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Consider a board game played on an m × n matrix. Player 1 has an unlimited supply of white chips, and player 2 has an unlimited supply of black chips. Starting with player 1, the players take turns claiming cells of the matrix. A player claims a cell by placing one of her chips in this cell. Once a cell is claimed, it cannot be altered. Players must claim exactly one cell in each round. The game ends after mn rounds, when all of the cells are claimed.
At the end of the game, each cell is evaluated as either a "victory cell" or a "loss cell." A cell is classified as a victory if it shares sides with at least two cells of the same color. That is, there are at least two cells of the same color that are immediately left, right, up, or down from (not diagonal to) the cell being evaluated. A player gets one point for each of the victory cells that she claimed and for each of the loss cells that her opponent claimed. The player with the most points wins the game; if the players have the same number of points, then a tie is declared.
(a) Under what conditions on m and n do you know that one of the players has a strategy that guarantees a win? Can you determine which player can guarantee a win? If so, provide some logic or a proof.
(b) Repeat the analysis for a version of this game in which a victory cell must share sides with at least three cells of the same color.
Player 1 has the following set of strategies {A1;A2;A3;A4}; player 2’s set of strategies are {B1;B2;B3;B4}. Use the best-response approach to find all Nash equilibria.
A supplier and a buyer, who are both risk neutral, play the following game, The buyer’s payoff is q^'-s^', and the supplier’s payoff is s^'-C(q^'), where C() is a strictly convex cost function with C(0)=C’(0)=0. These payoffs are commonly known.
Pertaining to the matrix need simple and short answers, Find (a) the strategies of the firm (b) where will the firm end up in the matrix equilibrium (c) whether the firm face the prisoner’s dilemma.
Consider the two-period repeated game in which this stage game is played twice and the repeated-game payo s are simply the sum of the payo s in each of the two periods.
Two players, Ben and Diana, can choose strategy X or Y. If both Ben and Diana choose strategy X, every earns a payoff of $1000.
The market for olive oil in new York City is controlled by 2-families, Sopranos and Contraltos. Both families will ruthlessly eliminate any other family that attempts to enter New York City olive oil market.
Following is a payoff matrix for Intel and AMD. In each cell, 1st number refers to AMD's profit, while second is Intel's.
Determine the solution to the given advertising decision game between Coke and Pepsi, assuming the companies act independently.
Little Kona is a small coffee corporation that is planning entering a market dominated through Big Brew. Each corporation's profit depends on whether Little Kona enters and whether Big Brew sets a high price or a low price.
Suppose you and your classmate are assigned a project on which you will earn one combined grade. You each wish to receive a good grade, but you also want to avoid hard work.
Consider trade relations in the United State and Mexico. Suppose that leaders of two countries believe the payoffs to alternative trade policies are as follows:
Use the given payoff matrix for a simultaneous move one shot game to answer the accompanying questions.
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