ECON414 Final Exam Study Notes

ECON414 FINAL EXAM – FALL, 2024. CANDIDATE SOLUTIONS

Instructor: Daniel R. Vincent
Exam Structure: There are FOUR questions in this exam. Please answer all questions. If you need to make additional assumptions to help you answer a question, please be explicit. Typically, the best answers use the weakest assumptions. This is a closed book, closed notes exam.

Question 1: True or False with Explanations

  • i) In the Bayesian Nash Equilibrium of an Independent Private Values auction model, the winning bidder is the same whether the auction is a second price auction or an English (ascending bid) auction.

    • Answer: True.

    • Explanation: In both auction types, the winning bidder is the player with the highest valuation.

  • ii) In the Bayesian Nash Equilibrium of an Independent Private Values auction model, the amount paid by the winning bidder is the same whether the auction is a second price auction or an English (ascending bid) auction.

    • Answer: True.

    • Explanation: In a second price auction, the winning bidder pays the amount bid of the second highest valuation bidder. In contrast, in an English auction, each bidder bids up to their valuation, and the second highest bidder drops out at its valuation, resulting in the same price being paid by the highest valuation bidder as in the second price auction.

  • iii) In the Bayesian Nash Equilibrium of an Independent Private Values auction model, the amount bid by a bidder of a given type is the same whether the auction is a first price auction or a Dutch (descending bid) auction.

    • Answer: True.

    • Explanation: The two games are strategically equivalent. Thus, any optimal strategy for a bidder in one auction type will be the same in the other.

  • iv) In a constant sum simultaneous move game with two players, the Maxmin solution yields the highest expected payoff for (say) player 1 assuming player 2 selected the strategy resulting in the worst outcome for player 1.

    • Answer: True.

    • Explanation: This is the motivation for using the Maxmin solution concept in game theory.

Question 2: Simultaneous Move Game

  • Game Setup: Each cell contains the payoff of the Row Player first, followed by the payoff of the Column Player. The value of $x$ will vary with subquestions below.

  • i) Suppose $x = 40$ and this game is played exactly once. What is the Nash Equilibrium of this game?

    • Answer: (R1, C1)

    • Explanation: Although Row Player does not possess a dominant strategy, Column Player has a dominant strategy (C2). Upon eliminating C1, the Row Player should optimally play R2.

  • ii) Suppose $x = 60$ and this game is played five times. After each game, firms learn the choices of the other. Each firm has a discount factor, $eta = rac{2}{3}$. Is (R1, C1) a Subgame Perfect Nash Equilibrium?

    • Answer: No.

    • Explanation: Under the new conditions, (R2, C2) becomes the unique Nash Equilibrium and equilibrium in dominant strategies of the stage game. Therefore, for any finite repetition of the game, playing (R2, C2) in all subsequent stages regardless of previous strategies is the only SPNE.

  • iii) Suppose $x = 60$ and this game is played an infinite number of times. What is the lowest value of $eta$ such that (R1, C1) is a Subgame Perfect Nash Equilibrium?

    • Answer: $eta ext{ must satisfy } eta ext{ ≥ } rac{1}{5}.$

    • Calculation: Using the Grim Trigger Strategy (GTS), the payoff for playing (R1, C1) is $50 + rac{50eta}{1 - eta}$, while deviating to R2 yields $60 + rac{10eta}{1 - eta}$. To prevent deviations:
      50 + rac{50eta}{1 - eta} ext{ ≥ } 60 + rac{10eta}{1 - eta}
      Simplifying gives:
      40 rac{eta}{1 - eta} ext{ ≥ } 10
      This leads to the conclusion that $eta ext{ ≥ } rac{1}{5}$ is required for no incentive to deviate.

  • iv) How does your answer change if $x = 80$?

    • Answer: The required value of $eta$ increases to $eta ext{ ≥ } rac{3}{7} > rac{1}{5}$.

    • Explanation: With a higher payoff of 80, the incentive for the Row Player to deviate is increased, necessitating a stricter requirement on $eta$.

Question 3: Sequential Game Analysis

  • Game representation: The payoffs of Shedeur are shown first, and those of Travis second in the bimatrix game.

  • i) Is this a game of perfect information?

    • Answer: False.

    • Explanation: Following Shedeur's first move, the players engage in a simultaneous move game, which indicates imperfect information.

  • ii) Find a Nash Equilibrium of the bimatrix game created following the Fast condition.

    • Answer: One mixed strategy Nash Equilibrium is $(0.5, 0.5)$ for each strategy and for both players.

    • Expected payoffs yield: Each player has an expected payoff of 3.

  • iii) Find two Subgame Perfect Nash equilibria of the full sequential game.

    • Answer: In the subgame following Slow, two equilibria identified are (Hard, Hard) and (Soft, Soft).

    • Explanation:

      • (Hard, Hard) results in payoffs (5,3).

      • (Soft, Soft) results in payoffs (2.5, 3).

      • The unique payoff found after Fast in ii) yields (3, 3). Depending on which equilibrium is chosen after Slow, Shedeur will select either Slow or Fast accordingly to maximize payoffs.

Question 4: Game of Incomplete Information

  • Game Setup: Nature selects from three outcomes for a stranger who might arrive in Dodge.

    • Outcomes are: 1/6 probability the stranger never arrives (payoffs (0,0)), 2/6 probability for a warmonger (W), and 3/6 probability for a pacifist (P).

  • i) Does the Marshall know whether the stranger is W or P?

    • Answer: False.

    • Explanation: The Marshall lacks the information regarding the identity of the stranger.

  • ii) Does the stranger know whether they are W or P?

    • Answer: True.

    • Explanation: The stranger is aware of their own identity in this scenario.

  • iii) In a pooling Perfect Bayesian Nash Equilibrium (PBNE) where both types of the stranger opt for the library, what must the Marshall believe?

    • Calculation: Use Bayes’ rule to compute:
      P(W∣Library)=racP(Library∣W)P(W)P(Library)P(W | Library) = rac{P(Library|W)P(W)}{P(Library)}

    • Where $P(Library|W) = 1$, $P(W) = 2/6$, and $P(Library) = 5/6$.

    • Thus: P(W∣Library)=rac2/6imes15/6=rac25.P(W|Library) = rac{2/6 imes 1}{5/6} = rac{2}{5}.

      • The Marshall should choose to Duel (D) upon seeing the stranger at the library.

  • iv) Characterize a complete Perfect Bayesian Nash Equilibrium (PBNE) that is a separating equilibrium.

    • Answer: In the chosen separating equilibrium, W selects Saloon and P selects Library.

    • Explanation:

    • After observing Saloon, Marshall’s beliefs are $(P(W|Library), P(P|Library)) = (1, 0)$; after observing Library, beliefs change to $(0, 1)$.

    • Consequently, Marshall selects N after Saloon and D after Library, as confirmed by the preferences of both types of strangers given Marshall's strategy.