Diversion Theory: Dynamic .

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Prerequisites. Almost essential Game Theory: Strategy and Equilibrium. Game Theory: Dynamic. MICROECONOMICS Principles and Analysis Frank Cowell . November 2006 . Overview. Game Theory: Dynamic. Game and subgame. Mapping the temporal structure of games. Equilibrium Issues.
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Requirements Almost vital Game Theory: Strategy and Equilibrium Game Theory: Dynamic MICROECONOMICS Principles and Analysis Frank Cowell November 2006

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Overview... Amusement Theory: Dynamic Game and subgame Mapping the fleeting structure of recreations Equilibrium Issues Applications

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Time Why present "time" into model of a diversion? Without it a few ideas useless… would we be able to truly talk about responses? a balance way? dangers? "Time" includes organizing monetary choices demonstrate the arrangement of basic leadership thought of rôle of data in that succession Be mindful so as to recognize procedures and activities see this in a straightforward setting

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A basic amusement Stage 1: Alf\'s choice Stage 2: Bill\'s choice after [LEFT] Stage 2: Bill\'s choice after [RIGHT] Alf The adjustments [RIGHT] [LEFT] Bill [left] [right] [left] [right] ( u 1 a , u 1 b ) ( u 3 a , u 3 b ) ( u 2 a , u 2 b ) ( u 4 a , u 4 b )

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A basic diversion: Normal shape Alf has two methodologies Always play [left] whatever Alf picks Play [left] if Alf plays [LEFT] . Play [right] if Alf plays [RIGHT] Play [right] if Alf plays [LEFT]. Play [left] if Alf plays [RIGHT] Always play [right] whatever Alf picks Bill has four procedures The adjustments Bill [left-left] [left-right] [right-left] [right-right] Alf [LEFT] ( u 1 a , u 1 b ) ( u 1 a , u 1 b ) ( u 2 a , u 2 b ) ( u 2 a , u 2 b ) [RIGHT] ( u 3 a , u 3 b ) ( u 4 a , u 4 b ) ( u 3 a , u 3 b ) ( u 4 a , u 4 b ) Alf moves first: methodology set contains only two components Bill moves second: technique set contains four components

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The setting Take an amusement in vital frame If every player has precisely one round of play… diversion is greatly basic… … concurrent or successive? Generally require a method for depicting structure envision a specific way through the tree outline describe unfurling choice issue Begin with a few updates

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Structure: 1 (updates) Decision hubs in the broad frame… … speak to focuses where a choice is made by a player Information set where is player (leader) found? might not have watched some past move in the diversion realizes that he is at one of various choice hubs accumulation of these hubs is the data set

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Structure: 2 (detail) Stage a specific point in the sensible time grouping of the amusement settlements come after the last stage Direct successor hubs take the choice branches (activities) that take after from hub * if the branches prompt other choice hubs at the following stage… … then these are immediate successor hubs to hub * Indirect successor hubs rehash the above through no less than one more stage… … to get roundabout successor hubs How would we be able to utilize this structure? separate general diversion… … into segment recreations?

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Subgames (1) A subgame of a broad shape diversion a subset of the amusement… … fulfilling three conditions Begins with a "singleton" data set contains a solitary choice hub simply like begin of general amusement Contains all the choice hubs that… are immediate or circuitous successors,… … and no others If a choice hub is in the subgame then whatever other hub in similar data set… … is likewise in the subgame.

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Subgames (2) Stage 1: (Alf\') Alf Stage 2: (Bill) Add a Stage (Alf again)... The adjustments [RIGHT] [LEFT] A subgame... Another subgame... Charge Bill [left] [right] [left] [right] Alf [LEFT] [RIGHT] [LEFT] [RIGHT] [LEFT] [RIGHT] [LEFT] [RIGHT] [ u 1 ] [ u 2 ] [ u 3 ] [ u 4 ] [ u 5 ] [ u 6 ] [ u 7 ] [ u 8 ]

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Alf [RIGHT] [LEFT] [RIGHT] [LEFT] Bill [left] [right] [left] [right] [left] [right] [left] [right] Alf [LEFT] [RIGHT] [LEFT] [RIGHT] [LEFT] [RIGHT] [LEFT] [RIGHT] [LEFT] [RIGHT] [LEFT] [RIGHT] [LEFT] [RIGHT] Subgames (3) The past structure Additional procedure for Alf Ambiguity at stage 3 A subgame [MID] Not a subgame (disregards 2) Not a subgame (damages 3) Bill [left] [right] Alf [ u 1 ] [ u 2 ] [ u 3 ] [ u 4 ] [ u 5 ] [ u 6 ] [ u 7 ] [ u 8 ] [ u 9 ] [ u 10 ] [ u 11 ] [ u 12 ]

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Game and subgame: lessons "Time" forces structure on basic leadership Representation of multistage recreations requires mind recognize activities and methodologies ordinary shape representation can be ungainly Identifying subgames three criteria cases with non-singleton data sets are dubious

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Overview... Diversion Theory: Dynamic Game and subgame Concepts and strategy Equilibrium Issues Applications

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Equilibrium raises issues of idea and technique both need some care… … as with the basic single-shot amusements Concept would we be able to utilize the Nash Equilibrium once more? unmistakably requires cautious determination of the system sets Method a basic inquiry procedure? however, will this dependably work? We begin with a framework of the strategy…

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Backwards acceptance Suppose the amusement has N stages Start with stage N assume there are m choice hubs at stage N Pick a subjective hub … assume h is player at this stage … decide h \'s decision, contingent on landing at that hub … note result to h and to each other player emerging from this decision Repeat for each of the other m −1 hubs this totally unravels organize N gives m vectors of [ u 1 ],… ,[ u m ] Re-utilize the qualities from understanding stage N gives the settlements for a round of N −1 stages Continue on up the tree… A case:

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Backwards enlistment: illustration Examine the last phase of the 3-arrange diversion utilized prior Suppose the eight result levels for Alf fulfill υ 1 a > υ 2 a (first hub) υ 3 a > υ 4 a (second hub) υ 5 a > υ 6 a (third hub) υ 7 a > υ 8 a (fourth hub) If the amusement had in certainty achieved the main hub: clearly Alf would pick [LEFT] so esteem to (Alf, Bill) of achieving first hub is [υ 1 ] = (υ 1 a ,υ 1 b ) Likewise the estimation of achieving different hubs at stage 3 is… [υ 3 ] (second hub) [υ 5 ] (third hub) [υ 7 ] (fourth hub) Backwards enlistment has lessened the 3-organize amusement… … to a two-arrange diversion … with adjustments [υ 1 ], [υ 3 ], [υ 5 ], [υ 7 ]

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Backwards enlistment: outline 3-organize amusement as before υ 1 a > υ 2 a υ 3 a > υ 4 a υ 5 a > υ 6 a υ 7 a > υ 8 an Alf Payoffs to 3-arrange amusement Alf would play [LEFT] at this hub [RIGHT] [LEFT] … and here The 2-arrange diversion got from the 3-arrange amusement Bill [left] [right] [left] [right] Alf [ u 1 ] [ u 3 ] [ u 5 ] [ u 7 ] [LEFT] [RIGHT] [LEFT] [RIGHT] [LEFT] [RIGHT] [LEFT] [RIGHT] [ u 2 ] [ u 1 ] [ u 4 ] [ u 6 ] [ u 7 ] [ u 8 ] [ u 5 ] [ u 3 ]

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Equilibrium: addresses Backwards enlistment is an effective technique agrees with instinct ordinarily prompts an answer But what is the suitable fundamental idea? Does it discover all the applicable equilibria? What is the part for the Nash Equilibrium (NE) idea? Start with the remainder of these… A straightforward case:

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Equilibrium illustration The broad frame Bill\'s decisions in conclusive stage Values found through in reverse enlistment Alf\'s decision in first stage The harmony way [RIGHT] [LEFT] Backwards acceptance discovers balance result of 2 for Alf, 1 for Bill (2,1) (1,2) Bill But what is/are the NE here? Take a gander at the diversion in typical frame… [left] [right] [left] [right] (1,2) (0,0) (2,1) (1,2) (2,1)

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Equilibrium illustration: Normal shape Alf\'s two techniques Bill\'s four systems Payoffs s 1 b s 2 b s 3 b s 4 b Best answers to s 1 a [left-left] [left-right] [right-left] [right-right] Best answer to s 3 b or to s 4 b Alf [LEFT] s 1 a 0,0 2,1 Best answers to s 2 a Best answer to s 1 b or to s 2 b [RIGHT] 1,2 s 2 a Nash equilibria: ( s 1 a , s 3 b ), ( s 1 a , s 4 b ) ( s 2 a , s 1 b ), ( s 2 a , s 2 b ),

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Equilibrium case: the arrangement of NE The arrangement of NE incorporate the arrangement officially discovered in reverse acceptance strategy… ( s 1 a , s 3 b ) yields result (2,1) ( s 1 a , s 4 b ) yields result (2,1) What of the other NE? ( s 2 a , s 1 b ) yields result (1,2) ( s 2 a , s 2 b ) yields result (1,2) These propose two comments First, Bill\'s balance system may prompt some odd conduct Second could such a NE be maintained by and by? We follow up each of these thusly

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Alf [RIGHT] [LEFT] Bill [left] [right] [left] [right] (1,2) (0,0) (2,1) (1,2) Equilibrium illustration: odd conduct? Take the Bill system s 1 b = [left-left] "Play [left] whatever Alf does" If Alf plays [RIGHT] on Monday On Tuesday it\'s sensible for Bill to play [left] But in the event that Alf plays [LEFT] on Monday what ought to Bill do on Tuesday? the above system says play [left] yet, from Tuesday\'s viewpoint, it\'s odd Given that the amusement achieves hub * Bill then improves playing [right] Yet… s 1 b is a piece of a NE?? Monday * Tuesday

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Equilibrium case: reasonableness Again consider the NE not found by in reverse acceptance Give a result of 1 to Alf, 2 to Bill Could Bill "drive" such a NE by a risk? Envision the accompanying pre-play discussion. Charge: "I will play methodology [left-left] whatever you do" Alf: "Which implies?" Bill: "To abstain from getting a result of 0 you would be advised to play [RIGHT]" The shortcoming of this is evident Suppose Alf feels free to plays [LEFT] Would Bill now truly do this risk? After all Bill would likewise endure (gets 0 rather than 1) Bill\'s risk appears to be staggering So the "harmony" that appears to depend on it is not exceptionally noteworthy

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Equilibrium idea Some NEs are odd in the dynamic setting So there\'s a need to refine balance con

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