16 October 2024
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 |
Each player gets their expected marginal contribution
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 |
Core: Section 5.14
Nucleolus: Section 5.15
Same idea as with the Nash bargaining solution
The value of being indispensable
Set of all players: \(N = \{P, A_1, \dots, A_n\}\)
Assumptions
\[ v(S) = \begin{cases} 0 & \text{if } P \notin S \\ f\left(\frac{\#_A (S)}{n}\right) & \text{otherwise} \end{cases} \]
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: \[ \scriptsize \varphi_P^n(f) = \frac{1}{n+1} \sum_{k=0}^{n} f(k/n) \]
\[ \scriptsize \to \int_0^1 f(t) \, \mathrm{d}t \]
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: \[ \scriptsize \varphi_P^n(f) = \sum_{k=0}^{n} {\color{RoyalBlue}\Pr(|\mathrm{prec}P| = k)} f(k/n) \]
\[ \scriptsize \to \int_0^1 f(t) \mathrm{d} {\color{RoyalBlue}G(t)} \]
Theorem 1.1.
Then
\[ \begin{aligned} \varphi_P^\infty &\coloneqq \lim_{n \to \infty} \varphi_P^n = \int_0^1 f(t) \mathrm{d}G(t) \\ &= \int_0^1 g(t) f(t) \mathrm{d}t. \end{aligned} \]
The big player gets a larger slice when
Theorem 1.2.
Then
\[ \varphi_P^{\infty, m} = \int_0^1 f(t) \mathrm{d}H(t) \] with \(H(t) = 1 - (1 - G(t))^m\).
Weighted value: Section 5.16
SV for \(m \in \{1, 2, 3\}\) big players
Weighted value: Section 5.16
\[ v(S) = \begin{cases} 0 \; \text{ if } P \notin S \\ f\left(\frac{\#_{A_1}(S)}{n}, \dots, \frac{\#_{A_L}(S)}{n}\right). \end{cases} \]
Theorem. 1.4. Assume that
\[ \implies \varphi_P^\infty = \int_0^1 \dots \int_0^1 f(t_1, \dots, t_L) \mathrm{d}G(t_1, \dots t_L) \]
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 \[ \varphi_{A^l}^\infty = \int_0^1 t \partial_l f(t, \dots, t) \mathrm{d}t. \]
General idea: Section 5.18
Shapley value (diagonal formula)
General idea: Section 5.18
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 \[ \varphi_{A^l}^\infty = \int_0^1 t^{\lambda_P} \lambda_l t^{\lambda_l - 1} \partial_l f(t^{\lambda_1}, \dots, t^{\lambda_L}) \mathrm{d}t. \]
Weighted value with \(\lambda_1 < \lambda_2\)
Hybrid platforms and bargaining power
Examine the bargaining power implications of hybrid platforms and their welfare consequences
Benchmark model
Bargaining model
Logit-like demand for each product \(T_i\)
\[ x_{Ti} = \frac{\exp\left( \frac{v_T - p_{T_i}}{\mu} \right)}{A} \]
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
Proposition 2.3.
The optimal price is an additive markup over marginal costs:
Proposition 2.4.
If \(I_F\) and \(K_F\) are low enough, the equilibrium size of the aggregate is \[ A = \mu \frac{V_F}{K_F + I_F} \]
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 the hybrid regime,
\[ \Pi(N_P, N_F) = \mu \frac{N_F V_F + N_P V_P}{N_F V_F + N_P V_P + 1} \]
\[ \pi^t_P = \int_0^1 \Pi(N_P, sN_F) \mathrm{d}s, \quad \pi^t_{F_i} = \frac{\Pi(N_P, N_F) - \pi^t_P}{N_F} \]
\[ K_F^{impl} = \pi^v_{F_i} - \pi^t_{F_i} \]
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 the hybrid regime
Other case: Section 5.20
Platform’s choice of product variety: Section 5.22
Extensions: Section 5.23
Other case: Section 5.20
Platform’s choice of product variety: Section 5.22
Extensions: Section 5.23
Characterizing multiplayer free-form bargaining
joint work with Mia Lu
How to divide the value between the players?
Timing of agreements: Section 5.24
Details: Section 5.25
All chat messages: Section 5.35
Example chat messages: Section 5.31
There are no profitable deviations
\[ \begin{aligned} \pi_P &\geq 0 \\ \pi_{A_i} &\geq 1 \\ \pi_{A_1} + \pi_{A_2} &\geq 2 \\ \pi_{P} + \pi_{A_i} &\geq 3 \end{aligned} \]
It may be multi-valued, e.g.:
| \(\pi_P\) | \(\pi_{A_1}\) | \(\pi_{A_2}\) |
|---|---|---|
| 2 | 2 | 2 |
| 0 | 3 | 3 |
| 3 | 2 | 1 |
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 |
Maximize the 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 |
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 for \(\lambda \in \{0.5, 1, 3\}\)
Lemma 1.2.. Assume that
Then \[ \begin{aligned} \varphi_P^\infty &= \int_0^1 f(a_1(s), \dots a_L(s)) \mathrm{d}H(s), \\ \varphi_{A^l}^\infty &= \int_0^1 H(s) a_l'(s) \partial_l f(a_1(s), \dots a_L(s)) \mathrm{d}s. \end{aligned} \]
Propositions 2.11, 2.12.