Bertrand Model Game Theory Prisoner’s Dilemma Dominant Strategies Repeated Games Oligopoly and Game Theory

Bertrand Model Competition based on setting prices— not quantities (like Cournot) Two variants: Homogeneous goods Differentiated goods

Bertrand Model with Homogeneous Products Market demand curve: P = 30 – Q Q = q 1 + q 2 MC 1 = MC 2 = 3 Good is homogeneousBuyers only care about price Outcome: Both firms will charge $3 Total output will be Q = 27q 1 = q 2 = 13.5 π 1 = π 2 = 0

Bertrand Model with Differentiated Products Firm 1’s Demand: Q 1 = 12 – 2P 1 + P 2 Firm 2’s Demand: Q 2 = 12 – 2P 2 + P 1 TFC = $20TVC = 0 π 1 = P 1 Q 1 – 20 = 12P 1 – 2P P 1 P 2 – 20 Δπ 1 / ΔP 1 =12 – 4P 1 + P 2 = 0 Firm 1’s reaction curve: P 1 = 3 + ¼P 2 Firm 2’s reaction curve: P 2 = 3 + ¼P 1

Bertrand Model with Differentiated Products Firm 1’s reaction curve: P 1 = 3 + ¼P 2 Firm 2’s reaction curve: P 2 = 3 + ¼P 1 To find the Nash Equilibrium: P 1 = 3 + ¼P 2 = 3 + ¼(3 + ¼P 1 )

Nash Equilibrium in a Bertrand Model with Differentiated Products P1P1 P2P2 $4 Firm 2’s reaction curve Firm 1’s reaction curve Nash Equilibrium

What if Firm 1 and 2 Could Collude? π 1 = 12P 1 – 2P P 1 P 2 – 20 π 2 = 12P 2 – 2P P 1 P 2 – 20 π T = 24P – 4P 2 + 2P 2 – 40 = 24P – 2P 2 – 40 Δπ T / ΔP = 24 – 4P= 0 P* = 6 π T = 24(6) – 2(6 2 ) – 40 πTπT = 32 π 1 = π 2 = 16

Components of a Game Players Example: Coke and Pepsi

Components of a Game Players Example: Coke and Pepsi Strategies for Each Player Example: Spend a little (small) or a lot (large) on advertising

Prisoner’s Dilemma Situation Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small Large

Components of a Game Players Example: Coke and Pepsi Strategies for Each Player Example: Spend a little (small) or a lot (large) on advertising Payoffs

Prisoner’s Dilemma Situation Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 Large

Prisoner’s Dilemma Situation Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 Large

Prisoner’s Dilemma Situation Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 Large

Prisoner’s Dilemma Situation Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 π P = +13 Large

Prisoner’s Dilemma Situation Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 π P = +13 Large π C = +13 π P = −2 π C = +3 π P = +3

Prisoner’s Dilemma Situation Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 π P = +13 Large π C = +13 π P = −2 π C = +3 π P = +3

Prisoner’s Dilemma Situation Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 π P = +13 Large π C = +13 π P = −2 π C = +3 π P = +3

Prisoner’s Dilemma Situation Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 π P = +13 Large π C = +13 π P = −2 π C = +3 π P = +3

Prisoner’s Dilemma Situation Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 π P = +13 Large π C = +13 π P = −2 π C = +3 π P = +3

Nash Equilibrium Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 π P = +13 Large π C = +13 π P = −2 π C = +3 π P = +3

Equilibrium if Players Could Collude Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 π P = +13 Large π C = +13 π P = −2 π C = +3 π P = +3

Coke’s Dominant Strategy Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 π P = +13 Large π C = +13 π P = −2 π C = +3 π P = +3

Pepsi’s Dominant Strategy Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 π P = +13 Large π C = +13 π P = −2 π C = +3 π P = +3

Outcome of the Game Pepsi’s Spending On Advertising SmallLarge Coke’s Spending on Advertising Small π C = +8 π P = +8 π C = −2 π P = +13 Large π C = +13 π P = −2 π C = +3 π P = +3

Repeated Games ● repeated game Game in which actions are taken and payoffs received over and over again. PRICING PROBLEM Firm 2 Low priceHigh price Firm 1 Low price 10, 10100, –50 High price –50, 10050, 50 Suppose this game is repeated over and over again—for example, you and your competitor simultaneously announce your prices on the first day of every month. Should you then play the game differently? 13.8 TIT-FOR-TAT STRATEGY In the pricing problem above, the repeated game strategy that works best is the tit-for-tat strategy. ● tit-for-tat strategy Repeated-game strategy in which a player responds in kind to an opponent’s previous play, cooperating with cooperative opponents and retaliating against uncooperative ones.