Problem Set 1 1. Games in Strategic Form: Do questions 1-4 and ...

however, since there are two firms, their prices determine what share of the market eachfirm gets. If the firms choose the same price, they split the market half-half. If one firmcharges a strictly higher price, it gets nothing, while the firm that charged less serves thewhole market. More formally, if p i is firm i’s price and p j is its opponent’s price, thenprofits for firm i are ⎧⎨ (4−p i )p i ,p i < p jπ i (p i ,p j ) = (4−p⎩ i )p i /2 ,p i = p j0 ,p i > p jWrite out the strategic form and solve by iterated elimination of weakly dominated strategies.Find all Nash equilibria. Were any Nash equilibria removed by elimination of weaklydominated strategies? Is there an economic argument to focus on any of the eliminatedstrategies that you can think of?If we fill out the strategic form, we get0 1 2 3 40 0,0 0,0 0,0 0,0 0,01 0,0 3/2,3/2 3,0 3,0 3,02 0,0 0,3 1,1 4,0 4,03 0,0 0,3 0,4 1/2,1/2 1,04 0,0 0,3 0,4 0,1 0,0If we do IDWDS, the following happens: 0, 3 and 4 are weakly dominated for bothplayers, by 1 and 2. That leaves us with a strategy set of {1,2} for both players. Butthen 1 weakly dominates 2 for both players, so the outcome of IDWDS is (1,1) — eachplayer uses a price of 1 in equilibrium.Note there are two Nash equilibria: (0,0) and (1,1). You might say the (0,0)equilibrium is silly, but remember that marginal costs here are 0: The model is reallysaying that charging marginal cost or marginal cost plus the smallest unit of currencyarethe equilibriaof the game. Thisis pretty surprising, sinceit says that two competingfirms will behave very similarly to a perfectly competitive market.The interpretation of the previous paragraph is one way of thinking about the game:As long as the players are pricing above marginal cost, the other player has an incentiveto undercut. IDWDS leads to the (1,1) equilibrium, making it a good candidate solution,because it has the appealing feature that the players can reason there separatelyby using strategy dominance arguments. However, like the previous paragraph said, it’sfair to think of the (0,0) equilibrium as one in which the firms are pricing at marginalcost. (Like most things in game theory, there are different arguments that supportdifferent predictions.)4. (a). Consider the following three player simultaneous-move game: Player A choosesthe strategic from {1,2}, player B chooses a row, and C chooses a column. A gets the firstnumber as payoff, B gets the second, C gets the third.4