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Appendix A - Game Theory Tools
- Ana Espinola-Arredondo, Washington State University, Felix Muñoz-Garcia, Washington State University
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Summary
We next provide a list of basic tools on game theory that are recurrently used throughout this book. For a more detailed presentation, see Tadelis (2013) and Muñoz-Garcia and Toro-Gonzalez (2019).
BACKGROUND
Consider a setting with N ≥ 2 players (e.g., firms exploiting a CPR), each choosing a strategy sifrom a set of available strategies Si(known as the “strategy set”). In a CPR context, strategy sirepresents the tons of fish that fisherman iappropriates out of the range of feasible appropriation levels. For instance, if it is technologically impossible to appropriate more than 10 tons of fish, the strategy set would be the real numbers between 0 and 10, i.e., Si= [0, 10]. Similarly, let sjrepresent player j 's strategy,where j_= i , from his strategy set Sj ,which may differ from player i ’s, Si , if each firm has access to different technologies; otherwise, Si= Sj= S. For compactness, we often use (si,s −i)to denote a strategy profile where player ichooses siwhile his rivals select s −i , defined as
Finally, we assume that players are rational,meaning that they seek to maximize their payoff function, and that this is common knowledge. In a two-player game, this entails that player 1 tries to maximize his payoff, that player 2 knows that player 1 seeks to maximize his payoff, that player 1 knows that player 2 knows that he seeks tomaximize his payoff…and, so on, ad infinitum. Intuitively, this will help each player put himself in his rival's shoes, anticipating the actions that his rival chooses. For compactness, we refer to this assumption as “common knowledge of rationality.”
The following subsections present different solution concepts that seek to predict how players behave (e.g., which specific appropriation levels they choose) by relying on different behavioral assumptions.
STRICTLY DOMINATED STRATEGIES
In this first solution concept, rather than focusing on which specific strategies each player chooses to maximize his payoff, we seek to delete those strategies that a rational player would never select.
4 - Entry Deterrence in the Commons
- Ana Espinola-Arredondo, Washington State University, Felix Muñoz-Garcia, Washington State University
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INTRODUCTION
In this chapter we relax one of our assumptions in Chapter 3, namely that the entrant joins the CPR in the second period. Importantly, we considered that the entrant exploits the CPR regardless of how depleted the resource becomes after the incumbent's first-period appropriation. However, in more realistic settings, the incumbent may strategically deplete the resource during the first period (choosing a relatively high x ), deterring the potential entrant from joining the CPR. We examine the incumbent's strategic exploitation in this chapter.
Such an entry-deterring behavior is, however, costly for the incumbent, since entry deterrence may require the resource to be exploited more intensively than in the absence of entry threats. This intense exploitation decreases the incumbent's profits in the first period and leaves the CPR relatively depleted, thus also reducing its profits in the second period. As we show in this chapter, the incumbent may find it worthwhile to deter entry when the benefit from doing so (i.e., facing no competition during the second period) offsets its associated cost, but in other contexts the incumbent may find the entry-deterring costs too high, inducing the firm to accommodate entry.
Our analysis is based on Mason and Polasky (1994, 2002) and Espinola-Arredondo andMuñoz-Garcia (2013a). An example of entrydeterring behavior by an incumbent facing entry threats is that of the Hudson's Bay Company. As described in McLean (1849), this furtrading company faced a threat from French traders considering entering the market, which responded by increasing its beaver harvests from 1736 to 1740, decreasing the estimated beaver population from 208,000 to 173,000. Other examples include the Navajo tribesmen increasing their cattle grazing communal meadows to deter other individuals from using this area, as reported in Johnson et al. (1980), or oil companies increasing their extraction rate to deter other companies from the same oil lease, as in Wiggins and Libecap (1985).
Technically, our setting is analogous to that inGilbert and Vives (1986), which considers an oligopolistic market with N incumbent firms, each of them simultaneously and independently committing, in the first period of the game, to a given production level sold in the second period (once the potential entrant has decided whether or not to enter).
Preface
- Ana Espinola-Arredondo, Washington State University, Felix Muñoz-Garcia, Washington State University
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Summary
This textbook offers an introduction to the analysis of common pool resources, such as fishing grounds, aquifers, and forests, using gametheory tools familiar for most undergraduate students in economics, business, and social sciences.
Since Gordon (1954) and Hardin (1968), a large body of literature has emerged – theoretical, but especially experimental and field studies – seeking to understand the main incentives behind individuals and firms exploiting a common pool resource (CPR). These studies also focus on identifying which institutions and information contexts help ameliorate the so-called tragedy of the commons, where every individual ignores the effect that their appropriation causes on other individuals exploiting the resource, leading to its overexploitation. While several authors develop literature reviews, they mostly focus on the institutional arrangements that induce individuals to reduce their appropriation in the commons; see Ostrom (1990,1994, and 2000), Carpenter (2000), Faysee (2005), or Araral (2014).
These are important points, but literature reviews often overlook (or significantly summarize) the mathematical representation of how to find equilibrium behavior in CPRs, how to identify the socially optimal appropriation, how to measure the inefficiencies that arise, and how incomplete information affects equilibrium behavior. This textbook seeks to fill this gap by providing a relatively brief introduction to CPR models and results, specifically targeted to upper-level undergraduate and graduate students.
Our presentation emphasizes the intuition behind each modeling assumption, the steps we need to follow to solve similar CPR problems, and the economic interpretation of each result. In addition, it assumes only a basic background in intermediate microeconomics, and perhaps some game theory, but does not require readers to have a good command of dynamic programming techniques and differential game theory, as opposed to Dockner et al. (2000).1 While CPR problems are often presented using these techniques, we believe that the main incentives behind players exploiting a resource can be discussed without the need to rely on advanced mathematical tools. As a result, we expect our text to be appropriate for undergraduate courses in environmental economics and in natural resource economics for students undergoing economics and business degrees, environmental policy for students undergoing public policy or political science degrees, or as a first introduction to the topic for graduate students.
1 - Introduction
- Ana Espinola-Arredondo, Washington State University, Felix Muñoz-Garcia, Washington State University
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WHAT ARE COMMON POOL RESOURCES?
If we ask you to find examples of common pool resources (CPRs), you may consider fishing grounds, hunting grounds, or forests, along with oil fields, pastures, irrigation systems, and aquifers. Other, more recent, examples may include the use of a computer facility or Wi-Fi internet connections that require no password. But, what are the distinctive features that these examples, andCPRs in general, exhibit? For us to qualify a good or service as a CPR, it needs to satisfy two properties:
1. It must exhibit rivalry(rival goods), that is, its consumption by one individual reduces the amount of the good available to other individuals. This property holds in all the above examples, where a larger fishing catch by one fisherman reduces the available stock that other fisherman can catch; or the internet browsing by one more individual reduces the Wi-Fi speed other individuals can enjoy.
2. It must be nonexcludable , which means that preventing an individual from enjoying the good is costly or impossible. Again, the above examples satisfy this property, since preventing a new fisherman from accessing a fishing ground is relatively costly.
DIFFERENCES BETWEEN CPRS AND OTHER GOODS
How do CPRs differ from the other types of goods and services we encounter every day? Table 1.1 classifies different types of goods according to whether they satisfy the above two properties: the rows consider whether the good is rival, while the columns evaluate whether the good is excludable. As suggested above, CPRs are rival in consumption but nonexcludable, leaving us with three other types of goods to discuss:
a. Private goods: Starting from private goods, such as an apple, we see that its consumption is rival (if you eat it, I cannot enjoy the same apple) and excludable (if you don't pay for an apple, you cannot eat it).
b. Club goods: We can then move on to club goods, such as a gym membership. Club goods are nonrival since the good can be enjoyed by several members without affecting each other's utility, unless the gym becomes congested. In addition, they are excludable since gym owners can easily prevent nonmembers from accessing the center by requiring users to show a membership card.
7 - Signaling in the Commons
- Ana Espinola-Arredondo, Washington State University, Felix Muñoz-Garcia, Washington State University
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INTRODUCTION
In this chapter, we continue our analysis of equilibrium behavior in settings where firms operate under incomplete information. We now examine CPRs that have been exploited by a single firm (the incumbent) and subject to entry threats by a potential entrant. The incumbent, after years of experience appropriating the resource, has access to more accurate information about the available stock than the potential entrant. However, the latter can observe the appropriation decisions of the incumbent, using them as signals that can help the potential entrant infer the stock. We analyze equilibria where the incumbent chooses a different appropriation for each stock level, and thus appropriation is an informative signal of the unobserved stock, i.e., separating equilibria. We then study equilibria where the incumbent selects the same appropriation level regardless of the stock they face. In this context, the entrant cannot infer the available stock upon observing the incumbent's appropriation. This type of equilibria are known as “pooling equilibria” since an incumbent facing different levels of stock pools into the same appropriation level, which conceals the underlying stock from the entrant.
For each of these equilibria, we investigate how their equilibrium appropriation levels differ from those arising under complete information, and evaluate how far away they are from the social optimum. This helps us understand if the static and dynamic inefficiencies that emerge under complete information (see Chapters 2–4) become emphasized when firms operate under incomplete information and face entry threats. We show that, while in some cases this information setting may lead to more severe inefficiencies, in other contexts firms’ incentives under incomplete information may induce them to alleviate part of the inefficiencies arising under complete information. In these situations, the uncertainty that the potential entrant faces about the available stock yields a welfare-improving outcome.
At the end of the chapter, we consider a regulatory agency that does not observe the available stock, thus being as poorly informed as the potential entrant, and evaluate under which conditions this agency may have incentives to invest in acquiring information about the stock value, subsequently distributing this information in media outlets, making the entrant informed; and in which cases it prefers, instead, to remain uninformed.
List of Matrices
- Ana Espinola-Arredondo, Washington State University, Felix Muñoz-Garcia, Washington State University
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Frontmatter
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2 - Common Pool Resources in a Static Setting
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INTRODUCTION
In this chapter, we start our analysis of CPRs from a stylized setting, namely, a one-shot interaction between the agents exploiting the resource, such as the fishermen operating in a fishing ground. This is, of course, a simplifying assumption, since a key feature of CPRs is that an initial stock can be depleted over time. Such depletion affects the intensity with which agents exploit the resource in each period, which we will examine in future chapters. By abstracting from the dynamic aspects of the CPR, however, we gain a clearer understanding of the incentives that agents face within a period, and which are not driven by the stock depletion.
We first investigate appropriation in a CPR such as a fishing ground, a forest, or an aquifer, e.g., how many tons of fish each fishing boat catches.We analyze how this appropriation is affected by the available stock and the number of agents exploiting the resource. Section 2.3 then identifies the socially optimal appropriation of the resource and how it differs from the equilibrium appropriation that agents choose when left unregulated. Section 2.4 elaborates on policies commonly used to induce these agents to exploit the CPR at a socially optimal level.
MODELING THE CPR
Assume that N firms (or individuals) have free access to the resource. Every unit of appropriation (e.g., a ton of fish) is sold in the international market that, for simplicity, is assumed to be perfectly competitive. Intuitively, every fisherman's appropriation (e.g., 20 tons of cod) represents a small share of industry catches, thus not affecting market prices for this variety of fish. As a result, every firm takes the market price pas given, whichwe normalize to p= $1 to facilitate our analysis. (We examine how our results are affected by this assumption in Subsection 2.3.2.)
Cost Function: In addition, every firm faces the following cost Function
where S > 0 denotes the stock of the resource, which reduces fisherman i 's cost when the resource becomes more abundant (intuitively, fish are easier to catch).
List of Figures
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Contents
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5 - Repeated Interaction in the Commons
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INTRODUCTION
Previous chapters discussed games where firms (e.g., fishing companies or farmers sharing an aquifer) interact only once. These games are also known as “one-shot games” or “unrepeated games,” and can help us model strategic settings in which players do not anticipate interacting again in future periods. In many settings, however, the same group of firms interact several times, facing the same game repeatedly. An interesting feature of repeated games is that they can help us rationalize players’ cooperation, even when such cooperation could not be sustained in the unrepeated version of the game.
Section 5.2 presents a simplemodel of a CPR game, highlighting its similarities with the canonical prisoner's dilemma game. This tractable model helps us in our presentation of repeated interaction in finitely repeated games (Section 5.3) or infinitely repeated games (Section 5.4). Cooperative outcomes, understood as firms exploiting the resource below what they would do in an unrepeated game, cannot be sustained in the equilibrium of the finitely repeated game. Intuitively, firms anticipate that they will be appropriating as much as possible in the last period of interaction, and that such behavior will not be affected by previous history of play. In the previousto-last period, they anticipate such exploitation in the subsequent period, which leads all firms to exploit the CPR at maximal levels on the previous-to-last period too. A similar argument extends to all previous periods until the first, implying that firms choose a high appropriation level during all periods; a big failure in our quest to use repeated games as a tool to promote cooperation in the commons! Fortunately, Section 5.4 considers infinitely repeated games, showing that, in this case, firms may have incentives to cooperate by selecting lower appropriation levels if they assign a sufficiently large weight to their future profits.
MODELING REPEATED INTERACTION
Consider the CPR game in Matrix 5.1, where both firms simultaneously and independently choose between a high and a low appropriation level. Firm 1 selects a row, while firm 2 chooses a column. The first payoff in every cell corresponds to firm 1 and the second payoff to firm 2. When both firms choose Low appropriation, at the bottom right-hand corner of the matrix, both earn a payoff of $a .
Common Pool Resources
- Strategic Behavior, Inefficiencies, and Incomplete Information
- Ana Espinola-Arredondo, Felix Muñoz-Garcia
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Common Pool Resources include, for instance, fishing grounds, irrigation systems, forests and the atmosphere. Now more than ever, how we responsibly share and use those goods is a vital issue. This textbook introduces students of economics, business and policy studies to the key issues in the field. It uses a game-theory approach to help readers understand the mathematical representation of how to find equilibrium behavior in CPRs, how to identify the socially optimal appropriation, and how to measure the inefficiencies that arise. Algebra and calculus steps are clearly explained, so students can more easily reproduce the analysis and apply it in their own research. Finally, the book also summarizes experimental studies that tested theoretical results in controlled environments, introducing readers to a literature that has expanded over the last decades, and provides references for further reading.
Bibliography
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3 - Common Pool Resources in a Dynamic Setting
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INTRODUCTION
In this chapter we examine another form of inefficiency in CPRs. While Chapter 2 analyzed a staticinefficiency – that emerged when firms ignore the negative externality that their current appropriation imposes on other firms’ current costs – this chapter explores a dynamicinefficiency, arising from firms ignoring the negative externality that their current appropriation imposes on other firms’ futurecosts.
For presentation purposes, we consider a sequential-move game where, in the first period, only one firm operates in the CPR (e.g., the incumbent), but in the second period two firms compete for the resource (the incumbent and an entrant). This stylized setting helps us isolate the dynamic inefficiency that the incumbent's firstperiod appropriation imposes on the entrant's second-period costs, separating it from the static inefficiency that arises from the two firms’ simultaneous appropriation of the CPR in the second period. Our analysis can be extended to settings with two or more firms operating in both the first and second period, which gives rise to static inefficiencies in each period (where equilibrium appropriation is socially excessive) and a dynamic inefficiency across periods (as every firmignores the negative externality that its first-period appropriation has on its rivals’ future costs), and to contexts where two or more firms interact during more than two periods.
This chapter considers, for simplicity, the entrant's decision as exogenous, that is, the entrant has either joined or not joined the CPR. In subsequent chapters, however, we make this entry decision endogenous, that is, we allow for the potential entrant to evaluate whether entry is profitable. Interestingly, we will also examine which actions the incumbent can take during the first period to prevent entry of potential competitors, and how these actions can, paradoxically, protect the resource from overexploitation or, at least, ameliorate aggregate appropriation.
MODELING CPRS IN A DYNAMIC SETTING
How can we alter the basic setting presented in Chapter 2 to account for dynamic effects in the CPR? In a two-period sequential-move model, every firm ichooses its individual appropriation xi , facing a cost function similar to that in Chapter 2:
Index
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6 - Commons under Incomplete Information
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Summary
INTRODUCTION
Previous chapters assumed that firms perfectly observe all relevant information necessary to operate in the CPR, such as the available stock or the cost externality that they suffer from their rivals’ appropriation. Firms interacted in a complete information game, either simultaneously or sequentially, facing no uncertainty about their own profit function or their rivals’. While such a model can be justified in commons where several firms have operated for long periods, and where technologies are well-known, it may not be a good description of CPRs where all (or some) firms have started to operate and/or have limited experience exploiting a similar resource. In that setting, firms face uncertainty about how abundant the resource is, since their technology does not provide a precise estimate of the stock, but instead a distribution of possible stocks, each with an associated probability (i.e., a probability distribution over stocks).
This chapter explores settings where all firms face uncertainty about the available stock, and must choose their appropriation decisions without observing the exact stock. Firms may, of course, have estimates of this stock, but do not know it with certainty. In this context, firms maximize their expected profits, taking into account the probability associated with each stock level. We find equilibrium appropriation in this setting, and compare it against that arising under complete information, identifying in which cases firms exploit the resource more or less intensively when they operate under uncertainty than otherwise.
We then study another context of incomplete information where, rather than having all firms being uninformed about the stock, only one firm is uninformed while its rival – having more experience exploiting the CPR – observes the available stock. This creates an information asymmetry that the privately informed firm may exploit to its benefit. Similarly, as in our previous discussion, we find equilibrium appropriation in this setting, which involves a stock-dependent appropriation for the informed firm but a stockindependent appropriation for the uninformed firm. Intuitively, the informed (uninformed) firm can (cannot) condition its exploitation decision on the stock level since it can (cannot, respectively) observe the available stock.We also compare equilibrium appropriation under complete and incomplete information, and evaluate if the static inefficiencies we found in Chapter 2 are ameliorated or augmented by the presence of incomplete information.We finish the chapterwith a review of the literature testing CPRs under incomplete information in controlled experiments.
Appendix B - Solutions to Selected End-of-Chapter Exercises
- Ana Espinola-Arredondo, Washington State University, Felix Muñoz-Garcia, Washington State University
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CHAPTER 2 – COMMON POOL EESOURCES IN A STATIC SETTING
Exercise 2.1 – Allowing for Different Cost Externalities
Consider the setting in Section 2.3 and, for simplicity, assume N = 2 firms. In addition, consider that every firm's cost function is
where θ≥ 0 denotes the severity of the cost externality. When θ= 0, firm i 's costs are unaffected by its rival's appropriation qj , whereas when θ >0, firm i 's costs are affected by qj . As a remark, note that our setting in Section 2.3 can be interpreted as a special case of this more general model, where θ= 1.
(a) Find every firm i 's best response function, qi(qj) .
• Each firm chooses its appropriation level qito solve
Following Section 2.3, we normalize the price of the good to $1. Differentiating with respect to qi , we obtain
To find the best response function, we first rearrange the equation to S = 2qi+ θqj , and solving for qiwe find
This indicates that appropriation levels are strategic substitutes, that is, an increase in firm j 's appropriation induces a reduction in firm i ’s.
(b) How is qi(qj)affected by an increase in parameter θ ? Interpret.
• Differentiating qi(qj)with respect to θ , we obtain that
Therefore, when firm jappropriates one more unit, it affects firm i 's cost more significantly as the severity of the cost externality, θ , increases. This intuition goes in line with our interpretation in part (a).
(c) Find the equilibrium appropriation q *i.
• In a symmetric equilibrium, appropriation levels satisfy qi= qj= q *i.
Therefore, as the stock becomes more abundant (higher S), equilibrium appropriation increases.
(d) How is q *iaffected by an increase in parameter θ ? Interpret.
• An increase in θdecreases the equilibrium appropriation. We can find the exact amount by which it decreases by taking a derivative:
A higher θincreases the negative externality from the other firm's exploitation of the resource, which increases the marginal cost for firm i . The increase in marginal cost (absent a change in the price, or marginal revenue, of the good) reduces the equilibrium appropriation from the firm.
Regulators and environmental groups: better together or apart?
- Ana Espinola-Arredondo, Eleni Stathopoulou, Felix Munoz-Garcia
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- Environment and Development Economics / Volume 27 / Issue 1 / February 2022
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- 09 March 2021, pp. 40-66
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This paper examines green alliances between environmental groups (EGs) and polluting firms, which have become more common in the last decades, and analyzes how they affect policy design. We first show that the activities of regulators and EGs are strategic substitutes, giving rise to free-riding incentives on both agents. Nonetheless, the presence of the EG yields smaller welfare benefits when firms are subject to regulation than when they are not. In addition, the introduction of environmental policy yields large welfare gains when the EG is absent but small benefits when the EG is already present.
International coordination of environmental policies: is it always worth the effort?
- Jude Bayham, Félix Muñoz-García, Ana Espínola-Arredondo
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- Environment and Development Economics / Volume 24 / Issue 3 / June 2019
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- 11 March 2019, pp. 294-316
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We study entry policies as an alternative form of environmental policy. Given the strong political opposition to standard output subsidies and taxes, several countries have recently used entry policies to promote renewable energy technology, such as solar panels and biofuels. We study a two-stage game in which two regulators choose an entry policy (i.e., tax, subsidy or permit) to maximize domestic welfare. Observing the policy, firms decide the region in which to enter and compete as Cournot oligopolists. We find that both domestic (uncoordinated) policies and internationally coordinated policies increase welfare relative to unregulated settings. However, the welfare gains from international policy coordination are only large when the product is extremely clean. These results indicate that the welfare gains of international policy coordination may only offset the costs of negotiation in relatively clean industries.
Why do firms oppose entry-deterring policies? Environmental regulation and entry deterrence
- Ana Espínola-Arredondo, Félix Muñoz-García
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- Environment and Development Economics / Volume 20 / Issue 2 / April 2015
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- 23 April 2014, pp. 141-160
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This paper investigates the design of environmental regulation under different regimes: flexible and inflexible policies. We analyze under which settings strict emission fees can be used as an entry-deterring tool, and become socially optimal. Furthermore, we demonstrate that the incentives of the social planner and the incumbent firm are aligned regarding policy regimes ifentry can be easily deterred by setting a stringent regulation. Their incentives, however, can bemisaligned when entry becomes more costly to deter, leading the incumbent to actually preferenvironmental policies that attract entry.