CHAPTER 15 Sequential rationality 1-1

Transcription

1 . CHAPTER 15 Sequential rationality 1-1

2 Sequential irrationality Industry has incumbent. Potential entrant chooses to go in or stay out. If in, incumbent chooses to accommodate (both get modest profits) or to trigger a price war, both suffering. Efg represented as a nfg. Latter has two NE: (I,A) and (O,P). (O,P): 1 is getting 0, by deviating he gets -1, given 2 s strategy 2 is getting 4, by deviating, no change, given 1 stays out Each is best responding, given other s strategy: NE Is (O,P) sequentially rational? What if 1 mistakenly enters? Then would benefit by accommodating (+2) than warring (-1). 1-2

3 Rationality: Ex ante vs Sequential NE is fine concept for nfg where players move independently When nfg is an efg translation with dependent moves, NE may fail to capture sequential rationality In the preceding efg, (O,P) is a NE and is ex anterational: from the standpoint of the initial node and strategies. (O,P) is not sequentially rational: from the standpoint of information set of player 2 (even as (O,P) does not lead there) Sequential rationality of a player s strategy: It specifies, at every information set of his, an action that is optimal conditionalon reaching this information set, even if ex ante he does not believe to reach this information set. In this sense, incumbent s Price War not sequentially rational 1-3

4 Backward induction & Perfect information Restrict attention to games of perfect information (singletons) Assume rationality is common knowledge, so each player during her decisions puts herself in shoes of players to follow A pre-terminalnode is one with terminal nodes for successors At each such, the relevant player is to choose an action that would determine everyone s payoff; let him choose one that maximizes his payoff. Having done so, trim the treeat this preterminal node: erase all but maximizing branch-action, and mark node with everyone s continuation value, ie the payoffs associated with the chosen maximizing action. Doing so for every pre-terminal node gets a trimmed tree. Backward induction: Trim the original tree. Given a trimmed tree, trim it. Stop when mark reaches the initial node. 1-4

5 Example Backward induction gives (DE,AC) Marks at nodes aka continuation values 1-5

6 Redrawing backward induction Arrows indicate optimal action at each info set. Backward induction still clear, also concise Backward induction equilibrium: A strategy profile so derived 1-6

7 Backward induction and NE Backward induction identifies a unique path of actions, provided there are no terminal nodes at which some player gets the same payoff. Zermelo s theorem Every finite efg of perfect information has a pure-strategy Nash equilibrium that can be identified by backward induction. If no player has the same payoffs at any two terminal nodes, then there is a unique Nash equilibrium that can be so identified. Corollary Chess has a pure-strategy NE (it is unknown) 1-7

8 Stackelberg example Recall the sequential game of two firms producing costlessly a common good, sold at market price p = 12 q 1 q 2 Firm 1 chooses s 1 =q 1 ε[0,12] first. Then firm 2 observes this choice and makes its own: Strategy is a fn [0,12] [0,12] Payoffs are profits p q. i Checked that this was a NE: q 1 =2, s 2 (q 1 )={5 if q 1 =2, 12-q 1 ow} Here, 2 threatens to crash the price to 0 unless 1 produces 2. Problem: Is 2 s threat sequentially rational?what if 1 gifts full market (produces 0), would 2 make it profitless with p=0? Backward induction says, Threat is not sequentially rational : Exercise: Backward induction gives s 2 (q 1 )=(12-q 1 )/2, q 1 = 6 In particular, s 2 (0)=6, not 12! 1-8

9 BIE might apply with imperfect information Here 3 has imperfect information on what 1,2 did. Optimal actionof 3 happens to be independent thereof: H dominated by G. So BIE applies here. In general, however, must acknowledge that what is optimal can depend on node. 1-9

10 Here imperfect information an obstacle to BIE Here optimal actionof 2 (up, down) depends on what 1 did (up, down), of which 2 is uninformed. So how can backward induction work (1 trying to predict2 s action )? Need somehow to simultaneously pin action & prediction! 1-10

11 Subgame A subgameof an efg is a subset such that: (1) It begins with an information set containing only one node, contains all its successors, and only these nodes (2) If the decision node x is in the subset, then so is every node in x s information set. Node initiates a subgame Neither Node does Nodedoes always does (original game always subgame) 1-11

12 Subgame perfect Nash equilibrium A strategy profile is a subgame perfect Nash equilibrium (SPE) it specifies a Nash equilibrium in every subgame. Assumes early players think ahead & expect NE rationality Note: A SPE is always a NE (for original game is a subgame) One proper subgame: It has two NE, (Up,Up), (Down,Down) Two SPEs: (Out&Up,Up) and (In&Down,Down) 1-12

13 Subgame A subgameof an efg is a subset such that: (1) It begins with an information set containing only one node, contains all its successors, and only these nodes (2) If the decision node x is in the subset, then so is every node in x s information set. y initiates a subgame x does not, (2) violated by z neither z nor w initiates a subgame initial node does always does (original game always a subgame) 1-13

14 Subgame perfect Nash equilibrium A strategy profile is a subgame perfect Nash equilibrium (SPE) if it specifies a Nash equilibrium in every subgame. Note: A SPE is always a NE (for original game is a subgame) Example has one proper subgame -(UA,X),(DA,Y), (DB,Y) are NE (see in first table) -Latter two are not SPE (sole NE in subgame is (A,X)) -(UA,X) is sole SPE 1-14

15 SPE x BIE BIE suited for efg s of perfect information. Has early players think ahead and expect individual optimality. SPE fine for efg s of imperfect information as well. Also has early players think ahead and expect simultaneous individual optimality (i.e. NE) SPE valid for efg s of perfect information. How do its predictions compare to BIE s? Preservation theorem Fix a a strategy profile in an efg of perfect information. It is a BIE if and only if it is a SPE. So SPE preserves and extends BIE to efg s of possibly imperfect info. 1-15

16 Sequential nfg s with unique SPE Imagine 1,,t,,T nfg s with same n players, each with a unique Nash equilibrium, denoted s t = ( s t 1,..., s t n ) Consider the following efg: Above nfg s are played sequentially, where at each stage each player observes the history of pure strategies played. At the end, each player s payoff is the sum of her payoffs in the nfg s. For this class of efg s, SPE makes a sharp prediction. Result Such an efg has a unique SPE, where each player i t plays strategy in stage t. s i No history dependence, in that no player can induce others to particular actions by promising rewards and punishments. Will see result often false if underlying nfg has multiple NE s. 1-16

17 Finiteness Some games involve infinitely many stages (examples soon) For these, the concept of backward induction cannot apply there is no end from which to start backwards but the concept of SPE remains valid. Threats and rewards by a player can induce others to change their actions dramatically (unlike previous result) for a long while. 1-17

18 BIE/SPE assumes individual rationality even in presence of contrary evidence Players start with $1 each. A player can say stop(game ends, players keep $) or continue($1 taken from him, $2 added to other). If continue, it is the other player s turn Unique BIE: (s,s) where s= Say stop whenever my turn So game ends immediately, each gets $1; they forego $100 How can sequential rationality lead to joint stupidity? Tension Point: BIE assumes, even at nodes only reachable through repeated individual sub-optimality, that players continue to act (and expect others to act) optimally. If node 2 is reached, must 2 really expect 1 to act optimally? 1-18