EC941 - Game Theory

1. EC941 - Game Theory Lecture 5 Prof. Francesco Squintani Email: f.squintani@warwick.ac.uk

2. Structure of the Lecture • Definition of Extensive-Form Games • Subgame Perfection and Backward Induction • Applications: Stackelberg Duopoly, Harris-Vickers Race.

3. 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.

4. 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.

5. Fight Share Let us model the entry game in strategic form. 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. In Out 1, 1 -1,-1 0, 2 0, 2

6. 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, 1

7. Backward Induction Solution 1 Starting from the end, player 2 prefers S to F, because 1 > -1. Carry the value [1, 1] to player 2 decision node. O I [1, 1] 2 0, 2 F S -1,-1 1, 1

8. 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, 1] O I [1, 1] 2 0, 2 F S -1,-1 1, 1

9. 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.

10. Extensive Form Representation 2 B N 1 1 O I O I 2 2 0, 2 0, 2 F S F S -1,-1 1, -2 -1,-1 1, 1

11. Backward Induction Solution 2 B N 1 1 O I O I 2 2 0, 2 0, 2 F S F S -1,-1 1, -2 -1,-1 1, 1

12. General Definitions Definition An extensive game with perfect information G 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, preferencesuiover the set of terminal histories.

13. 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.

14. Strategies and Outcomes DefinitionA strategy siof player iin 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.

15. A A a 1 2 1 • For example, in the game below, the strategies of player 1 are {AA, AD, DA, DD}, and the strategies of player 2 are {a, d}. 3,3 D D d 1,1 0,0 2,2

16. 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 G = (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.

17. a d • The strategic form representation is: The NashEquilibrium are (AA, a) and any mix between DA and DD, with player 2 playing d. DD DA 2, 2 2, 2 2, 2 2, 2 AD AA 1, 1 0, 0 3, 3 1, 1

18. In terms of outcomes in the extensive form, they correspond to: • The backward-induction solution is (AA, a). A A a 1 2 1 3,3 [3,3] [3,3] [3,3] D D d 1,1 0,0 2,2

19. 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.

20. Subgames Definition Let Γ be an extensive game with perfect information, with player function P. For any nonterminal history h of Γ, the subgame Γ(h) following the history h is the following extensive form game: • Players The players in Γ. • Terminal histories The set of all sequences h’ of actions such that (h, h’ ) is a terminal history of Γ. • Player function The player P(h, h’ ) is assigned to each proper subhistoryh’ of a terminal history. • Preferences Each player prefers h’ to h’’ if and only if she prefers (h, h’ ) to (h, h’’ ) in Γ.

21. A A a 1 2 1 Example: 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. 3,3 D D d 1,1 0,0 2,2

22. Subgame Perfect Equilibrium Definition The strategy profile s∗is a subgameperfect 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) whereplayer iplaysri while every other player j playss∗j.

23. A A a 1 2 1 • Solving for the subgame-perfect equilibrium • 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. 3,3 D D d 1,1 0,0 2,2

24. A A a 1 2 1 • 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. 3,3 D D d 1,1 0,0 2,2

25. 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.

26. 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 Cournotmodel. • 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.

27. 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 ito 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.

28. 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).

29. Suppose that Ci(qi) = cqi for i = 1, 2, and P(Q) = a − Q if Q ≤ a, P(Q) = 0 if Q > a, where c > 0 and c < a. • We know that firm 2’s best response to output q1 of firm 1 is b2(q1) = (a − c − q1)/2 if q1 ≤ a − c, b2(q1) = 0 if q1 > a − c. • In a subgame perfect equilibrium of Stackelberg’s game firm 2’s strategy is this function b2 and firm 1’s strategy q1* maximizes q1 (a − c − (q1 + (a − c − q1)/2)) = q1 (a − c − q1)/2. • The first order condition yields q1* =(a − c)/2. • The unique subgame perfect equilibrium is (q1*, b2).

30. The SPE outcome of the Stackelberg game is: q1S = (a − c)/2, q2S = (a − c − q1* )/2 = (a − c)/4. • Firm 1’s profit is q1S(P(q1S+q2S) − c) = (a − c)2/8, firm 2’s profit is q1S(P(q1S+q2S) − c) = (a − c)2/16. • Recall that in the unique Nash equilibrium of Cournot’s (simultaneous-move) game, q1C = q2C = (a − c)/3, so that each firm’s profit is (a − 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.

31. 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.

32. 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.

33. 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 tthturn.

34. 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 iloses, 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.

35. Backward Induction Solution k2 3 2 1 1 2 3 4 • Firm k wins game Gk(1, 1): whoever moves first takes one step towards the finish line and wins. f k1

36. k2 4 3 2 1 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). f f f k1

37. k2 4 3 2 1 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). f f f f k1

38. k2 4 3 2 1 1 Firm 1 wins games G1(1, 3) and G1(1, 4) with 1 step. 1 1 2 3 4 • 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. f f f f k1

39. k2 1 1 4 3 2 1 1 1 1 2 3 4 • Firm 1 wins games G1(2, 3) andG1(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. f f f f k1

40. k2 4 3 2 1 1 1 1 1 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. f f 2 2 f f 2 2 k1

41. k2 4 3 2 1 1 1 f f 1 1 f f 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. f f 2 2 f f 2 2 k1

42. k2 5 4 3 2 1 1 1 1 1 2 2 f f 1 1 f f 2 2 2 2 1 2 3 4 5 6 7 8 • Hence, the matrix can then be completed inductively. 1 1 f f 2 2 2 2 f f 2 2 2 2 2 2 f f 2 2 2 2 2 2 k1

43. Summary of the Lecture • Definition of Extensive-Form Games • Subgame Perfection and Backward Induction • Applications: Stackelberg Duopoly, Harris-Vickers Race.

44. Preview of the Next Lecture • Extensive-Form Games with Imperfect Information • Spence Signalling Game • Crawford and Sobel Cheap Talk