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N08-Nash-continuous[1]

# N08-Nash-continuous[1] - 2009 Calculus and Game Theory...

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Calculus and Game Theory 2009 Finding Nash equilibria in continuous strategies 1 Cournot duopoly with specific numbers Two firms compete in quantities. The market inverse demand curve is P = 56 - 2 Q , where Q = q 1 + q 2 . The cost functions are c 1 ( q 1 ) = 20 q 1 and c 2 ( q 2 ) = 20 q 2 . Step 1: formalise payoff functions Firm 1’s payoff is her profits; revenue minus costs: π 1 ( p 1 , p 2 ) = P × q 1 - 20 q 1 where P = 56 - q 1 - q 2 = (56 - 2 · ( q 1 - q 2 )) × q 1 - 20 q 1 = (56 - 20 - 2 q 1 - 2 q 2 ) × q 1 = (36 - 2 q 1 - 2 q 2 ) × q 1 Step 2: take a first-order condition Using the product rule: ∂π 1 ∂q 1 = ( - 2) × q 1 + (36 - 2 q 1 - 2 q 2 ) × 1 = 36 - 4 q 1 - 2 q 2 Setting this slope to zero gives us a first-order condition. Step 3: solve the first-order condition to get a best-response rela- tion When the slope of firm 1’s payoff function is zero as q 1 is increased, it must be true that: 0 = 36 - 4 q 1 - 2 q 2 4 q 1 = 36 - 2 q 2 q 1 = 36 - 2 q 2 4 = B 1 ( q 2 ) 1

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Calculus and Game Theory 2009 Note that this is a linear function with vertical intercept 36 / 4 = 9 and slope - 1 2 . Step 4: simultaneously solve the best-responses (or the first-order conditions) to get the Nash equilibrium If the two best response functions are: q 1 = 9 - q 2 2 , q 2 = 9 - q 1 2 then the Nash equilibrium will be the intersection of these best responses. We can substitute the second one into the first, and then solve for q 1 . q 1 = 9 - 1 2 bracketleftbigg 9 - 1 2 q 1 bracketrightbigg q 1 - q 1 1 4 = 9 - 9 2 q 1 × parenleftbigg 1 - 1 4 parenrightbigg = 9 × parenleftbigg 1 - 1 2 parenrightbigg q 1 3 4 = 9 2 q 1 = 4 3 · 9 2 = 2 × 3 = 6 The short cut The two firms have symmetric payoff functions, so it is a reasonable conclu-
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N08-Nash-continuous[1] - 2009 Calculus and Game Theory...

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