3. DON'T MESS WITH ME GAME TREE. Suppose Player #1 (P1) anticipates the possibility of being challenged by rival Player #2 (P2). To discourage P2 from challenging P1 tomorrow, P1 could invest in mixed martial arts training today, thereby making P1 a more formidable rival in case it has to battle tomorrow. Suppose the game tree below accurately models this dynamic game. TIP: Review the dynamic games slides to refresh your memory about analyzing games trees and finding SPNE. a) b) P1 Fight (F) (100,-50) Challenge (C) MMA Training (M) Give Up (G) P2 (-40, 100) Do Not (D) (60,0) fight (f) (-20, 150) challenge (c) Pl No Training (N) give up (g) P2 (0, 100) do not (d) (100, 0) Find all Subgame Perfect Nash Equilibria (SPNE) of this game. TIP: What is the definition of SPNE? In the subgames in the game tree directly following P1's choice to invest in MMA Training, identify ALL of the noncredible threats and explain why each is noncredible.

3. DON'T MESS WITH ME GAME TREE. Suppose Player #1 (P1) anticipates the possibility of being challenged by rival Player #2 (P2). To discourage P2 from challenging P1 tomorrow, P1 could invest in mixed martial arts training today, thereby making P1 a more formidable rival in case it has to battle tomorrow. Suppose the game tree below accurately models this dynamic game. TIP: Review the dynamic games slides to refresh your memory about analyzing games trees and finding SPNE. a) b) P1 Fight (F) (100,-50) Challenge (C) MMA Training (M) Give Up (G) P2 (-40, 100) Do Not (D) (60,0) fight (f) (-20, 150) challenge (c) Pl No Training (N) give up (g) P2 (0, 100) do not (d) (100, 0) Find all Subgame Perfect Nash Equilibria (SPNE) of this game. TIP: What is the definition of SPNE? In the subgames in the game tree directly following P1's choice to invest in MMA Training, identify ALL of the noncredible threats and explain why each is noncredible.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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8.8
Identify any strictly dominant strategies for the players.
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Suppose that, in an ultimatum game, the proposer may not propose less than $1 nor fractional amounts, and therefore must propose $1, $2, …, or $10 (see image attached below). The responder must Accept (A) or Reject (R). Suppose, first, that this game is played by two egoists, for whom u(x,y)=x. Find all subgame-perfect equilibria in this game. Suppose, second, that this game is played by two altruists, for whom u(x,y) = ⅔(x)1/2 + ⅓(y)1/2. Find all subgame-perfect equilibria in this game.
3. Two individuals are involved in a synergistic relationship. If both individuals devote more effort to the relationship, they are both better off. Specifically, an effort level is a nonnegative number, and player 1's payoff function is e₁(1+€₂-e₁), where ei is player i's effort level. For player 2 the cost of effort is either the same as that of player 1, and hence her payoff function is given by e2(1+e₁ - e₂), or effort is very costly for her in which case her payoff function is given by e₂(1 + e₁ — 2e2). 1 Player 2 knows player 1's payoff function and whether the cost of effort is high for herself or not. Player 1, however, is uncertain about player 2's cost of effort. He believes that the cost of effort is low with probability p, and high with probability 1 - p, where 0 < p < 1. Find the Bayesian Nash equilibrium of this game as a function of p.
In a three stage alternating offers bargaining game, player 1 demands a fraction x of $100. If this is accepted by player 2, the $100 is split between the players with outcome ($100x,$100(1 - x)). Otherwise, at stage 2, player 2 makes a demand for a fraction y. If this is accepted by player 1, the $100 is divided accordingly but otherwise player 1 can make another offer and demand a fraction z. If that offer is rejected by player 2, the outcome is (0,0). Both players have a discount factor of 1/2. (a) Find a perfect equilibrium for this game. (b) Is there a first-mover advantage?
2. Consider the following "centipede game." The game starts with player 1 choosing be- tween terminate (T) and continue (C). If player 1 chooses C, the game proceeds with player 2 choosing between terminate (t) and continue (c). The two players choose be- tween terminate and continue in turn if the other player chooses continue until the terminal nodes with (player l's payoff, player 2's payoff) are reached as shown below. TTTT Player 1 Player 2 Player 1 Player 2 (3, 3) t (1, 1) (0, 3) (2, 2) (1, 4) (a) List all possible strategies of each player. (b) Transform the game tree into a normal-form matrix representation. (c) Find all pure-strategy Nash equilibria. (d) Find the unique pure-strategy subgame-perfect equilibrium.
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Solve parts 1(d) and 1(e) ONLY
Consider the following static, single-shot game played between two cyclists, Chico and Loco, who find themselves riding towards each other down a narrow lane. Chico wants to pass by safely, and he can only do this by cycling on the opposite side of the lane to Loco. So, he has two options: to cycle on the Right of the lane, or to cycle on the Left. Loco meanwhile is having a bad day and wants to crash into Chico, and the way he can achieve this is by cycling on the same side of the lane as Chico. So again, he has two options: to cycle on the Right of the lane, or to cycle on the Left. He does not care at all about hurting himself if they collide. (c) By constructing the best response lines of both players, identify all Nash equilibrium outcomes in the above game i.e., in pure and mixed strategies. (d) Provide an intuitive interpretation of the equilibrium outcome(s) identified in part (c) above.
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