page.1 Game Theory Extensive Form Games: Applications Levent Ko¸ ckesen Ko¸ c University Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 1 / 23
page.2 A Simple Game You have 10 TL to share A makes an offer ◮ x for me and 10 − x for you If B accepts ◮ A’s offer is implemented If B rejects ◮ Both get zero Half the class will play A (proposer) and half B (responder) ◮ Proposers should write how much they offer to give responders ◮ I will distribute them randomly to responders ⋆ They should write Yes or No Click here for the EXCEL file Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 2 / 23
page.3 Ultimatum Bargaining A Two players, A and B, bargain over a cake of size 1 x Player A makes an offer x ∈ [0 , 1] to player B B If player B accepts the offer ( Y ) , agreement is reached Y N ◮ A receives x ◮ B receives 1 − x If player B rejects the offer ( N ) x, 1 − x 0 , 0 both receive zero Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 3 / 23
page.4 Subgame Perfect Equilibrium of Ultimatum Bargaining We can use backward induction B’s optimal action ◮ x < 1 → accept ◮ x = 1 → accept or reject 1. Suppose in equilibrium B accepts any offer x ∈ [0 , 1] ◮ What is the optimal offer by A? x = 1 ◮ The following is a SPE x ∗ = 1 s ∗ B ( x ) = Y for all x ∈ [0 , 1] 2. Now suppose that B accepts if and only if x < 1 ◮ What is A’s optimal offer? ⋆ x = 1? ⋆ x < 1? Unique SPE x ∗ = 1 , s ∗ B ( x ) = Y for all x ∈ [0 , 1] Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 4 / 23
page.5 Bargaining Bargaining outcomes depend on many factors ◮ Social, historical, political, psychological, etc. Early economists thought the outcome to be indeterminate John Nash introduced a brilliant alternative approach ◮ Axiomatic approach: A solution to a bargaining problem must satisfy certain “reasonable” conditions ⋆ These are the axioms ◮ How would such a solution look like? ◮ This approach is also known as cooperative game theory Later non-cooperative game theory helped us identify critical strategic considerations Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 5 / 23
page.6 Bargaining Two individuals, A and B, are trying to share a cake of size 1 If A gets x and B gets y ,utilities are u A ( x ) and u B ( y ) If they do not agree, A gets utility d A and B gets d B What is the most likely outcome? u B 1 d B u A 1 d A Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 6 / 23
page.7 Bargaining Let’s simplify the problem u A ( x ) = x , and u B ( x ) = x d A = d B = 0 A and B are the same in every other respect What is the most likely outcome? u B 1 45 ◦ 0 . 5 ( d A , d B ) u A 0 . 5 1 Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 7 / 23
page.8 Bargaining How about now? d A = 0 . 3 , d B = 0 . 4 u B 1 45 ◦ 0 . 55 0 . 4 u A 0 . 3 0 . 45 1 Let x be A’s share. Then Slope = 1 = 1 − x − 0 . 4 x − 0 . 3 or x = 0 . 45 So A gets 0 . 45 and B gets 0 . 55 Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 8 / 23
page.9 Bargaining In general A gets d A + 1 2(1 − d A − d B ) B gets d B + 1 2(1 − d A − d B ) But why is this reasonable? Two answers: 1. Axiomatic: Nash Bargaining Solution 2. Non-cooperative: Alternating offers bargaining game Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 9 / 23
page.10 Bargaining: Axiomatic Approach John Nash (1950): The Bargaining Problem, Econometrica 1. Efficiency ⋆ No waste 2. Symmetry ⋆ If bargaining problem is symmetric, shares must be equal 3. Scale Invariance ⋆ Outcome is invariant to linear changes in the payoff scale 4. Independence of Irrelevant Alternatives ⋆ If you remove alternatives that would not have been chosen, the solution does not change Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 10 / 23
page.11 Nash Bargaining Solution What if parties have different bargaining powers? Remove symmetry axiom Then A gets x A = d A + α (1 − d A − d B ) B gets x B = d B + β (1 − d A − d B ) α, β > 0 and α + β = 1 represent bargaining powers If d A = d B = 0 x A = α and x B = β Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 11 / 23
page.12 Alternating Offers Bargaining Two players, A and B, bargain over a cake of size 1 At time 0, A makes an offer x A ∈ [0 , 1] to B ◮ If B accepts, A receives x A and B receives 1 − x A ◮ If B rejects, then at time 1, B makes a counteroffer x B ∈ [0 , 1] ◮ If A accepts, B receives x B and A receives 1 − x B ◮ If A rejects, he makes another offer at time 2 This process continues indefinitely until a player accepts an offer If agreement is reached at time t on a partition that gives player i a share x i ◮ player i ’s payoff is δ t i x i ◮ δ i ∈ (0 , 1) is player i ’s discount factor If players never reach an agreement, then each player’s payoff is zero Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 12 / 23
page.13 A x A B Y N B x A , 1 − x A x B A Y N A δ A (1 − x B ) , δ B x B x A B Y N A δ 2 A x A , δ 2 B (1 − x A ) Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 13 / 23
page.14 Alternating Offers Bargaining Stationary No-delay Equilibrium A 1. No Delay: All equilibrium offers are accepted x A B 2. Stationarity: Equilibrium offers do not depend on time Y N Let equilibrium offers be ( x ∗ A , x ∗ B ) B x A , 1 − x A What does B expect to get if she rejects x ∗ A ? x B ◮ δ B x ∗ B A Therefore, we must have Y N A 1 − x ∗ A = δ B x ∗ δ A (1 − x B ) , δ B x B B x A B Similarly 1 − x ∗ B = δ A x ∗ Y N A A δ 2 A x A , δ 2 B (1 − x A ) Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 14 / 23
page.15 Alternating Offers Bargaining There is a unique solution 1 − δ B x ∗ A = 1 − δ A δ B 1 − δ A x ∗ B = 1 − δ A δ B There is at most one stationary no-delay SPE Still have to verify there exists such an equilibrium The following strategy profile is a SPE Player A: Always offer x ∗ A , accept any x B with 1 − x B ≥ δ A x ∗ A Player B: Always offer x ∗ B , accept any x A with 1 − x A ≥ δ B x ∗ B Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 15 / 23
page.16 Properties of the Equilibrium Bargaining Power Player A’s share 1 − δ B π A = x ∗ A = 1 − δ A δ B Player B’s share A = δ B (1 − δ A ) π B = 1 − x ∗ 1 − δ A δ B Share of player i is increasing in δ i and decreasing in δ j Bargaining power comes from patience Example δ A = 0 . 9 , δ B = 0 . 95 ⇒ π A = 0 . 35 , π B = 0 . 65 Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 16 / 23
page.17 Properties of the Equilibrium First mover advantage If players are equally patient: δ A = δ B = δ 1 δ π A = 1 + δ > 1 + δ = π B First mover advantage disappears as δ → 1 δ → 1 π B = 1 δ → 1 π i = lim lim 2 Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 17 / 23
page.18 Capacity Commitment: Stackelberg Duopoly Remember Cournot Duopoly model? ◮ Two firms simultaneously choose output (or capacity) levels ◮ What happens if one of them moves first? ⋆ or can commit to a capacity level? The resulting model is known as Stackelberg oligopoly ◮ After the German economist Heinrich von Stackelberg in Marktform und Gleichgewicht (1934) The model is the same except that, now, Firm 1 moves first Profit function of each firm is given by u i ( Q 1 , Q 2 ) = ( a − b ( Q 1 + Q 2 )) Q i − cQ i Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 18 / 23
page.19 Nash Equilibrium of Cournot Duopoly Best response correspondences: Q 1 = a − c − bQ 2 2 b Q 2 = a − c − bQ 1 2 b Nash equilibrium: � a − c � 3 b , a − c ( Q c 1 , Q c 2 ) = 3 b In equilibrium each firm’s profit is 2 = ( a − c ) 2 π c 1 = π c 9 b Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 19 / 23
page.20 b Cournot Best Response Functions Q 2 a b a − c b B 1 a − c 2 b a − c 3 b B 2 a Q 1 a − c a − c a − c b 3 b 2 b b Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 20 / 23
page.21 Stackelberg Model The game has two stages: 1. Firm 1 chooses a capacity level Q 1 ≥ 0 2. Firm 2 observes Firm 1’s choice and chooses a capacity Q 2 ≥ 0 1 Q 1 2 Q 2 u 1 , u 2 u i ( Q 1 , Q 2 ) = ( a − b ( Q 1 + Q 2 )) Q i − cQ i Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 21 / 23
page.22 Backward Induction Equilibrium of Stackelberg Game Sequential rationality of Firm 2 implies that for any Q 1 it must play a best response: Q 2 ( Q 1 ) = a − c − bQ 1 2 b Firms 1’s problem is to choose Q 1 to maximize: [ a − b ( Q 1 + Q 2 ( Q 1 ))] Q 1 − cQ 1 given that Firm 2 will best respond. Therefore, Firm 1 will choose Q 1 to maximize [ a − b ( Q 1 + a − c − bQ 1 )] Q 1 − cQ 1 2 b This is solved as Q 1 = a − c 2 b Levent Ko¸ ckesen (Ko¸ c University) Extensive Form Games: Applications 22 / 23
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