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Commons Dilemmas and Game Theory
This lecture explores commons dilemmas through the lens of game theory, examining how strategic decision-making leads to collective action problems and investigating potential solutions to these dilemmas.
Introduction to Game Theory and the Prisoner’s Dilemma
Game theory provides a powerful framework for understanding strategic interactions between rational decision-makers. While game theory has broad applications in microeconomics, particularly in analyzing competitive firm behavior and market strategies, this discussion emphasizes its applications to environmental and sustainability issues. The prisoner’s dilemma stands as the most famous game theory model and lies at the core of many sustainability challenges, including climate change.
The Classic Prisoner’s Dilemma Setup
The prisoner’s dilemma involves two suspects who have been charged with a crime. The police interrogate these prisoners in separate rooms, preventing any communication between them. The police are seeking to obtain a confession from at least one of the suspects.
To make the scenario concrete, suppose the two suspects allegedly robbed a bank while it was closed. The police have proof that both suspects were in the bank after hours and that they broke into it. However, the police have no proof that the suspects actually stole anything. Perhaps security cameras show them entering the building, but it remains unclear whether they or someone else actually committed the theft.
The Payoff Matrix Structure
In this game, Suspect A and Suspect B are the two players. Each player has two possible strategies: remain silent or blame the other. The payoffs under all possible strategy combinations must be carefully considered.
If both suspects remain silent, they can only be charged with the lesser crime of breaking and entering. In this case, each suspect receives two years in jail. When expressing these outcomes mathematically, a negative sign is applied because jail time represents a bad outcome. Thus, if both remain silent, Suspect A receives negative two (representing two years in jail) and Suspect B receives the same.
If both suspects blame each other, they can be convicted of bank robbery, resulting in seven years in jail for each. This outcome is represented as negative seven for both players.
The critical asymmetric outcomes occur when one suspect blames while the other remains silent. If Suspect A blames Suspect B while B remains silent, then A goes free with zero years in jail, achieving the best possible personal outcome. This is equivalent to receiving a plea deal. In this scenario, Suspect B faces severe consequences: because Player A blamed them, the police have strong evidence. B gets convicted of bank robbery and receives an additional year for lying under oath, totaling negative eight years.
The reverse situation applies if Suspect B blames while Suspect A remains silent. Suspect A receives negative eight years, while Suspect B goes free with zero years.
Classroom Demonstration of the Prisoner’s Dilemma
To demonstrate the prisoner’s dilemma with real stakes, volunteers can play the game for class participation points. Participants receive eight class points for volunteering, but will lose points according to the payoff matrix based on their decisions. If both blame each other, each loses seven points. If both remain silent, each loses only two points, keeping six points. If one blames while the other stays silent, the outcomes are zero and eight respectively.
A crucial modification to the standard prisoner’s dilemma can be introduced by allowing participants up to thirty seconds to communicate with each other. This fundamentally changes the game dynamics because communication represents one of the primary ways to overcome the prisoner’s dilemma.
When participants can communicate, they often achieve the cooperative outcome of mutual silence. Several factors contribute to this result that differ from the standard prisoner’s dilemma setup. First, the presence of an audience creates a commitment mechanism because participants may care about their reputation beyond just the game payoffs. Second, and perhaps most importantly, participants know they will see each other for the remainder of the semester, unlike two strangers who will never meet again.
Empirical evidence demonstrates that the more anonymous the participants and the greater the probability they will never encounter each other again, the more accurately people converge on the prisoner’s dilemma outcome of mutual blame. Conversely, when participants know who the other person is, outcomes tend toward mutual cooperation.
Solving the Prisoner’s Dilemma: Dominant Strategies
The standard economic approach to solving games like the prisoner’s dilemma involves identifying dominant strategies. A dominant strategy is a choice that is better regardless of what the other player does.
To identify dominant strategies mechanically, consider one player at a time and determine their best response given each possible action by the other player. Starting with Player A, examine what happens if Suspect B remains silent. Comparing the payoffs for Player A in this column, zero (from blaming) is greater than negative two (from staying silent). The better option receives a star. Repeating for the second column where Suspect B blames, negative seven is better than negative eight.
The analysis then flips to consider Suspect B’s perspective. If Suspect A stays silent, Suspect B’s payoffs are negative two (from staying silent) or zero (from blaming). Zero is better. If Suspect A blames, Suspect B’s payoffs are negative eight (from staying silent) or negative seven (from blaming). Negative seven is better.
The conclusion is that blame is a dominant strategy for Suspect A because regardless of what Suspect B does, A is better off blaming. Simultaneously, blame is a dominant strategy for Suspect B because regardless of what A does, B is better off blaming.
The fundamental behavioral statement is that rational agents will always play a dominating strategy.
Nash Equilibrium
When both players have a dominant strategy, a Nash equilibrium emerges. A Nash Equilibrium is a situation where no player can increase their payoff by unilaterally changing their strategy. The term “unilaterally” is crucial here: it means that if only one player changes their behavior while the other’s choice remains fixed, there is no way for that player to improve their outcome. This definition says nothing about cases where coordination might allow both players to simultaneously change their behavior.
A simpler operational definition of Nash equilibrium in a normal form game is any cell (or pair of choices) where both players have stars indicating their best responses. In the prisoner’s dilemma, there is only one Nash equilibrium: mutual blame.
To verify this is an equilibrium, consider whether either player would benefit from unilateral deviation. From the blame-blame outcome, if Suspect A considers switching to remain silent, they would receive negative eight instead of negative seven, making them worse off. Additionally, the other suspect would go free, which seems particularly unfair. The same logic applies to Suspect B.
The concept is called an equilibrium partly because economists favor equilibrium concepts, but more substantively because it provides powerful predictions. In situations involving rational decision-making, the Nash Equilibrium is very likely to occur. Understanding how to calculate Nash Equilibria enables strategic action. In business contexts, this knowledge improves profit maximization. More generally, like other equilibria in economics such as the supply-equals-demand equilibrium, Nash Equilibria are useful because forces push outcomes toward them. The marble-in-a-bowl analogy captures this: disturbances to the system tend to return to the equilibrium state.
The Normalized Prisoner’s Dilemma
The prisoner’s dilemma can be confusing when discussed in terms of years in prison with negative values. A normalization helps clarify the analysis by transforming outcomes into utility terms. Instead of years in prison, the payoffs represent utility levels: prison reduces utility below what would occur without imprisonment. This transformation connects back to basic preference theory. In the normalized version, the payoffs might show positive utilities that reflect how years in prison affect overall well-being. The structure remains a prisoner’s dilemma where mutual cooperation yields better outcomes than the Nash equilibrium of mutual defection.
Formal Definition of the Prisoner’s Dilemma
A prisoner’s dilemma is formally defined as a game with parameters (the numbers in the payoff matrix) such that two conditions hold. First, a dominant strategy must exist for both players. Second, both players end up worse off when they play their dominant strategies compared to if they had both played the alternative.
Connection to the Tragedy of the Commons
The prisoner’s dilemma connects directly to environmental commons problems. Resources with common access tend toward overuse, a market failure that occurs when there is no mechanism to limit access. This dynamic was famously analyzed as the Tragedy of the Commons by Garrett Hardin in 1968.
Hardin developed the most famous version of this scenario using the example of a commons where herders must decide how many goats or sheep to graze on communal pastureland. Every rancher has an incentive to increase their herd size because they capture all the benefits of additional animals while sharing the costs of overgrazing with all users. This incentive structure inevitably leads each herder to add animals until the commons collapses from overgrazing.
The tragedy of the commons is a tragedy precisely because it represents a prisoner’s dilemma that always drives outcomes toward the negative solution. Instead of silence versus blame, the choices become “limit your goats” versus “don’t limit your goats.” Every herder has an incentive not to limit their animals, and this collective behavior causes the collapse of the commons.
The Iterated Prisoner’s Dilemma
The standard prisoner’s dilemma contains unrealistic assumptions. Perhaps the most significant is that real-world interactions rarely involve one-shot games. Instead, people interact repeatedly over time.
In reality, reputation matters. Someone who consistently cheats, takes too much, or fails to contribute to shared responsibilities develops a reputation. Consider the mundane example of dishes in a shared living situation. If the game were truly one-shot, perhaps leaving dishes for a roommate to clean would be optimal. However, this game repeats every day, and reputation effects come into play.
Real-world situations more closely resemble the iterated prisoner’s dilemma, which unfolds over multiple rounds. This repetition fundamentally changes the strategic landscape because decisions in one round can depend on observations of opponent behavior in previous rounds. This creates information that is useful for predicting future opponent choices.
Laboratory experiments on the iterated prisoner’s dilemma reveal that outcomes depend on numerous factors including communication skills, trust between players, how well participants know each other, and whether they like each other. Personal feelings can enter utility functions: someone who dislikes their opponent might accept mutual defection outcomes that they would reject with a liked opponent.
Classroom Demonstration of the Iterated Prisoner’s Dilemma
Participants can be paired to play an iterated prisoner’s dilemma over six rounds. The payoff matrix maintains the prisoner’s dilemma structure with defection as a dominant strategy in any single round. Mutual cooperation yields ten points each, mutual defection yields zero each, and asymmetric outcomes yield twenty-five for the defector and negative five for the cooperator.
Each round, participants secretly write their strategy (cooperate or defect), then simultaneously reveal. They record individual scores and track cumulative totals. Brief communication periods between rounds allow for apology, persuasion, or strategic discussion.
Several strategic dynamics emerge from iterated play. The final round presents a unique strategic consideration because reputation costs disappear after the game ends. One possible strategy involves establishing cooperation through the early rounds, then defecting on the final round. If a pair cooperated five times, they would each have fifty points entering the final round. Mutual cooperation yields sixty total, but unilateral defection yields seventy-five for the defector (fifty plus twenty-five) with no reputational consequence since no seventh round exists.
When teams can divide winnings, a strategy emerges where one player consistently cooperates while the other consistently defects, maximizing total team points for later division. When payoffs cannot be shared, this strategy requires genuine altruism from the cooperating partner.
Real-World Applications of Iterated Prisoner’s Dilemmas
Iterated prisoner’s dilemmas appear throughout real-world strategic interactions. International trade provides a clear example. Tariffs and trade wars fundamentally involve tit-for-tat strategies in an iterated prisoner’s dilemma framework.
Countries face choices about reducing tariffs, which represents cooperation. If all countries reduce tariffs, freer trade benefits everyone. However, if all countries except one maintain free trade, the defecting country can extract significant benefits while free-riding on others’ openness. This creates a standard prisoner’s dilemma structure. When one country initiates trade restrictions, others typically respond in kind.
The actual strategies observed in international trade are quite complicated, reflecting the iterated nature of these interactions and the importance of reputation and future relationship considerations.
Beyond the Prisoner’s Dilemma: Other Game Structures
The prisoner’s dilemma represents just one possible game structure. By varying the payoff parameters in a two-player, two-strategy normal form game, at least seven distinct game types can be derived. Understanding these variations matters because not all strategic interactions are prisoner’s dilemmas, and different real-life negotiation scenarios correspond to different game structures.
The Stag Hunt (Coordination Game)
The stag hunt derives from a philosophical parable about two hunters who leave their separate cabins to hunt. They cannot coordinate before departing. Each hunter must choose between hunting stag or hunting rabbits.
Rabbit hunting requires only one person and essentially guarantees a meal. Stag hunting offers a much larger reward but requires two hunters working together. A single hunter cannot take down a stag.
The stag hunt differs from the prisoner’s dilemma because it contains two Nash equilibria. Both the mutual cooperation outcome (both hunt stag) and the mutual defection outcome (both hunt rabbit) are Nash equilibria. However, the stag hunt is much easier to solve through iteration because it primarily requires building trust. Once trust is established, both players prefer the superior equilibrium of mutual cooperation. This is why it is called a coordination game.
The key difference from the prisoner’s dilemma lies in the ordering of payoff parameters. When the reward for mutual cooperation exceeds the temptation payoff from unilateral defection, the game becomes a stag hunt rather than a prisoner’s dilemma.
Chicken (Hawk-Dove)
The game of Chicken gets its name from the dangerous driving game where two cars drive toward each other and the first driver to swerve loses. The decision-making involves weighing the benefit of not swerving (appearing to have stronger resolve, not looking “wimpy”) against the catastrophic outcome if neither swerves (crashing).
In the payoff structure, if one player swerves while the other does not, the non-swerving player gains status and the swerving player loses status. However, both players avoid the crash. The Nash equilibrium analysis reveals incentives that differ from both the prisoner’s dilemma and the stag hunt.
While literal car-based chicken games may be rare, the strategic structure appears frequently in international negotiations. The nuclear standoff during the Kennedy administration with Russia exemplifies Chicken dynamics. Both sides threatened nuclear escalation, essentially communicating “I’m moving forward with this threat, you need to back down.” The strategic question becomes how to manipulate parameters such that the opponent blinks first, or both sides blink simultaneously.
Strategic Diversity in Iterated Games
When playing iterated games, uncertainty about opponent type becomes strategically relevant. Reputation provides information about what type of player someone might be. Numerous researched strategies exist for iterated prisoner’s dilemma play, including various forms of tit-for-tat, always cooperate, always defect, and more sophisticated conditional strategies.
Solving for optimal play requires considering both known opponent strategies (where one can calculate best responses) and unknown opponent types (where optimal strategy must account for the distribution of possible opponent behaviors).
Solutions to Commons Dilemmas
Having established how commons dilemmas arise through prisoner’s dilemma dynamics, attention turns to potential solutions.
Government Ownership and Regulation
One straightforward solution involves government ownership of the commons resource with governmental decisions about access allocation. The government can pass laws requiring cooperation or, in the tragedy of the commons context, simply restrict access. Hunting permit requirements exemplify this approach.
Government solutions face significant challenges. When a small number of people receive rights to access a valuable resource, the winners receive enormous payoffs while others receive nothing. Equity concerns become paramount.
Two specific problems emerge from government-based solutions. First, lobbying becomes profitable when government policies create large potential gains for specific groups. The profit-maximizing choice for potential beneficiaries may be hiring lobbyists to influence policy. Economists call this behavior rent-seeking: attempting to manipulate government or laws for capture by a favored subset of people. Agricultural businesses and food industries statistically spend the most money lobbying government because they can benefit from agricultural subsidies that help them while imposing costs on broader populations.
Second, bureaucratic capture presents challenges independent of external lobbying. Public agencies demonstrate tendencies toward excessive bureaucratization, leading to slower processes compared to private sector alternatives. One theorized explanation is that public agencies behave as budget maximizers rather than utility maximizers or efficiency maximizers. Agencies resist any cuts to their budgets or redistribution to other purposes, creating perverse incentives that may not align with public benefit.
Privatization
An alternative to government ownership is privatization: arbitrarily assigning property rights to what was previously common property. This approach underlies programs like cap-and-trade systems.
Privatization faces its own challenges in commons contexts because the ownership of these newly created property rights itself becomes valuable. The government must choose winners and losers in the initial allocation, which creates its own political economy problems.
Building Robust Institutions
A third solution, sometimes underemphasized in standard environmental and natural resource economics, involves building robust institutions. Throughout human history, institutions have effectively solved commons dilemmas. Despite the theoretical predictions of prisoner’s dilemma dynamics always leading to tragedy, the actual historical record shows civilizations repeatedly solving commons dilemmas successfully.
Elinor Ostrom conducted foundational research in this area. She became the first woman to win the Nobel Prize in Economics, though this did not occur until 2009, which represents a troubling delay in recognition. Ostrom examined what actually happens in historical commons dilemmas rather than relying solely on theoretical predictions.
Across the history of commons dilemmas worldwide, Ostrom documented how civilizations develop their own solutions. Whether at the level of villages, tribal communities, or nation states, organizations tend to develop specific institutions that effectively manage common resources. Her research identified principles that successful institutions tend to share.
The key insight from Ostrom’s work is that outcomes are not as dire as the tragedy of the commons framework suggests. Success depends on the strength of institutions that communities can develop. These principles turn out to be highly relevant to contemporary policy frameworks including donut economics, where many policy precepts align closely with Ostrom’s institutional design principles.
The institutional approach provides conceptual bridges between local decision-making contexts and global-scale challenges. Understanding how communities have historically solved commons dilemmas through institutional development informs contemporary approaches to challenges ranging from local resource management to global climate change.