Introducing Game Theory

1. Game Theory: Understanding Strategic Interaction

Game theory is a set of tools used to help analyze situations where an individual’s best course of action depends on what others do or are expected to do.

Interdependence. Game theory provides a framework for analyzing situations where the outcome of your choices depends on the choices of others. This strategic interaction is pervasive in economics, politics, biology, computer science, and sociology. It helps us understand how individuals, firms, or even countries make decisions when their fates are intertwined.

Wide applicability. Game theory is useful in various fields. For example, in economics, firms consider competitors' pricing strategies; in political science, candidates adjust platforms based on rivals' announcements; and in biology, animals compete for resources. The core idea is that your best move is contingent on what you anticipate others will do.

Strategic thinking. Game theory emphasizes the importance of anticipating others' actions and planning accordingly. It's not just about making the best choice in isolation, but about considering how your choice will influence the behavior of others, and vice versa. This requires a deep understanding of incentives and expectations.

2. Rationality and Its Limits in Game Theory

Rationality refers to players understanding the setup of the game and exercising the ability to reason.

Rationality assumption. Game theory often assumes that players are rational, meaning they understand the rules of the game and make decisions to maximize their own payoffs. However, this assumption has limitations, as real-world behavior is often influenced by emotions, biases, and cognitive constraints.

Common knowledge of rationality. A more stringent requirement is "common knowledge of rationality," where everyone is rational, everyone knows that everyone is rational, and so on, ad infinitum. This assumption is often unrealistic, as demonstrated by Keynes' Beauty Contest and Thaler's Guessing Game, where people's choices deviate from what strict rationality would predict.

Bounded rationality. Human behavior is better approximated by bounded rationality, where cognitive limitations, time constraints, and the complexity of the decision problem limit rationality. People often use heuristics and rules of thumb rather than engaging in exhaustive calculations. This highlights the gap between theoretical models and real-world decision-making.

3. Simultaneous-Move Games and Strategic Form

Often players do not know the actions of the other players when they make their own decisions.

Uncertainty in decision-making. Simultaneous-move games model situations where players make decisions without knowing the choices of others. This can occur when choices are literally made at the same time or when players are unaware of each other's actions. The key is the lack of information about the other player's move.

Strategic form representation. These games are often represented in strategic form, also known as a payoff matrix. This table lists the possible actions for each player and the corresponding payoffs for all possible outcomes. It provides a clear overview of the strategic interaction and the potential consequences of each choice.

Movie release example. Consider two film studios deciding when to release their movies. Each studio's revenue depends not only on its own release date but also on the release date of its competitor. The payoff matrix shows the potential revenues for each studio based on whether they both release in October, both in December, or one in each month.

4. Nash Equilibrium: A Foundation for Prediction

The idea of Nash equilibrium is both simple and powerful: in equilibrium each rational player chooses his or her best response to the choice of the other player.

Best response. A Nash equilibrium occurs when each player's strategy is the best response to the strategies of the other players. In other words, no player can improve their payoff by unilaterally changing their strategy, assuming the other players' strategies remain the same. It's a stable state where everyone is doing the best they can, given what everyone else is doing.

Regret-free outcome. One characteristic of a Nash equilibrium is that it is regret-free. No player would benefit from deviating from their equilibrium strategy. It's also a rational expectations equilibrium, where players' expectations about each other's actions are correct.

Movie release example revisited. In the movie release game, the Nash equilibrium is for both studios to release in December. If one studio is releasing in December, it is optimal for the other to release in December as well. This is the only outcome where both studios have best responses to each other.

5. Prisoner's Dilemma: The Conflict Between Cooperation and Self-Interest

The game illustrates the difficulty of acting together for common or mutual benefit given that people pursue self-interest.

Classic paradox. The Prisoner's Dilemma is a famous game theory paradox that illustrates the tension between individual self-interest and collective well-being. Two prisoners, interrogated separately, must decide whether to remain silent or confess. The best outcome for both is to remain silent, but the dominant strategy for each is to confess, leading to a worse outcome for both.

Dominant strategy. In the Prisoner's Dilemma, confessing is a dominant strategy for both players, meaning it's the best choice regardless of what the other player does. This leads to a Nash equilibrium where both confess, even though they would both be better off if they had cooperated and remained silent.

Real-world applications. The Prisoner's Dilemma has broad applications, from competition between firms to social norms to environmental issues. For example, wireless network routers competing for bandwidth face a similar dilemma: each router benefits from broadcasting at high power, but if both do so, they interfere with each other, resulting in slower speeds for both.

6. Multiple Equilibria and Coordination Problems

In games with multiple Nash equilibria, the concept of Nash equilibrium by itself does not provide us with sufficient tools to predict what will happen.

Indeterminacy. Some games have multiple Nash equilibria, making it difficult to predict which outcome will occur. In these situations, the concept of Nash equilibrium alone is insufficient to guide behavior. Players face a coordination problem, where they need to align their expectations to achieve a desirable outcome.

Battle of the Sexes. The Battle of the Sexes is a classic example of a game with multiple Nash equilibria. A couple wants to spend the evening together, but one prefers football while the other prefers dancing. There are two equilibria: both go to football or both go dancing. However, if they fail to coordinate, they may end up going to different activities, resulting in a worse outcome for both.

Social norms and coordination devices. In environments with multiple equilibria, players may use social norms or coordination devices to align their expectations. For example, a couple might have a social norm where one partner usually gets their way, or they might use a shared observation, such as an advertisement, to coordinate on a particular activity.

7. Mixed Strategies: Randomization and Unpredictability

In order to win I have to be unpredictable.

Unpredictable behavior. Some games, like Rock-Paper-Scissors, have no pure-strategy Nash equilibrium, where players choose a particular action with certainty. In these games, the only equilibrium is in mixed strategies, where players randomize over their possible actions. This randomization makes their behavior unpredictable.

Indifference. In a mixed-strategy Nash equilibrium, players are indifferent between their different actions. This means that they get the same expected payoff from each action, given the mixed strategies of the other players. This indifference is what allows them to randomize.

Currency speculation. Mixed strategies can explain the element of surprise in speculative attacks. If a central bank could predict when investors would attack its currency, it could preemptively devalue the currency to avoid losses. However, speculators randomize the timing of their attacks to make them unpredictable, preventing the central bank from preempting them.

8. Repeated Interaction: Cooperation Over Time

In continuously competitive games, individual self-interest can dictate a kind of cooperative behaviour sustained due to the fear of punishment by the other players for failing to cooperate.

Beyond one-shot games. In one-shot games, where players interact only once, cooperation is often difficult to achieve due to the incentives for self-interest. However, in repeated interactions, where players play the same game again and again, cooperation becomes more likely.

Grim strategy. One strategy that can sustain cooperation in a repeated Prisoner's Dilemma is the grim strategy. A player starts by cooperating and continues to cooperate as long as the other player cooperates. However, if the other player ever defects, the player defects forever after.

Credible threats. For cooperation to be sustained, the threat of punishment must be credible. If players are impatient or expect to be able to renegotiate easily, the threat of punishment may not be enough to deter defection. However, if players are patient and expect renegotiation to take time, cooperation can be sustained in equilibrium.

9. Evolutionary Game Theory: Beyond Rationality

The question is not what choices will individuals make, but rather what genetic or social programming will survive in the long term.

Genetic or social programming. Evolutionary game theory examines interactions from a different perspective, focusing on which behavioral patterns will survive in the long term. Instead of assuming rationality, it considers that people or animals are socially or genetically programmed to engage in certain behaviors.

Hawk-Dove Game. The Hawk-Dove Game is a widely used model in evolutionary biology. It assumes two types of animals: "hawks," which fight for resources, and "doves," which make aggressive displays but avoid physical conflict. The game highlights the importance of evolutionary stability.

Evolutionarily stable equilibrium. An evolutionarily stable equilibrium is a state where, if a small number of animals with different conditioning are added, evolutionary forces will eventually restore the equilibrium. In the Hawk-Dove Game, the long-run steady state depends on the cost of conflict relative to the value of the prize.

10. Sequential-Move Games: Dynamics and Credibility

I knew having two shops here would scare away the competition. That’s why I opened them last year.

Order of actions. Sequential-move games model situations where players can observe the actions of others before making their own moves. This creates a dynamic environment where players can make decisions based on past actions and anticipate future actions.

Extensive form representation. These games are often represented in extensive form, also known as a game tree. This diagram shows the order of choices, decision nodes, and payoffs for each player. It provides a clear visualization of the sequential nature of the game.

Subgame perfection. Subgame perfection is a refinement of Nash equilibrium that requires players to have best responses to each other in each subgame of the original game. This eliminates equilibria that rely on non-credible threats or promises.

11. Information Asymmetry: Unequal Knowledge

Asymmetric information and unemployment

Unequal knowledge. Asymmetric information occurs when one player has superior information compared to others. This can create problems in markets, as those with less information may be exploited or make suboptimal decisions.

Adverse selection. One consequence of asymmetric information is adverse selection, where those with private information exploit it to their advantage. For example, in the car insurance market, drivers with poor driving habits are more likely to purchase insurance, leading to higher premiums for everyone.

Signaling. To overcome asymmetric information, informed parties may use signals to convey their private information to others. For example, firms may offer warranties to signal the quality of their products, or individuals may pursue education to signal their ability to potential employers.

12. Group Decision Making: The Challenge of Collective Rationality

I’m not really sure the best way to reconcile everyone’s preferences in the group…

Irrationality. Group decision-making presents a challenge for game theory because the group as a whole may seem irrational even when each member of the group is rational. This can occur because individual preferences may not align, leading to non-transitive group preferences.

Arrow's Impossibility Theorem. Arrow's Impossibility Theorem shows that, for groups which are not run by a dictator, there will always be the possibility of some situations where group preferences are non-transitive. This means that there is no perfect way to aggregate individual preferences into a collective decision.

Voting paradox. The voting paradox illustrates how group preferences can be non-transitive. In a series of votes, a group may prefer A to B, B to C, and C to A, creating a cycle where there is no clear winner. This highlights the inherent difficulties in group decision-making.

Last updated: March 12, 2025