Microeconomics. 2. Game Theory

Microeconomics 2. Game Theory Alex Gershkov http://www.econ2.uni-bonn.de/gershkov/gershkov.htm 18. November 2008 1 / 36

Dynamic games Time permitting we will cover 2.a Describing a game in extensive form (efg) 2.b Imperfect information 2.c Mixed and behavioural strategies; Kuhn s Theorem 2.d Nash equilibrium 2.e Credible strategies and Subgame Perfection 2.f Backward induction 2.g Applications 2 / 36

2.a Describing a game in Extensive form We want to capture dynamic aspects of a game where timing is important. Consider the following game 1. 2 parties {P1,P2} are trying to share two indivisible units of a good yielding positive utility (say one util each) 2. suppose that P1 makes a take-it-or-leave-it offer to P2 3. P2 after having observed this offer decides whether to accept or reject this offer 4. if no agreement is reached (ie. if P2 rejects the offer), both P1 and P2 get nothing. A game s Extensive form (efg) is a description of the sequential structure of dynamic games such as above. 3 / 36

2.a Describing a game in Extensive form Features: A1 moves occur in sequence, A2 all previous moves are observed before a move is chosen, and A3 payoffs and structure of the game are common knowledge. Definition Games satisfying (A1 A3) are called finite efg of perfect information. 4 / 36

2.a Describing a game in Extensive form The efg of a perfect information game consists of E1. the set of players N E2. the set of sequences (finite or infinite) H, that satisfies the following three properties: 0.1 H 0.2 if {a k } K k=1 H and K > L, then {ak } L k=1 H for every L 0.3 if an infinite sequence {a k } k=1 satisfies {ak } L k=1 H for all finite L, then {a k } k=1 H. 5 / 36

2.a Describing a game in Extensive form Each member of H is a history; each component of a history a k is an action taken by a player. A history {a l } k l=1 is terminal if it is infinite or if there is no a K+1 such that {a l } K+1 l=1 H. The set of terminal histories is denoted Z. E3. a player fn P : H Z N where P(h) is the label of the player who is supposed to choose an action after history h H E4. for each player i N, a payoff fn u i : Z R. 6 / 36

2.a Describing a game in Extensive form Definition A game is called finite if the number of stages is finite and the number of feasible actions at any stage is finite. We denote, for every h H Z such that P(h) = i, the set of actions available to i by A i (h) = {s i (h,s i ) H}. Definition A perfect information efg Γ consists of Γ = {N, H,P,u}. 7 / 36

2.a Describing a game in Extensive form Typically, an efg of perfect info can be described by the game tree The game tree T consisting of the set of nodes (including decision and terminal nodes) and the branches which are directed connections between nodes; T must satisfy the tree conditions there is one node without incoming branches called the initial node (the open circle) for any given node, there is a unique path connecting it to the initial node. any non-terminal node corresponds to the player that chooses the action there. Each branch from a node corresponds to an action a available to the player at this info set. Payoffs are given for all i N at the terminal nodes The tree captures the temporal structure of how events unfold over time. 8 / 36

2.a Describing a game in Extensive form The key definition of the efg is the strategy. It is a complete, contingent plan of action specifying a choice for the concerned player at every possible history. Definition A strategy for player i N is a sequence s i = {s i (h)} h Hi, where H i = {h h H Z and P(h) = i} and s i (h) A i (h). 9 / 36

2.g Backward induction Finite games satisfying (A1 A3) can be solved by Backward Induction. Roughly speaking this is replacing each choice set which only leads to terminal nodes in Z by the corresponding NE qm outcome notice that this creates a new set of terminal nodes Z in the shortened game now again replace all choice sets which only lead to terminal nodes in Z by the corresponding NE qm outcome repeat the above until the game is reduced to the initial node and a set of choices leading to terminal nodes only; the NE qm outcome of this game is the backward induction outcome of the original game. 10 / 36

2.g Backward induction Theorem Any perfect information game with a finite number of strategies and players can be solved backwards and therefore has a pure strategy eq m. 11 / 36

2.b Imperfect information To introduce information imperfections we require the following additions to the perfect information requirements (E1 E4) E3. the opportunity for chance moves by the additional player Nature (N) E3. a players fn P : H Z 2 N assigns to each nonterminal history a set of players E4. we assume that the u i : Z R satisfy the axioms of vn-m expected utility theory E4. for every h H Z such that N P(h), we assume that there exists a probability measure f N ( h) defined on A N (h), where each such probability measure is independent of every other such measure 12 / 36

2.b Imperfect information E5. for every i N, D i is a partition of {h H i P(h)} with the property that A i (h) = A i (h ) whenever h and h are in the same element d i D i Definition A set d i D i is called information set of player i. Interpretation: any given member of d i is indistinguishable to player i. Definition A game is of perfect recall if no player ever forgets any information he once knew and all players know the actions they have chosen previously. In perfect information games of perfect recall all information sets are singletons. 13 / 36

2.b Imperfect information Definition An imperfect information efg Γ consists of Γ = {N, H, P, f, D, u}. Definition The probability measures f N ( h) over Nature s moves A N (h) at h H are called prior probabilities over A(h). 14 / 36

2.c Mixed & behavior strategies Convexity considerations lead us to allow for mixed actions. Definition Player i s mixed strategy σ i is a probability distribution over a set of (pure) strategies. Definition A behavior strategy for player i, β i is an element of the Cartesian product di D i (A(d i )). So the difference is that a mixed strategy is a mixture over complete, contingent plans: a pure strategy is selected randomly before play starts a behavioral strategy specifies a probability distribution over actions at each d i and the probability distribution at different info sets are independent. 15 / 36

2.c Mixed & behavioural strategies These two objects are different the mixed strategy selects one pure complete, contingent plan randomly at the beginning of the game the behavioural strategy specifies a randomisation over the available actions for each point of choice but the difference only matters in games of imperfect recall. Definition Two strategies σ i and σ i are equivalent if they lead to the same prob distribution over outcomes for all σ i. Theorem (Kuhn 1953) In a game of perfect recall, mixed and behavior strategies are equivalent. 16 / 36

2.d Nash equilibrium Definition We denote the terminal histories when each player i follows s i,i N by o(s) Z. Definition A NE qm of an efg is a strategy profile σ such that for every player i N and all σ i (S i ) u i (o(σ )) u i (o(σ i,σ i )). We can solve an efg for NE qa by transforming the game into a reduced strategic form game (rsfg) and solving this game in the usual way. 17 / 36

2.d Nash equilibrium Consider the following efg l 1 (2,1) 2 u r 1 (0,0) 1 d l 2 (-1,1) 2 r 2 A 1 = {u,d} S 1 = {u,d} A 2 = {l 1,r 1,l 2,r 2 } S 2 = {(l 1,l 2 ),(l 1,r 2 ), (r 1,l 2 ),(r 1,r 2 )} The rsfg for this game is {N,S,u(o(s))} (3,2) N = [u,(l 1,l 2 )], [d,(l 1,r 2 )], [d,(r 1,r 2 )] l 1,l 2 l 1,r 2 r 1,l 2 r 1,r 2 u 2,1 2,1 0,0 0,0 d -1,1 3,2-1,1 3,2 18 / 36

2.d Nash equilibrium Notice that there are several efg s for the same sfg the set of NE qa of the efg can be found by looking at the set of NE qa of the sfg there is a serious problem with NE qa in efg: there may be actions in a strategy which do not affect an (e qm) outcome which are inconsistent with what the associated player would choose if moving at that node. 19 / 36

2.e Subgame Perfection Definition: A proper subgame G of an efg Γ consists of a single node and all its successors in Γ, with the following properties: 1. if node x G and for some i N x d i (x ), then x G. That is x and x are in the same information set in the subgame if and only if they are in the same information set in the original game 2. the payoff function of the subgame is just the restriction of the original payoff function to the terminal nodes of the subgame 20 / 36

2.e Subgame Perfection Less formally, part of an efg is called a subgame if it starts from a singleton information set d it contains all successors to d (until the end of the game) no node outside the set of successors to d is contained in any of the subgame s information sets. Notice that by definition, the entire game is a subgame of itself after different histories follow different subgames players know that they are in the same subgame. 21 / 36

2.e Subgame Perfection Definition (Selten): A NE qm s of an efg Γ is called subgame perfect e qm (SGPE qm) iff it induces a NE qm for every (proper) subgame Γ(h) for every h H Z. 22 / 36

2.e Subgame Perfection Consider the games such that 1. in each stage k, every player knows all the actions, including those by Nature, that were taken at any previous stage 2. each player moves at most once within a given stage 3. no information set contained in stage k provides any knowledge of play in that stage these games are called multi-stage games with observed actions. Notice that multi-stage games with observed actions can be both finite and infinite horizon games. 23 / 36

2.e Subgame Perfection Definition: A strategy profile s satisfies the one-stage-deviation condition if no player i can gain by deviating from s in a single stage and conforming to s thereafter. Theorem(one-stage-deviation principle): In a finite multi-stage game with observed actions, strategy profile s is subgame perfect if and only if it satisfies the one-stage-deviation condition. More precisely, profile s is subgame perfect iif there is no player i and strategy ŝ i that agrees with s i except at a single stage t and history h t, and such that ŝ i is a better response to s i than s i conditional on h t being realized. 24 / 36

2.e Subgame Perfection Definition: A game is continuous at infinity if for each player i the utility function u i satisfies sup u i (h) u i ( h) 0 as t h, h s.t. h t = h t Theorem(one-stage-deviation principle): In an infinite multi-stage game with observed actions that is continuous at infinity, strategy profile s is subgame perfect if and only if it satisfies the one-stage-deviation condition. 25 / 36

2.e Subgame Perfection Example: Bargaining model two players must agree on how to share a pie of size 1 in periods 0,2,4,... P1 proposes sharing rule (x,1 x) and P2 can accept or reject. If P2 accepts, the game ends if P2 rejects in period 2k, then in period 2k + 1 P2 proposes sharing rule (x,1 x) and P1 can accept or reject. if (x,1 x) is accepted at date t, the payoffs are (δ t x,δ t (1 x)), where δ < 1 is the players discount factor 26 / 36

2.e Subgame Perfection The SGPE qm of this game is: Pi always demands a share (1 δ)/(1 δ 2 ) when it is his turn to make an offer. He accepts any share greater or equal to δ(1 δ)/(1 δ 2 ). 27 / 36

2.f The value of commitment Tournament (Dixit 87) 2 players choose how much effort to exert (e 1,e 2 ) 0 player i wins the tournament with probability pr i (e i,e j ) = e i e i +e j where i {1,2} and j = 3 i if (e 1,e j ) 0 and 1 2 otherwise the winner gets the prize of K > 0 the costs of exerting effort e 0 is c 1 (e) = e for player 1 and c 2 (e) = 2e for player 2 agent i s expected utility is given by Kpr i (e i,e j ) c i (e i ) 28 / 36

2.f The value of commitment Without commitment: Nash equilibrium (e1,e 2 ) should satisfy e 1 = argmax e1 Kpr 1 (e 1,e 2) c 1 (e 1 ) e 2 = argmax e 2 Kpr 2 (e 2,e 1 ) c 2(e 2 ) Which is given by (e 1,e 2 ) = (2 9 K, 1 9 K) while equilibrium utilities are ( 2 9 K, 1 9 K) 29 / 36

2.f The value of commitment With commitment: Assume that player 1 is able to commit to some effort level e 1. P2 will choose effort e 2 = argmax e2 Kpr 2 (e 1,e 2 ) c 2 (e 2 ) K The solution is given by e 2 (e 1 ) = 2 e 1 e 1 Hence, P1 should commit to the effort level e 1 e 1 = argmax e1 Kpr 1 (e 1,e 2 (e 1 )) c 1 (e 1 ) The solution is given by e 1 = K 2. Hence, the vector of the chosen actions is ( K 2,0) and of utilities ( K 2,0) the ability to commit increases the utility. 30 / 36

2.f Repeated games Repeated prisoner s dilemma. Current payoffs g i (a ) are given by cooperate defect cooperate 1,1-1,2 defect 2,-1 0,0 players discount factor is δ the utility of a sequence (a 0,...,a T ) is 1 δ 1 δ T+1 T δ t g i (a t ) t=0 31 / 36

2.f Repeated games Stage game finite game (just for simplicity) N players simultaneous move (just for simplicity) game A i action space for i stage-game payoff function g i : A R with A = i A i Repeated game history h t = (a 0,...,a t 1 ) is the realized choice of actions before t with a τ = (a1 τ,...,aτ N ) a mixed (behavior) strategy σ i : H t (A i ) where H t = (A) t is the space of all possible period t histories payoff function, for instance 1 δ T 1 δ T+1 δ t g i (a t ) or 1 T g i (a t ) T t=0 t=0 32 / 36

2.f Repeated games Equilibria of repeated prisoner s dilemma The game is played only once, (defect,defect) is the only equilibrium. The game is repeated finite number of times. The unique subgame perfect eq. is for both players to defect every period. The game is played infinitely often. The profile (defect, defect) in all periods remains a subgame perfect equilibrium. 33 / 36

2.f Repeated games For infinitely repeated game with δ > 1/2 the following strategy profile is a subgame-perfect eq as well: cooperate in the first period and continue to cooperate as long as no player has ever defected. if any player has ever defected, then defect for the rest of the game 34 / 36

2.f Repeated games Two repetition of the stage game: L M R U 0,0 3,4 6,0 M 4,3 0,0 0,0 D 0,6 0,0 5,5 If played once, there are three equilibria: (M,L), (U,M), and mixed equilibrium ((3/7U, 4/7M),(3/7L, 4/7M)) with corresponding payoffs (4,3), (3,4) and (12/7,12/7). 35 / 36

2.f Repeated games Assume that δ > 7/9 Consider the following strategy profile: P1: Play D in the first stage. If the first stage outcome is (D,R), then play M in the second stage, otherwise, play (3/7U, 4/7M) P2: Play R in the first stage. If the first stage outcome is (D,R), then play L in the second stage, otherwise, play (3/7L, 4/7M) It constitutes a subgame-perfect equilibrium. 36 / 36