代写AS.440.684 (50) Game Theory Problem Set #10 Spring 2025代写C/C++语言

AS.440.684 (50)

Problem Set #10

Game Theory

Assigned 4/8

Spring 2025

Due 4/15

Please complete all questions. Submit via Canvas in a single PDF file.

1. Consider the following game, where nature first chooses types t1, t2, or t3 for player 1 (the sender), who chooses L or R, then player 2 (the receiver), who observes player 1’s action but not their type, chooses u or d.

a. Specify a pooling equilibrium where all three Sender types play R. Show it is a Perfect Bayesian Equilibrium.

b. Applying Bayes’ rule to this equilibrium, what are the Receiver’s beliefs about the Sender’s type on the equilibrium path? Explain and show your work.

c. Applying Bayes’ rule, what are the Receiver’s beliefs about the Sender’s type off the equilibrium path? Explain and show your work.

2. Consider the following simplified game of poker: There are two players. Each player begins by placing $1 into the pot as an ante. Then Player 1 is dealt a single card from a deck consisting of an equal number of Aces and Kings. Player 1 is allowed to see this card, but Player 2 cannot see it. Then Player 1 can either Bet another dollar, or Fold. If Player 1 folds, Player 2 wins the pot, thus winning $1 from Player 1 (i.e., Player 1’s ante). If Player 1 bets, then Player 2 has a choice: either Call, which requires Player 2 to match Player 1’s additional $1 bet, or Fold. If Player 2 folds, then Player 1 wins the pot, thus winning $1 from Player 2 (i.e., Player 2’s ante). But if Player 2 calls, then Player 1 must reveal his card; if the card is an Ace then Player 1 wins the entire pot (which has $2 of Player 2’s money), and if the card is a King then Player 2 wins the entire pot (which has $2 of Player 1’s money).

a. Draw this game in extensive form. List the strategy space for both players.

b. Assess whether there are any pure strategy pooling or separating Perfect Bayesian Equilibria by considering each of Player 1’s possible strategies. For each, explain Player 2’s updated beliefs at its information set, its sequentially rational best response, then whether Player 1 then has an incentive to defect from its initial strategy.

c. Depict the game in normal form. Apply iterated dominance to eliminate dominated strategies. What is the set of rationalizable strategies?

d. Find all Nash Equilibria of the rationalizable strategies from part (c).

e. Let us consider the expected value of this strategy:

i. Calculate player 1’s expected value from the equilibrium in part (d).

ii. Consider a “play the cards” strategy for player 1: always Bet with an Ace, always Fold with a King. Calculate the expected value of playing this strategy, assuming player 2 knows this and plays their best response. Compare Player 1’s expected value of this strategy to their expected payoff of the equilibrium you found in part (d).

iii. Consider an “always bet” strategy for player 1: play Bet regardless of what card you are dealt in the hopes of bluffing your opponent into always folding. Calculate the expected value of playing this strategy, assuming player 2 knows this and plays their best response. Compare Player 1’s expected value of this strategy to their expected payoff of the equilibrium you found in part (d).

f. While real poker is much more complex than the simple version above, we can still gain insights. Using the results above, explain in words your intuition regarding why we observe this equilibrium result. Be sure to reference your results from parts (b) and (d) in your response and explain why they are consistent with your intuition.

3. Consider the following entry deterrence game: there are two firms. Firm 1 is an incumbent, while Firm 2 is deciding whether to enter. They make a simultaneous decision: Firm 1 decides whether to fight the entry or not, while Firm 2 decides whether to enter or not. There is a cost c to firm 1 to fight entry: c=3 with probability 0.75, or c=0 with probability 0.25. Only the incumbent observes c, the entrant does not. All of this is common knowledge. The payoff matrix of the game can be shown as follows:

g. Draw the extensive form. of the game, create the Bayesian normal form, and solve for all Bayesian Nash equilibria.

h. Allow the incumbent to announce whether it wants to fight or not in advance of playing the game above. This is just a signal, it does not commit the incumbent to any specific action. Prove whether a separating Bayesian Perfect equilibrium exists where the incumbent truthfully announces its action in advance. First list the candidate strategy, then show the updated beliefs at all information sets using Bayes’ rule. Finally, assess whether all decisions are sequentially rational.

4. Consider the three-stage game where first a student is bestowed with either high ability (with probability p) or low ability (with probability 1-p). That student knows their ability, and decides whether to go to college. Finally, employers decide to hire people, observing whether they went to college, but not observing their ability. The extensive form. looks like this (payoffs omitted):

Employers pay wages of $90k to college grads, and $15k to non-college grads. It costs a high ability student $45k of effort to go to college, but the effort cost is $60k for low ability people. Employers get $150k of work product (before paying their salary) from a high ability person, $75k of work product from a low ability person, and zero work product if they do not hire someone.

a) Find all pure strategy Perfect Bayesian Equilibrium, and list if each is separating or pooling. Be sure to list any requirements (such as beliefs or values of p) for each equilibrium to hold.

b) Note that a high ability person is equally productive whether they go to college or not. Given this, and using the results from part (b), explain why it still makes sense for people to go to college, even though it may not improve their productivity.

c) Let the effort cost for low ability people be increased by $30k. Find all pure strategy Perfect Bayesian Equilibrium for this new game, and list if each is separating or pooling. Be sure to list any requirements (such as beliefs or values of p) for each equilibrium to hold.

d) Compare the results from parts (a) and (c). What is required for college to be an accurate signal of someone’s ability?