SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Welfare is an important quantity in policy analysis. Welfare reflects the cumulative benefits of all market participants. Thus, if redistribution between people is possible (e.g., through government transfers), increase in welfare means increase in utility for all people given the right redistribution scheme.
Now, we want to evaluate how data-sharing in duopoly case affects the expected welfare. This will help us to understand in which cases the firms should be incentivized to collaborate with each other. By direct substitution, we get
$$
W ^ {e} = q _ {1} ^ {*} + q _ {2} ^ {*} - \frac {\left(q _ {1} ^ {*}\right) ^ {2} + \left(q _ {2} ^ {*}\right) ^ {2} + 2 \gamma q _ {1} ^ {*} q _ {2} ^ {*}}{2} + B - c _ {1} ^ {e} q _ {1} ^ {*} - c _ {2} ^ {e} q _ {2} ^ {*}.
$$
We show that collaboration increases welfare in the Cournot case if the marginal costs of the firms are close enough:  $|c_1^e - c_2^e| \leq \frac{\gamma}{6}$ . To accomplish that, we will first demonstrate that  $\frac{\partial W}{\partial c_i^e} < 0$  if  $c_i^e > c_{-i}^e$  and then show that  $W$  decreases along the line  $c_1^e = c_2^e = c$ . These properties will show
that collaboration increases welfare since after collaboration the costs of the companies equalizes and become smaller.
The proof of the first part follows from the derivations below. W.l.o.g. assume that  $n_1 \geq n_2$ , then  $c_1^e \leq c_2^e$ . Direct computations show
$$
\frac {\partial W ^ {e}}{\partial c _ {2} ^ {e}} = \frac {\partial q _ {1} ^ {*}}{\partial c _ {2} ^ {e}} (1 - c _ {1} ^ {e}) + \frac {\partial q _ {2} ^ {*}}{\partial c _ {2} ^ {e}} (1 - c _ {2} ^ {e}) - \frac {\partial q _ {1} ^ {*}}{\partial c _ {2} ^ {e}} (q _ {1} ^ {*} + \gamma q _ {2} ^ {*}) - \frac {\partial q _ {2} ^ {*}}{\partial c _ {2} ^ {e}} (q _ {2} ^ {*} + \gamma q _ {1} ^ {*}) - q _ {2} ^ {*}.
$$
The results of Section 4.1.2 give
$$
q _ {i} ^ {*} = \frac {2 - \gamma - 2 c _ {i} ^ {e} + \gamma c _ {- i} ^ {e}}{4 - \gamma^ {2}}.
$$
Substituting the expressions above, we have
$$
\begin{array}{l} \frac {\partial W ^ {e}}{\partial c _ {2} ^ {e}} = \frac {(1 2 - \gamma^ {2}) c _ {2} ^ {e} - (8 \gamma - \gamma^ {3}) c _ {1} ^ {e} - (1 2 - 8 \gamma - \gamma^ {2} + \gamma^ {3})}{(4 - \gamma^ {2}) ^ {2}} \\ <   \frac {(1 2 - \gamma^ {2}) c _ {2} ^ {e} - (8 \gamma - \gamma^ {3}) \left(c _ {2} ^ {e} - \frac {\gamma}{6}\right) - (1 2 - 8 \gamma - \gamma^ {2} + \gamma^ {3})}{(4 - \gamma^ {2}) ^ {2}} \\ = \frac {- (1 2 - 8 \gamma - \gamma^ {2} + \gamma^ {3}) (1 - c _ {2} ^ {e}) + \frac {\gamma}{6} (8 \gamma - \gamma^ {3})}{(4 - \gamma^ {2}) ^ {2}}. \\ \end{array}
$$
The restriction  $c_{2} \leq 1 - \frac{\gamma}{2}$  gives
$$
\frac {\partial W ^ {e}}{\partial c _ {2} ^ {e}} \leq \frac {- (1 2 - 8 \gamma - \gamma^ {2} + \gamma^ {3}) \frac {\gamma}{2} + \frac {\gamma}{6} (8 \gamma - \gamma^ {3})}{(4 - \gamma^ {2}) ^ {2}}
$$
This inequality is equivalent to the following inequality
$$
3 6 - 2 4 \gamma - 3 \gamma^ {2} + 3 \gamma^ {3} - 8 \gamma + \gamma^ {3} > 0 \Longleftrightarrow (3 6 - 3 \gamma^ {2}) (1 - \gamma) + \gamma^ {3} > 0.
$$
The last inequality holds since  $\gamma < 1$ .
The proof of the second part comes from the computations below. Assume  $c_1^e = c_2^e = c$ , we want to show that  $W$  decreases in  $c$ . We have
$$
\begin{array}{l} W ^ {e} = \frac {(1 2 - 8 \gamma - \gamma^ {2} + \gamma^ {3}) c ^ {2} - (2 4 - 1 6 \gamma - 2 \gamma^ {2} + 2 \gamma^ {3}) c + 1 2 - 8 \gamma - \gamma^ {2} + \gamma^ {3}}{(4 - \gamma^ {2}) ^ {2}} + B \\ = \frac {(1 2 - 8 \gamma - \gamma^ {2} + \gamma^ {3}) (1 - c) ^ {2}}{(4 - \gamma^ {2}) ^ {2}} + B. \\ \end{array}
$$
Clearly, this function is decreasing in  $c$  for  $c < 1$ .
F Additional experiments for Section 7