The expected externality mechanism in a level-\(k\) environment
The author studies the expected externality (AVG) mechanism of \textit{C. d'Aspremont} and \textit{L.-A. Gerard-Varet} [Cah. Cent. Étud. Rech. Opér. 18, 167--172 (1976; Zbl 0324.90108)] in the level-\(k\) model, where agents maximize their payoffs against a non-equilibrium belief about other agents' strategies by performing at most \(k\) iterations of best reply. It is obtained that, if level-\(0\) agents truthfully reveal their preference types to the principal, then the AVG mechanism leads to efficient implementation, while, if level-\(0\) agents report distorted preferences, then the AVG mechanism does not induce truth-telling and produces a suboptimal allocation, and the deviation from the optimal strategy tends to zero as \(k\) tends to infinity. The author also shows that, by introducing a betting round -- where agents guess the reports of each other -- and an incentive transfer rewarding truth-telling, the AVG mechanism can be altered to produce close-to-efficient allocation even if level-\(0\) agents report distorted preferences and the principal does not know the deviation of reported preferences from the true preference distribution. The level-\(k\) model is a game of incomplete information where for each agent \(i \in I\), with type \(\theta_{i} \in \Theta\), the payoffs are given by \(u_{i}(s,\theta_{i})\) where \(s=(s_{1},s_{2},\dots,s_{n})\) is a strategy profile, i.e., \(s_{j}\) associates actions to types. The types of agents have a commonly known distribution \(F\). Agents engage in iterations of best reply to maximize their payoff. Their utility is of the form \(v_{i} (x,\theta_{i})+T_{i}\) where \(x\) is the allocation and \(T_{i}\) is the monetary transfer to agent \(i\). The functions \(v_{i}\) are assumed to satisfy the Spence-Mirrlees condition. The strategy \(s^{(k)}_{i}(\theta_{i})\) is recursively defined to maximize the expected payoff against opponents making exactly \(k-1\) iterations of best reply. Level-\(0\) agents are nonstrategic, modeled by assuming that either they reveal their type truthfully, or reports a type randomly according to a common -- known or unknown -- distribution \(\Phi\). The author illustrates some of the main results of the paper by the following example. Suppose that there are \(n\) agents, and \(v_{i} (x,\theta_{i}) = \theta_{i}x-x^{2}/2\). Then the efficient allocation is \(x = \sum_{i}\theta_{i}/n\), while the optimal level-\(k\) strategy is \[ s^{(k)}_{i}(\theta_{i}) = \theta_{i}+\Delta\left(-\frac{n-1}{n}\right)^{k+1} \] where \(\Delta = \int \theta d F(\theta) - \int s d \Phi(s)\) is the difference between the average type and the average random move of a level-\(0\) agent. The author obtains that, if in addition to \(F\), both \(\Phi\) and \(k\) are known by the principal, then by changing the incentive transfer to level-\(1\) agents, the principal can implement efficient allocation. If either \(k\) or \(\Phi\) is unknown by the principal, then a betting round can be added to the mechanism -- in which the agents guess the level of the other agents if \(k\) is unknown, or report \(\Phi\) if \(\Phi\) is unknown -- and the incentive transfer can be customized to induce truth-telling.
scientific article; zbMATH DE number 6858289