Eco 302 Microeconomic Theory Week 11 Homework Fall 2017 Due

Eco302microeconomictheoryweek11homeworkfall2017dueonweek1

Eco302 Microeconomic Theory Week 11 Homework Fall 2017 due on Week 12 class 1. What is a “strategy” in game theory? Provide its definition briefly. 2. What is a “Nash equilibrium”? 3. Find the unique Nash equilibrium in the following game (Prisoner’s Dilemma) Player 2 Deny Confess Player 1 Deny -1,-1 -20,0 Confess 0,-20 -5,-. Find two pure-strategy Nash equilibrium in the following game (Battle of Sexes) Wife Soccer Opera Husband Soccer 2,1 0,0 Opera 0,0 1,. Show that there does not exist a pure-strategy Nash equilibrium in the following game (Matching pennies) Player 2 Head Tail Player 1 Head 1,-1 -1,1 Tail -1,1 1,. In the game given in 5, show that it is a mixed-strategy equilibrium that both players play each action with probability 1/2. 7. What is a “subgame perfect Nash equilibrium”? Provide its definition briefly. 8. Find a subgame perfect Nash equilibrium of the following extensive-form game. 9. Find a subgame perfect Nash equilibrium of the following extensive-form game.

Paper For Above instruction

Game theory provides a foundational framework in microeconomic analysis, particularly through the concepts of strategies, Nash equilibrium, and subgame perfect equilibrium. These concepts help elucidate strategic interactions among rational agents and are instrumental in predicting outcomes in competitive and cooperative scenarios.

Strategies in Game Theory

A strategy in game theory is a comprehensive plan of action that specifies the choices a player will make at every possible decision point in the game, regardless of whether that decision point is reached. It encompasses all contingencies and ensures that a player has a predetermined course of action in response to any of the other players’ moves. Formally, a strategy can be viewed as a complete plan that accounts for every possible outcome in an extensive or strategic game, thereby guiding the player's decision-making process under uncertainty (Myerson, 1991).

Nash Equilibrium

The Nash equilibrium, named after John Nash, is a set of strategies—one for each player—such that no player can improve their payoff by unilaterally changing their own strategy, given the strategies of the others. In other words, each player's strategy is a best response to the strategies of the other players. The equilibrium represents a stable state of the game where no player has an incentive to deviate, reflecting a point of mutual consistency in strategic decision-making (Nash, 1950).

Analysis of Specific Games

Prisoner’s Dilemma

Given the payoff matrix for the Prisoner’s Dilemma:

The unique Nash equilibrium in this game occurs when both players confess. Despite mutual defection leading to worse outcomes collectively, each player’s dominant strategy is to confess because it yields a higher payoff regardless of the other player's choice.

Battle of the Sexes

Payoff matrix:

Two pure-strategy Nash equilibria exist: (Husband Soccer, Wife Soccer) and (Husband Opera, Wife Opera). Both players prefer coordinating their actions, though their preferences differ in the payoffs earned from each outcome.

Matching Pennies

Payoff matrix:

No pure-strategy Nash equilibrium exists in matching pennies because each player's optimal response depends on the opponent's choice, leading to an inherent strategic conflict. The game admits a mixed-strategy equilibrium where both players randomize equally over their strategies, each choosing head or tail with probability ½, making the opponent indifferent between their options.

Mixed-Strategy Equilibrium in Matching Pennies

When both players assign equal probabilities to their strategies, they effectively neutralize each other's incentives, establishing a mixed-strategy equilibrium where neither can improve their expected payoff by unilaterally deviating.

Subgame Perfect Nash Equilibrium (SPNE)

A subgame perfect Nash equilibrium is a refinement of the Nash equilibrium applicable to dynamic games with a sequential structure. It requires that players' strategies constitute a Nash equilibrium in every subgame of the original game. This concept eliminates non-credible threats and promises, ensuring rationality at every stage of the game (Selten, 1965).

Finding SPNE in Extensive-Form Games

Determining SPNE typically involves backward induction, where players reason from the end of the game backward to determine optimal strategies at every node. By doing so, players eliminate non-credible threats, and the equilibrium strategies are consistent throughout the entire game.

Conclusion

The exploration of strategies, Nash equilibrium, and subgame perfect equilibrium provides critical insights into rational decision-making within strategic contexts. These concepts underpin much of modern microeconomic theory and are essential for analyzing competitive, cooperative, and sequential interactions among economic agents.

References

• Myerson, R. B. (1991). Game Theory: Analysis of Conflict. Harvard University Press.
• Nash, J. F. (1950). Equilibrium points in N-person games. Proceedings of the National Academy of Sciences, 36(1), 48-49.
• Selten, R. (1965). Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. Zeitschrift für die gesamte Staatswissenschaft, 121(2), 301-324.
• Osborne, M. J., & Rubinstein, A. (1994). A Course in Game Theory. MIT Press.
• Fudenberg, D., & Tirole, J. (1991). Game Theory. MIT Press.
• Binmore, K. (2007). Playing for Keeps: A Game Theoretic Approach to Economics. Oxford University Press.
• Rubinstein, A. (1998). Economics and Language: An Analytical Approach. Cambridge University Press.
• Gibbons, R. (1992). Game Theory for Applied Economists. Princeton University Press.
• Kreps, D. M. (1990). A Course in Microeconomic Theory. Princeton University Press.
• Myerson, R. (2013). Game Theory: Analysis of Conflict. Harvard University Press.