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Unformatted text preview: GAME THEORY Econ 414 Lecture 1718 Sequential Games with Imperfect Information Instructor: Nuno Limão slides_414_17_18_2011_no_solutions 1 OUTLINE Backward induction and SPNE w/ imperfect information Existence and multiplicity of SPNE Application: OS/2 slides_414_17_18_2011_no_solutions 2 BACKWARD INDUCTION AND SPNE W/ IMPERFECT INFORMATION In many situations of strategic interaction at least one player does not know what the other(s) did before deciding how to act, so there is imperfect information. In sequential games with imperfect information there will be information sets with multiple decision nodes, e.g. if guy is being chased and must decide whether to release before knowing whether the ransom has been paid slides_414_17_18_2011_no_solutions 3 Solution of sequential games w/ imperfect information Still apply backward induction (similar to perfect information) Must be careful in defining what a subgame is to ensure that it encompasses all the decision nodes in any given information set Ransom example w/ perfect information we defined a subgame for guy starting at III; another at IV w/ imperfect information guy does not know if he is at III or IV. It is not clear what his beliefs should be then so, he would not know what to do at this situation slides_414_17_18_2011_no_solutions 4 But, if we go up the tree we do obtain a well defined subgame, as shown below This is simply a simultaneous move game of a nature that we have solved before after writing in strategic form Notice that the NE of the subgame is kill/do not pay. Very different from what we had in the perfect information. What may this capture? slides_414_17_18_2011_no_solutions 5 Is this outcome worse for the kidnapped? No, need to solve for the SPNE: (Do not kidnap/kill, do not pay ranson) slides_414_17_18_2011_no_solutions 6 Definitions: Subgame: a subset of a sequential game that begins at any non‐terminal node where each player knows all previous actions by others and himself. Or alternatively: a regular subtree and its associated with payoffs Regular subtree: a type of subtree that contains all nodes of an information set that has at least one node in the tree Example of subtrees that are not a “regular subtree” in ransom game: 3 and 4 Examples of regular subtrees in ransom game: 1 and 2 slides_414_17_18_2011_no_solutions 7 Exercise: Consider the 3 player game below and answer the following What (if any) are the “proper” subgames? How do we represent the game in strategic form? What are the SPNE (if any)? slides_414_17_18_2011_no_solutions 8 Solution … slides_414_17_18_2011_no_solutions 9 EXISTENCE AND MULTIPLICITY OF SPNE Existence of SPNE in sequential games of imperfect information There is no guarantee of existence of SPNE in pure strategies (unlike the perfect information case) If there is no NE then there is no SPNE (since it is a subset of NE) There is always a SPNE in mixed strategies (just as there was a NE strategic games in chapter 7) Multiplicity of SPNE in sequential games of imperfect information If there are multiple NE in pure strategies then there may also exist multiple SPNE To find these we proceed as before using backward induction and if at one node a player is indifferent between two actions we just choose one of them and continue backward induction. Then we use the other and do the same (see OS2 example that follows) slides_414_17_18_2011_no_solutions 10 APPLICATION: OS/2 Background IBM decided to enter the personal computer (PC) market in 1981. In trying to establish an early lead and become the standard it outsourced several parts The operating system was bought from Microsoft (MS‐DOS), but w/out the copyright since IBM forecasted that it would make most of the profits from the sale of hardware. As the competition in the PC market intensified, IBM wanted to build its own operating system, OS/2, but knew its value crucially depended on the number of programs available to operate with it (and knew that by then Microsoft had built on MS‐DOS to create MS‐windows and several programs had been written for it) slides_414_17_18_2011_no_solutions 11 Modeling the OS/2 development decision Assume there are 3 companies that might develop application software for OS/2. OS/2 is valuable to consumers if at least two companies develop application software for it (otherwise IBM gets negative payoffs if it develops). Each software developer (1,2,3) waits to see if OS/2 is developed but moves simultaneously relative to other two slides_414_17_18_2011_no_solutions 12 slides_414_17_18_2011_no_solutions 13 SPNE for the development game First subgame to consider: development decision by 1,2,3 conditional on OS/2 availability. Strategic form Two Nash equilibria in pure strategies: all develop and none develop. Not surprising given it is simply a coordination game. slides_414_17_18_2011_no_solutions 14 Second subgame (if all 3 developed) has IBM also developing (0<20) Third subgame (if none of the 3 developed) has IBM not developing (0>‐3) Thus there are 2 SPNE in pure strategies: all develop or none do. What did we observe? IBM introduced OS/2 in 1987 but there were few applications, so despite it being faster and more stable than the alternative MS‐DOS, OS/2 failed. Does this mean that the game above fails to capture this strategic interaction? slides_414_17_18_2011_no_solutions 15 Mixed strategy SPNE in OS/2 game Assume firms 1,2,3 are symmetric and choose develop with probability d For 1 to find it optimal to randomize its expected payoff from develop or not must be equal Expected payoff develop: 3d2+2d(1‐d) + d(1‐d) + (1‐d)d + (‐1) (1‐d2) = 4d‐1 Expected payoff do not develop: 0 Indifference requires: d=1/4 Expected payoff to IBM from developing 20(1/4)3 +15(3)(1/4)2(3/4) + (‐2)(3/4)2(1/4)+ (‐3)(3/4)3 =20/64 Thus IBM develops (20/64>0) but knows that it will only succeed with probability (1/4)3+3(1/4)2(3/4) =10/64 but will fail with probability 50/64, The latter case was the realization observed so the theory can predict it. slides_414_17_18_2011_no_solutions 16 Revised OS/2 game (Exercise 2, ch. 9 of Harrington) Different structure of information: companies 2 and 3 observe OS/2 AND company 1’s decision. 2 and 3 still act simultaneously Payoffs changed slides_414_17_18_2011_no_solutions 17 Exercise What are all the subgame perfect Nash equilibria? Derive a Nash equilibrium that is not a subgame perfect Nash equilibrium, and explain why it is not a SPNE slides_414_17_18_2011_no_solutions 18 Solution: SPNE … slides_414_17_18_2011_no_solutions 19 Solution: Nash equilibria that are not SPNE… slides_414_17_18_2011_no_solutions 20 THE VALUE OF COMMITMENT Sequential games generate an important role and potential value of “commitment” Commitment , i.e. limiting future options, would usually seem to reduce one’s welfare, but not always in a strategic setting with sequential actions since it can be used to affect others actions Basic intuition for gain from commitment: recall game of chicken Two Nash equilibria. (Straight, Swerve) and (Swerve, Straight) Each driver strictly prefers the equilibrium, in which he goes straight and the other swerves. slides_414_17_18_2011_no_solutions 21 Now, suppose that we allow driver 1 an additional action prior to steering, namely whether to remove the steering wheel or not If he removes it and it is common knowledge that he does so (e.g. in a sequential setting with perfect information) then he has committed not to swerve. The resulting subgame is the one shown below so the other drive swerves and the one that limited his options does not and ends up better off as a result slides_414_17_18_2011_no_solutions 22 Another example of the value of commitment: the Doomsday machine http://www.gametheory.net/media/Doomsday.wmv Key necessary conditions for commitment to be valuable It has to be common knowledge: otherwise other players will not change their actions It has to be credible slides_414_17_18_2011_no_solutions 23 Commitment and deterrence of entry into a product market Setup Incumbent monopolist earns a profit of 1000 lone potential entrant is considering entry at a cost of 350, otherwise gets 0 Monopolist & entrant can set price at low, moderate, or high & obtain symmetric gross profits Analysis: Conditional on entry, the NE is (moderate, moderate) slides_414_17_18_2011_no_solutions 24 Analysis (ctd): Entry decision: need to consider the extensive form Subtracting entry cost from (moderate, moderate) equilibrium we get 400 – 350 = 50 for entrant 50>0 thus potential entrant enters and lowers monopoly payoff (1000 under no entry) slides_414_17_18_2011_no_solutions 25 What can the monopolist do to deter entry and its high profit? If monopolist set low in post entry game, then the entrant’s gross profit will be at most 325 and its net profit is never positive so (Do not enter / Moderate, Low) is a Nash equilibrium (neither has incentive to deviate) but it is not a SPNE since low price threat is not credible (if entry occurs then monopolist will charge moderate, not a low price) Monopolist needs a credible commitment device that convinces entrant he will face a low price Is a cost reduction investment by monopolist a credible entry deterrent? Suppose monopolist can make sunk cost investment in technology that costs 500 Lowers variable cost of production so if entrant stays out the monopoly profit is 700 = 1200 – 500 If potential entrant is not deterred and enters then monopolist always gets lower net profit then if it did not invest (see payoffs below) slides_414_17_18_2011_no_solutions 26 Exercise: investment as deterrence game What is the NE of the post investment and entry subgame? What is the SPNE of the full investment game? Potential entrant slides_414_17_18_2011_no_solutions 27 Solution … slides_414_17_18_2011_no_solutions 28 The entry investment game where potential entrant is unaware of investment … slides_414_17_18_2011_no_solutions 29 ...
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