8.1. Independent Firms The downstream firm's problem is max(a-bxc-wT The upstream firm's problem is max(a-2b)3-cr The output is 8.2. Integrated Firm Suppose now that the two firms merge into one firm. This firms problem is max(a-by)y The output The integrated firm produces twice as much. Why? In general, an integrated monopolist will always produce more than an upstream- downstream pair of monopolists. With an upstream-downstream pair, the upstream monopolist raises its price above its MC and then the downstream monopolist raises its price again above its already inflated MC
8.1. Independent Firms The downstream firm’s problem is maxx (a − bx)x − wx. The upstream firm’s problem is maxx (a − 2bx) x − cx. The output is y∗ = a − c 4b . 8.2. Integrated Firm Suppose now that the two firms merge into one firm. This firm’s problem is max y (a − by)y − cy. The output is y¯ = a − c 2b . The integrated firm produces twice as much. Why? In general, an integrated monopolist will always produce more than an upstreamdownstream pair of monopolists. With an upstream-downstream pair, the upstream monopolist raises its price above its MC and then the downstream monopolist raises its price again above its already inflated MC. 2 — 16
Micro Theory, 2005 Chapter 3 Game Theory The materials are from MWG(1995), Chapters 7-9, 219-305 Two Game forms 1.1. Extensive-Form game The extensive form is a game tree Example 3.1. Matching Pennies(B). Two players. Player 1 puts a penny down first Then, after seeing player 1s choice, player 2 puts her penny down. If the two pennies match, player 1 pays Sl to player 2: otherwise player 2 pays Sl to player 1. Write down the extensive-form game Example 3.2. Matching Pennies(C). This game is just like version B except that when player 1 puts her penny down, she keeps it covered with her hand. Write down the extensive-form game. Definition 3.1. A game in extensive form consists of the following items 1. Sets. A finite set of nodes X, a finite set of possible actions A, a finite set of players N=f1,., nh, and a collection of information sets H 2. Sequence. A game starts from a single node. Except the initial node, each node follows from a single immediate predecessor node. The set of terminal nodes is T all other nodes in x are called decision nodes
Chapter 3 Game Theory Micro Theory, 2005 The materials are from MWG (1995), Chapters 7—9, 219—305. 1. Two Game Forms 1.1. Extensive-Form Game The extensive form is a game tree. Example 3.1. Matching Pennies (B). Two players. Player 1 puts a penny down first. Then, after seeing player 1’s choice, player 2 puts her penny down. If the two pennies match, player 1 pays $1 to player 2; otherwise player 2 pays $1 to player 1. Write down the extensive-form game. Example 3.2. Matching Pennies (C). This game is just like version B except that when player 1 puts her penny down, she keeps it covered with her hand. Write down the extensive-form game. Definition 3.1. A game in extensive form consists of the following items: 1. Sets. A finite set of nodes X, a finite set of possible actions A, a finite set of players N = {1,...,n}, and a collection of information sets H. 2. Sequence. A game starts from a single node. Except the initial node, each node follows from a single immediate predecessor node. The set of terminal nodes is T. All other nodes in X are called decision nodes. 3—1
3. Information Structure. Each node belongs to one and only one information set Denote H(a) as the information set that contains node a. When H() is a singleton, i.e.,H(a)=al, we often refer to H(a) as node a. Each information set is followed by a few branches. Each branch represents a possible action taken by the player who is to make a decision upon observing that information set, i. e, when the play reaches that information set. For H E H, let A(H)=all the branches following H. If it is player i's turn to make a move at an information set h, we call h player i's information set. Each information set belongs to one and only one player, including a special player called nature. 4. Nature. Sometimes nature is included. Let ho be the information set where nature makes a move. A function p: A(Ho)-0, 1 assigns probabilities to actions at information set Ho. Nature is like a player in the model. except that it does not have a payoff function and it does not optimize its choices Payoffs. A collection of payoff functions u=u10,.,un(1 assigns utilities to the players at each terminal node,vlz:T→武 Thus, a game in extensive form is specified by the collection TE={x,A,N,H,,p(),A(),H()} a game is of perfect information if each information set contains a single decisio ode. Otherwise, it is a game of imperfect information The game structure is common knowledge, meaning that all players know the structure of the game, know that their rivals know it, know their rivals know that they know it. and so on 1.2. Normal-Form Game A strategy is a complete contingent plan that specifies how a player will act in ever possible distinguishable circumstance. Thus, a player's strategy is a specification of how he plans to move at each of his information set Definition 3.2. A(pure) strategy for player i is a function s;: Hi-A such that S1(H)∈A(H) for all h∈H,■ Denote S; as the strategy space of player i, which contains all the possible strategie es of player i
3. Information Structure. Each node belongs to one and only one information set. Denote H(x) as the information set that contains node x. When H(x) is a singleton, i.e., H(x) = {x} , we often refer to H(x) as node x. Each information set is followed by a few branches. Each branch represents a possible action taken by the player who is to make a decision upon observing that information set, i.e., when the play reaches that information set. For H ∈ H, let A(H) = { all the branches following H}. If it is player i’s turn to make a move at an information set H, we call H player i’s information set. Each information set belongs to one and only one player, including a special player called nature. 4. Nature. Sometimes nature is included. Let H0 be the information set where nature makes a move. A function ρ : A(H0) → [0, 1] assigns probabilities to actions at information set H0. Nature is like a player in the model, except that it does not have a payoff function and it does not optimize its choices. 5. Payoffs. A collection of payoff functions U = {u1(·),...,un(·)} assigns utilities to the players at each terminal node, ui : T → R. Thus, a game in extensive form is specified by the collection ΓE = {X, A, N, H, U, ρ(·), A(·), H(·)}. A game is of perfect information if each information set contains a single decision node. Otherwise, it is a game of imperfect information. The game structure is common knowledge, meaning that all players know the structure of the game, know that their rivals know it, know their rivals know that they know it, and so on. 1.2. Normal-Form Game A strategy is a complete contingent plan that specifies how a player will act in every possible distinguishable circumstance. Thus, a player’s strategy is a specification of how he plans to move at each of his information set. Definition 3.2. A (pure) strategy for player i is a function si : Hi → A such that si(H) ∈ A(H) for all H ∈ Hi. Denote Si as the strategy space of player i, which contains all the possible strategies of player i. 3—2
Example 3. 3. Strategies for Matching Pennies(B). Write down the strategies. Example 3.4. Strategies for Matching Pennies(C). Write down the strategies Denote a profile of strategies as s=(s1,., Sn), where si E S is a strategy from player i. The normal form or strategic form of a game is to specify a game in terms of strategies and their associated payoffs Definition 3. 3. For a game with n players, the normal form of a game specifies for each player i a set of strategies Si and a payoff function ui(s1, .. Sn). Formally, we write the game as IN=N, S, u. Example 3.5. The Normal Form of Matching Pennies(B). Write down the normal-form game Example 3.6. The Normal Form of Matching Pennies(C). Write down the normal-form game In the normal form, the game looks like a simultaneous-move game For any extensive form of a game, there is a unique normal form. However, the converse is not true, because the condensed representation of a game in the normal form generally omits some of the details present in the extensive form 1.3. Mixed Strategy Definition 3.4. Given a normal-form game TN= N, S, uill, for Si ISlir., Sni, we denote a mixed strategy as ai=(oli,., on, i), where Oki is the probability that Ski is taken. That is, a mixed strategy oi is a probability distribution over the pure strategies in Si. Denote the mixed extension of Si as △(S)={( )≥0 We sometimes denote oki as ai (ski), i.e., oi (ski) is the probability that the mixed strategy oi assigns to the pure strategy Ski E Si For an extensive-form game, there is a simpler way that a player can randomize She could randomize separately over the possible actions at each of her information sets H EH. This is called a behavior strategy
Example 3.3. Strategies for Matching Pennies (B). Write down the strategies. Example 3.4. Strategies for Matching Pennies (C). Write down the strategies. Denote a profile of strategies as s = (s1,...,sn), where si ∈ Si is a strategy from player i. The normal form or strategic form of a game is to specify a game in terms of strategies and their associated payoffs. Definition 3.3. For a game with n players, the normal form of a game specifies for each player i a set of strategies Si and a payoff function ui(s1,...,sn). Formally, we write the game as ΓN = [N, {Si}, {ui}]. Example 3.5. The Normal Form of Matching Pennies (B). Write down the normal-form game. Example 3.6. The Normal Form of Matching Pennies (C). Write down the normal-form game. In the normal form, the game looks like a simultaneous-move game. For any extensive form of a game, there is a unique normal form. However, the converse is not true, because the condensed representation of a game in the normal form generally omits some of the details present in the extensive form. 1.3. Mixed Strategy Definition 3.4. Given a normal-form game ΓN = [N, {Si}, {ui}], for Si = {s1i,...,snii}, we denote a mixed strategy as σi = (σ1i,..., σnii), where σki is the probability that ski is taken. That is, a mixed strategy σi is a probability distribution over the pure strategies in Si. Denote the mixed extension of Si as 7(Si) = + (σ1i,..., σnii) ≥ 0 [ni k=1 σki = 1, . We sometimes denote σki as σi(ski), i.e., σi(ski) is the probability that the mixed strategy σi assigns to the pure strategy ski ∈ Si. For an extensive-form game, there is a simpler way that a player can randomize. She could randomize separately over the possible actions at each of her information sets H ∈ Hi. This is called a behavior strategy. 3—3
Definition 3.5. Given an extensive form game Te, a behavior strategy for player i specifies, for every information set H E H; and action a E A(H), a probability A(a,H)≥0,wth∑a∈A)A(a,H)=1 for all h∈.■ For games of perfect recall, the two types of randomization are equivalent. Because of his, we typically use behavior strategies for extensive-form games and mixed strategies for normal-form games. In fact, we will refer to behavior strategies as mixed strategies 2. Simultaneous-Move games We study simultaneous-move games using the normal form. We introduce four equi librium concepts: Nash equilibrium, dominant-strategy equilibrium, and trembling-hand NE The expected utility function is u4(,0-)=∑…∑ok1…mn(sx1 k1=1kn=1 ∑01(s1)…oa()u1(s,…,sn) s;∈S,i∈N 2.1. Nash Equilibrium Definition 3.6. A strategy profile s=(si,., s*)is a Nash equilibrium (NE) rN=,{S},{ ui if for every i∈N,(s,s)≥ta(s,s2) for all si∈S1 Definition 3. 7. A mixed strategy profile o*=(oi,., a* is a Nash equilibrium (NE)inIN=N,{△(s,)},{ui} if for every i∈N,u(o,0*)≥t(o02) for all ;∈△(S) Let Si=sl, S2i,..., Sn; i. Given a ne o', let St be the set of pure strategies that player i plays with a positive probability a*i >0 under o*, and let so be the set of pure strategies that player i plays with probability zero aki=0 under a Proposition 3. 1. Strategy profile o=oi,., of) is a NE in IN=N, A(Si), uinl iff for each i∈N ()a2(sk,0)=u1(sf1,02) for all ski,si∈时t, (i)ua(s,02)≥ua(si,o2) for all ski∈时 and sii∈.口(2)
Definition 3.5. Given an extensive form game ΓE, a behavior strategy for player i specifies, for every information set H ∈ Hi and action a ∈ A(H), a probability λi(a, H) ≥ 0, with S a∈A(H) λi(a, H)=1 for all H ∈ Hi. For games of perfect recall, the two types of randomization are equivalent. Because of this, we typically use behavior strategies for extensive-form games and mixed strategies for normal-form games. In fact, we will refer to behavior strategies as mixed strategies. 2. Simultaneous-Move Games We study simultaneous-move games using the normal form. We introduce four equilibrium concepts: Nash equilibrium, dominant-strategy equilibrium, and trembling-hand NE. The expected utility function is ui(σi, σ−i) = [n1 k1=1 ··· [nn kn=1 σk11 ··· σknnui(sk11,...,sknn) = [ si∈Si, i∈N σ1(s1)··· σn(sn)ui(s1,...,sn). 2.1. Nash Equilibrium Definition 3.6. A strategy profile s∗ = (s∗ 1,...,s∗ n) is a Nash equilibrium (NE) in ΓN = [N, {Si}, {ui}] if for every i ∈ N, ui(s∗ i , s∗ −i) ≥ ui(si, s∗ −i) for all si ∈ Si. Definition 3.7. A mixed strategy profile σ∗ = (σ∗ 1,..., σ∗ I ) is a Nash equilibrium (NE) in ΓN = [N, {7(Si)}, {ui}] if for every i ∈ N, ui(σ∗ i , σ∗ −i) ≥ ui(σi, σ∗ −i) for all σi ∈ 7(Si). Let Si = {s1i, s2i,...,snii}. Given a NE σ∗, let S+ i be the set of pure strategies that player i plays with a positive probability σ∗ ki > 0 under σ∗, and let S0 i be the set of pure strategies that player i plays with probability zero σ∗ ki = 0 under σ∗. Proposition 3.1. Strategy profile σ∗ = (σ∗ 1,..., σ∗ I ) is a NE in ΓN = [N, {7(Si)}, {ui}] iff for each i ∈ N, (i) ui(ski, σ∗ −i) = ui(sji, σ∗ −i) for all ski, sji ∈ S+ i , (2.1) (ii) ui(ski, σ∗ −i) ≥ ui(sji, σ∗ −i) for all ski ∈ S+ i and sji ∈ S0 i . (2.2) 3—4