBlue}{=2}} +\end{aligned} +\]

+
+
+
+

Unique and contained in the core: \[ +\pi_P = \pi_{A_1} = \pi_{A_2} \mathrel{\color{RoyalBlue}= 2} +\]

+
+
+ +
+

Players: \[N = \{P, A_1, A_2\}\]

+

Characteristic function \(v(S)\):

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

🔙 Section 2.2

+
+
+
+
+
+

Appendix – Chapter 1

+ +
+
+

Special case: weighted value

+
+
+
+
    +
  • Assume that +
      +
    • Player \(P\) has weight \(\lambda \geq 0\)
      • +
    • Small players have weight \(1\)
      • +
    • +
+
+
+
    +
  • In this case +
      +
    • The weighted value corresponds to a specific random order value
      • +
    • Probability distribution depends on the weights
      • +
    • +
+
+
+ +
+
+

Theorem 1.2

+

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

+

\[ +\begin{aligned} +\varphi_P^{\infty, \lambda} &= \int_0^1 f(t) \mathrm{d}G(t) \\ +&= \int_0^1 \lambda t^{\lambda-1} f(t) \mathrm{d}t. +\end{aligned} +\] with \(G(t) = t^\lambda\).

+
+
+
+
+

Weighted value: Section 3.7

+
+ +
+
+

Special case: weighted value

+
+
+
+
    +
  • Assume that +
      +
    • Player \(P\) has weight \(\lambda \geq 0\)
      • +
    • Small players have weight \(1\)
      • +
    • +
+
+
+
    +
  • In this case +
      +
    • The weighted value corresponds to a specific random order value
      • +
    • Probability distribution depends on the weights
      • +
    • +
+
+
+ +
+

Weighted value for \(\lambda \in \{0.5, 1, 3\}\)

+

+ + + + + 2024-10-06T15:08:32.808481 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
+
+
+

Weighted value: Section 3.7

+
+
+
+

Heterogeneity – general case

+
+

Lemma 1.2

+

Assume that

+
    +
  • \(X_n \xrightarrow[]{d} X\) where \(X\) with cdf. \(H\).
    • +
  • The whole probability mass of \(X\) is concentrated on the manifold \((a_1(s), \dots a_L(s)), t \in [0, 1]\).
    • +
+

Then \[ +\begin{aligned} + \varphi_P^\infty &= \int_0^1 f(a_1(s), \dots a_L(s)) \mathrm{d}H(s), \\ + \varphi_{A^l}^\infty &= \int_0^1 H(s) a_l'(s) \partial_l f(a_1(s), \dots a_L(s)) \mathrm{d}s. +\end{aligned} +\]

+
+
+

🔙 Section 3.10

+
+
+
+
+

Appendix – Chapter 2

+ +
+
+

Utility function

+
+
    +
  • 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
      • +
    • +
+
+
+

🔙 Section 4.6

+
+
+
+

Bargaining model – other case

+
+
+
+
+
+

+ + + + + 2024-10-06T15:08:34.903690 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
(Implied) Entry fee
+
+
+
+
+
+
+

+ + + + + 2024-10-06T15:08:37.451431 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
Number of fringe entrants
+
+
+
+
+
+
+

🔙 Section 4.14

+
+
+
+

Bargaining model – other case

+
+
+
+
+
+

+ + + + + 2024-10-06T15:08:30.576997 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
Consumer surplus
+
+
+
+
+
+
+

+ + + + + 2024-10-06T15:08:36.062119 + image/svg+xml + + + Matplotlib v3.8.4, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
Platform profits
+
+
+
+
+
+
+

🔙 Section 4.14

+
+
+
+

Platform’s choice of product variety

+
+
    +
  • Higher platform product variety → lower consumer surplus when platform is operating in hybrid mode
    • +
  • Does it want to have more products?
    • +
  • Assume that at time 0 the platform can invest in own products at cost \(I_P\) per product
    • +
+
+
+

Propositions 2.11, 2.12.

+
    +
  • In the benchmark model, \(N_P^* > 0 \implies \frac{V_P}{I_P} \geq \frac{V_F}{I_F}\).
    • +
  • In the bargaining model, it can happen that \(N_P^* > 0\) even if \(\frac{V_P}{I_P} < \frac{V_F}{I_F}\).
    • +
+
+
+

🔙 Section 4.14

+
+
+
+

Extensions

+
+
    +
  • Difference in innate bargaining power +
      +
    • Players get their weighted value
      • +
    • More flexible but no qualitative difference
      • +
    • +
+
+
+
    +
  • Consumers (customers) take part in the bargaining +
      +
    • C.f. heterogeneous small players (Section 3.10)
      • +
    • Platform and customers get the same share
      • +
    • What’s good for the platform is good for the customers
      • +
    • Otherwise similar results
      • +
    • +
+
+
+

🔙 Section 4.14

+
+
+
+
+

Appendix – Chapter 3

+ +
+
+

Time of accepting final allocation

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