{"id":1087,"date":"2023-06-01T01:57:43","date_gmt":"2023-05-31T16:57:43","guid":{"rendered":"https:\/\/saraheee.com\/?p=1087"},"modified":"2023-06-03T16:25:35","modified_gmt":"2023-06-03T07:25:35","slug":"game-theory-3-chap08-application","status":"publish","type":"post","link":"https:\/\/saraheee.com\/ko\/2023\/06\/game-theory-3-chap08-application\/","title":{"rendered":"Game Theory 3 \u2013 chap08. Application\u00a0I"},"content":{"rendered":"

How far can we go with rationalization?
1) ISD, strictly dominated strategies were predictable as they were erased one by one
2) cross out one for rationalizability, never best response<\/mark><\/p>\n\n\n\n

2-player games make the same predictions as ISD and rationalizability
The concept of Nash equilibrium emerged for people’s beliefs.<\/mark><\/p>\n\n\n\n

\\(BR_{1}(\\theta_{1})\\): 1’s belief about 2’s behavior<\/mark><\/p>\n\n\n\n

\\(\\theta_{1}\\ = s_{2}^{*}\\): correct belief<\/mark><\/p>\n\n\n\n

\\(BR_{1}(S_{2}^{*}) = S_{1}^{*}\\)<\/mark><\/p>\n\n\n\n

\\(BR_{2}(\\theta_{2}) = S_{2}^{*}, \\theta_{2} = s_{1}^{*}\\)<\/mark><\/p>\n\n\n\n

<\/p>\n\n\n\n

2. Normal-form games
2.5 Application 1: Oligpoly, Tariffs, and Crime (Chapter 10)<\/p>\n\n\n\n

Cournot Duopoly<\/h3>\n\n\n\n

EXAMPLE 2.17<\/mark>. Two firms compete by choosing quantity produced in a market, where the demand function is given by \\(P(q_{1}, q_{2}) = 12 – q_{1} – q_{2}\\). The two firms have an identical cost function \\(C(q_{i}) = 4q_{i}\\). Find the Nash equilibrium of this game.<\/p>\n\n\n\n

Firm 1’s profit function,<\/p>\n\n\n\n

\\(u_{1}(q_{1}, q_{2}) = (12 – q_{1} – q_{2})q_{1} – 4q_{1}\\)<\/p>\n\n\n\n

From the FOC, we obtain \\(BR_{1}(q_{2})\\)<\/mark><\/strong> = \\(\\frac{8-q_{2}}{2}\\)<\/p>\n\n\n\n

By symmetry,<\/p>\n\n\n\n

\\(BR_{2}(q_{1})\\)<\/mark><\/strong> = \\(\\frac{8-q_{1}}{2}\\)<\/p>\n\n\n\n

N = {1, 2}<\/mark><\/p>\n\n\n\n

\\(S_{1}\\) = [0, \u221e) = \\(S_{2}\\)<\/mark><\/p>\n\n\n\n

\\(\\Pi_{1}\\) = Rev – cost = p\u2219q – 4\u2219\\(q_{1}\\) = \\((12 – q_{1} – q_{2})q_{1} – 4q_{1}\\)<\/mark><\/p>\n\n\n\n

\\(\\frac{\\partial \\Pi_{1}}{\\partial q_{1}} = 12 – 2q_{1} – q_{2} – 4 = 0\\) \u21d4 \\(q_{1} = \\frac{8-q_{2}}{2}\\)<\/mark><\/p>\n\n\n\n

Slutsky’s equation<\/p>\n\n\n\n

f(q) = (8-q)\/2, f(q) = 8-2q, q=(0,8)\nx-axis: q1, y-axis: q2<\/code><\/pre>\n\n\n
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\"\"<\/figure><\/div>\n\n\n
<\/p>\n\n\n\n