Discrete Dynamics in Nature and Society

Volume 2017, Article ID 2768045, 8 pages

https://doi.org/10.1155/2017/2768045

## Strategic Uncertainty in Markets for Nonrenewable Resources: A Level- Approach

^{1}Helmut Schmidt University/University of the Federal Armed Forces, Hamburg, Germany^{2}WZB Berlin Social Science Center, Berlin, Germany

Correspondence should be addressed to Armin Fügenschuh; ed.hh-ush@huhcsnegeuf

Received 20 June 2017; Revised 16 September 2017; Accepted 18 September 2017; Published 30 October 2017

Academic Editor: Chris Goodrich

Copyright © 2017 Ingmar Vierhaus et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Existing models of nonrenewable resources assume that sophisticated agents compete with other sophisticated agents. This study instead uses a level- approach to examine cases where the focal agent is uncertain about the strategy of his opponent or predicts that the opponent will act in a nonsophisticated manner. Level-0 players are randomized uniformly across all possible actions, and level- players best respond to the action of player . We study a dynamic nonrenewable resource game with a large number of actions. We are able to solve for the level-1 strategy by reducing the averaging problem to an optimization problem against a single action. We show that lower levels of strategic reasoning are close to the Walras and collusive benchmark, whereas higher level strategies converge to the Nash-Hotelling equilibrium. These results are then fitted to experimental data, suggesting that the level of sophistication of participants increased over the course of the experiment.

#### 1. Introduction

Existing models of nonrenewable resource markets assume that optimizing agents compete against other optimizing agents in environments characterized by Cournot or Stackelberg competition (see [1–5] and many others). This implies that agents will, in the Nash equilibrium, have a perfect understanding of what their opponents will do. In reality, however, agents may experience considerable uncertainty about the strategy the opponent will follow. For example, agents may not have all the information (such as cost functions, stock sizes) required to calculate an opponent’s Nash equilibrium strategy [6, 7]. In such cases, they must form beliefs about the possible strategies chosen by the opponent and maximize their expected profits given these beliefs.

Similarly, even with perfect information, agents may in some cases expect their opponent not to follow the Nash-Hotelling equilibrium. For example, the opponent may face political pressure to maximize current revenue and produce at maximal capacity or otherwise be too focused on maximizing present revenue rather than discounted profits (see, e.g., [8–10]). In such cases, agents must again form beliefs about the possible strategies chosen by the opponent and maximize their expected profits given these beliefs.

We model the ensuing uncertainty about the opponent’s chosen strategy using a level- framework (see, e.g., [11–16]), a commonly used approach to model nonequilibrium opponents in behavioral economics. The framework starts by specifying the strategy for a level- player, who is argued to choose a random production trajectory from the set of all possible trajectories. Level- players then best respond to the strategy of a level- opponent. We use a standard linear demand function that allows us to compute the Nash, collusive, and Walrasian benchmark. We then show that higher level strategies converge to the Nash-Hotelling equilibrium, while lower level strategies may closely approximate the Walras and collusive benchmark. Finally, we fit the results to experimental data from van Veldhuizen and Sonnemans [17] to empirically estimate the distribution of types and find that participants appear to be using higher level strategies in latter parts of the experiment.

The contribution of this paper is threefold. First, we add to the literature on nonrenewable resources by suggesting a novel way to analyze nonequilibrium behavior. Second, we contribute to the literature on level- reasoning in behavioral economics by applying level- to the novel setting of nonrenewable resources. Third, we fit our theoretical results to data from an existing laboratory experiment.

#### 2. Model and Benchmarks

We assume that two players, player and player , are active in a nonrenewable resource market characterized by linear demand and Cournot competition. (The reason for giving players integer numbers as identifiers is that we later will assume player plays the level-.) Both players are assumed to start with a fixed stock of resources () which they can allocate over a discrete number of time periods (). The quantity of the resource extracted by player in period is denoted as . Further, let the set of quantities player allocated over the periods be denoted by . We will refer to as player ’s trajectory in the remainder of the text.

Given two trajectories of players and , respectively, and assuming a linear demand function, the profit for player is defined asHere, and are parameters of the demand function and is the discount factor. In line with the lab experiment, we will use symmetric firms and take , , , , and .

Player ’s problem is to choose a strategy that maximizes the sum of discounted profits, subject to the resource constraint:

Player solves an analogous problem. There are three relevant benchmarks to consider: the Nash equilibrium, collusion, and Walras. For the Nash equilibrium, both players maximize their own profits given the strategy of the opponent. In the collusive case, the two players maximize joint profits, and in the Walras case, each player maximizes its profits while taking prices as given. Table 1 shows the relevant trajectories; for a more detailed derivation see van Veldhuizen and Sonnemans [17].