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Find Pure Nash Equilibria
1. Consider a two-player game in which player 1 chooses the strategy x1 from the closed interval [-1, 1] while player 2 chooses the strategy x2 from the same closed interval [-1, 1]. Player 1's utility function is x21/2 + x1x2 and player 2's utility function is x2 2/2 - x1x2. Find and plot the best- response function of each player (against any pure strategy of the opponent). Is there a pure strategy Nash equilibrium of the game?
2. Consider a game in which player 1 chooses rows, player 2 chooses columns and player 3 chooses matrices. Only Player 3's payoffs are given below. Show that D is not a best response for player 3 against any combination of (mixed) strategies of players 1 and 2. However, prove that D is not dominated by any (mixed) strategies of player 3.
3. Consider the following three-player game and find all pure Nash Equilibria. Can you find any Nash equilibrium in which exactly two of the three players play a pure strategy while the other plays a mixed strategy (such as (B, R, ½X ½Y)). Explain by considering all possible cases.
4. Show that the following game has two types of NE: (i) player 1 chooses D, 2 chooses C with probability at least 1/3 and player 3 chooses L, and (ii) where player 1 chooses C, player 2 chooses C and 3 chooses R with probability at least ¾.
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Equilibrium payoffs are (4, 5). Player A’s equilibrium strategy is “S then S if n and then N if n again.” Player B’s equilibrium strategy is “n if S and then n if S again and then
please compute this number 885 for the swertres lotto game.
A form of a Japanese auction (which is a form of an English auction) in which bidders hold down a button as the auctioneer frequently increases the current price. Bidders irrevocab
A reserve worth is that the minimum acceptable bid in an auction. If no bidder submits a bid higher than the reserve worth, the auctioneer keeps the item offered for sale. Alternat
A proxy bidder represents the interests of a bidder not physically gift at the auction. Typically, the bidder can inform his proxy of the most quantity he's willing to pay, and als
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