Notes on Extensive Form Games y

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1 Notes on Extensive Form Games y ECON 01B - Game Theory Guillermo Ordoñez UCLA January 5, Extensive Form Games So far e have restricted attention to normal form games, here a player s strategy is just a choice of a single uncontingent action, exactly as in games of simultaneous moves. The extensive form of a game conveys more information than the strategic form since it shos a particular sequence of moves timing. This is hy every game presented in extensive form can be expressed in a strategic form and analyzed ith the methods seen in previous notes. Formally, the extensive form of a game contains the folloing information: 1) The set of players ) The order of moves (ho moves hen, represented in a "game tree") 3) Players payo s as a function of the moves that ere made. 4) What the players choices are hen they move 5) What each player knos hen he makes his choices 6) Probability distributions over any exogenous events (moves by "Nature") Since the extensive structure conveys more information it allos for an equiibrium concept that re nes Nash Equilibrium, hich is called Subgame Perfect Equilibrium (SGPE), hich basically consists on the elimination of non-credible threats. Subgame Perfect Equilibrium (SGPE) To discuss subgame perfection let me rst discuss the idea of a subgame. A subgame is every subset of the game tree that looks like a game itself. In ords, it consists on an initial single node and all the nodes that follos it. Naturally the original game is a subgame in itself (the biggest subgame in the 0 y These notes ere prepared as a back up material for TA session. If you have any questions or comments, or notice any errors or typos, please drop me a line at guilord@ucla.edu 1

2 game). Subgames that are not the original game sometimes are called "proper subgames". The main idea and importance of a subgame is that ALL previous actions and history are knon by ALL players at the start of the subgame. De nition 1 A subgame perfect equilibrium is a pro le of strategies that are a Nash Equilibrium in every subgame (including the game as a hole). Because any game is a subgame in itself, a subgame perfect equilibrium pro le is necessarily a Nash Equilibrium. If the only subgame in the game is the game as a hole, the sets of Nash and subgame perfect equilibria coincide. If there is any "proper" subgame, some Nash Equilibria may fail to be subgame perfect. It s easy to see that subgame perfection coincides ith backard induction in nite games of perfect information. (Hoever you have to remind that subgame perfection is more general than backard induction). In fact. Theorem (Zermelo 1913; Khun 1953) A nite extensive form game of perfect information has a pure-strategy Nash Equilibrium. 3 Examples Check examples in Rolf s notes as ell. 3.1 Wage Bargaining 1 An important aspect of the bargaining beteen a Union and a Firm is that the Firm interacts ith the market. A very simpli ed model might go like this: 1) The Union makes a take-it-or-leave-it age o er ; ) if the Firm rejects the o er, both the Union and Firm make 0 3) if the Firm accepts the o er, the Firm then chooses a production plan to maximize pro ts given the age : Suppose i) the objective of the Union is to maximize total income of all orkers, ii) the objective of the Firm is to maximize pro ts, iii) the production plan of the Firm is Q = L; iv) the market demand function is P = 1 Q. a) Sketching the game tree The game states the folloing: 1) The Union makes a take-it-or-leave-it age o er (its objective is to maximize total income I = L()). ) The rm accept or reject the o er (its objective is to maximize pro ts = ()).

3 Strategies: Union: [0; 1) Firm:! fa; rg The payo s (pro ts) for the Firm (()) are determined for each age o er by a pro t maximization for hich the Firm decides the labor to hire (L()) The payo s (orkers income) for the Union (I() = L()) depends on the age o ered considering the hiring decision rule of the Firm L(). Since L() rule follos just a maximization problem by the rm for each, it only helps us to get the payo s in each node. The game tree and the respective payo s are sketched in the folloing gure ( 1 < < ::: < n < :::) (Naturally there is a continuum of o ers. In the tree I m only representing ve of the in nite of them). Union Firm r a r a r a r a r a (0,0) (L 1 1, ( 1 )) (0,0)(L, ( )) (0,0)(L 3 3, ( 3 )) (0,0) (L 4 4, ( 4 )) (0,0) (L 5 5, ( 5 )) b) Finding the subgame perfect equilibrium (SGPE) (Recall this ill be unique since this a perfect information game). In order to nd the SGPE, rst e need to compute the rm reaction to an arbitrary age level set by the Union (e need to solve by backardsinduction). First, in order to get the payo s the Firm ill consider to determine its best response e need to maximize pro ts given a level of choosing the optimal level of employment L the rm ill choose once a age is set. max = P Q L = (1 Q)Q L The problem can be formulated as: s:t: Q = L max = (1 L)L L 3

4 = 1 L L = 0 L = Q = 1 1 = Therefore, the rm ill operate as long as 1 (since at = 1 pro ts are zero). Otherise, the rm ill not produce and ill reject the age o er. No, e turn to the problem of the Union hose objective is to maximize total orkers income. The Union should anticipate that the rm s reaction to the age o er ill be to choose the employment level L () and the best response ill be to accept the o er hen 1 The maximization problem of the Union is then the folloing one, 1 max I = L () = = 0 = 1 The unique subgame perfect equilibrium (SGPE) is: Union: = 1 Firm: < 1! a = 1! xa + (1 x)r ith x [0; 1] > 1! r The outcome of this SGPE is: Union: I ( = 1 ) = 1 8 (since L = 1 4 ) Firm: ( = 1 ) = Wage Bargaining The story goes the same as above. Hoever no suppose Suppose i) the objective of the Union is to maximize total income of all orkers, ii) the objective of the Firm is to maximize pro ts, iii) the production plan of the Firm is Q = (LM) 1= ; iv) the market demand function is P = 1 Q;v) machines cost P M per unit. 4

5 Find the unique subgame perfect equilibrium (SGPE) As can be seen the strategies and game tree do not change. The di erence arises only on the objective function of the rm that ill determine hen the rm accepts or rejects the o er. In order to nd the SGPE, rst e need to compute the rm reaction to an arbitrary age level set by the Union given an exogenous P M (e need to solve also using backards-induction). In this case, the rm must maximize pro ts given a level of and P M choosing the optimal level of employment L and machines M. max = P Q L P M M = (1 Q)Q L P M M The problem can be formulated as: s:t: Q = (LM) 1= max = (LM) 1= LM L P = 1 1= M L M = 0 1= L M L P M = 0 hich gives the folloing ratio and optimal levels: M L = P M 1= PM L = 1 P M M = 1 P M 1= As can be seen, L 0 and M 0 (hence Q 0) as long as 1 Otherise, the rm ill not operate and ill reject the age o er. The optimal production and maximum pro ts are the folloing: Q = (" 1 PM # " 1= 1= 1 P M P M #) 1= = 1 (P M ) 1= 5

6 1 = (P M ) 1= No, e turn to the problem of the Union hose objective is to maximize total income of the orkers. The Union should anticipate that the rm s reaction to the age o er ill be to reject all o ers here > 1 (and also that the Firm ill choose an employment level L (; P M )). The maximization problem of the Union is the folloing: max I = L (; P M ) = 1! 1= PM P M = 1 (P M ) 1= = 1 4 PM 1= P M = 0 = 1 16P M The unique subgame perfect equilibrium (SGPE) is: Union: = 1 16P M Firm: < 1! a = 1! xa + (1 x)r ith x [0; 1] > 1! r The outcome of this SGPE is: Union: I ( = 1 16P M ; P M ) = 1 16 (since L = P M ) Firm: ( = 1 16P M ; P M ) = 1 16 Notice that in this case the production function Q = (LM) 1= has elasticity of substitution equal to 1. Hence, given the possibility for the rm to substitute L by M, the Union is in disadvantage in relation to the result obtained in the rst example. Recall the outcome for the Union is orst hile the outcome for the Firm in equilibrium is the same irrespective of the machines price P M: 3.3 Sequential Bargaining Consider the folloing game: - Game of three stages here players are trying to share a dollar. - The discount rate per period is 0 < < 1 - At period 1 6

7 - Player 1 o ers to take a share s 1 of the dollar, leaving 1 s 1 to player - Player either accepts (and the game ends) or rejects (and the game continues another period). - At period - Player o ers that player 1 takes a share s, leaving 1 s for player. - Player 1 either accepts (and the game ends) or rejects (and the game continues another period). - At period 3 - Player 1 receives a share s and player receives 1 s (being s exogenous and 0 < s < 1). Finding all SGPE We have to solve this problem using backard induction. Period 3: If this period is reached the payo s are (s; 1 s). Expressed from period 1 s stand point (i.e. discounted tice) the payo ill be ( s; (1 s)) or hich is the same ( s; s)) Period : Player proposes and player 1 either accepts or rejects. By backard induction e should start analyzing hat player 1 (the last one in playing at this period) ill do. If player 1 rejects, he gets s (hat he gets in the last period discounted because he or she has to ait until period 3 to get the money). Hence: - Player 1 accepts if s s - Player 1 rejects if s < s No, considering this best response by player 1, if player o ers s < s, he or she ill be rejected and the payo ill be (1 s). If player o ers s = s, then the o er ill be accepted and the payo ill be 1 s. (Naturally, if player o ers s > s the o er ould be accepted but the payo for player ould be smaller than 1 s, so hy he ould do that?) Considering these payo s, player ill prefer to o er s = s and get 1 s > (1 s). Hence - Player o ers s = s Given this equilibrium, period 3 ill not be reached and the payo s ill be (s; 1 s). Expressed from period 1 s stand point (i.e. discounted once) the payo ill be ( s; (1 s)) or hich is the same ( s; s) Period 1: Player 1 proposes and player either accepts or rejects. By backard induction e should start analyzing hat player (the last one in playing at this period) ill do. If player rejects, he gets (1 s) (hat he gets in period discounted because he or she has to ait until period to get the money). Hence: - Player accepts if 1 s 1 (1 s) - Player rejects if 1 s 1 < (1 s) No, considering this best response by player, if player 1 o ers 1 s 1 < (1 s), he or she ill be rejected and the payo ill be s. If player 1 7

8 o ers 1 s 1 = (1 s), then the o er ill be accepted and the payo ill be 1 (1 s). (Naturally, if player 1 o ers 1 s 1 > (1 s) the o er ould be accepted but the payo for player 1 ould be smaller than 1 (1 s), so hy he ould do that?) Considering these payo s, player 1 ill prefer to o er s 1 = 1 (1 s) and get 1 (1 s) > s. Hence - Player 1 o ers s 1 = 1 (1 s) Given this equilibrium, period ill not be reached and the payo s ill be (1 (1 s); (1 s)) or hich is the same (1 + s; s) 3.4 Prison Bargaining Three prisoners are arguing over the last bottle of beer. The arden, tired of the noise, imposes the folloing bargaining scheme. Prisoner 1 must propose a division of beer. If Prisoner and 3 agree, the proposed division is implemented and otherise the arden pours out half the beer (or the arden drinks that half of the beer, hatever you prefer) and send Prisoner 1 back to his cell and Prisoner must propose a division of the remaining beer. If Prisoner 3 agrees the proposal is implemented and otherise arden pours out half the remaining beer and gives the rest to Prisoner 3 ho drinks it. Find the unique SGPE in pure strategies assuming prisoners alays agree hen they are indi erent and there is no discounting ( = 1). Solving backards. 1) If the amount of beer remaining is y and the proposal of Prisoner is (y z; z) then Prisoner 3 accepts if z y and reject otherise. Given this best response by Prisoner 3, Prisoner ill make the loest o er Prisoner 3 ill accept (since otherise Prisoner ould just receive a ride to his cell), this is z = y ) If Prisoner or 3 reject the o er by Prisoner 1, half the beer ill be eliminated and (considering the strategies discussed above) both Prisoners and 3 ould get 1 4 each. Hence Prisoner and 3 ill accept if the share assigned is greater or equal than 1 4 and reject otherise. 3) Prisoner 1 ill propose the orst division that Prisoner and 3 ill accept, so Prisoner 1 ill propose 1 for himself and 1 4 to each of the other prisoners. 8