Topic VI: Extensive Form Games

1. Topic VI: Extensive Form Games

2. Dynamic Decision Making and Resource Allocation Problems • The dynamic timing of actions is a key component in many economic institutions and allocation problems • Retails firms set prices, and consumers set quantities • Governments set tax rates, and individuals choose work levels and reported incomes. • Negotiations often involve offers and counter offers • Bidders in auctions observe prices and then can improve their own prices • We want to expand our tools of analysis to incorporate dynamic timing of actions, and dynamic information structures.

3. Basic Elements of an Extensive Form Game • The set of players, • Who moves when and under what circumstances, • What actions are available to a player when she is called upon to move, • What she knows when she is called upon to move, and • What payoff each player receives when the game is played in a particular way. • Elements 2) and 4) are additions to the normal form • Element 3) needs more detail than in normal form

4. Extensive-form games trees with perfect information • The tree formation consists of the following elements • Decision nodes –indicate places where a player takes an action. So each decision node has a unique associated player • Branches: These are the sets of possible actions at each decision node. • Information Sets: These are nodes that have the same observed previous history. Perfect Information every information set consists of a single node Player 1 L R Player 2 Player 2 R S M N Player 1 3 2 2 4 5 3 G P Leaves of the tree show player 1’s utility first, then player 2’s utility 1 0 0 1

5. Extensive-form games strategies • A strategy for a player is a complete plan of action for the player in every contingency in which the player might be called to act. • Player 1’s set of possible strategies {LP, LG, RP, RG} • Player 2’s set of possible Strategies {MR, MS, NR, NS} Player 1 L R Player 2 Player 2 R S M N • A strategy profile defines a unique path through the tree and contingencies for “off the path” information sets • Example: Strategy profile (LP,NR) • p1(LP,NR) =5 and p1(LP,NR)=3 Player 1 3 2 2 4 5 3 G P 1 0 0 1

6. Every Extensive Form Game has an Equivalent Normal Form MR MS NR NS Player 1 LP L R LG RP Player 2 Player 2 R S M N RG • So we can use all of our previous solution concepts • Nash equilibria of this game are (LP, MS), (LG, MS), (RP, MR), and (RG, MR) Player 1 3 2 2 4 5 3 G P 1 0 0 1

7. Chain Store Paradox (Selten 1978) – Credible vs Incredible Nash Equilibrium • Chain store has branches in many towns • Each town also has a small Mom & Pop store • M & P owners will eventually have sufficient capital to: • Establish a 2nd store • Sell, exit the market & retire to Hainan • The chain store prefers M & P owner exit; it can • Accept the competition, and split the local market • Launch a price war that is costly to both sides to drive out M & P • If chain accepts, M & P owners prefer to establish 2nd store • Can chain establish reputation for driving out M & P owners?

8. Chain Store Paradox –Two Kinds of Nash Equilibrium Game Tree Normal Form Price War (0, 0) Chain Accept Enter (2, 2) M & P Exit Two NE: (Exit, War) and (Enter, Accept) (1, 5) Consider the NE (Exit, War) – It relies upon the Chain store making a threat to make War. However, if M&P entered and Chain observes this, then it’s better of Accepting. This raises several issues What are the Reputational benefits of being a “fighter” Should we think of how we can select NE that only involve credible claims?

9. Subgame perfect equilibrium • Each node in a (perfect-information) game tree, together with the remainder of the game after that node is reached, is called a subgame • Four sub-games • Backward Induction Algorithm • Indentify the Subgames closest to the terminal nodes. Find the NE of these games • Roll the NE payoffs up to predecessor node, solve for nash • Roll up until you solve for initial node Player 1 L R Player 2 Player 2 R S M N Player 1 3 2 2 4 5 3 G P 1 0 0 1

10. Subgame perfect equilibrium • Solving by backward induction: Yields the Subgame Perfect Equilibrium (RP, MR) The set of SGP is a subset of the NE So we call the set of SGP equilibrium a refinement of the set of Nash Equilibrium Player 1 L R L Player 2 Player 2 R R S M N R M N S P Player 1 G 3 2 2 4 5 3 G P 1 0 0 1

11. Stackelberg Competition – First mover Advantage • Quantity competition but firms take turns choosing quantities • Two identical firms who can grow rice at MC = 2 and FC = 0. • They face the inverse demand function p = 102 - .5(q1 + q2), where qi is firms i’s chosen level of production (strategy) • Suppose Firm 1 chooses quantity first, then Firm 2 Observe q1, and then chooses q2. • What are the set of strategies for the two firms? • Q1 = [0, ∞] • Q1 = {f:[0, ∞] [0, ∞] } The set of strategies that map q1 to q2 qi*(qj) = 100 - .5qj. q1 q2 q1[P(Q) – c], q2[P(Q) – c]

12. Stackelberg Competition – First mover Advantage • Solve for the SGP equilibrium by backward induction. • Firm 2 problem maximize profit for each possible level of Firm 1 production p2((q1, q2)=(102 -.5 (q1+q2)) q2 – 2q2 • Take the derivative with respect to q2 and set equal to zero q2*(q1) = 100 - .5q1 • This is Firm 2’s SGP strategy • We roll this up and Firm 1 will maximize profit assuming firm 2 follows this strategy. p2((q1, q2)=(102 -.5 ((q1+(100-.5q1)) q1 – 2q1 • First Order Condition is MR= MC, or 52 - .5q1 = 2  q1* = 100 • The SGP equilibrium is (q1*, q2* )= (100 q2*(q1) = 100 - .5q1) • The Firm 1 production level is 100, and Firm 2 is 50; P = 27; Firm 1 profit is 2500, and Frim 2 profit is 1750. • Compare this to the Cournot Equilibrium NE is (q1*, q2* )= (200/3. 200/3) P= 35.33; Each firm profit is $2,222,22

13. What is imperfect information? • Imperfect information is when a player may not have observed all of the previous relevant decisions by other plays. • Prior to engaging in Cournot competition, your opponent may have engaged in a marginal cost reducing investment. • When you evaluate potential job applicants, they may or may not have provided false information on their application. • When dating to find a spouse, they could be dating someone else. • When buying an apartment in a new development, you don’t know how may other units are really sold. • How do we expand our extensive game form description to reflect situations imperfect information, and how we define strategies?

14. Imperfect information • Dotted lines indicate that a player cannot distinguish between two (or more) states • A set of states that are connected by dotted lines is called an information set • Reflected in the normal-form representation Player 1 L R L R Player 2 Player 2 0, 0 -1, 1 1, -1 -5, -5 • Any normal-form game can be transformed into an imperfect-information extensive-form game this way

15. Information Set • A subset of a player’s decision nodes • Each decision node has the same observable history preceding it from the decision maker’s perspective. • Each decision node belongs to exactly one information set • Each decision node within a information set has the same set of available actions. • We assume perfect recall; that is a player never forget any move he has previously made.

16. Strategies • A strategy calls for an action at each of a player’s action set.

17. Wife Football Ballet Husband Football 3, 1 0, 0 Ballet 0, 0 1, 3 Husband Home Out (2, 2) Extensive-form games with imperfect information • This game can be represented as...

18. Husband Home Out Wife (2, 2) Football Ballet Husband Husband Football Ballet Football Ballet (3, 1) (0, 0) (1, 1) (1, 3) Extensive-form games with imperfect information • Husband set of possible strategies is {(H,F), (H,B), (O,F), (O,B)}; Wife {F, B}

19. Husband Home Out Wife (2, 2) Football Ballet Husband Husband Football Ballet Football Ballet (3, 1) (0, 0) (1, 1) (1, 3) Extensive-form games with imperfect information • A strategy profile will define a unique path through the decision tree, and also define a what can happen at each “off the path” information set • Eg. (H,F), B

20. Nash Equilibrium • To find the set of Nash equilibrium we analyze the normal form game version • NE are highlighted in blue

21. Subgame Perfection • We need to modify our backward induction principle. • Proper Subgame • A proper subset of the whole game • makes sense in that it preserves the information structure in the whole game (we don’t break information sets) • As if today was the first day of the rest of your life

22. Husband Home Out (2, 2) Subgame Perfection • Backward induct from the lowest subgames and work back up the tree. Now we have to worry about multiple equilibrium • At SG following out 2 NE • If they play the BB NE in the SG Husband would choose home • If they play FF, then Husband will choose go out • There are two SGP equibrium (HB, B) and (OF, F)