lecture 6.pptx

This document provides an in‑depth exploration of game theory fundamentals and social preferences as they apply to economic interactions and various applications. # Introduction to game theory and social preferences Game theory examines situations where individuals’ choices affect one another, emphasizing strategic interactions. In this lecture, we explore the foundations of game theory, its applications in real‑world scenarios, and how social preferences influence behavior beyond material incentives. ## Elements of a normal form game A normal form (or strategic form) game is characterized by three key components: 1. **Players:** Decision makers such as individuals, firms, or governments. 2. **Strategies (Actions):** The set of possible actions available to each player. These can be finitely many or infinitely many. 3. **Payoff functions:** Functions that assign outcomes or payoffs to each player for every combination of strategies chosen. **Tip:** Understanding these elements is crucial because they form the backbone of analyzing any strategic interaction. ### Best response functions Instead of checking every possible strategy combination—especially as games grow in size—best response functions simplify the process by identifying the optimal strategy for each player given the strategies chosen by others. In essence, a player’s best response to the other players’ strategies helps in finding the Nash equilibrium. **Example:** In a two-player game, if player one’s optimal strategy against a given strategy by player two yields the highest payoff, this relation constitutes a best response function for player one. ## Representing a game in normal form Normal form games are frequently represented using a matrix. For a two-player game, the rows represent the strategies of player one and the columns those of player two. Each cell in the matrix contains an ordered pair that represents the payoffs for player one and player two, respectively. ## The prisoner’s dilemma The prisoner’s dilemma is a fundamental game used to illustrate strategic interactions: - **Scenario:** Two suspects are interrogated separately. Each has the option to confess or remain silent. - **Outcomes:** - If both remain silent, they receive moderate penalties (e.g., 2 years). - If one confesses while the other does not, the confessor receives a light sentence (e.g., 1 year) and the silent party a heavy sentence (e.g., 8 years). - If both confess, both receive an intermediate sentence (e.g., 5 years). The dilemma lies in the fact that while mutual silence is collectively better, each player’s dominant strategy is to confess, leading to the Nash equilibrium where both confess. **Tip:** Always check whether the Nash equilibrium outcome is optimal for all players, or if it represents a suboptimal collective outcome—as exemplified by the prisoner’s dilemma. ## Other game examples ### Arms race game In the arms race game, two countries decide on nuclear armament. Their payoff preferences might be ordered as follows: - Having nuclear bombs while the opponent does not (highest payoff). - Both countries not having bombs. - Both having bombs. - Not having bombs while the opponent does. Although treaties might aim to prevent escalation in reality, the game model demonstrates how strategic considerations lead to decisions that escalate armament. ### Battle of the sexes This game captures a coordination problem where two players (e.g., Robert and Beth) prefer different outcomes (Bach for Robert, Stravinsky for Beth) but value coordinating over being apart. This game features two Nash equilibria: - (Bach, Bach) - (Stravinsky, Stravinsky) ### Matching pennies In matching pennies, two players each choose heads or tails: - If both choices match, player one wins. - If they differ, player two wins. The game has no Nash equilibrium in pure strategies because any fixed strategy can be exploited by the other. ### Stag hunt game In the stag hunt game: - Both players can hunt stag together, yielding a high payoff if both coordinate. - If a player defects to hunt hare alone, they receive a lower payoff. This game illustrates the importance of coordination and the risk involved in high-reward cooperation. ## Nash equilibrium and its intuition **Definition:** A Nash equilibrium is a set of strategies where no player has an incentive to deviate unilaterally after considering the strategies of the others. **Intuition:** - Each player’s strategy is optimal given the strategies of others. - The equilibrium is strategically stable because any unilateral deviation would lower a player’s payoff. - With common knowledge of rationality, each player’s belief about another’s strategy is correct. **Example:** In the prisoner’s dilemma, even though mutual cooperation would yield a better collective outcome, the Nash equilibrium is for both players to confess. ## Social preferences and social motives While game theory traditionally assumes material self-interest, experimental evidence shows that individuals often care about fairness, reciprocity, and inequality: - **Ultimatum game:** Here, a proposer offers a division of a fixed prize. Despite the standard prediction (offering a tiny fraction), many responders reject low offers due to fairness concerns. - **Dictator game:** Even though the dictator is expected to keep the entire endowment, many share a positive amount, reflecting altruism and fairness. - **Trust game:** A sender transfers part of an endowment which is then tripled, after which the responder decides how much to return. High levels of trust lead to greater cooperation and larger transfers. - **Public good game:** Each group member receives an endowment (e.g., an amount written as ten dollars in full letters) and decides how much to contribute to a public project. The contributions are multiplied by a factor $k$ (such as $1.6$) and then divided among the players. For example, if one person contributes one unit: $$ \text{Total return} = 1.6 \times 1 = 1.6 $$ Each person receives: $$ \frac{1.6}{4} = 0.4 $$ In a purely self-interested scenario, the Nash equilibrium is for no one to contribute. However, real experiments show that contributions are positive, especially when punishment for free riding is possible. **Example:** In the public good game with punishment, individuals who contribute less are punished by their peers, leading to higher contributions over time. ## Applications in economics and finance The lecture extends game theory and social preference insights to various applied settings: ### Corporate finance: Trust and corporate cash holdings Studies show that societal trust can positively affect corporate cash holdings. In environments where formal institutions are weak, higher trust correlates with higher cash reserves, as corporate insiders may be less inclined to divert resources. ### Financial markets 1. **Trust and stock market participation:** Surveys indicate that individuals with higher trust are more likely to participate in stock markets. However, when additional factors are controlled, the relationship may change, although trust remains a core factor. 2. **Sin stocks:** Companies involved in vices (alcohol, tobacco, gaming) tend to have lower institutional ownership due to social norms against supporting such activities. These norms create financial costs like reduced analyst coverage, leading to potential undervaluation. ### Banking applications 1. **Mortgage choices:** Research shows strong peer effects in mortgage refinancing decisions. For instance, a one standard deviation increase in peer refinancing activity can increase an individual teacher’s likelihood of refinancing by approximately twenty point seven percent. 2. **Mortgage default (strategic default):** In cases where the difference between the market value of a house and the mortgage balance is large and negative, some homeowners may strategically default even if they can afford the payments. Social factors such as morality and the influence of knowing other defaulters play a key role in such decisions. **Tip:** In applied settings, always consider how social motives and external factors (like trust and peer influence) might cause deviations from purely materialistic predictions of standard economic models. # General mistakes to avoid - **Omitting currency rules:** Always write monetary amounts in full letters (for example, "ten dollars" instead of using symbols). - **Ignoring strategic stability:** Failing to verify that a proposed strategy is a best response can lead to misidentifying a Nash equilibrium. - **Overlooking social preferences:** Disregarding fairness, altruism, and inequality aversion can result in incomplete analysis of experiments such as the ultimatum and dictator games. - **Misinterpreting game outcomes:** Assuming that the Nash equilibrium always corresponds to the best collective outcome can be misleading, as seen in the prisoner’s dilemma and public good games. - **Neglecting model assumptions:** Always check and state the assumptions inherent in each game model, particularly when extending theory to real-world applications. By mastering these concepts and avoiding these common mistakes, you will be well-prepared for exam questions on game theory, strategic interactions, and social preferences.

# Glossary | Term | Definition | |--------------------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| | Game theory | A framework for analyzing situations where decision-makers (players) interact strategically, considering that the outcome for each depends on the actions of all. | | Normal form game | A representation of a game using a matrix, where each player’s strategy set and their corresponding payoff functions are listed; it includes the players, available actions (strategies), and the resulting payoffs for every combination of strategies. | | Players | The decision makers in a game, who can be individuals, firms, or governments, that choose strategies to maximize their payoffs. | | Strategies (actions) | The set of choices available to each player in a game; strategies can be finite or infinite in number. | | Payoff functions | Functions that assign a specific outcome or payoff to each player for every possible combination of strategies. These functions represent the preferences of the players. | | Best response function | A function that determines the optimal strategy for a player, given the strategies chosen by the other players; useful for finding equilibria in games. | | Nash equilibrium | A set of strategies where no player has an incentive to unilaterally deviate from their chosen strategy, given the choices of the other players. | | Prisoner’s dilemma | A classic game in which two suspects choose whether to confess or remain silent; the equilibrium outcome demonstrates that individually rational decisions can lead to collectively suboptimal outcomes. | | Arms race | A strategic interaction modeled as a game where two countries decide on nuclear armament, with the equilibrium reflecting the strategic tension between security and the risks of escalation. | | Battle of the sexes | A game that describes a coordination problem where players prefer different outcomes but still prioritize coordinating with each other, resulting in multiple Nash equilibria. | | Matching pennies | A two-player game in which one player wins if the pennies match and the other wins if they do not; this game typically has no Nash equilibrium in pure strategies. | | Stag hunt game | A coordination game that shows the trade-off between risk and high reward, where cooperation (hunting stag) yields higher payoffs if both players coordinate, but there is a risk if one defects to hunt a hare. | | Ultimatum game | A bargaining game where one player (the proposer) offers a division of a fixed prize to a responder, who can accept or reject the offer; rejection results in zero payoffs for both, highlighting concerns of fairness. | | Dictator game | A game in which one player (the dictator) unilaterally decides how to split a fixed endowment between themselves and a passive recipient, thereby testing notions of altruism and fairness. | | Trust game | A game in which a sender transfers a portion of their endowment that is then multiplied before a responder decides how much to return, illustrating the importance of trust and reciprocity in strategic interactions. | | Public good game | A game where group members decide how much of their individual endowment to contribute to a collective project, with contributions being multiplied by a factor before being split equally among the group; it often exhibits a free rider problem. | | Public good game with punishment | A variant of the public good game where players can pay a cost to punish those who contribute less, thus encouraging higher contributions and cooperation. | | Social preferences | Preferences that include concerns for fairness, reciprocity, and inequality aversion, often driving behavior that deviates from strict material self-interest. | | Coordination | The process by which players adjust their strategies based on expectations of others' actions, essential for achieving mutually beneficial outcomes in games like stag hunt or battle of the sexes. | | Strategic default | A decision by a borrower to willingly stop paying a mortgage even when they can afford it because the financial cost of repayment outweighs the perceived benefits, often influenced by social and moral considerations. | | Corporate cash holding | The level of liquid assets maintained by firms, which can be affected by factors such as trust in the environment and the quality of formal institutions; high cash holdings may indicate a reluctance to invest due to agency concerns. | | Sin stocks | Stocks of companies engaged in activities considered vices (e.g., alcohol, tobacco, gaming) that may experience lower institutional ownership and analyst coverage, potentially leading to undervaluation due to social norm pressures. |