Game Theory and Strategy Quiz 1: 7 July 2024 (May 2024 term)

An item is up for auction. Player 1 values the item at 3 while player 2 values the item at 5. Each player can bid either 0, 1 or 2. If player i bids more than player j then i wins the good and pays his bid, while the loser does not pay. If both players bid the same amount, then a coin is tossed to determine who the winner is, who gets the good and pays his bid while the loser pays nothing. Formulate the game in matrix form and answer the given subquestions.

Which player does not have strictly dominated strategy?

An item is up for auction. Player 1 values the item at 3 while player 2 values the item at 5. Each player can bid either 0, 1 or 2. If player i bids more than player j then i wins the good and pays his bid, while the loser does not pay. If both players bid the same amount, then a coin is tossed to determine who the winner is, who gets the good and pays his bid while the loser pays nothing. Formulate the game in matrix form and answer the given subquestions.

Which strategies survive Iterated Elimination of Strictly Dominated Strategy?

(Hawk-Dove) The following game has been widely used in evolutionary biology to understand how “fighting” and “display” strategies by animals could coexist in a population. For a typical Hawk- Dove game there are resources to be gained (i.e. food, mates, territories, etc.) denoted as v. Each of two players can chooses to be aggressive, called “Hawk” (H), or can be compromising, called “Dove” (D). If both players choose H then they split the resources, but loose some payoff from injuries, denoted as k. Assume that k > v/2. If both choose D then they split the resources, but engage in some display of power that a display cost d, with d < v/2. Finally, if player i chooses H while j chooses D, then i gets all the resources while j leaves with no benefits and no costs. Describe the game as a game matrix and answer the following question.
The outcomes cannot be supported as pure strategy Nash equilibrium, given v = 10, k = 6, and d = 4, is (are)