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Chapter-6

# Chapter-6 - Chapter 6 Continuous Games Players strategy...

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Chapter 6: Continuous Games Players’ strategy sets are an interval of real numbers: continuous strategy set . First we cover Bertrand (price) competition and elected office competition. Then we show how to find Nash equilibria by using calculus. We need to get “inside the head” of a player to figure out his incentives. Players’ strategy sets are an interval of real numbers: continuous strategy set . First we cover Bertrand (price) competition and elected office competition. Then we show how to find Nash equilibria by using calculus. We need to get “inside the head” of a player to figure out his incentives.

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Bertrand (Price) Competition Two gift shops sell the same souvenir (Ted Bear), which costs \$10 per unit. Each shop charges a price (p 1 and p 2 ) within the interval [0,100]. If the lower price is p, then the total demand is 100 - p. Consumers will only buy from the shop who charges a lower price. If two shops charge the same price, then they split the demand equally. Two gift shops sell the same souvenir (Ted Bear), which costs \$10 per unit. Each shop charges a price (p 1 and p 2 ) within the interval [0,100]. If the lower price is p, then the total demand is 100 - p. Consumers will only buy from the shop who charges a lower price. If two shops charge the same price, then they split the demand equally.
Bertrand (Price) Competition Shop 1's demand curve : Shop 2’s demand curve is defined analogously. Shop 1's demand curve : Shop 2’s demand curve is defined analogously. ( ) 112 , Dpp () 11 2 1 2 1 1 2 2 21 100 if ,1 0 0 i f 0i f pp p D p p p < = −= <

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Bertrand (Price) Competition Price (\$) # of units 40 100 60 112 ( , 40) Dpp = 30 0 0 Suppose Shop 2 charges a price of 40, the demand curve for Shop 1 looks like: Suppose Shop 2 charges a price of 40, the demand curve for Shop 1 looks like:
Bertrand (Price) Competition Each Shop’s objective is to maximize its profit (given the other Shop’s price). Shop 1’s payoff (or profit) function: Each Shop’s objective is to maximize its profit (given the other Shop’s price). Shop 1’s payoff (or profit) function: () ( ) ( ) 11 1 2 1 1 2 2 21 10 100 if 10 100 if 0i f p pp p p p p p −− < −= <

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Bertrand (Price) Competition Suppose Shop 2 charges a price of 40, the payoff function for Shop 1 looks like: Suppose Shop 2 charges a price of 40, the payoff function for Shop 1 looks like: Payoff 40 100 0 10 1 p
Bertrand (Price) Competition First we rule out any price strictly below 10. - If a shop make any sells for a price strictly below 10, it ends up with a negative payoff. - But it can always ensure a zero payoff. Suppose both shops charge prices > 10, and (say ) - Under this strategy pair, shop 1 gets 0 and shop 2 gets a positive payoff.

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Chapter-6 - Chapter 6 Continuous Games Players strategy...

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