Extensive-Form Games and Backward Induction: Strategic Insights
Explore the application of extensive-form games in strategic decision-making, focusing on Subgame Perfection and Backward Induction. Understand the dynamics of entry games, such as the interplay between a potential entrant and a monopolist. Delve into the implications of commitment in game theory through insightful analyses of various scenarios.
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EC941 - Game Theory Lecture 5 Prof. Francesco Squintani Email: f.squintani@warwick.ac.uk 1
Structure of the Lecture Definition of Extensive-Form Games Subgame Perfection and Backward Induction Applications: Stackelberg Duopoly, Harris-Vickers Race. 2
Extensive Form Games The strategic form representation is appropriate to describe simultaneous move games. In order to account for the sequence of moves in a game, we introduce the extensive form representation. This will allow to refine the predictions of the Nash Equilibrium concept, and formulate more precise and reasonable predictions. 3
An Entry Game A potential entrant chooses whether to enter a market controlled by a monopolist. If the entrant enters the market, the monopolist can either start a price war, or share the market with the entrant. Intuitively, the incumbent cannot credibly commit to a price war the entrant, and hence cannot deter entry. 4
Let us model the entry game in strategic form. Fight Share In 1, 1 -1,-1 Out 0, 2 0, 2 The game has two Nash Equilibria: (In, Share) and (Out, Fight). Only the first one is reasonable . The second one contains a non-credible threat. 5
Extensive Form Representation 1 The extensive form underlines that the incumbent choice takes place only if the entrant enters, and after entry. O I 2 0, 2 F S 1 -1,-1 6
Backward Induction Solution Starting from the end, player 2 prefers S to F, because 1 > -1. Carry the value [1, 1] to player 2 decision node. 1 O I 1] 2 0, 2 F S 1 -1,-1 7
1 -1,-1 Given that player 2 plays S if player plays I, player 1 chooses I over O, because 1 > 0. Carry the value [1, 1] to player 1 decision node. This is the BI value of the game. The BI solution is (I, S). 1] O I 1] 2 0, 2 F S 1 8
The Value of Commitment Before a potential competitor chooses whether to enter a market or not, the incumbent may or may not burn bridges , i.e. make an investment that reduces her payoff if sharing the market with the competitor. By burning bridges, she successfully deters entry. The competitor correctly anticipates that if entering the market, she will be fought by the incumbent. As a result, the competitor chooses not to enter the market. 9
Extensive Form Representation 2 B N 1 1 O I O I 2 2 0, 2 0, 2 F S F S -2 1 -1,-1 -1,-1 10
Backward Induction Solution 2 B N 1 1 O I O I 2 2 0, 2 0, 2 F S F S -2 1 -1,-1 -1,-1 11
General Definitions Definition An extensive game with perfect information is: a set of players I, a set of sequences E (terminal histories) with the property that no sequence is a proper subhistory of any other sequence, a function P (the player function) that assigns a player to every sequence that is a proper subhistory of some terminal history, for each player i, preferences ui over the set of terminal histories. 12
For example, in the entry game, Players: The challenger and the incumbent. Terminal histories: (In, Share), (In, Fight), and Out. Player function: P( ) = Challenger,P(In) = Incumbent. Preferences The challenger s preferences are given by the payoff function u1 with u1(I, S) = 1, u1(O) = 0, u1(I, F) = -1. The incumbent s preferences are given by payoff function u2 with u2(O) = 2, u2(I, S) = 1, u2(I, F) = -1. 13
Strategies and Outcomes Definition A strategy siof player i in an extensive game with perfect information is a function that assigns to each history h after which it is player i s turn to move (i.e., P(h) = i) an action in A(h), the set of actions available after h. A player s strategy specifies the action the player chooses for every history after which it is her turn to move. 14
For example, in the game below, A A a 1 2 1 3,3 D D d 1,1 0,0 2,2 the strategies of player 1 are {AA, AD, DA, DD}, and the strategies of player 2 are {a, d}. 15
Each strategy profile s uniquely determines a terminal history e that is reached through the strategy s. Because each terminal history e is associated with payoffs u(e), from each extensive form game = (I, E, P, u), we obtain a normal form game G=(I, S, u), by assigning the utility u(e) to the strategy profiles s identifying terminal node e. 16
The strategic form representation is: a d DD The Nash Equilibrium are (AA, a) and any mix between DA and DD, with player 2 playing d. 2, 2 2, 2 DA 2, 2 2, 2 AD 1, 1 0, 0 AA 3, 3 1, 1 17
In terms of outcomes in the extensive form, they correspond to: A A a 1 [3,3] 2 1 3,3 [3,3] [3,3] D D d 1,1 0,0 2,2 The backward-induction solution is (AA, a). 18
Theorem In any extensive form game of perfect information, the backward-induction solution is a Nash Equilibrium. PropositionIn any extensive form game of perfect information in which there are no ties in payoffs, the backward-induction solution is unique. PropositionThere exists Nash Equilibria of extensive form games of perfect information that do not correspond to a backward induction solution. 19
Subgames Definition Let be an extensive game with perfect information, with player function P. For any nonterminal history h of , the subgame (h) following the history h is the following extensive form game: Players The players in . Terminal histories The set of all sequences h of actions such that (h, h ) is a terminal history of . Player function The player P(h, h ) is assigned to each proper subhistory h of a terminal history. Preferences Each player prefers h to h if and only if she prefers (h, h ) to (h, h ) in . 20
Example: A A a 1 2 1 3,3 D D d 1,1 0,0 2,2 These are the 3 subgames of the game: The whole game The game starting with player 2 s decision The final decision of player 1. 1. 2. 3. 21
Subgame Perfect Equilibrium Definition The strategy profile s is a subgame perfect equilibrium if it induces a Nash equilibrium in every subgame. Theorem The strategy profile s is a subgame perfect equilibriumif, for every player i, every history h after which it is player i s turn to move (i.e. P(h) = i), and every strategy ri of player i, the terminal history Oh(s ) generated by s after the history h yields utility to player i no less than the terminal history Oh (ri, s i) generated by the strategy profile (ri, s i) where player i plays ri while every other player j plays s j. 22
Solving for the subgame-perfect equilibrium A A a 1 2 1 3,3 D D d 1,1 0,0 2,2 The unique Nash equilibrium of the smallest subgame is A. The middle subgame has 2 Nash equilibria (D,d) and (A,a), but only (A, a) induces a N.E. in the smaller subgame. 23
A A a 1 2 1 3,3 D D d 1,1 0,0 2,2 The game has 3 (pure-strategy) Nash equilibria (DD, d), (DA, d) and (AA, a), but only (AA, a) induces a Nash Equilibrium in every subgame. 24
In games of perfect information, the subgame perfect equilibrium coincides with the backward-induction solution. But subgame-perfect equilibrium is a more general concept, defined also for extensive form games without perfect information. 25
Application: Stackelberg duopoly There are two firms in the market. Firm i s cost of producing qi units of the good is Ci (qi); the price at which output is sold when the total output is Q is P(Q). Each firm s strategy is the output, as in Cournot model. But the firms make their decisions sequentially, rather than simultaneously. One firm chooses its output, then the other firm does so, knowing the output chosen by the first firm. 26
Extensive Form Representation Players: The two firms. Terminal histories: The set of all sequences (q1, q2) of outputs for the firms (where qi, the output of firm i, is a nonnegative number). Player function: P( ) = 1 and P(q1) = 2 for all q1. Preferences: The payoff of firm i to the terminal history (q1, q2) is its profit qiP(q1 + q2) Ci(qi), for i = 1, 2. A strategy of firm 1 is an output choice q1. A strategy of firm 2 is a function that associates an output q2 to each possible output q1 of firm 1. 27
Backward Induction Solution For any q1 output of firm 1, we find the output b2(q1) of firm 2 that maximize its profit q2 P(q1 + q2) C2 (q2). In any subgame perfect equilibrium, firm 2 s strategy is b2. We then find the output q1 of firm 1 that maximize its profit, given the strategy b2(q1) of firm 2. Firm 1 s output in a subgame perfect equilibrium is the value q1 that maximizes q1P(q1 + b2(q1)) C1 (q1). 28
Suppose that Ci(qi) = cqi for i = 1, 2, and P(Q) = Q if Q , P(Q) = 0 if Q > , where c > 0 and c < . We know that firm 2 s best response to output q1 of firm 1 is b2(q1) = ( c q1)/2 if q1 c, b2(q1) = 0 if q1 > c. In a subgame perfect equilibrium of Stackelberg s game firm 2 s strategy is this function b2and firm 1 s strategy q1* maximizes q1 ( c (q1 + ( c q1)/2)) = q1 ( c q1)/2. The first order condition yields q1* =( c)/2. The unique subgame perfect equilibrium is (q1*, b2). 29
The SPE outcome of the Stackelberg game is: q1S = ( c)/2, q2S = ( c q1* )/2 = ( c)/4. Firm 1 s profit is q1S(P(q1S+q2S) c) = ( c)2/8, firm 2 s profit is q1S(P(q1S+q2S) c) = ( c)2/16. Recall that in the unique Nash equilibrium of Cournot s (simultaneous-move) game, q1C = q2C = ( c)/3, so that each firm s profit is ( c)2/9. First-Mover Advantage: The first firm to move produces more output and obtains more profit than if firms move simultaneously, the second firm produces less output and obtains less profit. 30
Application: Harris-Vickers Race Two firms compete to develop new technologies. How does competition affect the pace of activity? How does a leading advantage translates into a final outcome? We model this race as an extensive game where players alternately choose how many steps to take towards a finish line. 31
Player i = 1, 2 is initially ki > 0 steps from the finish line. On each of her turns, a player can either take 0 steps (at zero cost), 1 step, at a cost of c(1), or 2 steps, at a cost of c(2) > 2c(1). The first player to reach the finish line wins a prize, worth vi > 0 to player i; the losing player s payoff is 0. If, on successive turns, neither player takes any step, the game ends and neither player obtains the prize. 32
Extensive Form Representation The game G1(k1, k2) is defined as follows. Players: The two parties. Player function: P( ) = 1, P(x1) = 2 for all x1, P(x1, y1) = 1 for all (x1, y1), P(x1, y1, x2) = 2 for all (x1, y1, x2), etc. For all t, xtis the number of steps taken by player 1 on her tth turn, and ytare the steps taken by 2 on tth turn. 33
Terminal histories: The set of sequences of the form (x1, y1, x2, y2, . . . , xT) or (x1, y1, x2, y2, . . . , yT) for some integer T, where xtand ytare 0, 1, or 2, there are never two successive 0 s except possibly at the end of a sequence, and either x1+ +xT= k1, y1+ +yT < k2 (1 wins the race), or x1+ +xT < k1 and y1+ +yT= k2 (2 wins the race). Preferences: For an end history in which player i loses, her payoff is minus the sum of the costs of all her moves; for an end history in which she wins it is viminus the sum of these costs. 34
Backward Induction Solution k2 3 2 f 1 k1 1 2 3 4 Firm k wins game Gk(1, 1): whoever moves first takes one step towards the finish line and wins. 35
k2 4 3 f 2 f f 1 k1 1 2 3 4 Firm 1 wins game G1(1, 2) by taking 1 step, and wins game G1(2, 1) by taking 2 steps. (If she took 1 only step, the next round would be equivalent to game G2(1, 1), where 2 wins.) Similarly, firm 2 wins games G2(1, 2) and G2(2, 1). 36
k2 4 3 f f 2 f f 1 k1 1 2 3 4 Firm 1 wins game G1(2, 2) by taking 2 steps. (If she took 1 step, the next round would be equivalent to game G2(1, 2), where 2 wins.) Similarly, firm 2 wins game G2(2, 2). 37
k2 4 1 Firm 1 wins games G1(1, 3) and G1(1, 4) with 1 step. 1 3 f f 2 f f 1 k1 1 2 In games G2(1, 3) and G2(1, 4), firm 2 loses. With 1 step, the next round is G1(1, 2) and G1(1, 3), with 2 steps, the next round is G1(1, 1) and G1(1, 2): in all such games, firm 1 wins. Hence 2 takes 0 steps. 3 4 38
k2 1 1 4 1 1 3 f f 2 f f 1 k1 1 2 Firm 1 wins games G1(2, 3) and G1(2, 4): with 1 step, the next round is G2(1, 3) and G2(1, 4), where firm 2 loses. Firm 2 loses in G2(2, 3) and G2(2, 4): with j > 0 steps, the next round are G1(2, 3-j) and G1(2, 4-j), where 1 wins. 3 4 39
k2 4 1 1 3 1 1 f f 2 2 2 f f 2 2 1 k1 1 2 3 4 Analogously to the previous steps, firm 2 wins games Gk(3, 1), Gk(4, 1), Gk(3, 2) and Gk(4, 2) for k=1, 2. 40
k2 4 1 1 f f 3 1 1 f f f f 2 2 2 f f 2 2 1 k1 1 2 3 4 Consider games Gk(3, 3), Gk(3, 4), Gk(4, 3) and Gk(4, 4). Reaching regions where one player wins for sure is the same as reaching the finish line. So, player k wins all these games. 41
k2 5 1 1 1 1 2 2 f f 4 1 1 f f 2 2 2 2 3 1 1 f f 2 2 2 2 2 f f 2 2 2 2 2 2 1 f f 2 2 2 2 2 2 k1 1 2 3 4 5 6 7 8 Hence, the matrix can then be completed inductively. 42
Summary of the Lecture Definition of Extensive-Form Games Subgame Perfection and Backward Induction Applications: Stackelberg Duopoly, Harris-Vickers Race. 43
Preview of the Next Lecture Extensive-Form Games with Imperfect Information Spence Signalling Game Crawford and Sobel Cheap Talk 44
