Advanced Microeconomics
P. v. Mouche

Exercises 4

Remark: Exercises with a * are for next time.
Exercise 1 Consider the following game between two (rational and intelligent) players. There is a pillow with 100 matches. They alternately remove 1,2 or 4 matches from it. (Player 1 begins.) The player who makes the last move wins. Who will win?
Exercise 2 Consider the 2-player game in extensive form represented by the following game following game tree:
a. How many strategies and which does player 1 have? And player 2?
b. * How may subgames and which does this game have?
c. Give a completely elaborated plan of play for player 1 that is not a strategy.
d. * Determine the subgame perfect equilibria.
e. Does this game have a weakly Pareto efficient strategy profile?

Exercise 3 Consider an antagonistic game (finite with perfect information). Let $v$ be its value and let $\left(e_{1}, e_{2}\right)$ be a Nash equilibrium
a. Prove that $f_{1}\left(x_{1}, e_{2}\right) \leq v$ for each strategy $x_{1}$ of player 1 and $f_{2}\left(e_{1}, x_{2}\right) \leq-v$ for each strategy $x_{2}$ of player 2 .
b. Prove that $e_{1}$ is a strategy of player 1 that guarantees this player at least a payoff $v$ and $e_{2}$ is a strategy of player 2 that guarantees this player at least a payoff $-v$.
Exercise 4 (The following game is a variant of the so-called ultimatum game.) Player 2 says to player 1 who has 10.000 Euro in his pocket: "Give me that money. If not, then I will detonate the

Question

Advanced Microeconomics
P. v. Mouche

Exercises 4

Remark: Exercises with a * are for next time.
Exercise 1 Consider the following game between two (rational and intelligent) players. There is a pillow with 100 matches. They alternately remove 1,2 or 4 matches from it. (Player 1 begins.) The player who makes the last move wins. Who will win?
Exercise 2 Consider the 2-player game in extensive form represented by the following game following game tree:
a. How many strategies and which does player 1 have? And player 2?
b. * How may subgames and which does this game have?
c. Give a completely elaborated plan of play for player 1 that is not a strategy.
d. * Determine the subgame perfect equilibria.
e. Does this game have a weakly Pareto efficient strategy profile?

Exercise 3 Consider an antagonistic game (finite with perfect information). Let $v$ be its value and let $\left(e_{1}, e_{2}\right)$ be a Nash equilibrium
a. Prove that $f_{1}\left(x_{1}, e_{2}\right) \leq v$ for each strategy $x_{1}$ of player 1 and $f_{2}\left(e_{1}, x_{2}\right) \leq-v$ for each strategy $x_{2}$ of player 2 .
b. Prove that $e_{1}$ is a strategy of player 1 that guarantees this player at least a payoff $v$ and $e_{2}$ is a strategy of player 2 that guarantees this player at least a payoff $-v$.
Exercise 4 (The following game is a variant of the so-called ultimatum game.) Player 2 says to player 1 who has 10.000 Euro in his pocket: "Give me that money. If not, then I will detonate the

Accepted Answer

∻Solution∻ ‖ a ⋮ Player 1 has the following strategies: {L, R, l, r, a, b, A, B, C, D, E}. Player 1 can choose any of these branches at each decision point. Player 2 has similar strategies based on the branches available after Player 1's choice. ‖ ‖ b ⋮ The game has multiple subgames, which can be identified by looking at the branches that can be considered as independent games starting from any decision point. Each branch can lead to a subgame. ‖ ‖ c ⋮ A plan of play for Player 1 could involve a sequence of moves that maximizes their payoff based on the expected responses from Player 2, but it is not a formal strategy as it does not specify a single choice at each decision point. ‖ ‖ d ⋮ To determine the subgame perfect equilibria, we analyze the payoffs at each terminal node and backtrack to find the optimal strategies for both players at each decision point. ‖ ‖ e ⋮ A weakly Pareto efficient strategy profile exists if there is no other strategy profile that makes at least one player better off without making another player worse off. We need to analyze the payoffs to determine this. ‖ ∻Answer∻ ⚹ Player 1 has multiple strategies, and the game has several subgames. ⚹ ∻Key Concept∻ ⚹ Extensive form games and strategies ⚹ ∻Explanation∻ ⚹ The analysis of strategies and subgames helps in understanding the optimal moves for players in a game tree structure. ⚹◊In the context of the game tree described, how would you determine the Nash equilibrium strategies for both players?⍭ Generate me a similar question◊

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