Cooperation and efficiency in markets
The book under review studies the theory of repeated games in oligopoly theory and analyzes countably infinite repetition (or supergame) of interactions between firms on both sides of a market with dicounting of future profits. The author proves (1) a sufficient condition for the existence of an equilibrium, (2) efficiencies of the equilibria (no waste of resources), (3) a condition for welfare economics. The book shows that all continuation equilibrium payoff vectors are strictly Pareto efficient and natural oligopoly. The grand coalition can weak Pareto improve the vector of payoffs in the repeated game by abandoning the punishment. Thus, the equilibrium strategy profile proposed in this book would not be renegotiation-proof. \textit{A. Rubinstein} [``Strong perfect equilibrium in supergames'', Int. J. Game Theory 9, 1--12 (1980; Zbl 0433.90093)] developed a strong perfect equilibrium solution for infinite horizon noncooperative games that take into account deviations by all coalitions without imposing restrictions either on the strategies of individual player or on the strategy profiles to which a coalition can deviate. This is an application of Aumann's concept of a strong Nash equilibrium for strategic form noncooperative games to supergames. A strong perfect equilibrium fails to exist if the weak Pareto efficient frontier of the set of individual rational stage game payoff vectors has no sufficiently large flat portion. The model in this book does not have this weak Pareto efficient frontier, it is strictly Pareto efficient. The solution concepts for supergames take into account only the deviations by the grand coalition. The strict renegotiation-proof equilibrium is a subgame perfect equilibrium with the property that the grand coalition cannot weakly Pareto improve the vector of continuation payoffs in any subgame. \textit{J. Farrell} and \textit{E. Maskin} [``Renegotiation in repeated games'', Games Econ. Behav. 1, No. 4, 327--360 (1989; Zbl 0754.90082)] introduced a weakly renegotiation-proof equilibrium. Chapter 1 gives an introduction to the book and background as well as mathematical notations. Chapter 2 demonstrates the stage game between firms in the model and the supergame with discounting of payoffs. There is a nonempty finite set \(J\) of producers and a nonempty finite set \(I\) of buyers. The cardinality of \(J\) is denoted by \(|J|\). Then \(1\leq |J| < \infty, 1\leq |I| < \infty\). A coalition is a nonempty subset of \(J \cup I\). Thus, the set of all coalitions equals \(2^{J\cup I} \setminus \{\emptyset\}\), with cardinality \(2^{|I|+|J|}-1\). For each \(j\in J\), there exists the upper bound \(\chi_j\) on the output of good \(j\), and the producer \(j\in J\) has the cost function \(c_j: Y_j = [0, \chi_j] \to {\mathbb R}_+\). For each buyer \(i\in I, U_i: X \to {\mathbb R}_+\) is the buyer's revenue function, where \(X =\{ (x_i)_{i\in I}= ( (x_{ji})_{j\in J})_{i\in I} \in Y^{| I|} \}\) with quantity \(x_{ji}\) of good \(j\in J\) purchased by \(i\in I\). Let \(\theta_j(x) = \sum_{i\in I} x_{ji}\) and \(\theta (x) = (\theta_j(x))_{j\in J}\) be the output vector (each producer's output equals the sum of his/her deliveries to buyers specified by \(x\)). The model requires the following basic assumptions. {\parindent=4mm \begin{itemize}\item[--] Assumption 2.1. For each \(j, J\), \(c_j\) is (i) continuous, (ii) strictly increasing, and (iii) \(c_j(0) > 0\). \item[--] Assumption 2.2. For each \(i\in I\), \(c_i\) is (i) continuous and (ii) \(c_i(0)>0\). \item[--] Assumption 2.3. For each \(i\in I, U_i\) is (i) continuous and (ii) \(x_i=0\) implies \(U_i(x) =0\). \item[--] Assumption 2.4. There exists \(x^+\in X\) such that \[ \sum_{i\in I, x_i^+>0} ( U_i(x) - c_i(x)) > \sum_{j\in J, \theta_j(x^+)>0} c_j(\theta_j(x^+)). \] \end{itemize}} Define \[ X^{\max}= \text{arg}\max \bigg\{\sum_{i\in I} U_i(x) - \sum_{j\in J, \theta_j(x)>0} c_j(\theta_j(x))- \sum_{i\in I, x_i>0} c_i(x_i)\mid x\in X\bigg\}. \] Each \(x^{\max}\in X^{\max}\) is a vector of traded quantities that maximizes the surplus from the trade in the market, and \(x^{\max}\neq 0\) vector by Assumption 2.4. The maximization problem can be solved in two steps: {\parindent=6,5mm \begin{itemize}\item[(1)] for each coalition \(D\) with \(D\cap I \neq \emptyset\), \(D\cap J \neq \emptyset\), solve the maximization problem \[ \begin{multlined} \xi_D = \max \bigg\{\sum_{i\in D\cap I} U_i(x) - \sum_{j\in D\cap J, \theta_j(x) > 0}c_j(\theta_j(x)) - \sum_{i\in D\cap I, x_i > 0} c_i(x_i)\,\Big| \\ x\in X,\;x_{ij}=0,\;(j, i)\not \in (D\cap J)\times (D\cap I)\bigg\};\end{multlined} \] and denote the set of its solutions by \(X^{(D)\max}\); \item[(2)] for each \(F\in \text{arg}\max\{\xi_D\mid D\cap J\neq \emptyset\), \(D\cap I \neq \emptyset\}\), set \(X^{(\max)}= \cup \{X^{(F)\max}: F\in \text{arg}\max \{\xi_D\mid D\cap J\neq \emptyset\),\( D\cap I \neq \emptyset\}\}\). \end{itemize}} Let \(V\) be the set of payoff vectors in the game that can result from the pure strategy profiles. The model allows objectively correlated strategies and a strict Pareto efficient frontier. The behavior strategy of the firm \(k\in J\cup I\) is the function \(s_k: H_f = \bigcup_{t\in N}H^{(t)} \to S_k\), where \(H^{(t)}\) is the set of histories leading to period \(t\), \(S_k\) is the set of the behavior strategies of the firm \(k\). The function \(\pi_k: S_k \to {\mathbb R}\) is the firm \(k\)'s payoff function, \(\pi = (\pi_k)_{k\in J\cup I}\) is the payoff function for the market. A strict renegotiation-proof equilibrium (SRPE) is a strategy profile \(s^*\in S= \prod_{k\in J\cup I}S_k\) such that (i) there is no history \(h, k\in J\cup I, s_k\in S_{k(h)}\) with \[ \pi_{k(h)} \big((s_k, s^*_{-k(h)})\big) > \pi_{k(h)}(s^*_{(h)}), \] and (ii) there does not exist a nonterminal history \(h\), and a strategy \(s\in S_{(h)}\) such that \(\pi_{k(h)}(s) \geq \pi_{k(h)}(s^*_{(h)})\) for all \(k\in J\cup I\) and there exists a \(k\in J\cup I\) with \(\pi_{k(h)}(s) > \pi_{k(h)}(s^*_{(h)})\). Similarly, a strict strong perfect equilibrium (SSPE) is a strategy profile \(s^*\in S\) that no coalition in no subgame can increase the expected average discounted profile of at least one of its members without decreasing the expected average discounted profit of any other member by a deviation from the \(s^*\). Both assumptions, setups and definitions are given in Chapter 2. Proposition 3.1 gives a sufficient condition for the existence of a strict renegotiation-proof equilibrium (SRPE), Proposition 3.2 gives a condition for the existence of a strict strong perfect equilibrium (SSPE). In a strict renegotiation the maximized sum of the firm's expected average discounted profits, along the equilibrium path, the active producer of each type of good can produce their total output prescribed by the equilibrium with lower or the same total production cost as other producers. These active producers form a natual oligopoly, and the efficiency of the strict renegotiation-proof equilibrium (SRPE) in Section 4.1. In Section 4.2, when all buyers are retailers, in the same situation as in Section 4.1, the active buyers can sell the vector of their sales prescribed by the equilibrium with lower or the same total selling cost as other buyers. The groups of these active buyers form a natural oligopsony or monopsony. The impact of collusion in a strict renegotiation-proof equilibrium (SRPE) on consumer welfare is discussed in Section 4.3. The first comparison is done with a monopsonist choosing the traded quantities on the demand side and price taking behavior on the supply side of the analyzed market, the second comparison with price taking behavior on the demand side and a Cournot oligopoly on the supply side of the analyzed market. In the last Chapter 5, the author summarizes the results obtained in this book over 20 years research. The possible positive impact of collusion on consumer welfare suggests that the antitrust policy in the US and the competition policy in the European Union should not view collusion that involves firms on both sides of a market as illegal. The collusive scheme studied in this book represent the agreements about the sequences of contracts between firms, and they are beneficial for consumers. If a collusive scheme can be legally implemented, then the firms would be in favor to implementation to a merger, from points of view on both participating firms and antitrust authorities. The results of the book may be useful in adjusting antitrust policy and competition policy in the world. It would be interesting to further explore the necessary and sufficient conditions for the existence of a strict renegotiation-proof equilibrium (SRPE), and the uniqueness. Applying the theoretical results in this book into the real world would be challenging for both producers and buyers.
scientific article; zbMATH DE number 5872940